Subgroups — C₂₃×#22756 #19170382

Order 5888.

858 conjugacy classes of subgroups.

The subgroup lattice has 680 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#433	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₂₃×#424	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#2293	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#1923	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂₃×#1923	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂₃×#1917	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₃×#1916	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂₃×#1870	C₂	1	1	yes	maximal, characteristic
9	C₂₃×#1869	C₂	1	1	yes	maximal, characteristic
10	C₂₃×#1831	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂₃×#1798	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂₃×#1798	C₂	1	1	yes	maximal, characteristic, complemented
13	C₄×C₂₃×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂₃×#1753	C₂	1	1	yes	maximal, characteristic, complemented
15	C₈×C₂²⋊C₄×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂²×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
17	C₂³×C₈×C₂₃	C₂²	1	1	yes	characteristic
18	C₂×C₄×D₄×C₂₃	C₂²	1	1	yes	characteristic
19	C₂×C₂₃×#117	C₂²	1	1	yes	characteristic
20	C₂×C₂₃×#117	C₂²	1	1	yes	characteristic
21	C₂₃×#433	—	2	4	no	—
22	C₂₃×#424	—	2	4	no	—
23	C₂×C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
24	C₂×C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
25	C₂×C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
26	C₂×C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
27	C₂×C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
28	C₂×C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
29	C₂×C₂₃×C₂²⋊C₈	C₂²	1	1	yes	characteristic
30	C₂×C₂₃×C₂²⋊C₈	C₂²	1	1	yes	characteristic
31	C₂×C₂₃×C₂²⋊C₈	C₂²	1	1	yes	characteristic
32	C₂×C₂₃×C₂²⋊C₈	C₂²	1	1	yes	characteristic
33	C₂×C₂₃×C₂²⋊C₈	C₂²	1	1	yes	characteristic
34	C₂×C₂₃×C₂²⋊C₈	C₂²	1	1	yes	characteristic
35	C₂×C₂₃×#97	C₂²	1	1	yes	characteristic
36	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
37	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
38	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
39	C₂₃×#390	C₄	1	1	yes	characteristic
40	C₂₃×#390	C₄	1	1	yes	characteristic
41	C₂₃×#389	C₄	1	1	yes	characteristic
42	C₂₃×#389	C₄	1	1	yes	characteristic
43	C₂₃×#388	C₄	1	1	yes	characteristic
44	C₂₃×#388	C₄	1	1	yes	characteristic
45	C₂₃×#388	C₄	1	1	yes	characteristic
46	C₂₃×#388	C₄	1	1	yes	characteristic
47	C₄×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
48	C₄×C₂²⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
49	C₄×C₂²⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
50	C₄³×C₂₃	C₂²	1	1	yes	characteristic
51	C₂₃×#336	C₂²	1	1	yes	characteristic
52	C₂₃×#336	C₂²	1	1	yes	characteristic
53	C₂₃×#336	C₂²	1	1	yes	characteristic
54	C₂₃×#336	C₂²	1	1	yes	characteristic
55	C₂₃×#336	C₂²	1	1	yes	characteristic
56	C₂₃×#336	C₂²	1	1	yes	characteristic
57	C₂₃×#336	C₂²	1	1	yes	characteristic
58	C₂₃×#336	C₂²	1	1	yes	characteristic
59	C₂₃×#336	C₂²	1	1	yes	characteristic
60	C₂₃×#336	C₂²	1	1	yes	characteristic
61	C₂²×D₄×C₂₃	C₂×C₄	1	1	yes	characteristic
62	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
63	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
64	C₂×M₁₆×C₂₃	—	2	2	no	—
65	C₂×M₁₆×C₂₃	—	2	2	no	—
66	C₂×M₁₆×C₂₃	—	4	1	no	—
67	C₂²×C₈×C₂₃	—	2	2	no	—
68	C₂²×C₈×C₂₃	—	2	2	no	—
69	C₂²×C₈×C₂₃	—	4	1	no	—
70	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
71	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
72	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
73	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
74	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
75	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
76	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
77	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
78	C₄×D₄×C₂₃	—	2	2	no	—
79	C₄×D₄×C₂₃	—	2	2	no	—
80	C₂₃×#117	—	2	2	no	—
81	C₂₃×#117	—	2	2	no	—
82	C₂₃×#117	—	2	2	no	—
83	C₂₃×#117	—	2	2	no	—
84	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
85	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
86	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
87	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
88	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
89	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
90	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
91	C₂×C₂²⋊C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
92	C₂×C₂²⋊C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
93	C₂×C₂²⋊C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
94	C₂×C₂²⋊C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
95	C₂×C₂²⋊C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
96	C₂×C₂²⋊C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
97	C₂×C₂²⋊C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
98	C₂×C₂²⋊C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
99	C₂×C₂²⋊C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
100	C₂×C₂²⋊C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
101	C₂×C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
102	C₂×C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
103	C₂×C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
104	C₂×C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
105	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
106	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
107	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
108	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
109	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
110	C₂₃×C₄⋊C₈	—	2	2	no	—
111	C₂₃×C₄⋊C₈	—	2	2	no	—
112	C₂₃×C₄⋊C₈	—	2	2	no	—
113	C₂₃×C₄⋊C₈	—	2	2	no	—
114	C₂₃×C₄⋊C₈	—	2	2	no	—
115	C₂₃×C₄⋊C₈	D₄	1	4	yes	complemented
116	C₂₃×C₂²⋊C₈	—	2	2	no	—
117	C₂₃×C₂²⋊C₈	—	2	2	no	—
118	C₂₃×C₂²⋊C₈	—	2	2	no	—
119	C₂₃×C₂²⋊C₈	—	2	2	no	—
120	C₂₃×C₂²⋊C₈	—	4	1	no	—
121	C₂₃×C₂²⋊C₈	—	4	1	no	—
122	C₂₃×#97	—	2	2	no	—
123	C₄×C₈×C₂₃	—	2	2	no	—
124	C₄×C₈×C₂₃	—	4	1	no	—
125	C₄×C₈×C₂₃	—	4	1	no	—
126	C₂₃×#95	C₂×C₄	1	1	yes	characteristic
127	C₂₃×#95	C₂×C₄	1	1	yes	characteristic
128	C₂₃×#95	C₂×C₄	1	1	yes	characteristic
129	C₂₃×#95	C₂×C₄	1	1	yes	characteristic
130	C₂₃×#95	C₂×C₄	1	1	yes	characteristic
131	C₂₃×#95	C₂×C₄	1	1	yes	characteristic
132	C₂⁴×C₂₃	C₂²×C₄	1	1	yes	characteristic
133	C₂⁴×C₂₃	C₂²×C₄	1	1	yes	characteristic
134	C₂×D₄×C₂₃	—	2	2	no	—
135	C₂×D₄×C₂₃	—	2	2	no	—
136	C₂×D₄×C₂₃	—	4	1	no	—
137	C₂²×C₄×C₂₃	—	2	1	no	—
138	C₂²×C₄×C₂₃	—	2	1	no	—
139	C₂²×C₄×C₂₃	—	2	1	no	—
140	C₂²×C₄×C₂₃	—	2	1	no	—
141	C₂²×C₄×C₂₃	—	2	1	no	—
142	C₂²×C₄×C₂₃	—	2	1	no	—
143	C₂²×C₄×C₂₃	—	2	1	no	—
144	C₂²×C₄×C₂₃	—	2	1	no	—
145	C₂²×C₄×C₂₃	—	2	1	no	—
146	C₂²×C₄×C₂₃	—	2	1	no	—
147	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
148	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
149	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
150	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
151	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
152	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
153	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
154	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
155	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
156	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
157	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
158	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
159	C₂²×C₄×C₂₃	#41	1	2	yes	—
160	C₂²×C₄×C₂₃	#41	1	2	yes	—
161	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
162	M₁₆×C₂₃	—	4	4	no	—
163	C₂×C₈×C₂₃	—	2	1	no	—
164	C₂×C₈×C₂₃	—	2	1	no	—
165	C₂×C₈×C₂₃	—	2	1	no	—
166	C₂×C₈×C₂₃	—	2	1	no	—
167	C₂×C₈×C₂₃	—	2	1	no	—
168	C₂×C₈×C₂₃	—	2	1	no	—
169	C₂×C₈×C₂₃	—	2	1	no	—
170	C₂×C₈×C₂₃	—	2	1	no	—
171	C₂×C₈×C₂₃	—	2	1	no	—
172	C₂×C₈×C₂₃	—	2	1	no	—
173	C₂×C₈×C₂₃	—	2	1	no	—
174	C₂×C₈×C₂₃	—	2	1	no	—
175	C₂×C₈×C₂₃	—	2	1	no	—
176	C₂×C₈×C₂₃	—	2	1	no	—
177	C₂×C₈×C₂₃	—	2	1	no	—
178	C₂×C₈×C₂₃	—	2	1	no	—
179	C₂×C₈×C₂₃	—	2	1	no	—
180	C₂×C₈×C₂₃	—	2	1	no	—
181	C₂×C₈×C₂₃	—	2	1	no	—
182	C₂×C₈×C₂₃	—	2	1	no	—
183	C₂×C₈×C₂₃	—	2	1	no	—
184	C₂×C₈×C₂₃	—	2	1	no	—
185	C₂×C₈×C₂₃	—	4	4	no	—
186	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
187	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
188	C₄⋊C₄×C₂₃	—	2	2	no	—
189	C₄⋊C₄×C₂₃	—	2	2	no	—
190	C₄⋊C₄×C₂₃	—	2	2	no	—
191	C₄⋊C₄×C₂₃	—	2	2	no	—
192	C₄⋊C₄×C₂₃	—	2	2	no	—
193	C₄⋊C₄×C₂₃	—	2	2	no	—
194	C₄⋊C₄×C₂₃	—	4	1	no	—
195	C₂²⋊C₄×C₂₃	—	2	2	no	—
196	C₂²⋊C₄×C₂₃	—	2	2	no	—
197	C₂²⋊C₄×C₂₃	—	2	2	no	—
198	C₂²⋊C₄×C₂₃	—	2	2	no	—
199	C₂²⋊C₄×C₂₃	—	2	2	no	—
200	C₂²⋊C₄×C₂₃	—	2	2	no	—
201	C₂²⋊C₄×C₂₃	—	2	2	no	—
202	C₂²⋊C₄×C₂₃	—	2	2	no	—
203	C₂²⋊C₄×C₂₃	—	2	2	no	—
204	C₂²⋊C₄×C₂₃	—	2	2	no	—
205	C₄²×C₂₃	—	2	2	no	—
206	C₄²×C₂₃	—	2	2	no	—
207	C₄²×C₂₃	—	2	2	no	—
208	C₄²×C₂₃	—	2	1	no	—
209	C₄²×C₂₃	—	2	1	no	—
210	C₄²×C₂₃	—	2	1	no	—
211	C₄²×C₂₃	—	2	1	no	—
212	C₄²×C₂₃	—	2	1	no	—
213	C₄²×C₂₃	—	2	1	no	—
214	C₄²×C₂₃	—	2	1	no	—
215	C₄²×C₂₃	—	2	1	no	—
216	C₄²×C₂₃	—	4	1	no	—
217	C₄²×C₂₃	C₂×D₄	1	2	yes	—
218	C₄²×C₂₃	#41	1	2	yes	—
219	#22756	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
220	C₂³×C₂₃	—	2	1	no	—
221	C₂³×C₂₃	—	2	1	no	—
222	C₂³×C₂₃	—	2	1	no	—
223	C₂³×C₂₃	—	2	1	no	—
224	C₂³×C₂₃	—	2	1	no	—
225	C₂³×C₂₃	—	2	1	no	—
226	C₂³×C₂₃	—	2	1	no	—
227	C₂³×C₂₃	—	2	1	no	—
228	C₂³×C₂₃	—	2	1	no	—
229	C₂³×C₂₃	—	2	1	no	—
230	C₂³×C₂₃	—	2	1	no	—
231	C₂³×C₂₃	—	2	1	no	—
232	C₂³×C₂₃	—	2	1	no	—
233	C₂³×C₂₃	#117	1	2	yes	—
234	C₂³×C₂₃	C₂³×C₄	1	1	yes	characteristic
235	D₄×C₂₃	—	4	4	no	—
236	C₂×C₄×C₂₃	—	2	1	no	—
237	C₂×C₄×C₂₃	—	2	1	no	—
238	C₂×C₄×C₂₃	—	2	1	no	—
239	C₂×C₄×C₂₃	—	2	1	no	—
240	C₂×C₄×C₂₃	—	2	1	no	—
241	C₂×C₄×C₂₃	—	2	1	no	—
242	C₂×C₄×C₂₃	—	2	1	no	—
243	C₂×C₄×C₂₃	—	2	1	no	—
244	C₂×C₄×C₂₃	—	2	1	no	—
245	C₂×C₄×C₂₃	—	2	1	no	—
246	C₂×C₄×C₂₃	—	2	1	no	—
247	C₂×C₄×C₂₃	—	2	1	no	—
248	C₂×C₄×C₂₃	—	2	2	no	—
249	C₂×C₄×C₂₃	—	2	2	no	—
250	C₂×C₄×C₂₃	—	2	2	no	—
251	C₂×C₄×C₂₃	—	2	2	no	—
252	C₂×C₄×C₂₃	—	2	1	no	—
253	C₂×C₄×C₂₃	—	2	1	no	—
254	C₂×C₄×C₂₃	—	2	1	no	—
255	C₂×C₄×C₂₃	—	2	1	no	—
256	C₂×C₄×C₂₃	—	2	1	no	—
257	C₂×C₄×C₂₃	—	2	1	no	—
258	C₂×C₄×C₂₃	—	2	1	no	—
259	C₂×C₄×C₂₃	—	2	1	no	—
260	C₂×C₄×C₂₃	—	2	1	no	—
261	C₂×C₄×C₂₃	—	2	1	no	—
262	C₂×C₄×C₂₃	—	2	1	no	—
263	C₂×C₄×C₂₃	—	2	1	no	—
264	C₂×C₄×C₂₃	—	2	1	no	—
265	C₂×C₄×C₂₃	—	2	1	no	—
266	C₂×C₄×C₂₃	—	2	1	no	—
267	C₂×C₄×C₂₃	—	2	1	no	—
268	C₂×C₄×C₂₃	—	2	1	no	—
269	C₂×C₄×C₂₃	—	2	1	no	—
270	C₂×C₄×C₂₃	—	4	1	no	—
271	C₂×C₄×C₂₃	—	4	1	no	—
272	C₂×C₄×C₂₃	—	4	1	no	—
273	C₂×C₄×C₂₃	—	4	1	no	—
274	C₂×C₄×C₂₃	#117	1	2	yes	—
275	C₂×C₄×C₂₃	C₄×D₄	1	2	yes	—
276	C₂×C₄×C₂₃	C₄×D₄	1	2	yes	—
277	C₂×C₄×C₂₃	#131	1	2	yes	—
278	C₂×C₄×C₂₃	#131	1	2	yes	—
279	C₂×C₄×C₂₃	#131	1	2	yes	—
280	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
281	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
282	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
283	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
284	C₂×C₄×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
285	C₂×C₄×C₂₃	2₋¹⁺⁴	1	1	yes	characteristic
286	C₈×C₂₃	—	2	2	no	—
287	C₈×C₂₃	—	2	2	no	—
288	C₈×C₂₃	—	4	1	no	—
289	C₈×C₂₃	—	4	1	no	—
290	C₈×C₂₃	—	4	1	no	—
291	C₈×C₂₃	—	4	1	no	—
292	C₈×C₂₃	—	4	1	no	—
293	C₈×C₂₃	—	4	1	no	—
294	C₂×#433	C₂×C₂₃	1	1	yes	characteristic, complemented
295	C₂×#424	C₂×C₂₃	1	1	yes	characteristic, complemented
296	#2293	C₂×C₂₃	1	1	yes	characteristic
297	#1923	C₂×C₂₃	1	1	yes	characteristic, complemented
298	#1923	C₂×C₂₃	1	1	yes	characteristic, complemented
299	#1917	C₂×C₂₃	1	1	yes	characteristic, complemented
300	#1916	C₂×C₂₃	1	1	yes	characteristic, complemented
301	#1870	C₂×C₂₃	1	1	yes	characteristic
302	#1869	C₂×C₂₃	1	1	yes	characteristic
303	#1831	C₂×C₂₃	1	1	yes	characteristic, complemented
304	#1798	C₂×C₂₃	1	1	yes	characteristic, complemented
305	#1798	C₂×C₂₃	1	1	yes	characteristic, complemented
306	C₄×C₄⋊C₈	C₂×C₂₃	1	1	yes	characteristic, complemented
307	#1753	C₂×C₂₃	1	1	yes	characteristic, complemented
308	C₈×C₂²⋊C₄	C₂×C₂₃	1	1	yes	characteristic, complemented
309	C₂²×C₂₃	—	2	2	no	—
310	C₂²×C₂₃	—	2	2	no	—
311	C₂²×C₂₃	—	2	2	no	—
312	C₂²×C₂₃	—	4	1	no	—
313	C₂²×C₂₃	—	4	1	no	—
314	C₂²×C₂₃	—	4	1	no	—
315	C₂²×C₂₃	—	4	1	no	—
316	C₂²×C₂₃	—	4	1	no	—
317	C₂²×C₂₃	—	4	1	no	—
318	C₂²×C₂₃	—	4	1	no	—
319	C₂²×C₂₃	—	4	1	no	—
320	C₂²×C₂₃	—	4	1	no	—
321	C₂²×C₂₃	—	4	1	no	—
322	C₂²×C₂₃	—	4	1	no	—
323	C₂²×C₂₃	C₂×#117	1	1	yes	characteristic
324	C₂²×C₂₃	C₂×C₄×D₄	1	1	yes	characteristic
325	C₂²×C₂₃	#520	1	1	yes	characteristic
326	C₂²×C₂₃	C₂×#131	1	1	yes	characteristic
327	C₂²×C₂₃	C₂×#131	1	1	yes	characteristic
328	C₂²×C₂₃	C₂×#131	1	1	yes	characteristic
329	C₂²×C₂₃	#568	1	1	yes	characteristic
330	C₄×C₂₃	—	2	2	no	—
331	C₄×C₂₃	—	2	2	no	—
332	C₄×C₂₃	—	2	2	no	—
333	C₄×C₂₃	—	2	2	no	—
334	C₄×C₂₃	—	2	2	no	—
335	C₄×C₂₃	—	2	2	no	—
336	C₄×C₂₃	—	4	1	no	—
337	C₄×C₂₃	—	4	1	no	—
338	C₄×C₂₃	—	4	1	no	—
339	C₄×C₂₃	—	4	1	no	—
340	C₄×C₂₃	—	4	1	no	—
341	C₄×C₂₃	—	4	1	no	—
342	C₄×C₂₃	#546	1	1	yes	characteristic
343	C₄×C₂₃	#548	1	1	yes	characteristic
344	C₄×C₂₃	#552	1	1	yes	characteristic
345	C₄×C₂₃	#553	1	1	yes	characteristic
346	C₂²×M₁₆	C₂²×C₂₃	1	1	yes	characteristic
347	C₂³×C₈	C₂²×C₂₃	1	1	yes	characteristic
348	C₂×C₄×D₄	C₂²×C₂₃	1	1	yes	characteristic
349	C₂×#117	C₂²×C₂₃	1	1	yes	characteristic
350	C₂×#117	C₂²×C₂₃	1	1	yes	characteristic
351	#433	—	2	4	no	—
352	#424	—	2	4	no	—
353	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
354	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
355	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
356	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
357	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
358	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
359	C₂×C₂²⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
360	C₂×C₂²⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
361	C₂×C₂²⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
362	C₂×C₂²⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
363	C₂×C₂²⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
364	C₂×C₂²⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
365	C₂×#97	C₂²×C₂₃	1	1	yes	characteristic
366	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
367	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
368	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
369	#390	C₄×C₂₃	1	1	yes	characteristic
370	#390	C₄×C₂₃	1	1	yes	characteristic
371	#389	C₄×C₂₃	1	1	yes	characteristic
372	#389	C₄×C₂₃	1	1	yes	characteristic
373	#388	C₄×C₂₃	1	1	yes	characteristic
374	#388	C₄×C₂₃	1	1	yes	characteristic
375	#388	C₄×C₂₃	1	1	yes	characteristic
376	#388	C₄×C₂₃	1	1	yes	characteristic
377	C₄×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
378	C₄×C₂²⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
379	C₄×C₂²⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
380	C₄³	C₂²×C₂₃	1	1	yes	characteristic
381	#336	C₂²×C₂₃	1	1	yes	characteristic
382	#336	C₂²×C₂₃	1	1	yes	characteristic
383	#336	C₂²×C₂₃	1	1	yes	characteristic
384	#336	C₂²×C₂₃	1	1	yes	characteristic
385	#336	C₂²×C₂₃	1	1	yes	characteristic
386	#336	C₂²×C₂₃	1	1	yes	characteristic
387	#336	C₂²×C₂₃	1	1	yes	characteristic
388	#336	C₂²×C₂₃	1	1	yes	characteristic
389	#336	C₂²×C₂₃	1	1	yes	characteristic
390	#336	C₂²×C₂₃	1	1	yes	characteristic
391	C₂×C₂₃	—	4	2	no	—
392	C₂×C₂₃	—	8	1	no	—
393	C₂×C₂₃	#2352	1	1	yes	characteristic
394	C₂×C₂₃	#2922	1	1	yes	characteristic
395	C₂×C₂₃	#2924	1	1	yes	characteristic
396	C₂×C₂₃	#2931	1	1	yes	characteristic
397	C₂×C₂₃	#2935	1	1	yes	characteristic
398	C₂×C₂₃	#2992	1	1	yes	characteristic
399	C₂×C₂₃	#2994	1	1	yes	characteristic
400	C₂²×D₄	C₂×C₄×C₂₃	1	1	yes	characteristic
401	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
402	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
403	C₂×M₁₆	—	2	2	no	—
404	C₂×M₁₆	—	2	2	no	—
405	C₂×M₁₆	—	4	1	no	—
406	C₂²×C₈	—	2	2	no	—
407	C₂²×C₈	—	2	2	no	—
408	C₂²×C₈	—	4	1	no	—
409	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
410	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
411	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
412	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
413	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
414	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
415	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
416	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
417	C₄×D₄	—	2	2	no	—
418	C₄×D₄	—	2	2	no	—
419	#117	—	2	2	no	—
420	#117	—	2	2	no	—
421	#117	—	2	2	no	—
422	#117	—	2	2	no	—
423	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
424	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
425	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
426	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
427	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
428	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
429	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
430	C₂×C₂²⋊C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
431	C₂×C₂²⋊C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
432	C₂×C₂²⋊C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
433	C₂×C₂²⋊C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
434	C₂×C₂²⋊C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
435	C₂×C₂²⋊C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
436	C₂×C₂²⋊C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
437	C₂×C₂²⋊C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
438	C₂×C₂²⋊C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
439	C₂×C₂²⋊C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
440	C₂×C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
441	C₂×C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
442	C₂×C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
443	C₂×C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
444	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
445	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
446	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
447	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
448	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
449	C₄⋊C₈	—	2	2	no	—
450	C₄⋊C₈	—	2	2	no	—
451	C₄⋊C₈	—	2	2	no	—
452	C₄⋊C₈	—	2	2	no	—
453	C₄⋊C₈	—	2	2	no	—
454	C₄⋊C₈	D₄×C₂₃	1	4	yes	complemented
455	C₂²⋊C₈	—	2	2	no	—
456	C₂²⋊C₈	—	2	2	no	—
457	C₂²⋊C₈	—	2	2	no	—
458	C₂²⋊C₈	—	2	2	no	—
459	C₂²⋊C₈	—	4	1	no	—
460	C₂²⋊C₈	—	4	1	no	—
461	#97	—	2	2	no	—
462	C₄×C₈	—	2	2	no	—
463	C₄×C₈	—	4	1	no	—
464	C₄×C₈	—	4	1	no	—
465	#95	C₂×C₄×C₂₃	1	1	yes	characteristic
466	#95	C₂×C₄×C₂₃	1	1	yes	characteristic
467	#95	C₂×C₄×C₂₃	1	1	yes	characteristic
468	#95	C₂×C₄×C₂₃	1	1	yes	characteristic
469	#95	C₂×C₄×C₂₃	1	1	yes	characteristic
470	#95	C₂×C₄×C₂₃	1	1	yes	characteristic
471	C₂₃	#22756	1	1	yes	characteristic, direct product factor, Sylow
472	C₂⁴	C₂²×C₄×C₂₃	1	1	yes	characteristic
473	C₂⁴	C₂²×C₄×C₂₃	1	1	yes	characteristic
474	C₂×D₄	—	2	2	no	—
475	C₂×D₄	—	2	2	no	—
476	C₂×D₄	—	4	1	no	—
477	C₂²×C₄	—	2	1	no	—
478	C₂²×C₄	—	2	1	no	—
479	C₂²×C₄	—	2	1	no	—
480	C₂²×C₄	—	2	1	no	—
481	C₂²×C₄	—	2	1	no	—
482	C₂²×C₄	—	2	1	no	—
483	C₂²×C₄	—	2	1	no	—
484	C₂²×C₄	—	2	1	no	—
485	C₂²×C₄	—	2	1	no	—
486	C₂²×C₄	—	2	1	no	—
487	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
488	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
489	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
490	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
491	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
492	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
493	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
494	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
495	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
496	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
497	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
498	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
499	C₂²×C₄	#41×C₂₃	1	2	yes	—
500	C₂²×C₄	#41×C₂₃	1	2	yes	—
501	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
502	M₁₆	—	4	4	no	—
503	C₂×C₈	—	2	1	no	—
504	C₂×C₈	—	2	1	no	—
505	C₂×C₈	—	2	1	no	—
506	C₂×C₈	—	2	1	no	—
507	C₂×C₈	—	2	1	no	—
508	C₂×C₈	—	2	1	no	—
509	C₂×C₈	—	2	1	no	—
510	C₂×C₈	—	2	1	no	—
511	C₂×C₈	—	2	1	no	—
512	C₂×C₈	—	2	1	no	—
513	C₂×C₈	—	2	1	no	—
514	C₂×C₈	—	2	1	no	—
515	C₂×C₈	—	2	1	no	—
516	C₂×C₈	—	2	1	no	—
517	C₂×C₈	—	2	1	no	—
518	C₂×C₈	—	2	1	no	—
519	C₂×C₈	—	2	1	no	—
520	C₂×C₈	—	2	1	no	—
521	C₂×C₈	—	2	1	no	—
522	C₂×C₈	—	2	1	no	—
523	C₂×C₈	—	2	1	no	—
524	C₂×C₈	—	2	1	no	—
525	C₂×C₈	—	4	4	no	—
526	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
527	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
528	C₄⋊C₄	—	2	2	no	—
529	C₄⋊C₄	—	2	2	no	—
530	C₄⋊C₄	—	2	2	no	—
531	C₄⋊C₄	—	2	2	no	—
532	C₄⋊C₄	—	2	2	no	—
533	C₄⋊C₄	—	2	2	no	—
534	C₄⋊C₄	—	4	1	no	—
535	C₂²⋊C₄	—	2	2	no	—
536	C₂²⋊C₄	—	2	2	no	—
537	C₂²⋊C₄	—	2	2	no	—
538	C₂²⋊C₄	—	2	2	no	—
539	C₂²⋊C₄	—	2	2	no	—
540	C₂²⋊C₄	—	2	2	no	—
541	C₂²⋊C₄	—	2	2	no	—
542	C₂²⋊C₄	—	2	2	no	—
543	C₂²⋊C₄	—	2	2	no	—
544	C₂²⋊C₄	—	2	2	no	—
545	C₄²	—	2	2	no	—
546	C₄²	—	2	2	no	—
547	C₄²	—	2	2	no	—
548	C₄²	—	2	1	no	—
549	C₄²	—	2	1	no	—
550	C₄²	—	2	1	no	—
551	C₄²	—	2	1	no	—
552	C₄²	—	2	1	no	—
553	C₄²	—	2	1	no	—
554	C₄²	—	2	1	no	—
555	C₄²	—	2	1	no	—
556	C₄²	—	4	1	no	—
557	C₄²	C₂×D₄×C₂₃	1	2	yes	—
558	C₄²	#41×C₂₃	1	2	yes	—
559	C₂³	—	2	1	no	—
560	C₂³	—	2	1	no	—
561	C₂³	—	2	1	no	—
562	C₂³	—	2	1	no	—
563	C₂³	—	2	1	no	—
564	C₂³	—	2	1	no	—
565	C₂³	—	2	1	no	—
566	C₂³	—	2	1	no	—
567	C₂³	—	2	1	no	—
568	C₂³	—	2	1	no	—
569	C₂³	—	2	1	no	—
570	C₂³	—	2	1	no	—
571	C₂³	—	2	1	no	—
572	C₂³	C₂₃×#117	1	2	yes	—
573	C₂³	C₂³×C₄×C₂₃	1	1	yes	characteristic
574	D₄	—	4	4	no	—
575	C₂×C₄	—	2	1	no	—
576	C₂×C₄	—	2	1	no	—
577	C₂×C₄	—	2	1	no	—
578	C₂×C₄	—	2	1	no	—
579	C₂×C₄	—	2	1	no	—
580	C₂×C₄	—	2	1	no	—
581	C₂×C₄	—	2	1	no	—
582	C₂×C₄	—	2	1	no	—
583	C₂×C₄	—	2	1	no	—
584	C₂×C₄	—	2	1	no	—
585	C₂×C₄	—	2	1	no	—
586	C₂×C₄	—	2	1	no	—
587	C₂×C₄	—	2	2	no	—
588	C₂×C₄	—	2	2	no	—
589	C₂×C₄	—	2	2	no	—
590	C₂×C₄	—	2	2	no	—
591	C₂×C₄	—	2	1	no	—
592	C₂×C₄	—	2	1	no	—
593	C₂×C₄	—	2	1	no	—
594	C₂×C₄	—	2	1	no	—
595	C₂×C₄	—	2	1	no	—
596	C₂×C₄	—	2	1	no	—
597	C₂×C₄	—	2	1	no	—
598	C₂×C₄	—	2	1	no	—
599	C₂×C₄	—	2	1	no	—
600	C₂×C₄	—	2	1	no	—
601	C₂×C₄	—	2	1	no	—
602	C₂×C₄	—	2	1	no	—
603	C₂×C₄	—	2	1	no	—
604	C₂×C₄	—	2	1	no	—
605	C₂×C₄	—	2	1	no	—
606	C₂×C₄	—	2	1	no	—
607	C₂×C₄	—	2	1	no	—
608	C₂×C₄	—	2	1	no	—
609	C₂×C₄	—	4	1	no	—
610	C₂×C₄	—	4	1	no	—
611	C₂×C₄	—	4	1	no	—
612	C₂×C₄	—	4	1	no	—
613	C₂×C₄	C₂₃×#117	1	2	yes	—
614	C₂×C₄	C₄×D₄×C₂₃	1	2	yes	—
615	C₂×C₄	C₄×D₄×C₂₃	1	2	yes	—
616	C₂×C₄	C₂₃×#131	1	2	yes	—
617	C₂×C₄	C₂₃×#131	1	2	yes	—
618	C₂×C₄	C₂₃×#131	1	2	yes	—
619	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
620	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
621	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
622	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
623	C₂×C₄	C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
624	C₂×C₄	C₂₃×2₋¹⁺⁴	1	1	yes	characteristic
625	C₈	—	2	2	no	—
626	C₈	—	2	2	no	—
627	C₈	—	4	1	no	—
628	C₈	—	4	1	no	—
629	C₈	—	4	1	no	—
630	C₈	—	4	1	no	—
631	C₈	—	4	1	no	—
632	C₈	—	4	1	no	—
633	C₂²	—	2	2	no	—
634	C₂²	—	2	2	no	—
635	C₂²	—	2	2	no	—
636	C₂²	—	4	1	no	—
637	C₂²	—	4	1	no	—
638	C₂²	—	4	1	no	—
639	C₂²	—	4	1	no	—
640	C₂²	—	4	1	no	—
641	C₂²	—	4	1	no	—
642	C₂²	—	4	1	no	—
643	C₂²	—	4	1	no	—
644	C₂²	—	4	1	no	—
645	C₂²	—	4	1	no	—
646	C₂²	—	4	1	no	—
647	C₂²	C₂×C₂₃×#117	1	1	yes	characteristic
648	C₂²	C₂×C₄×D₄×C₂₃	1	1	yes	characteristic
649	C₂²	C₂₃×#520	1	1	yes	characteristic
650	C₂²	C₂×C₂₃×#131	1	1	yes	characteristic
651	C₂²	C₂×C₂₃×#131	1	1	yes	characteristic
652	C₂²	C₂×C₂₃×#131	1	1	yes	characteristic
653	C₂²	C₂₃×#568	1	1	yes	characteristic
654	C₄	—	2	2	no	—
655	C₄	—	2	2	no	—
656	C₄	—	2	2	no	—
657	C₄	—	2	2	no	—
658	C₄	—	2	2	no	—
659	C₄	—	2	2	no	—
660	C₄	—	4	1	no	—
661	C₄	—	4	1	no	—
662	C₄	—	4	1	no	—
663	C₄	—	4	1	no	—
664	C₄	—	4	1	no	—
665	C₄	—	4	1	no	—
666	C₄	C₂₃×#546	1	1	yes	characteristic
667	C₄	C₂₃×#548	1	1	yes	characteristic
668	C₄	C₂₃×#552	1	1	yes	characteristic
669	C₄	C₂₃×#553	1	1	yes	characteristic
670	C₂	—	4	2	no	—
671	C₂	—	8	1	no	—
672	C₂	C₂₃×#2352	1	1	yes	characteristic
673	C₂	C₂₃×#2922	1	1	yes	characteristic
674	C₂	C₂₃×#2924	1	1	yes	characteristic
675	C₂	C₂₃×#2931	1	1	yes	characteristic
676	C₂	C₂₃×#2935	1	1	yes	characteristic
677	C₂	C₂₃×#2992	1	1	yes	characteristic
678	C₂	C₂₃×#2994	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₂₃	C₂⁴
commutator	C₂³	C₂³×C₄×C₂₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₂₃
socle	C₂³×C₂₃	C₂³×C₄
prime radical	C₂²×D₄×C₂₃	C₂×C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₂₃ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. C₂×C₂₃×#433	C₂	semidirect
2. C₂×C₂₃×#424	C₂	semidirect
4. C₂₃×#1923	C₂	semidirect
5. C₂₃×#1923	C₂	semidirect
6. C₂₃×#1917	C₂	semidirect
7. C₂₃×#1916	C₂	semidirect
10. C₂₃×#1831	C₂	semidirect
11. C₂₃×#1798	C₂	semidirect
12. C₂₃×#1798	C₂	semidirect
13. C₄×C₂₃×C₄⋊C₈	C₂	semidirect
14. C₂₃×#1753	C₂	semidirect
15. C₈×C₂²⋊C₄×C₂₃	C₂	semidirect
115. C₂₃×C₄⋊C₈	D₄	semidirect
219. #22756	C₂₃	direct (N × H)
294. C₂×#433	C₂×C₂₃	semidirect
295. C₂×#424	C₂×C₂₃	semidirect
297. #1923	C₂×C₂₃	semidirect
298. #1923	C₂×C₂₃	semidirect
299. #1917	C₂×C₂₃	semidirect
300. #1916	C₂×C₂₃	semidirect
303. #1831	C₂×C₂₃	semidirect
304. #1798	C₂×C₂₃	semidirect
305. #1798	C₂×C₂₃	semidirect
306. C₄×C₄⋊C₈	C₂×C₂₃	semidirect
307. #1753	C₂×C₂₃	semidirect
308. C₈×C₂²⋊C₄	C₂×C₂₃	semidirect
454. C₄⋊C₈	D₄×C₂₃	semidirect
471. C₂₃	#22756	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₂₃×#433	2
C₂×C₂₃×#424	2
C₂₃×#2293	2
C₂₃×#1923	2
C₂₃×#1917	2
C₂₃×#1916	2
C₂₃×#1870	2
C₂₃×#1869	2
C₂₃×#1831	2
C₂₃×#1798	2
C₄×C₂₃×C₄⋊C₈	2
C₂₃×#1753	2
C₈×C₂²⋊C₄×C₂₃	2
#22756	23

Subgroups — C23×#22756 #19170382

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#22756 #19170382