Subgroups — C₂₃×#22600 #19170226

Order 5888.

800 conjugacy classes of subgroups.

The subgroup lattice has 472 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	2	yes	maximal
2	C₂₃×#2285	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#1939	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×#1939	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂₃×#1917	C₂	1	2	yes	maximal, complemented
6	C₂₃×#1916	C₂	1	2	yes	maximal, complemented
7	C₂₃×#1818	C₂	1	2	yes	maximal
8	C₂₃×#1797	C₂	1	1	yes	maximal, characteristic
9	C₂₃×#1770	C₂	1	1	yes	maximal, characteristic, complemented
10	C₈×C₄⋊C₄×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×C₂₃×C₂²⋊C₈	C₂	1	1	yes	maximal, characteristic
12	C₂×C₂₃×#117	C₂²	1	2	yes	—
13	C₂²×C₄²×C₂₃	C₂²	1	1	yes	characteristic
14	C₂₃×#433	—	2	8	no	—
15	C₂×C₂₃×C₄⋊C₈	C₂²	1	2	yes	—
16	C₂×C₂₃×C₄⋊C₈	C₂²	1	2	yes	—
17	C₂×C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
18	C₂×C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
19	C₂×C₂₃×C₂²⋊C₈	C₂²	1	2	yes	—
20	C₂×C₂₃×C₂²⋊C₈	C₂²	1	1	yes	characteristic
21	C₂×C₂₃×C₂²⋊C₈	C₂²	1	1	yes	characteristic
22	C₂×C₂₃×#97	C₂²	1	2	yes	—
23	C₂×C₄×C₈×C₂₃	C₂²	1	2	yes	—
24	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
25	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
26	C₂₃×#384	C₄	1	1	yes	characteristic
27	C₂₃×#384	C₄	1	1	yes	characteristic
28	C₂₃×#382	C₄	1	2	yes	—
29	C₂₃×#380	C₄	1	1	yes	characteristic
30	C₂₃×#380	C₄	1	1	yes	characteristic
31	C₂₃×#380	C₄	1	2	yes	—
32	C₄×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
33	C₄×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
34	C₄×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
35	C₂₃×#336	C₂²	1	1	yes	characteristic
36	C₂₃×#336	C₂²	1	2	yes	—
37	C₂₃×#336	C₂²	1	2	yes	—
38	C₂₃×#336	C₂²	1	2	yes	—
39	C₂₃×#336	C₂²	1	1	yes	characteristic
40	C₂₃×#336	C₂²	1	2	yes	—
41	C₂₃×#336	C₂²	1	2	yes	—
42	C₂³×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
43	C₂³×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
44	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
45	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
46	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
47	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
48	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
49	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
50	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
51	C₂₃×#117	—	2	4	no	—
52	C₂₃×#117	—	2	4	no	—
53	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	2	yes	—
54	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
55	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
56	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
57	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
58	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	2	yes	—
59	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	2	yes	—
60	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	2	yes	—
61	C₂×C₂²⋊C₄×C₂₃	C₂×C₄	1	2	yes	—
62	C₂×C₂²⋊C₄×C₂₃	C₂×C₄	1	2	yes	—
63	C₂×C₄²×C₂₃	—	2	2	no	—
64	C₂×C₄²×C₂₃	—	2	2	no	—
65	C₂×C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
66	C₂×C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
67	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
68	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
69	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
70	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
71	C₂₃×C₄⋊C₈	—	2	4	no	—
72	C₂₃×C₄⋊C₈	—	2	4	no	—
73	C₂₃×C₄⋊C₈	—	2	2	no	—
74	C₂₃×C₄⋊C₈	—	2	2	no	—
75	C₂₃×C₂²⋊C₈	—	2	4	no	—
76	C₂₃×C₂²⋊C₈	—	2	2	no	—
77	C₂₃×C₂²⋊C₈	D₄	1	4	yes	—
78	C₂₃×#97	—	2	4	no	—
79	C₄×C₈×C₂₃	—	2	4	no	—
80	C₄×C₈×C₂₃	—	2	2	no	—
81	C₄×C₈×C₂₃	—	2	2	no	—
82	C₂₃×#95	C₂×C₄	1	2	yes	—
83	C₂₃×#95	C₂×C₄	1	2	yes	—
84	C₂₃×#95	C₂×C₄	1	2	yes	—
85	C₂₃×#95	C₂×C₄	1	2	yes	—
86	C₂⁴×C₂₃	C₂²×C₄	1	1	yes	characteristic
87	C₂²×C₄×C₂₃	—	2	1	no	—
88	C₂²×C₄×C₂₃	—	2	1	no	—
89	C₂²×C₄×C₂₃	—	2	1	no	—
90	C₂²×C₄×C₂₃	—	2	1	no	—
91	C₂²×C₄×C₂₃	—	2	2	no	—
92	C₂²×C₄×C₂₃	—	2	2	no	—
93	C₂²×C₄×C₂₃	—	2	2	no	—
94	C₂²×C₄×C₂₃	—	2	2	no	—
95	C₂²×C₄×C₂₃	—	2	2	no	—
96	C₂²×C₄×C₂₃	—	4	1	no	—
97	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
98	C₂²×C₄×C₂₃	C₂²×C₄	1	2	yes	—
99	C₂²×C₄×C₂₃	C₂²×C₄	1	2	yes	—
100	C₂²×C₄×C₂₃	C₂²×C₄	1	2	yes	—
101	C₂²×C₄×C₂₃	C₂²×C₄	1	2	yes	—
102	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
103	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
104	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
105	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
106	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
107	C₂²×C₄×C₂₃	#41	1	2	yes	—
108	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
109	C₂×C₈×C₂₃	—	2	1	no	—
110	C₂×C₈×C₂₃	—	2	1	no	—
111	C₂×C₈×C₂₃	—	2	1	no	—
112	C₂×C₈×C₂₃	—	2	1	no	—
113	C₂×C₈×C₂₃	—	2	2	no	—
114	C₂×C₈×C₂₃	—	2	2	no	—
115	C₂×C₈×C₂₃	—	2	2	no	—
116	C₂×C₈×C₂₃	—	2	2	no	—
117	C₂×C₈×C₂₃	—	2	1	no	—
118	C₂×C₈×C₂₃	—	2	1	no	—
119	C₂×C₈×C₂₃	—	2	1	no	—
120	C₂×C₈×C₂₃	—	2	1	no	—
121	C₂×C₈×C₂₃	—	2	2	no	—
122	C₂×C₈×C₂₃	—	2	2	no	—
123	C₂×C₈×C₂₃	—	2	1	no	—
124	C₂×C₈×C₂₃	—	2	1	no	—
125	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
126	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
127	C₄⋊C₄×C₂₃	—	2	4	no	—
128	C₄⋊C₄×C₂₃	—	2	4	no	—
129	C₄⋊C₄×C₂₃	—	2	4	no	—
130	C₄⋊C₄×C₂₃	—	2	2	no	—
131	C₄⋊C₄×C₂₃	—	2	2	no	—
132	C₄⋊C₄×C₂₃	—	2	2	no	—
133	C₄⋊C₄×C₂₃	—	2	2	no	—
134	C₄⋊C₄×C₂₃	—	4	2	no	—
135	C₂²⋊C₄×C₂₃	—	2	4	no	—
136	C₂²⋊C₄×C₂₃	—	2	4	no	—
137	C₄²×C₂₃	—	2	2	no	—
138	C₄²×C₂₃	—	2	2	no	—
139	C₄²×C₂₃	—	2	2	no	—
140	C₄²×C₂₃	—	2	2	no	—
141	C₄²×C₂₃	—	2	1	no	—
142	C₄²×C₂₃	—	2	1	no	—
143	C₄²×C₂₃	—	2	2	no	—
144	C₄²×C₂₃	—	2	2	no	—
145	C₄²×C₂₃	#41	1	2	yes	—
146	C₄²×C₂₃	#41	1	2	yes	—
147	#22600	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
148	C₂³×C₂₃	—	2	1	no	—
149	C₂³×C₂₃	—	2	1	no	—
150	C₂³×C₂₃	—	2	1	no	—
151	C₂³×C₂₃	—	2	1	no	—
152	C₂³×C₂₃	—	2	1	no	—
153	C₂³×C₂₃	—	2	1	no	—
154	C₂³×C₂₃	C₄×D₄	1	2	yes	—
155	C₂³×C₂₃	C₂³×C₄	1	1	yes	characteristic
156	C₂×C₄×C₂₃	—	2	1	no	—
157	C₂×C₄×C₂₃	—	2	1	no	—
158	C₂×C₄×C₂₃	—	2	1	no	—
159	C₂×C₄×C₂₃	—	2	1	no	—
160	C₂×C₄×C₂₃	—	2	1	no	—
161	C₂×C₄×C₂₃	—	2	1	no	—
162	C₂×C₄×C₂₃	—	2	2	no	—
163	C₂×C₄×C₂₃	—	2	2	no	—
164	C₂×C₄×C₂₃	—	2	2	no	—
165	C₂×C₄×C₂₃	—	2	2	no	—
166	C₂×C₄×C₂₃	—	2	1	no	—
167	C₂×C₄×C₂₃	—	2	2	no	—
168	C₂×C₄×C₂₃	—	2	2	no	—
169	C₂×C₄×C₂₃	—	2	2	no	—
170	C₂×C₄×C₂₃	—	2	2	no	—
171	C₂×C₄×C₂₃	—	2	2	no	—
172	C₂×C₄×C₂₃	—	2	2	no	—
173	C₂×C₄×C₂₃	—	2	2	no	—
174	C₂×C₄×C₂₃	—	2	2	no	—
175	C₂×C₄×C₂₃	—	2	2	no	—
176	C₂×C₄×C₂₃	—	2	2	no	—
177	C₂×C₄×C₂₃	—	2	1	no	—
178	C₂×C₄×C₂₃	—	2	1	no	—
179	C₂×C₄×C₂₃	—	2	2	no	—
180	C₂×C₄×C₂₃	—	2	2	no	—
181	C₂×C₄×C₂₃	—	4	4	no	—
182	C₂×C₄×C₂₃	—	4	4	no	—
183	C₂×C₄×C₂₃	#117	1	2	yes	—
184	C₂×C₄×C₂₃	#117	1	2	yes	—
185	C₂×C₄×C₂₃	C₄×D₄	1	2	yes	—
186	C₂×C₄×C₂₃	#131	1	2	yes	—
187	C₂×C₄×C₂₃	#131	1	2	yes	—
188	C₂×C₄×C₂₃	#131	1	2	yes	—
189	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
190	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
191	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
192	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
193	C₂×C₄×C₂₃	2₋¹⁺⁴	1	1	yes	characteristic
194	C₂×C₄×C₂₃	2₋¹⁺⁴	1	1	yes	characteristic
195	C₈×C₂₃	—	2	2	no	—
196	C₈×C₂₃	—	2	2	no	—
197	C₈×C₂₃	—	4	2	no	—
198	C₈×C₂₃	—	4	2	no	—
199	C₈×C₂₃	—	4	1	no	—
200	C₈×C₂₃	—	4	1	no	—
201	C₂×#433	C₂×C₂₃	1	2	yes	—
202	#2285	C₂×C₂₃	1	1	yes	characteristic
203	#1939	C₂×C₂₃	1	1	yes	characteristic, complemented
204	#1939	C₂×C₂₃	1	1	yes	characteristic, complemented
205	#1917	C₂×C₂₃	1	2	yes	complemented
206	#1916	C₂×C₂₃	1	2	yes	complemented
207	#1818	C₂×C₂₃	1	2	yes	—
208	#1797	C₂×C₂₃	1	1	yes	characteristic
209	#1770	C₂×C₂₃	1	1	yes	characteristic, complemented
210	C₈×C₄⋊C₄	C₂×C₂₃	1	1	yes	characteristic, complemented
211	C₄×C₂²⋊C₈	C₂×C₂₃	1	1	yes	characteristic
212	C₂²×C₂₃	—	2	2	no	—
213	C₂²×C₂₃	—	2	2	no	—
214	C₂²×C₂₃	—	2	2	no	—
215	C₂²×C₂₃	—	4	1	no	—
216	C₂²×C₂₃	—	4	1	no	—
217	C₂²×C₂₃	—	4	1	no	—
218	C₂²×C₂₃	—	4	1	no	—
219	C₂²×C₂₃	C₂×#117	1	1	yes	characteristic
220	C₂²×C₂₃	C₂×C₄×D₄	1	1	yes	characteristic
221	C₂²×C₂₃	#519	1	1	yes	characteristic
222	C₂²×C₂₃	C₂×#131	1	1	yes	characteristic
223	C₂²×C₂₃	C₂×#131	1	1	yes	characteristic
224	C₂²×C₂₃	C₂×#131	1	1	yes	characteristic
225	C₂²×C₂₃	#568	1	1	yes	characteristic
226	C₄×C₂₃	—	2	2	no	—
227	C₄×C₂₃	—	2	2	no	—
228	C₄×C₂₃	—	2	2	no	—
229	C₄×C₂₃	—	4	1	no	—
230	C₄×C₂₃	—	4	2	no	—
231	C₄×C₂₃	—	4	2	no	—
232	C₄×C₂₃	—	4	2	no	—
233	C₄×C₂₃	—	4	2	no	—
234	C₄×C₂₃	#433	1	4	yes	complemented
235	C₄×C₂₃	#547	1	1	yes	characteristic
236	C₄×C₂₃	#547	1	1	yes	characteristic
237	C₄×C₂₃	#552	1	1	yes	characteristic
238	C₄×C₂₃	#552	1	1	yes	characteristic
239	C₂×#117	C₂²×C₂₃	1	2	yes	—
240	C₂²×C₄²	C₂²×C₂₃	1	1	yes	characteristic
241	#433	—	2	8	no	—
242	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
243	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
244	C₂×C₄⋊C₈	C₂²×C₂₃	1	2	yes	—
245	C₂×C₄⋊C₈	C₂²×C₂₃	1	2	yes	—
246	C₂×C₂²⋊C₈	C₂²×C₂₃	1	2	yes	—
247	C₂×C₂²⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
248	C₂×C₂²⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
249	C₂×#97	C₂²×C₂₃	1	2	yes	—
250	C₂×C₄×C₈	C₂²×C₂₃	1	2	yes	—
251	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
252	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
253	#384	C₄×C₂₃	1	1	yes	characteristic
254	#384	C₄×C₂₃	1	1	yes	characteristic
255	#382	C₄×C₂₃	1	2	yes	—
256	#380	C₄×C₂₃	1	1	yes	characteristic
257	#380	C₄×C₂₃	1	1	yes	characteristic
258	#380	C₄×C₂₃	1	2	yes	—
259	C₄×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
260	C₄×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
261	C₄×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
262	#336	C₂²×C₂₃	1	2	yes	—
263	#336	C₂²×C₂₃	1	2	yes	—
264	#336	C₂²×C₂₃	1	2	yes	—
265	#336	C₂²×C₂₃	1	2	yes	—
266	#336	C₂²×C₂₃	1	2	yes	—
267	#336	C₂²×C₂₃	1	1	yes	characteristic
268	#336	C₂²×C₂₃	1	1	yes	characteristic
269	C₂×C₂₃	—	4	2	no	—
270	C₂×C₂₃	#2344	1	1	yes	characteristic
271	C₂×C₂₃	C₂×#433	1	1	yes	characteristic
272	C₂×C₂₃	#2924	1	1	yes	characteristic
273	C₂×C₂₃	#2931	1	1	yes	characteristic
274	C₂×C₂₃	#2935	1	1	yes	characteristic
275	C₂×C₂₃	#2985	1	1	yes	characteristic
276	C₂×C₂₃	#2986	1	1	yes	characteristic
277	C₂³×C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
278	C₂³×C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
279	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
280	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
281	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
282	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
283	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
284	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
285	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
286	#117	—	2	4	no	—
287	#117	—	2	4	no	—
288	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	2	yes	—
289	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	2	yes	—
290	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	2	yes	—
291	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
292	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
293	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
294	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
295	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	2	yes	—
296	C₂×C₂²⋊C₄	C₂×C₄×C₂₃	1	2	yes	—
297	C₂×C₂²⋊C₄	C₂×C₄×C₂₃	1	2	yes	—
298	C₂×C₄²	—	2	2	no	—
299	C₂×C₄²	—	2	2	no	—
300	C₂×C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
301	C₂×C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
302	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
303	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
304	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
305	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
306	C₄⋊C₈	—	2	4	no	—
307	C₄⋊C₈	—	2	4	no	—
308	C₄⋊C₈	—	2	2	no	—
309	C₄⋊C₈	—	2	2	no	—
310	C₂²⋊C₈	—	2	4	no	—
311	C₂²⋊C₈	—	2	2	no	—
312	C₂²⋊C₈	D₄×C₂₃	1	4	yes	—
313	#97	—	2	4	no	—
314	C₄×C₈	—	2	4	no	—
315	C₄×C₈	—	2	2	no	—
316	C₄×C₈	—	2	2	no	—
317	#95	C₂×C₄×C₂₃	1	2	yes	—
318	#95	C₂×C₄×C₂₃	1	2	yes	—
319	#95	C₂×C₄×C₂₃	1	2	yes	—
320	#95	C₂×C₄×C₂₃	1	2	yes	—
321	C₂₃	#22600	1	1	yes	characteristic, direct product factor, Sylow
322	C₂⁴	C₂²×C₄×C₂₃	1	1	yes	characteristic
323	C₂²×C₄	—	2	1	no	—
324	C₂²×C₄	—	2	1	no	—
325	C₂²×C₄	—	2	1	no	—
326	C₂²×C₄	—	2	1	no	—
327	C₂²×C₄	—	2	2	no	—
328	C₂²×C₄	—	2	2	no	—
329	C₂²×C₄	—	2	2	no	—
330	C₂²×C₄	—	2	2	no	—
331	C₂²×C₄	—	2	2	no	—
332	C₂²×C₄	—	4	1	no	—
333	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
334	C₂²×C₄	C₂²×C₄×C₂₃	1	2	yes	—
335	C₂²×C₄	C₂²×C₄×C₂₃	1	2	yes	—
336	C₂²×C₄	C₂²×C₄×C₂₃	1	2	yes	—
337	C₂²×C₄	C₂²×C₄×C₂₃	1	2	yes	—
338	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
339	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
340	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
341	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
342	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
343	C₂²×C₄	#41×C₂₃	1	2	yes	—
344	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
345	C₂×C₈	—	2	1	no	—
346	C₂×C₈	—	2	1	no	—
347	C₂×C₈	—	2	1	no	—
348	C₂×C₈	—	2	1	no	—
349	C₂×C₈	—	2	2	no	—
350	C₂×C₈	—	2	2	no	—
351	C₂×C₈	—	2	1	no	—
352	C₂×C₈	—	2	1	no	—
353	C₂×C₈	—	2	1	no	—
354	C₂×C₈	—	2	1	no	—
355	C₂×C₈	—	2	2	no	—
356	C₂×C₈	—	2	2	no	—
357	C₂×C₈	—	2	2	no	—
358	C₂×C₈	—	2	2	no	—
359	C₂×C₈	—	2	1	no	—
360	C₂×C₈	—	2	1	no	—
361	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
362	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
363	C₄⋊C₄	—	2	4	no	—
364	C₄⋊C₄	—	2	4	no	—
365	C₄⋊C₄	—	2	4	no	—
366	C₄⋊C₄	—	2	2	no	—
367	C₄⋊C₄	—	2	2	no	—
368	C₄⋊C₄	—	2	2	no	—
369	C₄⋊C₄	—	2	2	no	—
370	C₄⋊C₄	—	4	2	no	—
371	C₂²⋊C₄	—	2	4	no	—
372	C₂²⋊C₄	—	2	4	no	—
373	C₄²	—	2	2	no	—
374	C₄²	—	2	2	no	—
375	C₄²	—	2	2	no	—
376	C₄²	—	2	2	no	—
377	C₄²	—	2	1	no	—
378	C₄²	—	2	1	no	—
379	C₄²	—	2	2	no	—
380	C₄²	—	2	2	no	—
381	C₄²	#41×C₂₃	1	2	yes	—
382	C₄²	#41×C₂₃	1	2	yes	—
383	C₂³	—	2	1	no	—
384	C₂³	—	2	1	no	—
385	C₂³	—	2	1	no	—
386	C₂³	—	2	1	no	—
387	C₂³	—	2	1	no	—
388	C₂³	—	2	1	no	—
389	C₂³	C₄×D₄×C₂₃	1	2	yes	—
390	C₂³	C₂³×C₄×C₂₃	1	1	yes	characteristic
391	C₂×C₄	—	2	1	no	—
392	C₂×C₄	—	2	1	no	—
393	C₂×C₄	—	2	1	no	—
394	C₂×C₄	—	2	1	no	—
395	C₂×C₄	—	2	1	no	—
396	C₂×C₄	—	2	1	no	—
397	C₂×C₄	—	2	2	no	—
398	C₂×C₄	—	2	2	no	—
399	C₂×C₄	—	2	2	no	—
400	C₂×C₄	—	2	2	no	—
401	C₂×C₄	—	2	2	no	—
402	C₂×C₄	—	2	2	no	—
403	C₂×C₄	—	2	2	no	—
404	C₂×C₄	—	2	2	no	—
405	C₂×C₄	—	2	2	no	—
406	C₂×C₄	—	2	2	no	—
407	C₂×C₄	—	2	1	no	—
408	C₂×C₄	—	2	1	no	—
409	C₂×C₄	—	2	2	no	—
410	C₂×C₄	—	2	2	no	—
411	C₂×C₄	—	2	1	no	—
412	C₂×C₄	—	2	2	no	—
413	C₂×C₄	—	2	2	no	—
414	C₂×C₄	—	2	2	no	—
415	C₂×C₄	—	2	2	no	—
416	C₂×C₄	—	4	4	no	—
417	C₂×C₄	—	4	4	no	—
418	C₂×C₄	C₂₃×#117	1	2	yes	—
419	C₂×C₄	C₂₃×#117	1	2	yes	—
420	C₂×C₄	C₄×D₄×C₂₃	1	2	yes	—
421	C₂×C₄	C₂₃×#131	1	2	yes	—
422	C₂×C₄	C₂₃×#131	1	2	yes	—
423	C₂×C₄	C₂₃×#131	1	2	yes	—
424	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
425	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
426	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
427	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
428	C₂×C₄	C₂₃×2₋¹⁺⁴	1	1	yes	characteristic
429	C₂×C₄	C₂₃×2₋¹⁺⁴	1	1	yes	characteristic
430	C₈	—	2	2	no	—
431	C₈	—	2	2	no	—
432	C₈	—	4	2	no	—
433	C₈	—	4	2	no	—
434	C₈	—	4	1	no	—
435	C₈	—	4	1	no	—
436	C₂²	—	2	2	no	—
437	C₂²	—	2	2	no	—
438	C₂²	—	2	2	no	—
439	C₂²	—	4	1	no	—
440	C₂²	—	4	1	no	—
441	C₂²	—	4	1	no	—
442	C₂²	—	4	1	no	—
443	C₂²	C₂×C₂₃×#117	1	1	yes	characteristic
444	C₂²	C₂×C₄×D₄×C₂₃	1	1	yes	characteristic
445	C₂²	C₂₃×#519	1	1	yes	characteristic
446	C₂²	C₂×C₂₃×#131	1	1	yes	characteristic
447	C₂²	C₂×C₂₃×#131	1	1	yes	characteristic
448	C₂²	C₂×C₂₃×#131	1	1	yes	characteristic
449	C₂²	C₂₃×#568	1	1	yes	characteristic
450	C₄	—	2	2	no	—
451	C₄	—	2	2	no	—
452	C₄	—	2	2	no	—
453	C₄	—	4	1	no	—
454	C₄	—	4	2	no	—
455	C₄	—	4	2	no	—
456	C₄	—	4	2	no	—
457	C₄	—	4	2	no	—
458	C₄	C₂₃×#433	1	4	yes	complemented
459	C₄	C₂₃×#547	1	1	yes	characteristic
460	C₄	C₂₃×#547	1	1	yes	characteristic
461	C₄	C₂₃×#552	1	1	yes	characteristic
462	C₄	C₂₃×#552	1	1	yes	characteristic
463	C₂	—	4	2	no	—
464	C₂	C₂₃×#2344	1	1	yes	characteristic
465	C₂	C₂×C₂₃×#433	1	1	yes	characteristic
466	C₂	C₂₃×#2924	1	1	yes	characteristic
467	C₂	C₂₃×#2931	1	1	yes	characteristic
468	C₂	C₂₃×#2935	1	1	yes	characteristic
469	C₂	C₂₃×#2985	1	1	yes	characteristic
470	C₂	C₂₃×#2986	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	C₂⁴×C₂₃	C₂²×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
3. C₂₃×#1939	C₂	semidirect
4. C₂₃×#1939	C₂	semidirect
5. C₂₃×#1917	C₂	semidirect
6. C₂₃×#1916	C₂	semidirect
9. C₂₃×#1770	C₂	semidirect
10. C₈×C₄⋊C₄×C₂₃	C₂	semidirect
147. #22600	C₂₃	direct (N × H)
203. #1939	C₂×C₂₃	semidirect
204. #1939	C₂×C₂₃	semidirect
205. #1917	C₂×C₂₃	semidirect
206. #1916	C₂×C₂₃	semidirect
209. #1770	C₂×C₂₃	semidirect
210. C₈×C₄⋊C₄	C₂×C₂₃	semidirect
234. C₄×C₂₃	#433	semidirect
321. C₂₃	#22600	direct (N × H)
458. C₄	C₂₃×#433	semidirect

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
C₂₃×#2285	2
C₂₃×#1939	2
C₂₃×#1917	2
C₂₃×#1916	2
C₂₃×#1818	2
C₂₃×#1797	2
C₂₃×#1770	2
C₈×C₄⋊C₄×C₂₃	2
C₄×C₂₃×C₂²⋊C₈	2
#22600	23

Subgroups — C23×#22600 #19170226

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#22600 #19170226