Subgroups — C₂₃×#22064 #19169690

Order 5888.

850 conjugacy classes of subgroups.

The subgroup lattice has 462 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈×D₄×C₂₃	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂₃×#424	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#2395	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#1939	C₂	1	2	yes	maximal, complemented
5	C₂₃×#1923	C₂	1	2	yes	maximal, complemented
6	C₂₃×#1916	C₂	1	2	yes	maximal, complemented
7	C₂₃×#1870	C₂	1	1	yes	maximal, characteristic
8	C₂₃×#1851	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂₃×#1850	C₂	1	2	yes	maximal, complemented
10	C₂₃×#1818	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈×C₂²⋊C₄×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
13	C₂³×C₈×C₂₃	C₂²	1	1	yes	characteristic
14	C₂³×C₈×C₂₃	C₂²	1	1	yes	characteristic
15	C₂×C₄×D₄×C₂₃	C₂²	1	1	yes	characteristic
16	C₂×C₂₃×#117	C₂²	1	1	yes	characteristic
17	C₈×D₄×C₂₃	—	2	4	no	—
18	C₂₃×#424	—	2	4	no	—
19	C₂×C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
20	C₂×C₂₃×C₄⋊C₈	C₂²	1	2	yes	—
21	C₂×C₂₃×C₄⋊C₈	C₂²	1	2	yes	—
22	C₂×C₂₃×C₄⋊C₈	C₂²	1	2	yes	—
23	C₂×C₂₃×C₂²⋊C₈	C₂²	1	1	yes	characteristic
24	C₂×C₂₃×C₂²⋊C₈	C₂²	1	1	yes	characteristic
25	C₂×C₂₃×C₂²⋊C₈	C₂²	1	1	yes	characteristic
26	C₂×C₂₃×C₂²⋊C₈	C₂²	1	2	yes	—
27	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
28	C₂×C₄×C₈×C₂₃	C₂²	1	2	yes	—
29	C₂₃×#400	C₄	1	1	yes	characteristic
30	C₂₃×#399	C₄	1	1	yes	characteristic
31	C₂₃×#398	C₄	1	1	yes	characteristic
32	C₂₃×#397	C₄	1	1	yes	characteristic
33	C₂₃×#397	C₄	1	2	yes	—
34	C₂₃×#394	C₄	1	2	yes	—
35	C₄×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
36	C₄×C₂²⋊C₄×C₂₃	C₂²	1	2	yes	—
37	C₄×C₂²⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
38	C₂₃×#336	C₂²	1	2	yes	—
39	C₂₃×#336	C₂²	1	2	yes	—
40	C₂₃×#336	C₂²	1	2	yes	—
41	C₂₃×#336	C₂²	1	2	yes	—
42	C₂₃×#336	C₂²	1	2	yes	—
43	C₂²×D₄×C₂₃	C₂×C₄	1	1	yes	characteristic
44	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
45	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
46	C₂×M₁₆×C₂₃	—	2	2	no	—
47	C₂×M₁₆×C₂₃	—	4	2	no	—
48	C₂²×C₈×C₂₃	—	2	2	no	—
49	C₂²×C₈×C₂₃	—	4	2	no	—
50	C₂²×C₈×C₂₃	—	4	2	no	—
51	C₂²×C₈×C₂₃	D₄	1	4	yes	—
52	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
53	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
54	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
55	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
56	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
57	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
58	C₄×D₄×C₂₃	—	2	2	no	—
59	C₄×D₄×C₂₃	—	2	2	no	—
60	C₂₃×#117	—	2	2	no	—
61	C₂₃×#117	—	2	2	no	—
62	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	2	yes	—
63	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	2	yes	—
64	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
65	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
66	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
67	C₂×C₂²⋊C₄×C₂₃	C₂×C₄	1	2	yes	—
68	C₂×C₂²⋊C₄×C₂₃	C₂×C₄	1	2	yes	—
69	C₂×C₂²⋊C₄×C₂₃	C₂×C₄	1	2	yes	—
70	C₂×C₂²⋊C₄×C₂₃	C₂×C₄	1	2	yes	—
71	C₂×C₂²⋊C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
72	C₂×C₂²⋊C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
73	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
74	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
75	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
76	C₂₃×C₄⋊C₈	—	2	4	no	—
77	C₂₃×C₄⋊C₈	—	2	4	no	—
78	C₂₃×C₄⋊C₈	—	2	2	no	—
79	C₂₃×C₄⋊C₈	—	4	2	no	—
80	C₂₃×C₂²⋊C₈	—	2	2	no	—
81	C₂₃×C₂²⋊C₈	—	2	2	no	—
82	C₂₃×C₂²⋊C₈	—	2	2	no	—
83	C₂₃×C₂²⋊C₈	—	4	2	no	—
84	C₄×C₈×C₂₃	—	2	2	no	—
85	C₄×C₈×C₂₃	—	4	2	no	—
86	C₂₃×#95	C₂×C₄	1	2	yes	—
87	C₂₃×#95	C₂×C₄	1	2	yes	—
88	C₂₃×#95	C₂×C₄	1	1	yes	characteristic
89	C₂₃×#95	C₂×C₄	1	1	yes	characteristic
90	C₂₃×#95	C₂×C₄	1	2	yes	—
91	C₂₃×#95	C₂×C₄	1	2	yes	—
92	C₂⁴×C₂₃	C₂²×C₄	1	1	yes	characteristic
93	C₂⁴×C₂₃	C₂²×C₄	1	1	yes	characteristic
94	C₂×D₄×C₂₃	—	2	2	no	—
95	C₂×D₄×C₂₃	—	2	2	no	—
96	C₂×D₄×C₂₃	—	2	2	no	—
97	C₂²×C₄×C₂₃	—	2	2	no	—
98	C₂²×C₄×C₂₃	—	2	2	no	—
99	C₂²×C₄×C₂₃	—	2	1	no	—
100	C₂²×C₄×C₂₃	—	2	1	no	—
101	C₂²×C₄×C₂₃	—	2	2	no	—
102	C₂²×C₄×C₂₃	—	2	2	no	—
103	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
104	C₂²×C₄×C₂₃	C₂²×C₄	1	2	yes	—
105	C₂²×C₄×C₂₃	C₂²×C₄	1	2	yes	—
106	C₂²×C₄×C₂₃	C₂²×C₄	1	2	yes	—
107	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
108	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
109	C₂²×C₄×C₂₃	C₂²×C₄	1	2	yes	—
110	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
111	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
112	C₂²×C₄×C₂₃	#41	1	2	yes	—
113	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
114	M₁₆×C₂₃	—	4	4	no	—
115	C₂×C₈×C₂₃	—	2	2	no	—
116	C₂×C₈×C₂₃	—	2	2	no	—
117	C₂×C₈×C₂₃	—	2	2	no	—
118	C₂×C₈×C₂₃	—	2	1	no	—
119	C₂×C₈×C₂₃	—	2	1	no	—
120	C₂×C₈×C₂₃	—	2	2	no	—
121	C₂×C₈×C₂₃	—	2	2	no	—
122	C₂×C₈×C₂₃	—	2	2	no	—
123	C₂×C₈×C₂₃	—	2	2	no	—
124	C₂×C₈×C₂₃	—	2	2	no	—
125	C₂×C₈×C₂₃	—	2	2	no	—
126	C₂×C₈×C₂₃	—	2	2	no	—
127	C₂×C₈×C₂₃	—	4	4	no	—
128	C₂×C₈×C₂₃	—	4	4	no	—
129	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
130	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
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	—	2	4	no	—
135	C₄⋊C₄×C₂₃	—	2	4	no	—
136	C₂²⋊C₄×C₂₃	—	2	4	no	—
137	C₂²⋊C₄×C₂₃	—	2	2	no	—
138	C₂²⋊C₄×C₂₃	—	2	4	no	—
139	C₂²⋊C₄×C₂₃	—	2	2	no	—
140	C₂²⋊C₄×C₂₃	—	2	4	no	—
141	C₂²⋊C₄×C₂₃	—	2	4	no	—
142	C₄²×C₂₃	—	2	2	no	—
143	C₄²×C₂₃	—	2	1	no	—
144	C₄²×C₂₃	—	2	2	no	—
145	C₄²×C₂₃	—	2	1	no	—
146	C₄²×C₂₃	—	2	2	no	—
147	C₄²×C₂₃	—	2	2	no	—
148	#22064	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
149	C₂³×C₂₃	—	2	1	no	—
150	C₂³×C₂₃	—	2	2	no	—
151	C₂³×C₂₃	—	2	2	no	—
152	C₂³×C₂₃	—	2	2	no	—
153	C₂³×C₂₃	—	2	2	no	—
154	C₂³×C₂₃	—	2	1	no	—
155	C₂³×C₂₃	—	2	1	no	—
156	C₂³×C₂₃	—	2	1	no	—
157	C₂³×C₂₃	C₄×D₄	1	2	yes	—
158	C₂³×C₂₃	#131	1	2	yes	—
159	C₂³×C₂₃	C₂³×C₄	1	1	yes	characteristic
160	D₄×C₂₃	—	2	8	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	1	no	—
166	C₂×C₄×C₂₃	—	2	2	no	—
167	C₂×C₄×C₂₃	—	2	2	no	—
168	C₂×C₄×C₂₃	—	2	4	no	—
169	C₂×C₄×C₂₃	—	2	1	no	—
170	C₂×C₄×C₂₃	—	2	2	no	—
171	C₂×C₄×C₂₃	—	2	2	no	—
172	C₂×C₄×C₂₃	—	2	1	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	2	no	—
178	C₂×C₄×C₂₃	—	2	2	no	—
179	C₂×C₄×C₂₃	—	2	2	no	—
180	C₂×C₄×C₂₃	—	2	2	no	—
181	C₂×C₄×C₂₃	—	2	2	no	—
182	C₂×C₄×C₂₃	—	4	1	no	—
183	C₂×C₄×C₂₃	—	4	1	no	—
184	C₂×C₄×C₂₃	—	4	2	no	—
185	C₂×C₄×C₂₃	C₄×D₄	1	2	yes	—
186	C₂×C₄×C₂₃	#131	1	4	yes	—
187	C₂×C₄×C₂₃	#131	1	2	yes	—
188	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
189	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
190	C₂×C₄×C₂₃	H(5,𝔽₂)	1	2	yes	—
191	C₂×C₄×C₂₃	2₋¹⁺⁴	1	2	yes	—
192	C₈×C₂₃	—	2	2	no	—
193	C₈×C₂₃	—	2	2	no	—
194	C₈×C₂₃	—	4	1	no	—
195	C₈×C₂₃	—	4	2	no	—
196	C₈×C₂₃	—	4	1	no	—
197	C₈×C₂₃	—	4	2	no	—
198	C₂×C₈×D₄	C₂×C₂₃	1	1	yes	characteristic
199	C₂×#424	C₂×C₂₃	1	1	yes	characteristic, complemented
200	#2395	C₂×C₂₃	1	1	yes	characteristic
201	#1939	C₂×C₂₃	1	2	yes	complemented
202	#1923	C₂×C₂₃	1	2	yes	complemented
203	#1916	C₂×C₂₃	1	2	yes	complemented
204	#1870	C₂×C₂₃	1	1	yes	characteristic
205	#1851	C₂×C₂₃	1	1	yes	characteristic, complemented
206	#1850	C₂×C₂₃	1	2	yes	complemented
207	#1818	C₂×C₂₃	1	1	yes	characteristic, complemented
208	C₈×C₂²⋊C₄	C₂×C₂₃	1	1	yes	characteristic, complemented
209	C₂²×C₂₃	—	2	2	no	—
210	C₂²×C₂₃	—	2	2	no	—
211	C₂²×C₂₃	—	2	2	no	—
212	C₂²×C₂₃	—	2	2	no	—
213	C₂²×C₂₃	—	2	4	no	—
214	C₂²×C₂₃	—	4	1	no	—
215	C₂²×C₂₃	—	4	1	no	—
216	C₂²×C₂₃	—	4	2	no	—
217	C₂²×C₂₃	—	4	2	no	—
218	C₂²×C₂₃	—	4	2	no	—
219	C₂²×C₂₃	C₂×C₄×D₄	1	1	yes	characteristic
220	C₂²×C₂₃	#520	1	2	yes	—
221	C₂²×C₂₃	C₂×#131	1	1	yes	characteristic
222	C₂²×C₂₃	C₂×#131	1	1	yes	characteristic
223	C₂²×C₂₃	C₂×#131	1	2	yes	—
224	C₄×C₂₃	—	2	2	no	—
225	C₄×C₂₃	—	2	2	no	—
226	C₄×C₂₃	—	2	4	no	—
227	C₄×C₂₃	—	4	1	no	—
228	C₄×C₂₃	—	4	2	no	—
229	C₄×C₂₃	—	4	1	no	—
230	C₄×C₂₃	—	4	2	no	—
231	C₄×C₂₃	—	4	2	no	—
232	C₄×C₂₃	#534	1	1	yes	characteristic
233	C₄×C₂₃	#536	1	1	yes	characteristic
234	C₄×C₂₃	#539	1	2	yes	—
235	C₂²×M₁₆	C₂²×C₂₃	1	1	yes	characteristic
236	C₂³×C₈	C₂²×C₂₃	1	1	yes	characteristic
237	C₂³×C₈	C₂²×C₂₃	1	1	yes	characteristic
238	C₂×C₄×D₄	C₂²×C₂₃	1	1	yes	characteristic
239	C₂×#117	C₂²×C₂₃	1	1	yes	characteristic
240	C₈×D₄	—	2	4	no	—
241	#424	—	2	4	no	—
242	C₂×C₄⋊C₈	C₂²×C₂₃	1	2	yes	—
243	C₂×C₄⋊C₈	C₂²×C₂₃	1	2	yes	—
244	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
245	C₂×C₄⋊C₈	C₂²×C₂₃	1	2	yes	—
246	C₂×C₂²⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
247	C₂×C₂²⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
248	C₂×C₂²⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
249	C₂×C₂²⋊C₈	C₂²×C₂₃	1	2	yes	—
250	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
251	C₂×C₄×C₈	C₂²×C₂₃	1	2	yes	—
252	#400	C₄×C₂₃	1	1	yes	characteristic
253	#399	C₄×C₂₃	1	1	yes	characteristic
254	#398	C₄×C₂₃	1	1	yes	characteristic
255	#397	C₄×C₂₃	1	2	yes	—
256	#397	C₄×C₂₃	1	1	yes	characteristic
257	#394	C₄×C₂₃	1	2	yes	—
258	C₄×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
259	C₄×C₂²⋊C₄	C₂²×C₂₃	1	2	yes	—
260	C₄×C₂²⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
261	#336	C₂²×C₂₃	1	2	yes	—
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	C₂×C₂₃	—	4	2	no	—
267	C₂×C₂₃	—	4	2	no	—
268	C₂×C₂₃	#2322	1	1	yes	characteristic
269	C₂×C₂₃	#2929	1	1	yes	characteristic
270	C₂×C₂₃	#2931	1	1	yes	characteristic
271	C₂×C₂₃	#2994	1	2	yes	—
272	C₂×C₂₃	#2994	1	2	yes	—
273	C₂²×D₄	C₂×C₄×C₂₃	1	1	yes	characteristic
274	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
275	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
276	C₂×M₁₆	—	2	2	no	—
277	C₂×M₁₆	—	4	2	no	—
278	C₂²×C₈	—	2	2	no	—
279	C₂²×C₈	—	4	2	no	—
280	C₂²×C₈	—	4	2	no	—
281	C₂²×C₈	D₄×C₂₃	1	4	yes	—
282	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
283	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
284	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
285	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
286	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
287	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
288	C₄×D₄	—	2	2	no	—
289	C₄×D₄	—	2	2	no	—
290	#117	—	2	2	no	—
291	#117	—	2	2	no	—
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×C₄⋊C₄	C₂×C₄×C₂₃	1	2	yes	—
297	C₂×C₂²⋊C₄	C₂×C₄×C₂₃	1	2	yes	—
298	C₂×C₂²⋊C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
299	C₂×C₂²⋊C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
300	C₂×C₂²⋊C₄	C₂×C₄×C₂₃	1	2	yes	—
301	C₂×C₂²⋊C₄	C₂×C₄×C₂₃	1	2	yes	—
302	C₂×C₂²⋊C₄	C₂×C₄×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	2	yes	—
306	C₄⋊C₈	—	2	4	no	—
307	C₄⋊C₈	—	2	4	no	—
308	C₄⋊C₈	—	2	2	no	—
309	C₄⋊C₈	—	4	2	no	—
310	C₂²⋊C₈	—	2	2	no	—
311	C₂²⋊C₈	—	2	2	no	—
312	C₂²⋊C₈	—	2	2	no	—
313	C₂²⋊C₈	—	4	2	no	—
314	C₄×C₈	—	2	2	no	—
315	C₄×C₈	—	4	2	no	—
316	#95	C₂×C₄×C₂₃	1	2	yes	—
317	#95	C₂×C₄×C₂₃	1	1	yes	characteristic
318	#95	C₂×C₄×C₂₃	1	2	yes	—
319	#95	C₂×C₄×C₂₃	1	1	yes	characteristic
320	#95	C₂×C₄×C₂₃	1	2	yes	—
321	#95	C₂×C₄×C₂₃	1	2	yes	—
322	C₂₃	#22064	1	1	yes	characteristic, direct product factor, Sylow
323	C₂⁴	C₂²×C₄×C₂₃	1	1	yes	characteristic
324	C₂⁴	C₂²×C₄×C₂₃	1	1	yes	characteristic
325	C₂×D₄	—	2	2	no	—
326	C₂×D₄	—	2	2	no	—
327	C₂×D₄	—	2	2	no	—
328	C₂²×C₄	—	2	2	no	—
329	C₂²×C₄	—	2	2	no	—
330	C₂²×C₄	—	2	1	no	—
331	C₂²×C₄	—	2	1	no	—
332	C₂²×C₄	—	2	2	no	—
333	C₂²×C₄	—	2	2	no	—
334	C₂²×C₄	C₂²×C₄×C₂₃	1	2	yes	—
335	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
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	2	yes	—
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	M₁₆	—	4	4	no	—
346	C₂×C₈	—	2	2	no	—
347	C₂×C₈	—	2	2	no	—
348	C₂×C₈	—	2	1	no	—
349	C₂×C₈	—	2	1	no	—
350	C₂×C₈	—	2	2	no	—
351	C₂×C₈	—	2	2	no	—
352	C₂×C₈	—	2	2	no	—
353	C₂×C₈	—	2	2	no	—
354	C₂×C₈	—	2	2	no	—
355	C₂×C₈	—	2	2	no	—
356	C₂×C₈	—	2	2	no	—
357	C₂×C₈	—	2	2	no	—
358	C₂×C₈	—	4	4	no	—
359	C₂×C₈	—	4	4	no	—
360	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
361	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
362	C₄⋊C₄	—	2	2	no	—
363	C₄⋊C₄	—	2	2	no	—
364	C₄⋊C₄	—	2	2	no	—
365	C₄⋊C₄	—	2	4	no	—
366	C₄⋊C₄	—	2	4	no	—
367	C₂²⋊C₄	—	2	4	no	—
368	C₂²⋊C₄	—	2	2	no	—
369	C₂²⋊C₄	—	2	4	no	—
370	C₂²⋊C₄	—	2	2	no	—
371	C₂²⋊C₄	—	2	4	no	—
372	C₂²⋊C₄	—	2	4	no	—
373	C₄²	—	2	1	no	—
374	C₄²	—	2	2	no	—
375	C₄²	—	2	2	no	—
376	C₄²	—	2	1	no	—
377	C₄²	—	2	2	no	—
378	C₄²	—	2	2	no	—
379	C₂³	—	2	2	no	—
380	C₂³	—	2	2	no	—
381	C₂³	—	2	1	no	—
382	C₂³	—	2	1	no	—
383	C₂³	—	2	1	no	—
384	C₂³	—	2	1	no	—
385	C₂³	—	2	2	no	—
386	C₂³	—	2	2	no	—
387	C₂³	C₄×D₄×C₂₃	1	2	yes	—
388	C₂³	C₂₃×#131	1	2	yes	—
389	C₂³	C₂³×C₄×C₂₃	1	1	yes	characteristic
390	D₄	—	2	8	no	—
391	C₂×C₄	—	2	2	no	—
392	C₂×C₄	—	2	1	no	—
393	C₂×C₄	—	2	1	no	—
394	C₂×C₄	—	2	2	no	—
395	C₂×C₄	—	2	2	no	—
396	C₂×C₄	—	2	2	no	—
397	C₂×C₄	—	2	2	no	—
398	C₂×C₄	—	2	4	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	1	no	—
407	C₂×C₄	—	2	2	no	—
408	C₂×C₄	—	2	2	no	—
409	C₂×C₄	—	2	1	no	—
410	C₂×C₄	—	2	2	no	—
411	C₂×C₄	—	2	2	no	—
412	C₂×C₄	—	4	1	no	—
413	C₂×C₄	—	4	1	no	—
414	C₂×C₄	—	4	2	no	—
415	C₂×C₄	C₄×D₄×C₂₃	1	2	yes	—
416	C₂×C₄	C₂₃×#131	1	4	yes	—
417	C₂×C₄	C₂₃×#131	1	2	yes	—
418	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
419	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
420	C₂×C₄	C₂₃×H(5,𝔽₂)	1	2	yes	—
421	C₂×C₄	C₂₃×2₋¹⁺⁴	1	2	yes	—
422	C₈	—	2	2	no	—
423	C₈	—	2	2	no	—
424	C₈	—	4	1	no	—
425	C₈	—	4	2	no	—
426	C₈	—	4	1	no	—
427	C₈	—	4	2	no	—
428	C₂²	—	2	2	no	—
429	C₂²	—	2	2	no	—
430	C₂²	—	2	2	no	—
431	C₂²	—	2	4	no	—
432	C₂²	—	2	2	no	—
433	C₂²	—	4	1	no	—
434	C₂²	—	4	2	no	—
435	C₂²	—	4	2	no	—
436	C₂²	—	4	2	no	—
437	C₂²	—	4	1	no	—
438	C₂²	C₂×C₄×D₄×C₂₃	1	1	yes	characteristic
439	C₂²	C₂₃×#520	1	2	yes	—
440	C₂²	C₂×C₂₃×#131	1	1	yes	characteristic
441	C₂²	C₂×C₂₃×#131	1	1	yes	characteristic
442	C₂²	C₂×C₂₃×#131	1	2	yes	—
443	C₄	—	2	2	no	—
444	C₄	—	2	4	no	—
445	C₄	—	2	2	no	—
446	C₄	—	4	2	no	—
447	C₄	—	4	2	no	—
448	C₄	—	4	1	no	—
449	C₄	—	4	2	no	—
450	C₄	—	4	1	no	—
451	C₄	C₂₃×#534	1	1	yes	characteristic
452	C₄	C₂₃×#536	1	1	yes	characteristic
453	C₄	C₂₃×#539	1	2	yes	—
454	C₂	—	4	2	no	—
455	C₂	—	4	2	no	—
456	C₂	C₂₃×#2322	1	1	yes	characteristic
457	C₂	C₂₃×#2929	1	1	yes	characteristic
458	C₂	C₂₃×#2931	1	1	yes	characteristic
459	C₂	C₂₃×#2994	1	2	yes	—
460	C₂	C₂₃×#2994	1	2	yes	—

	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
2. C₂×C₂₃×#424	C₂	semidirect
4. C₂₃×#1939	C₂	semidirect
5. C₂₃×#1923	C₂	semidirect
6. C₂₃×#1916	C₂	semidirect
8. C₂₃×#1851	C₂	semidirect
9. C₂₃×#1850	C₂	semidirect
10. C₂₃×#1818	C₂	semidirect
11. C₈×C₂²⋊C₄×C₂₃	C₂	semidirect
148. #22064	C₂₃	direct (N × H)
199. C₂×#424	C₂×C₂₃	semidirect
201. #1939	C₂×C₂₃	semidirect
202. #1923	C₂×C₂₃	semidirect
203. #1916	C₂×C₂₃	semidirect
205. #1851	C₂×C₂₃	semidirect
206. #1850	C₂×C₂₃	semidirect
207. #1818	C₂×C₂₃	semidirect
208. C₈×C₂²⋊C₄	C₂×C₂₃	semidirect
322. C₂₃	#22064	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×C₈×D₄×C₂₃	2
C₂×C₂₃×#424	2
C₂₃×#2395	2
C₂₃×#1939	2
C₂₃×#1923	2
C₂₃×#1916	2
C₂₃×#1870	2
C₂₃×#1851	2
C₂₃×#1850	2
C₂₃×#1818	2
C₈×C₂²⋊C₄×C₂₃	2
#22064	23

Subgroups — C23×#22064 #19169690

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#22064 #19169690