Subgroups — C₂₃×#22664 #19170290

Order 5888.

978 conjugacy classes of subgroups.

The subgroup lattice has 414 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#436	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂₃×#428	C₂	1	2	yes	maximal, complemented
3	C₂₃×#2352	C₂	1	2	yes	maximal
4	C₂₃×#1949	C₂	1	2	yes	maximal, complemented
5	C₂₃×#1854	C₂	1	2	yes	maximal, complemented
6	C₂₃×#1852	C₂	1	2	yes	maximal, complemented
7	C₂₃×#1846	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂₃×#1809	C₂	1	2	yes	maximal, complemented
9	C₄×C₂₃×#107	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×M₁₆×C₂₃	C₂²	1	2	yes	—
11	C₂×C₄×D₄×C₂₃	C₂²	1	1	yes	characteristic
12	C₂×C₂₃×#117	C₂²	1	2	yes	—
13	C₂²×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
14	C₂₃×#436	C₄	1	8	yes	complemented
15	C₂₃×#428	—	2	8	no	—
16	C₂×C₂₃×#107	C₂²	1	2	yes	—
17	C₂×C₂₃×#107	C₂²	1	1	yes	characteristic
18	C₂×C₂₃×#107	C₂²	1	1	yes	characteristic
19	C₂×C₂₃×#107	C₂²	1	1	yes	characteristic
20	C₂×C₂₃×#107	C₂²	1	1	yes	characteristic
21	C₂×C₂₃×#106	C₂²	1	2	yes	—
22	C₂×C₂₃×#106	C₂²	1	2	yes	—
23	C₂×C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
24	C₂×C₂₃×C₂²⋊C₈	C₂²	1	2	yes	—
25	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
26	C₂₃×#388	—	2	2	no	—
27	C₂₃×#387	—	2	2	no	—
28	C₂₃×#386	—	2	2	no	—
29	C₂₃×#382	C₂²	1	2	yes	—
30	C₂₃×#380	C₂²	1	2	yes	—
31	C₂₃×#380	C₂²	1	2	yes	—
32	C₄×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
33	C₄×C₂²⋊C₄×C₂₃	—	2	2	no	—
34	C₂₃×#340	C₂²	1	2	yes	—
35	C₂₃×#340	C₂²	1	2	yes	—
36	C₂₃×#340	C₂²	1	2	yes	—
37	C₂²×D₄×C₂₃	D₄	1	1	yes	characteristic
38	C₂³×C₄×C₂₃	—	2	2	no	—
39	C₂³×C₄×C₂₃	C₂³	1	2	yes	—
40	C₂×M₁₆×C₂₃	—	2	4	no	—
41	C₂×M₁₆×C₂₃	—	2	4	no	—
42	C₂×M₁₆×C₂₃	C₂×C₄	1	8	yes	complemented
43	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
44	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
45	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
46	C₄×D₄×C₂₃	C₂×C₄	1	4	yes	—
47	C₄×D₄×C₂₃	Q₈	1	4	yes	—
48	C₂₃×#117	—	2	4	no	—
49	C₂₃×#117	—	4	2	no	—
50	C₂×C₄⋊C₄×C₂₃	—	2	4	no	—
51	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
52	C₂×C₄⋊C₄×C₂₃	—	4	2	no	—
53	C₂×C₄⋊C₄×C₂₃	D₄	1	1	yes	characteristic
54	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
55	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
56	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
57	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
58	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
59	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
60	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
61	C₂×C₂²⋊C₄×C₂₃	D₄	1	2	yes	—
62	C₂×C₄²×C₂₃	—	2	2	no	—
63	C₂×C₄²×C₂₃	—	2	2	no	—
64	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
65	C₂₃×#107	—	2	2	no	—
66	C₂₃×#107	—	2	4	no	—
67	C₂₃×#107	—	4	1	no	—
68	C₂₃×#107	—	4	1	no	—
69	C₂₃×#107	D₄	1	4	yes	complemented
70	C₂₃×#106	—	2	4	no	—
71	C₂₃×#106	—	2	4	no	—
72	C₂₃×C₄⋊C₈	C₂×C₄	1	4	yes	complemented
73	C₂₃×C₂²⋊C₈	C₂×C₄	1	8	yes	complemented
74	C₄×C₈×C₂₃	C₂×C₄	1	4	yes	complemented
75	C₂₃×#95	—	2	2	no	—
76	C₂₃×#95	—	2	2	no	—
77	C₂₃×#95	—	2	2	no	—
78	C₂₃×#95	—	2	2	no	—
79	C₂₃×#95	—	2	2	no	—
80	C₂⁴×C₂₃	C₂×D₄	1	2	yes	—
81	C₂×D₄×C₂₃	—	4	1	no	—
82	C₂×D₄×C₂₃	—	4	1	no	—
83	C₂×D₄×C₂₃	C₄⋊C₄	1	4	yes	—
84	C₂²×C₄×C₂₃	—	2	2	no	—
85	C₂²×C₄×C₂₃	—	2	2	no	—
86	C₂²×C₄×C₂₃	—	2	2	no	—
87	C₂²×C₄×C₂₃	—	2	2	no	—
88	C₂²×C₄×C₂₃	—	2	2	no	—
89	C₂²×C₄×C₂₃	—	2	2	no	—
90	C₂²×C₄×C₂₃	—	2	2	no	—
91	C₂²×C₄×C₂₃	—	2	2	no	—
92	C₂²×C₄×C₂₃	—	4	4	no	—
93	C₂²×C₄×C₂₃	—	4	4	no	—
94	C₂²×C₄×C₂₃	—	8	2	no	—
95	C₂²×C₄×C₂₃	C₂²×C₄	1	4	yes	—
96	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
97	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
98	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
99	C₂²×C₄×C₂₃	C₂×Q₈	1	4	yes	—
100	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
101	M₁₆×C₂₃	—	2	16	no	—
102	C₂×C₈×C₂₃	—	2	2	no	—
103	C₂×C₈×C₂₃	—	2	2	no	—
104	C₂×C₈×C₂₃	—	2	1	no	—
105	C₂×C₈×C₂₃	—	2	1	no	—
106	C₂×C₈×C₂₃	C₂²×C₄	1	4	yes	—
107	C₂×C₈×C₂₃	C₂²×C₄	1	2	yes	—
108	C₂×C₈×C₂₃	C₂²×C₄	1	2	yes	—
109	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
110	C₂×C₈×C₂₃	#41	1	2	yes	—
111	C₄⋊C₄×C₂₃	—	2	2	no	—
112	C₄⋊C₄×C₂₃	—	2	2	no	—
113	C₄⋊C₄×C₂₃	—	2	2	no	—
114	C₄⋊C₄×C₂₃	—	2	2	no	—
115	C₄⋊C₄×C₂₃	—	2	2	no	—
116	C₄⋊C₄×C₂₃	—	2	2	no	—
117	C₄⋊C₄×C₂₃	—	2	2	no	—
118	C₄⋊C₄×C₂₃	—	4	4	no	—
119	C₄⋊C₄×C₂₃	C₄⋊C₄	1	4	yes	—
120	C₄⋊C₄×C₂₃	C₂×D₄	1	4	yes	—
121	C₂²⋊C₄×C₂₃	—	4	4	no	—
122	C₂²⋊C₄×C₂₃	—	4	4	no	—
123	C₂²⋊C₄×C₂₃	—	8	2	no	—
124	C₂²⋊C₄×C₂₃	C₄⋊C₄	1	8	yes	—
125	C₄²×C₂₃	—	4	4	no	—
126	C₄²×C₂₃	—	8	2	no	—
127	C₄²×C₂₃	C₂²×C₄	1	2	yes	—
128	C₄²×C₂₃	C₂×Q₈	1	2	yes	—
129	#22664	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
130	C₂³×C₂₃	—	2	2	no	—
131	C₂³×C₂₃	—	2	2	no	—
132	C₂³×C₂₃	—	2	2	no	—
133	C₂³×C₂₃	—	2	2	no	—
134	C₂³×C₂₃	—	2	2	no	—
135	C₂³×C₂₃	—	2	2	no	—
136	C₂³×C₂₃	C₂×C₄⋊C₄	1	4	yes	—
137	C₂³×C₂₃	C₂²×D₄	1	1	yes	characteristic
138	D₄×C₂₃	—	4	4	no	—
139	C₂×C₄×C₂₃	—	2	2	no	—
140	C₂×C₄×C₂₃	—	2	2	no	—
141	C₂×C₄×C₂₃	—	2	1	no	—
142	C₂×C₄×C₂₃	—	2	1	no	—
143	C₂×C₄×C₂₃	—	2	1	no	—
144	C₂×C₄×C₂₃	—	2	1	no	—
145	C₂×C₄×C₂₃	—	2	4	no	—
146	C₂×C₄×C₂₃	—	2	4	no	—
147	C₂×C₄×C₂₃	—	2	4	no	—
148	C₂×C₄×C₂₃	—	2	4	no	—
149	C₂×C₄×C₂₃	—	2	4	no	—
150	C₂×C₄×C₂₃	—	4	2	no	—
151	C₂×C₄×C₂₃	—	4	2	no	—
152	C₂×C₄×C₂₃	—	4	2	no	—
153	C₂×C₄×C₂₃	—	4	2	no	—
154	C₂×C₄×C₂₃	—	4	2	no	—
155	C₂×C₄×C₂₃	—	4	2	no	—
156	C₂×C₄×C₂₃	—	4	2	no	—
157	C₂×C₄×C₂₃	—	4	2	no	—
158	C₂×C₄×C₂₃	—	4	2	no	—
159	C₂×C₄×C₂₃	—	4	2	no	—
160	C₂×C₄×C₂₃	—	4	2	no	—
161	C₂×C₄×C₂₃	—	8	8	no	—
162	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	2	yes	—
163	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	2	yes	—
164	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	4	yes	—
165	C₂×C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
166	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
167	C₂×C₄×C₂₃	C₂²×Q₈	1	1	yes	characteristic
168	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
169	C₂×C₄×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
170	C₂×C₄×C₂₃	2₋¹⁺⁴	1	1	yes	characteristic
171	C₈×C₂₃	—	2	4	no	—
172	C₈×C₂₃	—	2	2	no	—
173	C₈×C₂₃	C₄×D₄	1	4	yes	—
174	C₂×#436	C₂×C₂₃	1	1	yes	characteristic
175	C₂×#428	C₂×C₂₃	1	2	yes	complemented
176	#2352	C₂×C₂₃	1	2	yes	—
177	#1949	C₂×C₂₃	1	2	yes	complemented
178	#1854	C₂×C₂₃	1	2	yes	complemented
179	#1852	C₂×C₂₃	1	2	yes	complemented
180	#1846	C₂×C₂₃	1	1	yes	characteristic, complemented
181	#1809	C₂×C₂₃	1	2	yes	complemented
182	C₄×#107	C₂×C₂₃	1	1	yes	characteristic, complemented
183	C₂²×C₂₃	—	2	4	no	—
184	C₂²×C₂₃	—	2	4	no	—
185	C₂²×C₂₃	—	2	4	no	—
186	C₂²×C₂₃	—	4	2	no	—
187	C₂²×C₂₃	—	4	2	no	—
188	C₂²×C₂₃	—	4	2	no	—
189	C₂²×C₂₃	—	4	2	no	—
190	C₂²×C₂₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
191	C₂²×C₂₃	D₄²	1	1	yes	characteristic
192	C₂²×C₂₃	#547	1	1	yes	characteristic
193	C₂²×C₂₃	#575	1	1	yes	characteristic
194	C₂²×C₂₃	#575	1	1	yes	characteristic
195	C₂²×C₂₃	#576	1	1	yes	characteristic
196	C₂²×C₂₃	#578	1	1	yes	characteristic
197	C₄×C₂₃	—	2	4	no	—
198	C₄×C₂₃	—	2	2	no	—
199	C₄×C₂₃	—	2	2	no	—
200	C₄×C₂₃	—	4	4	no	—
201	C₄×C₂₃	—	8	2	no	—
202	C₄×C₂₃	—	8	2	no	—
203	C₄×C₂₃	—	8	2	no	—
204	C₄×C₂₃	C₂×C₄×D₄	1	1	yes	characteristic
205	C₄×C₂₃	#520	1	1	yes	characteristic
206	C₄×C₂₃	D₄×Q₈	1	1	yes	characteristic
207	C₄×C₂₃	#554	1	1	yes	characteristic
208	C₂²×M₁₆	C₂²×C₂₃	1	2	yes	—
209	C₂×C₄×D₄	C₂²×C₂₃	1	1	yes	characteristic
210	C₂×#117	C₂²×C₂₃	1	2	yes	—
211	C₂²×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
212	#436	C₄×C₂₃	1	8	yes	complemented
213	#428	—	2	8	no	—
214	C₂×#107	C₂²×C₂₃	1	1	yes	characteristic
215	C₂×#107	C₂²×C₂₃	1	2	yes	—
216	C₂×#107	C₂²×C₂₃	1	1	yes	characteristic
217	C₂×#107	C₂²×C₂₃	1	1	yes	characteristic
218	C₂×#107	C₂²×C₂₃	1	1	yes	characteristic
219	C₂×#106	C₂²×C₂₃	1	2	yes	—
220	C₂×#106	C₂²×C₂₃	1	2	yes	—
221	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
222	C₂×C₂²⋊C₈	C₂²×C₂₃	1	2	yes	—
223	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
224	#388	—	2	2	no	—
225	#387	—	2	2	no	—
226	#386	—	2	2	no	—
227	#382	C₂²×C₂₃	1	2	yes	—
228	#380	C₂²×C₂₃	1	2	yes	—
229	#380	C₂²×C₂₃	1	2	yes	—
230	C₄×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
231	C₄×C₂²⋊C₄	—	2	2	no	—
232	#340	C₂²×C₂₃	1	2	yes	—
233	#340	C₂²×C₂₃	1	2	yes	—
234	#340	C₂²×C₂₃	1	2	yes	—
235	C₂×C₂₃	—	4	4	no	—
236	C₂×C₂₃	D₄×C₄⋊C₄	1	1	yes	characteristic
237	C₂×C₂₃	#2911	1	1	yes	characteristic
238	C₂×C₂₃	#2913	1	1	yes	characteristic
239	C₂×C₂₃	#3273	1	1	yes	characteristic
240	C₂×C₂₃	#3277	1	1	yes	characteristic
241	C₂×C₂₃	#3352	1	1	yes	characteristic
242	C₂×C₂₃	#3355	1	1	yes	characteristic
243	C₂²×D₄	D₄×C₂₃	1	1	yes	characteristic
244	C₂³×C₄	—	2	2	no	—
245	C₂³×C₄	C₂³×C₂₃	1	2	yes	—
246	C₂×M₁₆	—	2	4	no	—
247	C₂×M₁₆	—	2	4	no	—
248	C₂×M₁₆	C₂×C₄×C₂₃	1	8	yes	complemented
249	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
250	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
251	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
252	C₄×D₄	C₂×C₄×C₂₃	1	4	yes	—
253	C₄×D₄	Q₈×C₂₃	1	4	yes	—
254	#117	—	2	4	no	—
255	#117	—	4	2	no	—
256	C₂×C₄⋊C₄	—	2	4	no	—
257	C₂×C₄⋊C₄	—	2	2	no	—
258	C₂×C₄⋊C₄	—	4	2	no	—
259	C₂×C₄⋊C₄	D₄×C₂₃	1	1	yes	characteristic
260	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
261	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
262	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
263	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
264	C₂×C₂²⋊C₄	—	2	2	no	—
265	C₂×C₂²⋊C₄	—	2	2	no	—
266	C₂×C₂²⋊C₄	—	2	2	no	—
267	C₂×C₂²⋊C₄	D₄×C₂₃	1	2	yes	—
268	C₂×C₄²	—	2	2	no	—
269	C₂×C₄²	—	2	2	no	—
270	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
271	#107	—	2	2	no	—
272	#107	—	2	4	no	—
273	#107	—	4	1	no	—
274	#107	—	4	1	no	—
275	#107	D₄×C₂₃	1	4	yes	complemented
276	#106	—	2	4	no	—
277	#106	—	2	4	no	—
278	C₄⋊C₈	C₂×C₄×C₂₃	1	4	yes	complemented
279	C₂²⋊C₈	C₂×C₄×C₂₃	1	8	yes	complemented
280	C₄×C₈	C₂×C₄×C₂₃	1	4	yes	complemented
281	#95	—	2	2	no	—
282	#95	—	2	2	no	—
283	#95	—	2	2	no	—
284	#95	—	2	2	no	—
285	#95	—	2	2	no	—
286	C₂₃	#22664	1	1	yes	characteristic, direct product factor, Sylow
287	C₂⁴	C₂×D₄×C₂₃	1	2	yes	—
288	C₂×D₄	—	4	1	no	—
289	C₂×D₄	—	4	1	no	—
290	C₂×D₄	C₄⋊C₄×C₂₃	1	4	yes	—
291	C₂²×C₄	—	2	2	no	—
292	C₂²×C₄	—	2	2	no	—
293	C₂²×C₄	—	2	2	no	—
294	C₂²×C₄	—	2	2	no	—
295	C₂²×C₄	—	2	2	no	—
296	C₂²×C₄	—	2	2	no	—
297	C₂²×C₄	—	2	2	no	—
298	C₂²×C₄	—	2	2	no	—
299	C₂²×C₄	—	4	4	no	—
300	C₂²×C₄	—	4	4	no	—
301	C₂²×C₄	—	8	2	no	—
302	C₂²×C₄	C₂²×C₄×C₂₃	1	4	yes	—
303	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
304	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
305	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
306	C₂²×C₄	C₂×Q₈×C₂₃	1	4	yes	—
307	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
308	M₁₆	—	2	16	no	—
309	C₂×C₈	—	2	2	no	—
310	C₂×C₈	—	2	2	no	—
311	C₂×C₈	—	2	1	no	—
312	C₂×C₈	—	2	1	no	—
313	C₂×C₈	C₂²×C₄×C₂₃	1	2	yes	—
314	C₂×C₈	C₂²×C₄×C₂₃	1	4	yes	—
315	C₂×C₈	C₂²×C₄×C₂₃	1	2	yes	—
316	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
317	C₂×C₈	#41×C₂₃	1	2	yes	—
318	C₄⋊C₄	—	2	2	no	—
319	C₄⋊C₄	—	2	2	no	—
320	C₄⋊C₄	—	2	2	no	—
321	C₄⋊C₄	—	2	2	no	—
322	C₄⋊C₄	—	2	2	no	—
323	C₄⋊C₄	—	2	2	no	—
324	C₄⋊C₄	—	2	2	no	—
325	C₄⋊C₄	—	4	4	no	—
326	C₄⋊C₄	C₄⋊C₄×C₂₃	1	4	yes	—
327	C₄⋊C₄	C₂×D₄×C₂₃	1	4	yes	—
328	C₂²⋊C₄	—	4	4	no	—
329	C₂²⋊C₄	—	4	4	no	—
330	C₂²⋊C₄	—	8	2	no	—
331	C₂²⋊C₄	C₄⋊C₄×C₂₃	1	8	yes	—
332	C₄²	—	4	4	no	—
333	C₄²	—	8	2	no	—
334	C₄²	C₂²×C₄×C₂₃	1	2	yes	—
335	C₄²	C₂×Q₈×C₂₃	1	2	yes	—
336	C₂³	—	2	2	no	—
337	C₂³	—	2	2	no	—
338	C₂³	—	2	2	no	—
339	C₂³	—	2	2	no	—
340	C₂³	—	2	2	no	—
341	C₂³	—	2	2	no	—
342	C₂³	C₂×C₄⋊C₄×C₂₃	1	4	yes	—
343	C₂³	C₂²×D₄×C₂₃	1	1	yes	characteristic
344	D₄	—	4	4	no	—
345	C₂×C₄	—	2	2	no	—
346	C₂×C₄	—	2	1	no	—
347	C₂×C₄	—	2	1	no	—
348	C₂×C₄	—	2	1	no	—
349	C₂×C₄	—	2	1	no	—
350	C₂×C₄	—	2	2	no	—
351	C₂×C₄	—	2	4	no	—
352	C₂×C₄	—	2	4	no	—
353	C₂×C₄	—	2	4	no	—
354	C₂×C₄	—	2	4	no	—
355	C₂×C₄	—	2	4	no	—
356	C₂×C₄	—	4	2	no	—
357	C₂×C₄	—	4	2	no	—
358	C₂×C₄	—	4	2	no	—
359	C₂×C₄	—	4	2	no	—
360	C₂×C₄	—	4	2	no	—
361	C₂×C₄	—	4	2	no	—
362	C₂×C₄	—	4	2	no	—
363	C₂×C₄	—	4	2	no	—
364	C₂×C₄	—	4	2	no	—
365	C₂×C₄	—	4	2	no	—
366	C₂×C₄	—	4	2	no	—
367	C₂×C₄	—	8	8	no	—
368	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	2	yes	—
369	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	2	yes	—
370	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	4	yes	—
371	C₂×C₄	C₂³×C₄×C₂₃	1	1	yes	characteristic
372	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
373	C₂×C₄	C₂²×Q₈×C₂₃	1	1	yes	characteristic
374	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
375	C₂×C₄	C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
376	C₂×C₄	C₂₃×2₋¹⁺⁴	1	1	yes	characteristic
377	C₈	—	2	4	no	—
378	C₈	—	2	2	no	—
379	C₈	C₄×D₄×C₂₃	1	4	yes	—
380	C₂²	—	2	4	no	—
381	C₂²	—	2	4	no	—
382	C₂²	—	2	4	no	—
383	C₂²	—	4	2	no	—
384	C₂²	—	4	2	no	—
385	C₂²	—	4	2	no	—
386	C₂²	—	4	2	no	—
387	C₂²	C₂²×C₄⋊C₄×C₂₃	1	1	yes	characteristic
388	C₂²	D₄²×C₂₃	1	1	yes	characteristic
389	C₂²	C₂₃×#547	1	1	yes	characteristic
390	C₂²	C₂₃×#575	1	1	yes	characteristic
391	C₂²	C₂₃×#575	1	1	yes	characteristic
392	C₂²	C₂₃×#576	1	1	yes	characteristic
393	C₂²	C₂₃×#578	1	1	yes	characteristic
394	C₄	—	2	2	no	—
395	C₄	—	2	2	no	—
396	C₄	—	2	4	no	—
397	C₄	—	4	4	no	—
398	C₄	—	8	2	no	—
399	C₄	—	8	2	no	—
400	C₄	—	8	2	no	—
401	C₄	C₂×C₄×D₄×C₂₃	1	1	yes	characteristic
402	C₄	C₂₃×#520	1	1	yes	characteristic
403	C₄	D₄×Q₈×C₂₃	1	1	yes	characteristic
404	C₄	C₂₃×#554	1	1	yes	characteristic
405	C₂	—	4	4	no	—
406	C₂	D₄×C₄⋊C₄×C₂₃	1	1	yes	characteristic
407	C₂	C₂₃×#2911	1	1	yes	characteristic
408	C₂	C₂₃×#2913	1	1	yes	characteristic
409	C₂	C₂₃×#3273	1	1	yes	characteristic
410	C₂	C₂₃×#3277	1	1	yes	characteristic
411	C₂	C₂₃×#3352	1	1	yes	characteristic
412	C₂	C₂₃×#3355	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
2. C₂×C₂₃×#428	C₂	semidirect
4. C₂₃×#1949	C₂	semidirect
5. C₂₃×#1854	C₂	semidirect
6. C₂₃×#1852	C₂	semidirect
7. C₂₃×#1846	C₂	semidirect
8. C₂₃×#1809	C₂	semidirect
9. C₄×C₂₃×#107	C₂	semidirect
14. C₂₃×#436	C₄	semidirect
42. C₂×M₁₆×C₂₃	C₂×C₄	semidirect
69. C₂₃×#107	D₄	semidirect
72. C₂₃×C₄⋊C₈	C₂×C₄	semidirect
73. C₂₃×C₂²⋊C₈	C₂×C₄	semidirect
74. C₄×C₈×C₂₃	C₂×C₄	semidirect
129. #22664	C₂₃	direct (N × H)
175. C₂×#428	C₂×C₂₃	semidirect
177. #1949	C₂×C₂₃	semidirect
178. #1854	C₂×C₂₃	semidirect
179. #1852	C₂×C₂₃	semidirect
180. #1846	C₂×C₂₃	semidirect
181. #1809	C₂×C₂₃	semidirect
182. C₄×#107	C₂×C₂₃	semidirect
212. #436	C₄×C₂₃	semidirect
248. C₂×M₁₆	C₂×C₄×C₂₃	semidirect
275. #107	D₄×C₂₃	semidirect
278. C₄⋊C₈	C₂×C₄×C₂₃	semidirect
279. C₂²⋊C₈	C₂×C₄×C₂₃	semidirect
280. C₄×C₈	C₂×C₄×C₂₃	semidirect
286. C₂₃	#22664	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×#436	2
C₂×C₂₃×#428	2
C₂₃×#2352	2
C₂₃×#1949	2
C₂₃×#1854	2
C₂₃×#1852	2
C₂₃×#1846	2
C₂₃×#1809	2
C₄×C₂₃×#107	2
#22664	23

Subgroups — C23×#22664 #19170290

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#22664 #19170290