Subgroups — C₂₃×#3551 #15671170

Order 2944.

1050 conjugacy classes of subgroups.

The subgroup lattice has 428 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#563	C₂	1	2	yes	maximal, complemented
2	C₂₃×#560	C₂	1	2	yes	maximal, complemented
3	C₂₃×#555	C₂	1	2	yes	maximal, complemented
4	C₂₃×#553	C₂	1	2	yes	maximal, complemented
5	C₂₃×#552	C₂	1	2	yes	maximal, complemented
6	C₂₃×#551	C₂	1	2	yes	maximal, complemented
7	C₂₃×#548	C₂	1	2	yes	maximal, complemented
8	C₂₃×#546	C₂	1	2	yes	maximal, complemented
9	C₂₃×#540	C₂	1	4	yes	maximal, complemented
10	C₂₃×#538	C₂	1	4	yes	maximal, complemented
11	C₂₃×#537	C₂	1	2	yes	maximal, complemented
12	C₂₃×#536	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂₃×#535	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂₃×#532	C₂	1	1	yes	maximal, characteristic, complemented
15	C₂₃×#529	C₂	1	2	yes	maximal, complemented
16	C₂×#41×C₂₃	C₂²	1	2	yes	—
17	C₂×#41×C₂₃	C₂²	1	1	yes	characteristic
18	C₂×#41×C₂₃	C₂²	1	1	yes	characteristic, complemented
19	C₂²×Q₈×C₂₃	C₂²	1	1	yes	characteristic, complemented
20	C₂²×D₄×C₂₃	C₂²	1	1	yes	characteristic
21	C₂₃×C₄⋊Q₈	C₂²	1	2	yes	complemented
22	C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic, complemented
23	C₂₃×#127	C₂²	1	1	yes	characteristic
24	C₂₃×#127	C₂²	1	2	yes	—
25	C₂₃×#126	C₂²	1	4	yes	—
26	C₂₃×#126	C₂²	1	4	yes	complemented
27	C₂₃×#126	C₂²	1	4	yes	complemented
28	C₂₃×#125	C₂²	1	2	yes	complemented
29	C₂₃×#125	C₂²	1	4	yes	complemented
30	C₂₃×#124	C₂²	1	2	yes	—
31	C₂₃×#124	C₂²	1	1	yes	characteristic
32	C₂₃×#124	C₂²	1	2	yes	—
33	C₂₃×#124	C₂²	1	2	yes	—
34	C₂₃×#124	C₂²	1	4	yes	complemented
35	C₂₃×#124	C₂²	1	1	yes	characteristic, complemented
36	C₂₃×#123	C₂²	1	2	yes	—
37	C₂₃×#123	C₂²	1	4	yes	—
38	C₂₃×#123	C₂²	1	4	yes	—
39	C₂₃×#123	C₂²	1	4	yes	complemented
40	C₂₃×#123	C₂²	1	4	yes	complemented
41	C₂₃×#123	C₂²	1	2	yes	complemented
42	C₂₃×C₂²⋊Q₈	C₂²	1	2	yes	—
43	C₂₃×C₂²⋊Q₈	C₂²	1	2	yes	complemented
44	C₂₃×C₂²⋊Q₈	C₂²	1	4	yes	complemented
45	C₂₃×C₂²⋊Q₈	C₂²	1	4	yes	complemented
46	C₂₃×C₂²⋊Q₈	C₂²	1	2	yes	complemented
47	C₂₃×#121	C₂²	1	2	yes	—
48	C₂₃×#121	C₂²	1	2	yes	—
49	C₂₃×#121	C₂²	1	4	yes	—
50	C₂₃×#121	C₂²	1	2	yes	—
51	C₂₃×#121	C₂²	1	2	yes	—
52	C₂₃×#121	C₂²	1	4	yes	complemented
53	C₂₃×#121	C₂²	1	4	yes	complemented
54	C₂₃×#121	C₂²	1	4	yes	complemented
55	C₂₃×#121	C₂²	1	2	yes	complemented
56	C₂₃×#121	C₂²	1	4	yes	complemented
57	C₂₃×#121	C₂²	1	4	yes	complemented
58	C₂₃×#120	C₂²	1	2	yes	—
59	C₂₃×#120	C₂²	1	4	yes	—
60	C₂₃×#120	C₂²	1	2	yes	—
61	C₄×Q₈×C₂₃	C₂²	1	2	yes	complemented
62	C₄×D₄×C₂₃	C₂²	1	2	yes	—
63	C₄×D₄×C₂₃	C₂²	1	2	yes	—
64	C₄×D₄×C₂₃	C₂²	1	4	yes	complemented
65	C₄×D₄×C₂₃	C₂²	1	2	yes	complemented
66	C₄×D₄×C₂₃	C₂²	1	4	yes	complemented
67	C₄×D₄×C₂₃	C₂²	1	4	yes	complemented
68	C₂₃×#117	C₂²	1	1	yes	characteristic
69	C₂₃×#117	C₂²	1	4	yes	complemented
70	C₂₃×#117	C₂²	1	4	yes	complemented
71	C₂₃×#117	C₂²	1	1	yes	characteristic, complemented
72	C₂×C₄⋊C₄×C₂₃	C₂²	1	2	yes	complemented
73	C₂×C₂²⋊C₄×C₂₃	C₂²	1	4	yes	—
74	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic, complemented
75	C₂⁴×C₂₃	C₂³	1	2	yes	—
76	#41×C₂₃	—	4	2	no	—
77	#41×C₂₃	—	4	2	no	—
78	#41×C₂₃	—	4	4	no	—
79	C₂×Q₈×C₂₃	—	4	2	no	—
80	C₂×Q₈×C₂₃	C₂³	1	2	yes	—
81	C₂×Q₈×C₂₃	C₂³	1	1	yes	characteristic
82	C₂×Q₈×C₂₃	C₂³	1	2	yes	—
83	C₂×Q₈×C₂₃	C₂³	1	1	yes	characteristic
84	C₂×D₄×C₂₃	—	4	2	no	—
85	C₂×D₄×C₂₃	C₂³	1	2	yes	—
86	C₂×D₄×C₂₃	C₂³	1	2	yes	—
87	C₂×D₄×C₂₃	C₂³	1	4	yes	—
88	C₂×D₄×C₂₃	C₂³	1	4	yes	—
89	C₂×D₄×C₂₃	C₂³	1	4	yes	—
90	C₂×D₄×C₂₃	C₂³	1	2	yes	—
91	C₂×D₄×C₂₃	C₂³	1	1	yes	characteristic
92	C₂×D₄×C₂₃	C₂³	1	1	yes	characteristic
93	C₂×D₄×C₂₃	C₂³	1	4	yes	—
94	C₂×D₄×C₂₃	C₂³	1	2	yes	—
95	C₂×D₄×C₂₃	C₂³	1	2	yes	—
96	C₂×D₄×C₂₃	C₂³	1	1	yes	characteristic
97	C₂×D₄×C₂₃	C₂³	1	1	yes	characteristic
98	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
99	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
100	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
101	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
102	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
103	C₂²×C₄×C₂₃	C₂³	1	4	yes	—
104	C₂²×C₄×C₂₃	C₂³	1	4	yes	—
105	C₂²×C₄×C₂₃	C₂³	1	4	yes	—
106	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
107	C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
108	C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
109	C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
110	C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
111	C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
112	C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
113	C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
114	C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
115	C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
116	C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
117	C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
118	C₂²⋊C₄×C₂₃	C₂³	1	2	yes	—
119	C₂²⋊C₄×C₂₃	C₂³	1	4	yes	—
120	C₂²⋊C₄×C₂₃	C₂³	1	4	yes	—
121	C₂²⋊C₄×C₂₃	C₂³	1	4	yes	—
122	C₂²⋊C₄×C₂₃	C₂³	1	2	yes	—
123	C₂²⋊C₄×C₂₃	C₂³	1	2	yes	—
124	C₂²⋊C₄×C₂₃	C₂³	1	4	yes	—
125	C₂²⋊C₄×C₂₃	C₂³	1	4	yes	—
126	C₂²⋊C₄×C₂₃	C₂³	1	2	yes	—
127	C₂²⋊C₄×C₂₃	C₂³	1	4	yes	—
128	C₂²⋊C₄×C₂₃	C₂³	1	2	yes	—
129	C₂²⋊C₄×C₂₃	C₂³	1	2	yes	—
130	C₂²⋊C₄×C₂₃	C₂³	1	4	yes	—
131	C₂²⋊C₄×C₂₃	C₂³	1	4	yes	—
132	C₂²⋊C₄×C₂₃	C₂³	1	4	yes	—
133	C₄²×C₂₃	C₂³	1	1	yes	characteristic
134	C₄²×C₂₃	C₂³	1	2	yes	—
135	C₄²×C₂₃	C₂³	1	1	yes	characteristic
136	C₄²×C₂₃	C₂³	1	2	yes	—
137	C₄²×C₂₃	C₂³	1	4	yes	—
138	C₄²×C₂₃	C₂³	1	2	yes	—
139	C₂³×C₂₃	—	4	2	no	—
140	C₂³×C₂₃	—	4	4	no	—
141	C₂³×C₂₃	C₂⁴	1	1	yes	characteristic
142	C₂³×C₂₃	C₂⁴	1	4	yes	—
143	C₂³×C₂₃	C₂⁴	1	1	yes	characteristic
144	C₂³×C₂₃	C₂⁴	1	2	yes	—
145	C₂³×C₂₃	C₂⁴	1	1	yes	characteristic
146	Q₈×C₂₃	—	2	4	no	—
147	Q₈×C₂₃	—	4	2	no	—
148	Q₈×C₂₃	—	4	1	no	—
149	Q₈×C₂₃	—	4	1	no	—
150	D₄×C₂₃	—	2	4	no	—
151	D₄×C₂₃	—	4	2	no	—
152	D₄×C₂₃	—	4	4	no	—
153	D₄×C₂₃	—	4	2	no	—
154	D₄×C₂₃	—	4	4	no	—
155	D₄×C₂₃	—	4	4	no	—
156	D₄×C₂₃	—	4	2	no	—
157	D₄×C₂₃	—	4	2	no	—
158	D₄×C₂₃	—	4	1	no	—
159	D₄×C₂₃	—	4	1	no	—
160	D₄×C₂₃	—	4	1	no	—
161	D₄×C₂₃	—	4	1	no	—
162	D₄×C₂₃	—	4	4	no	—
163	C₂×C₄×C₂₃	—	2	2	no	—
164	C₂×C₄×C₂₃	—	2	2	no	—
165	C₂×C₄×C₂₃	—	2	4	no	—
166	C₂×C₄×C₂₃	—	2	2	no	—
167	C₂×C₄×C₂₃	—	4	4	no	—
168	C₂×C₄×C₂₃	—	4	2	no	—
169	C₂×C₄×C₂₃	—	4	4	no	—
170	C₂×C₄×C₂₃	—	4	2	no	—
171	C₂×C₄×C₂₃	—	4	4	no	—
172	C₂×C₄×C₂₃	C₂⁴	1	2	yes	—
173	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
174	C₂×C₄×C₂₃	C₂⁴	1	2	yes	—
175	C₂×C₄×C₂₃	C₂⁴	1	4	yes	—
176	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
177	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
178	C₂×C₄×C₂₃	C₂⁴	1	4	yes	—
179	C₂×C₄×C₂₃	C₂⁴	1	4	yes	—
180	C₂×C₄×C₂₃	C₂⁴	1	2	yes	—
181	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
182	#3551	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
183	C₂²×C₂₃	—	2	1	no	—
184	C₂²×C₂₃	—	2	2	no	—
185	C₂²×C₂₃	—	2	1	no	—
186	C₂²×C₂₃	—	2	1	no	—
187	C₂²×C₂₃	—	2	4	no	—
188	C₂²×C₂₃	—	2	2	no	—
189	C₂²×C₂₃	—	2	2	no	—
190	C₂²×C₂₃	—	2	4	no	—
191	C₂²×C₂₃	—	2	2	no	—
192	C₂²×C₂₃	—	2	2	no	—
193	C₂²×C₂₃	—	2	4	no	—
194	C₂²×C₂₃	—	2	2	no	—
195	C₂²×C₂₃	—	8	4	no	—
196	C₂²×C₂₃	C₂⁵	1	1	yes	characteristic
197	C₄×C₂₃	—	2	2	no	—
198	C₄×C₂₃	—	2	1	no	—
199	C₄×C₂₃	—	2	1	no	—
200	C₄×C₂₃	—	2	1	no	—
201	C₄×C₂₃	—	2	4	no	—
202	C₄×C₂₃	—	2	2	no	—
203	C₄×C₂₃	—	2	4	no	—
204	C₄×C₂₃	—	2	2	no	—
205	C₄×C₂₃	—	2	4	no	—
206	C₄×C₂₃	H(5,𝔽₂)	1	2	yes	—
207	#563	C₂×C₂₃	1	2	yes	complemented
208	#560	C₂×C₂₃	1	2	yes	complemented
209	#555	C₂×C₂₃	1	2	yes	complemented
210	#553	C₂×C₂₃	1	2	yes	complemented
211	#552	C₂×C₂₃	1	2	yes	complemented
212	#551	C₂×C₂₃	1	2	yes	complemented
213	#548	C₂×C₂₃	1	2	yes	complemented
214	#546	C₂×C₂₃	1	2	yes	complemented
215	#540	C₂×C₂₃	1	4	yes	complemented
216	#538	C₂×C₂₃	1	4	yes	complemented
217	#537	C₂×C₂₃	1	2	yes	complemented
218	#536	C₂×C₂₃	1	1	yes	characteristic, complemented
219	#535	C₂×C₂₃	1	1	yes	characteristic, complemented
220	#532	C₂×C₂₃	1	1	yes	characteristic, complemented
221	#529	C₂×C₂₃	1	2	yes	complemented
222	C₂×C₂₃	—	4	1	no	—
223	C₂×C₂₃	—	4	2	no	—
224	C₂×C₂₃	—	4	1	no	—
225	C₂×C₂₃	—	4	4	no	—
226	C₂×C₂₃	—	4	1	no	—
227	C₂×C₂₃	C₂×H(5,𝔽₂)	1	1	yes	characteristic
228	C₂×C₂₃	#585	1	2	yes	—
229	C₂×#41	C₂²×C₂₃	1	1	yes	characteristic
230	C₂×#41	C₂²×C₂₃	1	2	yes	—
231	C₂×#41	C₂²×C₂₃	1	1	yes	characteristic, complemented
232	C₂²×Q₈	C₂²×C₂₃	1	1	yes	characteristic, complemented
233	C₂²×D₄	C₂²×C₂₃	1	1	yes	characteristic
234	C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic, complemented
235	C₄⋊Q₈	C₂²×C₂₃	1	2	yes	complemented
236	#127	C₂²×C₂₃	1	1	yes	characteristic
237	#127	C₂²×C₂₃	1	2	yes	—
238	#126	C₂²×C₂₃	1	4	yes	—
239	#126	C₂²×C₂₃	1	4	yes	complemented
240	#126	C₂²×C₂₃	1	4	yes	complemented
241	#125	C₂²×C₂₃	1	2	yes	complemented
242	#125	C₂²×C₂₃	1	4	yes	complemented
243	#124	C₂²×C₂₃	1	2	yes	—
244	#124	C₂²×C₂₃	1	1	yes	characteristic
245	#124	C₂²×C₂₃	1	2	yes	—
246	#124	C₂²×C₂₃	1	2	yes	—
247	#124	C₂²×C₂₃	1	1	yes	characteristic, complemented
248	#124	C₂²×C₂₃	1	4	yes	complemented
249	#123	C₂²×C₂₃	1	2	yes	—
250	#123	C₂²×C₂₃	1	4	yes	—
251	#123	C₂²×C₂₃	1	4	yes	—
252	#123	C₂²×C₂₃	1	4	yes	complemented
253	#123	C₂²×C₂₃	1	2	yes	complemented
254	#123	C₂²×C₂₃	1	4	yes	complemented
255	C₂²⋊Q₈	C₂²×C₂₃	1	2	yes	—
256	C₂²⋊Q₈	C₂²×C₂₃	1	2	yes	complemented
257	C₂²⋊Q₈	C₂²×C₂₃	1	2	yes	complemented
258	C₂²⋊Q₈	C₂²×C₂₃	1	4	yes	complemented
259	C₂²⋊Q₈	C₂²×C₂₃	1	4	yes	complemented
260	#121	C₂²×C₂₃	1	2	yes	—
261	#121	C₂²×C₂₃	1	2	yes	—
262	#121	C₂²×C₂₃	1	2	yes	—
263	#121	C₂²×C₂₃	1	2	yes	—
264	#121	C₂²×C₂₃	1	4	yes	—
265	#121	C₂²×C₂₃	1	2	yes	complemented
266	#121	C₂²×C₂₃	1	4	yes	complemented
267	#121	C₂²×C₂₃	1	4	yes	complemented
268	#121	C₂²×C₂₃	1	4	yes	complemented
269	#121	C₂²×C₂₃	1	4	yes	complemented
270	#121	C₂²×C₂₃	1	4	yes	complemented
271	#120	C₂²×C₂₃	1	2	yes	—
272	#120	C₂²×C₂₃	1	2	yes	—
273	#120	C₂²×C₂₃	1	4	yes	—
274	C₄×Q₈	C₂²×C₂₃	1	2	yes	complemented
275	C₄×D₄	C₂²×C₂₃	1	2	yes	—
276	C₄×D₄	C₂²×C₂₃	1	2	yes	—
277	C₄×D₄	C₂²×C₂₃	1	4	yes	complemented
278	C₄×D₄	C₂²×C₂₃	1	2	yes	complemented
279	C₄×D₄	C₂²×C₂₃	1	4	yes	complemented
280	C₄×D₄	C₂²×C₂₃	1	4	yes	complemented
281	#117	C₂²×C₂₃	1	1	yes	characteristic
282	#117	C₂²×C₂₃	1	4	yes	complemented
283	#117	C₂²×C₂₃	1	1	yes	characteristic, complemented
284	#117	C₂²×C₂₃	1	4	yes	complemented
285	C₂×C₄⋊C₄	C₂²×C₂₃	1	2	yes	complemented
286	C₂×C₂²⋊C₄	C₂²×C₂₃	1	4	yes	—
287	C₂×C₄²	C₂²×C₂₃	1	1	yes	characteristic, complemented
288	C₂₃	#3551	1	1	yes	characteristic, direct product factor, Sylow
289	C₂⁴	C₂³×C₂₃	1	2	yes	—
290	#41	—	4	2	no	—
291	#41	—	4	2	no	—
292	#41	—	4	4	no	—
293	C₂×Q₈	—	4	2	no	—
294	C₂×Q₈	C₂³×C₂₃	1	2	yes	—
295	C₂×Q₈	C₂³×C₂₃	1	1	yes	characteristic
296	C₂×Q₈	C₂³×C₂₃	1	2	yes	—
297	C₂×Q₈	C₂³×C₂₃	1	1	yes	characteristic
298	C₂×D₄	—	4	2	no	—
299	C₂×D₄	C₂³×C₂₃	1	2	yes	—
300	C₂×D₄	C₂³×C₂₃	1	2	yes	—
301	C₂×D₄	C₂³×C₂₃	1	4	yes	—
302	C₂×D₄	C₂³×C₂₃	1	4	yes	—
303	C₂×D₄	C₂³×C₂₃	1	4	yes	—
304	C₂×D₄	C₂³×C₂₃	1	2	yes	—
305	C₂×D₄	C₂³×C₂₃	1	1	yes	characteristic
306	C₂×D₄	C₂³×C₂₃	1	1	yes	characteristic
307	C₂×D₄	C₂³×C₂₃	1	4	yes	—
308	C₂×D₄	C₂³×C₂₃	1	2	yes	—
309	C₂×D₄	C₂³×C₂₃	1	2	yes	—
310	C₂×D₄	C₂³×C₂₃	1	1	yes	characteristic
311	C₂×D₄	C₂³×C₂₃	1	1	yes	characteristic
312	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
313	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
314	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
315	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
316	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
317	C₂²×C₄	C₂³×C₂₃	1	4	yes	—
318	C₂²×C₄	C₂³×C₂₃	1	4	yes	—
319	C₂²×C₄	C₂³×C₂₃	1	4	yes	—
320	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
321	C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
322	C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
323	C₄⋊C₄	C₂³×C₂₃	1	4	yes	—
324	C₄⋊C₄	C₂³×C₂₃	1	4	yes	—
325	C₄⋊C₄	C₂³×C₂₃	1	4	yes	—
326	C₄⋊C₄	C₂³×C₂₃	1	4	yes	—
327	C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
328	C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
329	C₄⋊C₄	C₂³×C₂₃	1	4	yes	—
330	C₄⋊C₄	C₂³×C₂₃	1	4	yes	—
331	C₄⋊C₄	C₂³×C₂₃	1	4	yes	—
332	C₂²⋊C₄	C₂³×C₂₃	1	2	yes	—
333	C₂²⋊C₄	C₂³×C₂₃	1	4	yes	—
334	C₂²⋊C₄	C₂³×C₂₃	1	4	yes	—
335	C₂²⋊C₄	C₂³×C₂₃	1	4	yes	—
336	C₂²⋊C₄	C₂³×C₂₃	1	2	yes	—
337	C₂²⋊C₄	C₂³×C₂₃	1	2	yes	—
338	C₂²⋊C₄	C₂³×C₂₃	1	4	yes	—
339	C₂²⋊C₄	C₂³×C₂₃	1	4	yes	—
340	C₂²⋊C₄	C₂³×C₂₃	1	2	yes	—
341	C₂²⋊C₄	C₂³×C₂₃	1	4	yes	—
342	C₂²⋊C₄	C₂³×C₂₃	1	2	yes	—
343	C₂²⋊C₄	C₂³×C₂₃	1	2	yes	—
344	C₂²⋊C₄	C₂³×C₂₃	1	4	yes	—
345	C₂²⋊C₄	C₂³×C₂₃	1	4	yes	—
346	C₂²⋊C₄	C₂³×C₂₃	1	4	yes	—
347	C₄²	C₂³×C₂₃	1	1	yes	characteristic
348	C₄²	C₂³×C₂₃	1	2	yes	—
349	C₄²	C₂³×C₂₃	1	1	yes	characteristic
350	C₄²	C₂³×C₂₃	1	2	yes	—
351	C₄²	C₂³×C₂₃	1	4	yes	—
352	C₄²	C₂³×C₂₃	1	2	yes	—
353	C₂³	—	4	2	no	—
354	C₂³	—	4	4	no	—
355	C₂³	C₂⁴×C₂₃	1	1	yes	characteristic
356	C₂³	C₂⁴×C₂₃	1	4	yes	—
357	C₂³	C₂⁴×C₂₃	1	1	yes	characteristic
358	C₂³	C₂⁴×C₂₃	1	2	yes	—
359	C₂³	C₂⁴×C₂₃	1	1	yes	characteristic
360	Q₈	—	2	4	no	—
361	Q₈	—	4	2	no	—
362	Q₈	—	4	1	no	—
363	Q₈	—	4	1	no	—
364	D₄	—	2	4	no	—
365	D₄	—	4	2	no	—
366	D₄	—	4	2	no	—
367	D₄	—	4	1	no	—
368	D₄	—	4	1	no	—
369	D₄	—	4	1	no	—
370	D₄	—	4	1	no	—
371	D₄	—	4	4	no	—
372	D₄	—	4	4	no	—
373	D₄	—	4	2	no	—
374	D₄	—	4	4	no	—
375	D₄	—	4	4	no	—
376	D₄	—	4	2	no	—
377	C₂×C₄	—	2	2	no	—
378	C₂×C₄	—	2	2	no	—
379	C₂×C₄	—	2	4	no	—
380	C₂×C₄	—	2	2	no	—
381	C₂×C₄	—	4	2	no	—
382	C₂×C₄	—	4	4	no	—
383	C₂×C₄	—	4	4	no	—
384	C₂×C₄	—	4	2	no	—
385	C₂×C₄	—	4	4	no	—
386	C₂×C₄	C₂⁴×C₂₃	1	2	yes	—
387	C₂×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
388	C₂×C₄	C₂⁴×C₂₃	1	2	yes	—
389	C₂×C₄	C₂⁴×C₂₃	1	4	yes	—
390	C₂×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
391	C₂×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
392	C₂×C₄	C₂⁴×C₂₃	1	4	yes	—
393	C₂×C₄	C₂⁴×C₂₃	1	4	yes	—
394	C₂×C₄	C₂⁴×C₂₃	1	2	yes	—
395	C₂×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
396	C₂²	—	2	1	no	—
397	C₂²	—	2	2	no	—
398	C₂²	—	2	1	no	—
399	C₂²	—	2	1	no	—
400	C₂²	—	2	4	no	—
401	C₂²	—	2	2	no	—
402	C₂²	—	2	2	no	—
403	C₂²	—	2	4	no	—
404	C₂²	—	2	2	no	—
405	C₂²	—	2	2	no	—
406	C₂²	—	2	4	no	—
407	C₂²	—	2	2	no	—
408	C₂²	—	8	4	no	—
409	C₂²	C₂⁵×C₂₃	1	1	yes	characteristic
410	C₄	—	2	2	no	—
411	C₄	—	2	1	no	—
412	C₄	—	2	1	no	—
413	C₄	—	2	1	no	—
414	C₄	—	2	4	no	—
415	C₄	—	2	2	no	—
416	C₄	—	2	4	no	—
417	C₄	—	2	2	no	—
418	C₄	—	2	4	no	—
419	C₄	C₂₃×H(5,𝔽₂)	1	2	yes	—
420	C₂	—	4	1	no	—
421	C₂	—	4	4	no	—
422	C₂	—	4	1	no	—
423	C₂	—	4	2	no	—
424	C₂	—	4	1	no	—
425	C₂	C₂×C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
426	C₂	C₂₃×#585	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₂₃	C₂⁵
commutator, Frattini, cosocle	C₂²	C₂⁵×C₂₃
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂₃×#563	C₂	semidirect
2. C₂₃×#560	C₂	semidirect
3. C₂₃×#555	C₂	semidirect
4. C₂₃×#553	C₂	semidirect
5. C₂₃×#552	C₂	semidirect
6. C₂₃×#551	C₂	semidirect
7. C₂₃×#548	C₂	semidirect
8. C₂₃×#546	C₂	semidirect
9. C₂₃×#540	C₂	semidirect
10. C₂₃×#538	C₂	semidirect
11. C₂₃×#537	C₂	semidirect
12. C₂₃×#536	C₂	semidirect
13. C₂₃×#535	C₂	semidirect
14. C₂₃×#532	C₂	semidirect
15. C₂₃×#529	C₂	semidirect
18. C₂×#41×C₂₃	C₂²	semidirect
19. C₂²×Q₈×C₂₃	C₂²	semidirect
21. C₂₃×C₄⋊Q₈	C₂²	semidirect
22. C₂₃×C₄⋊Q₈	C₂²	semidirect
26. C₂₃×#126	C₂²	semidirect
27. C₂₃×#126	C₂²	semidirect
28. C₂₃×#125	C₂²	semidirect
29. C₂₃×#125	C₂²	semidirect
34. C₂₃×#124	C₂²	semidirect
35. C₂₃×#124	C₂²	semidirect
39. C₂₃×#123	C₂²	semidirect
40. C₂₃×#123	C₂²	semidirect
41. C₂₃×#123	C₂²	semidirect
43. C₂₃×C₂²⋊Q₈	C₂²	semidirect
44. C₂₃×C₂²⋊Q₈	C₂²	semidirect
45. C₂₃×C₂²⋊Q₈	C₂²	semidirect
46. C₂₃×C₂²⋊Q₈	C₂²	semidirect
52. C₂₃×#121	C₂²	semidirect
53. C₂₃×#121	C₂²	semidirect
54. C₂₃×#121	C₂²	semidirect
55. C₂₃×#121	C₂²	semidirect
56. C₂₃×#121	C₂²	semidirect
57. C₂₃×#121	C₂²	semidirect
61. C₄×Q₈×C₂₃	C₂²	semidirect
64. C₄×D₄×C₂₃	C₂²	semidirect
65. C₄×D₄×C₂₃	C₂²	semidirect
66. C₄×D₄×C₂₃	C₂²	semidirect
67. C₄×D₄×C₂₃	C₂²	semidirect
69. C₂₃×#117	C₂²	semidirect
70. C₂₃×#117	C₂²	semidirect
71. C₂₃×#117	C₂²	semidirect
72. C₂×C₄⋊C₄×C₂₃	C₂²	semidirect
74. C₂×C₄²×C₂₃	C₂²	semidirect
182. #3551	C₂₃	direct (N × H)
207. #563	C₂×C₂₃	semidirect
208. #560	C₂×C₂₃	semidirect
209. #555	C₂×C₂₃	semidirect
210. #553	C₂×C₂₃	semidirect
211. #552	C₂×C₂₃	semidirect
212. #551	C₂×C₂₃	semidirect
213. #548	C₂×C₂₃	semidirect
214. #546	C₂×C₂₃	semidirect
215. #540	C₂×C₂₃	semidirect
216. #538	C₂×C₂₃	semidirect
217. #537	C₂×C₂₃	semidirect
218. #536	C₂×C₂₃	semidirect
219. #535	C₂×C₂₃	semidirect
220. #532	C₂×C₂₃	semidirect
221. #529	C₂×C₂₃	semidirect
231. C₂×#41	C₂²×C₂₃	semidirect
232. C₂²×Q₈	C₂²×C₂₃	semidirect
234. C₄⋊Q₈	C₂²×C₂₃	semidirect
235. C₄⋊Q₈	C₂²×C₂₃	semidirect
239. #126	C₂²×C₂₃	semidirect
240. #126	C₂²×C₂₃	semidirect
241. #125	C₂²×C₂₃	semidirect
242. #125	C₂²×C₂₃	semidirect
247. #124	C₂²×C₂₃	semidirect
248. #124	C₂²×C₂₃	semidirect
252. #123	C₂²×C₂₃	semidirect
253. #123	C₂²×C₂₃	semidirect
254. #123	C₂²×C₂₃	semidirect
256. C₂²⋊Q₈	C₂²×C₂₃	semidirect
257. C₂²⋊Q₈	C₂²×C₂₃	semidirect
258. C₂²⋊Q₈	C₂²×C₂₃	semidirect
259. C₂²⋊Q₈	C₂²×C₂₃	semidirect
265. #121	C₂²×C₂₃	semidirect
266. #121	C₂²×C₂₃	semidirect
267. #121	C₂²×C₂₃	semidirect
268. #121	C₂²×C₂₃	semidirect
269. #121	C₂²×C₂₃	semidirect
270. #121	C₂²×C₂₃	semidirect
274. C₄×Q₈	C₂²×C₂₃	semidirect
277. C₄×D₄	C₂²×C₂₃	semidirect
278. C₄×D₄	C₂²×C₂₃	semidirect
279. C₄×D₄	C₂²×C₂₃	semidirect
280. C₄×D₄	C₂²×C₂₃	semidirect
282. #117	C₂²×C₂₃	semidirect
283. #117	C₂²×C₂₃	semidirect
284. #117	C₂²×C₂₃	semidirect
285. C₂×C₄⋊C₄	C₂²×C₂₃	semidirect
287. C₂×C₄²	C₂²×C₂₃	semidirect
288. C₂₃	#3551	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂₃×#563	2
C₂₃×#560	2
C₂₃×#555	2
C₂₃×#553	2
C₂₃×#552	2
C₂₃×#551	2
C₂₃×#548	2
C₂₃×#546	2
C₂₃×#540	2
C₂₃×#538	2
C₂₃×#537	2
C₂₃×#536	2
C₂₃×#535	2
C₂₃×#532	2
C₂₃×#529	2
#3551	23

Subgroups — C23×#3551 #15671170

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#3551 #15671170