Subgroups — #15676694

Order 2944.

432 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃⋊#401	C₂	1	1	yes	maximal, characteristic, complemented
2	#13092359	C₂	1	1	yes	maximal, characteristic, complemented
3	#13092358	C₂	1	1	yes	maximal, characteristic, complemented
4	#13092355	C₂	1	1	yes	maximal, characteristic, complemented
5	#13092345	C₂	1	1	yes	maximal, characteristic
6	#13092345	C₂	1	1	yes	maximal, characteristic
7	#13092340	C₂	1	1	yes	maximal, characteristic
8	C₄⋊C₄×Dic₂₃	C₂	1	1	yes	maximal, characteristic, complemented
9	#13092268	C₂	1	1	yes	maximal, characteristic, complemented
10	#13092268	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂²⋊C₄×Dic₂₃	C₂	1	1	yes	maximal, characteristic
12	C₂²⋊C₄×Dic₂₃	C₂	1	1	yes	maximal, characteristic
13	C₂²⋊C₄×Dic₂₃	C₂	1	1	yes	maximal, characteristic
14	#13092262	C₂	1	1	yes	maximal, characteristic, complemented
15	C₂₃×#397	C₂	1	1	yes	maximal, characteristic
16	C₂³×Dic₂₃	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
18	C₂×C₂²⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
19	C₂×C₂²⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
20	C₂×C₂²⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
21	C₂×C₂²⋊Dic₂₃	C₂²	1	1	yes	characteristic
22	C₂×C₂²⋊Dic₂₃	C₂²	1	1	yes	characteristic
23	C₂×C₂²⋊Dic₂₃	C₂²	1	1	yes	characteristic
24	C₂×C₄⋊Dic₂₃	C₂²	1	1	yes	characteristic
25	C₂×Dic₂₃⋊C₄	C₂²	1	1	yes	characteristic
26	C₂×Dic₂₃⋊C₄	C₂²	1	1	yes	characteristic
27	C₂×Dic₂₃⋊C₄	C₂²	1	1	yes	characteristic
28	C₂×Dic₂₃⋊C₄	C₂²	1	1	yes	characteristic
29	C₂×C₄×Dic₂₃	C₂²	1	1	yes	characteristic
30	C₂×C₄×Dic₂₃	C₂²	1	1	yes	characteristic
31	C₂×C₄×Dic₂₃	C₂²	1	1	yes	characteristic
32	C₂×C₄×Dic₂₃	C₂²	1	1	yes	characteristic
33	C₂×C₄×Dic₂₃	C₂²	1	1	yes	characteristic
34	C₂×C₄×Dic₂₃	C₂²	1	1	yes	characteristic
35	C₂₃×#95	C₂²	1	1	yes	characteristic
36	C₂₃×#95	C₂²	1	1	yes	characteristic
37	C₂₃×#95	C₂²	1	1	yes	characteristic
38	C₂₃⋊#95	C₂²	1	1	yes	characteristic
39	C₂₃⋊#95	C₂²	1	1	yes	characteristic
40	C₂₃⋊#95	C₂²	1	1	yes	characteristic
41	C₂₃⋊#95	C₂²	1	1	yes	characteristic
42	C₂₃⋊#95	C₂²	1	1	yes	characteristic
43	C₂₃⋊#95	C₂²	1	1	yes	characteristic
44	C₂₃⋊#95	C₂²	1	1	yes	characteristic
45	C₂₃⋊#95	C₂²	1	1	yes	characteristic
46	C₂₃⋊#95	C₂²	1	1	yes	characteristic
47	C₂₃⋊#95	C₂²	1	1	yes	characteristic
48	C₂₃⋊#95	C₂²	1	1	yes	characteristic
49	C₂₃⋊#95	C₂²	1	1	yes	characteristic
50	C₂₃⋊#95	C₂²	1	1	yes	characteristic
51	C₂⁴×C₂₃	C₂³	1	1	yes	characteristic
52	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
53	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
54	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
55	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
56	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
57	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
58	C₂²×Dic₂₃	—	2	2	no	—
59	C₂²×Dic₂₃	—	2	2	no	—
60	C₂²×Dic₂₃	—	2	2	no	—
61	C₂²×Dic₂₃	C₂³	1	1	yes	characteristic
62	C₂²×Dic₂₃	C₂³	1	1	yes	characteristic
63	C₂²×Dic₂₃	C₂³	1	1	yes	characteristic
64	C₂²×Dic₂₃	C₂³	1	1	yes	characteristic
65	C₂²×Dic₂₃	C₂³	1	1	yes	characteristic
66	C₂²×Dic₂₃	C₂³	1	1	yes	characteristic
67	C₂²×Dic₂₃	C₂³	1	1	yes	characteristic
68	C₂²×Dic₂₃	C₂³	1	1	yes	characteristic
69	C₄⋊C₄×C₂₃	—	2	2	no	—
70	C₂²⋊C₄×C₂₃	—	2	2	no	—
71	C₂²⋊C₄×C₂₃	—	2	2	no	—
72	C₂²⋊C₄×C₂₃	—	2	2	no	—
73	C₂²⋊Dic₂₃	—	2	2	no	—
74	C₂²⋊Dic₂₃	—	2	2	no	—
75	C₂²⋊Dic₂₃	—	2	2	no	—
76	C₄⋊Dic₂₃	—	2	2	no	—
77	Dic₂₃⋊C₄	—	2	2	no	—
78	Dic₂₃⋊C₄	—	2	2	no	—
79	Dic₂₃⋊C₄	—	2	2	no	—
80	Dic₂₃⋊C₄	—	2	2	no	—
81	C₄×Dic₂₃	—	2	2	no	—
82	C₄×Dic₂₃	—	2	2	no	—
83	C₄×Dic₂₃	—	2	2	no	—
84	C₄×Dic₂₃	—	2	2	no	—
85	C₄×Dic₂₃	—	2	2	no	—
86	C₄×Dic₂₃	—	2	2	no	—
87	C₂³×C₂₃	—	2	1	no	—
88	C₂³×C₂₃	—	2	1	no	—
89	C₂³×C₂₃	—	2	1	no	—
90	C₂³×C₂₃	—	2	1	no	—
91	C₂³×C₂₃	—	2	1	no	—
92	C₂³×C₂₃	—	2	1	no	—
93	C₂³×C₂₃	#41	1	2	yes	—
94	C₂³×C₂₃	C₂⁴	1	1	yes	characteristic
95	C₂×C₄×C₂₃	—	2	1	no	—
96	C₂×C₄×C₂₃	—	2	1	no	—
97	C₂×C₄×C₂₃	—	2	1	no	—
98	C₂×C₄×C₂₃	—	2	1	no	—
99	C₂×C₄×C₂₃	—	2	1	no	—
100	C₂×C₄×C₂₃	—	2	1	no	—
101	C₂×C₄×C₂₃	—	2	1	no	—
102	C₂×C₄×C₂₃	—	2	1	no	—
103	C₂×C₄×C₂₃	—	2	1	no	—
104	C₂×C₄×C₂₃	—	2	1	no	—
105	C₂×C₄×C₂₃	—	2	1	no	—
106	C₂×C₄×C₂₃	—	2	1	no	—
107	C₂×C₄×C₂₃	—	2	1	no	—
108	C₂×C₄×C₂₃	—	2	1	no	—
109	C₂×C₄×C₂₃	—	2	1	no	—
110	C₂×C₄×C₂₃	—	2	1	no	—
111	C₂×C₄×C₂₃	—	2	1	no	—
112	C₂×C₄×C₂₃	#41	1	2	yes	—
113	C₂×Dic₂₃	—	2	1	no	—
114	C₂×Dic₂₃	—	2	1	no	—
115	C₂×Dic₂₃	—	2	1	no	—
116	C₂×Dic₂₃	—	2	1	no	—
117	C₂×Dic₂₃	—	2	1	no	—
118	C₂×Dic₂₃	—	2	1	no	—
119	C₂×Dic₂₃	—	2	1	no	—
120	C₂×Dic₂₃	—	2	1	no	—
121	C₂×Dic₂₃	—	2	1	no	—
122	C₂×Dic₂₃	—	2	1	no	—
123	C₂×Dic₂₃	—	2	1	no	—
124	C₂×Dic₂₃	—	2	1	no	—
125	C₂×Dic₂₃	—	2	1	no	—
126	C₂×Dic₂₃	—	2	1	no	—
127	C₂×Dic₂₃	—	2	1	no	—
128	C₂×Dic₂₃	—	2	1	no	—
129	C₂×Dic₂₃	—	2	1	no	—
130	C₂×Dic₂₃	—	2	1	no	—
131	C₂×Dic₂₃	—	2	1	no	—
132	C₂×Dic₂₃	—	4	4	no	—
133	C₂×Dic₂₃	#41	1	2	yes	—
134	C₂×Dic₂₃	#41	1	2	yes	—
135	C₂×Dic₂₃	#41	1	2	yes	—
136	C₂×Dic₂₃	#41	1	2	yes	—
137	C₂×Dic₂₃	#41	1	2	yes	—
138	#2401	—	23	1	no	maximal, Sylow
139	C₂²×C₂₃	—	2	2	no	—
140	C₂²×C₂₃	—	2	2	no	—
141	C₂²×C₂₃	—	2	2	no	—
142	C₂²×C₂₃	—	4	1	no	—
143	C₂²×C₂₃	—	4	1	no	—
144	C₂²×C₂₃	—	4	1	no	—
145	C₂²×C₂₃	—	4	1	no	—
146	C₂²×C₂₃	C₂×#41	1	1	yes	characteristic
147	C₂²×C₂₃	C₂×#41	1	1	yes	characteristic
148	C₂²×C₂₃	C₂×#41	1	1	yes	characteristic
149	C₂²×C₂₃	C₂×#41	1	1	yes	characteristic
150	C₂²×C₂₃	C₂×#41	1	1	yes	characteristic
151	C₂²×C₂₃	C₂×#41	1	1	yes	characteristic
152	C₂²×C₂₃	C₂×#41	1	1	yes	characteristic
153	C₄×C₂₃	—	2	2	no	—
154	C₄×C₂₃	—	4	1	no	—
155	C₄×C₂₃	—	4	1	no	—
156	C₄×C₂₃	—	4	1	no	—
157	C₄×C₂₃	—	4	1	no	—
158	C₄×C₂₃	—	4	1	no	—
159	Dic₂₃	—	2	2	no	—
160	Dic₂₃	—	2	2	no	—
161	Dic₂₃	—	4	1	no	—
162	Dic₂₃	—	4	1	no	—
163	Dic₂₃	—	4	1	no	—
164	Dic₂₃	—	4	1	no	—
165	Dic₂₃	—	4	1	no	—
166	Dic₂₃	#126	1	4	yes	—
167	#401	—	23	1	no	—
168	#400	—	23	1	no	—
169	#400	—	23	1	no	—
170	#400	—	23	1	no	—
171	#397	—	23	1	no	—
172	#397	—	23	1	no	—
173	#397	D₂₃	1	1	yes	characteristic
174	#396	—	23	1	no	—
175	C₄×C₄⋊C₄	—	23	1	no	—
176	C₄×C₄⋊C₄	—	23	1	no	—
177	C₄×C₄⋊C₄	—	23	1	no	—
178	C₄×C₂²⋊C₄	—	23	1	no	—
179	C₄×C₂²⋊C₄	—	23	1	no	—
180	C₄×C₂²⋊C₄	—	23	1	no	—
181	#376	—	23	1	no	—
182	C₂×C₂₃	—	4	2	no	—
183	C₂×C₂₃	C₂×#126	1	1	yes	characteristic
184	C₂×C₂₃	#529	1	1	yes	characteristic
185	C₂×C₂₃	#529	1	1	yes	characteristic
186	C₂×C₂₃	#529	1	1	yes	characteristic
187	C₂×C₂₃	#529	1	1	yes	characteristic
188	C₂×C₂₃	#529	1	1	yes	characteristic
189	C₂×C₂₃	#529	1	1	yes	characteristic
190	C₂³×C₄	—	23	1	no	—
191	C₂×C₄⋊C₄	—	23	1	no	—
192	C₂×C₄⋊C₄	—	23	1	no	—
193	C₂×C₄⋊C₄	—	23	1	no	—
194	C₂×C₄⋊C₄	—	23	1	no	—
195	C₂×C₄⋊C₄	—	23	1	no	—
196	C₂×C₄⋊C₄	C₂×D₂₃	1	1	yes	characteristic
197	C₂×C₂²⋊C₄	—	23	1	no	—
198	C₂×C₂²⋊C₄	—	23	1	no	—
199	C₂×C₂²⋊C₄	—	23	1	no	—
200	C₂×C₂²⋊C₄	C₂×D₂₃	1	1	yes	characteristic
201	C₂×C₂²⋊C₄	C₂×D₂₃	1	1	yes	characteristic
202	C₂×C₂²⋊C₄	C₂×D₂₃	1	1	yes	characteristic
203	C₂×C₄²	—	23	1	no	—
204	C₂×C₄²	—	23	1	no	—
205	C₂×C₄²	—	23	1	no	—
206	C₂×C₄²	—	23	1	no	—
207	C₂×C₄²	—	23	1	no	—
208	C₂×C₄²	—	23	1	no	—
209	#95	—	23	1	no	—
210	#95	—	23	1	no	—
211	#95	—	23	1	no	—
212	#95	—	23	1	no	—
213	#95	—	23	1	no	—
214	#95	—	23	1	no	—
215	#95	—	23	1	no	—
216	#95	—	23	1	no	—
217	#95	—	23	1	no	—
218	#95	—	23	1	no	—
219	#95	—	23	1	no	—
220	#95	—	23	1	no	—
221	#95	—	23	1	no	—
222	#95	C₂×D₂₃	1	1	yes	characteristic
223	#95	C₂×D₂₃	1	1	yes	characteristic
224	#95	C₂×D₂₃	1	1	yes	characteristic
225	C₂₃	#2401	1	1	yes	characteristic, complemented, Sylow
226	C₂⁴	C₂²×D₂₃	1	1	yes	characteristic
227	C₂²×C₄	—	23	1	no	—
228	C₂²×C₄	—	23	1	no	—
229	C₂²×C₄	—	23	1	no	—
230	C₂²×C₄	—	23	1	no	—
231	C₂²×C₄	—	23	1	no	—
232	C₂²×C₄	—	23	1	no	—
233	C₂²×C₄	—	23	1	no	—
234	C₂²×C₄	—	23	1	no	—
235	C₂²×C₄	—	46	2	no	—
236	C₂²×C₄	—	46	2	no	—
237	C₂²×C₄	—	46	2	no	—
238	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
239	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
240	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
241	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
242	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
243	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
244	C₄⋊C₄	—	2	2	no	—
245	C₄⋊C₄	—	46	2	no	—
246	C₄⋊C₄	—	46	2	no	—
247	C₄⋊C₄	—	46	2	no	—
248	C₄⋊C₄	—	46	2	no	—
249	C₄⋊C₄	—	46	2	no	—
250	C₂²⋊C₄	—	2	2	no	—
251	C₂²⋊C₄	—	2	2	no	—
252	C₂²⋊C₄	—	2	2	no	—
253	C₂²⋊C₄	—	46	2	no	—
254	C₂²⋊C₄	—	46	2	no	—
255	C₂²⋊C₄	—	46	2	no	—
256	C₄²	—	46	2	no	—
257	C₄²	—	46	2	no	—
258	C₄²	—	46	2	no	—
259	C₄²	—	46	2	no	—
260	C₄²	—	46	2	no	—
261	C₄²	—	46	2	no	—
262	C₂³	—	2	1	no	—
263	C₂³	—	2	1	no	—
264	C₂³	—	2	1	no	—
265	C₂³	—	2	1	no	—
266	C₂³	—	2	1	no	—
267	C₂³	—	2	1	no	—
268	C₂³	#67290	1	2	yes	—
269	C₂³	C₂³×D₂₃	1	1	yes	characteristic
270	C₂×C₄	—	2	1	no	—
271	C₂×C₄	—	2	1	no	—
272	C₂×C₄	—	2	1	no	—
273	C₂×C₄	—	2	1	no	—
274	C₂×C₄	—	2	1	no	—
275	C₂×C₄	—	2	1	no	—
276	C₂×C₄	—	2	1	no	—
277	C₂×C₄	—	2	1	no	—
278	C₂×C₄	—	2	1	no	—
279	C₂×C₄	—	2	1	no	—
280	C₂×C₄	—	2	1	no	—
281	C₂×C₄	—	2	1	no	—
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	2	1	no	—
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	2	1	no	—
286	C₂×C₄	—	2	1	no	—
287	C₂×C₄	—	23	2	no	—
288	C₂×C₄	—	23	2	no	—
289	C₂×C₄	—	23	2	no	—
290	C₂×C₄	—	23	2	no	—
291	C₂×C₄	—	23	2	no	—
292	C₂×C₄	—	46	1	no	—
293	C₂×C₄	—	46	1	no	—
294	C₂×C₄	—	46	1	no	—
295	C₂×C₄	—	46	1	no	—
296	C₂×C₄	—	46	1	no	—
297	C₂×C₄	—	46	1	no	—
298	C₂×C₄	—	46	1	no	—
299	C₂×C₄	—	46	1	no	—
300	C₂×C₄	—	46	1	no	—
301	C₂×C₄	—	46	1	no	—
302	C₂×C₄	—	46	1	no	—
303	C₂×C₄	—	46	1	no	—
304	C₂×C₄	—	46	1	no	—
305	C₂×C₄	—	46	1	no	—
306	C₂×C₄	—	46	1	no	—
307	C₂×C₄	—	46	1	no	—
308	C₂×C₄	—	46	1	no	—
309	C₂×C₄	—	46	1	no	—
310	C₂×C₄	—	46	1	no	—
311	C₂×C₄	—	92	4	no	—
312	C₂×C₄	#67290	1	2	yes	—
313	C₂²	—	2	2	no	—
314	C₂²	—	2	2	no	—
315	C₂²	—	2	2	no	—
316	C₂²	—	4	1	no	—
317	C₂²	—	4	1	no	—
318	C₂²	—	4	1	no	—
319	C₂²	—	4	1	no	—
320	C₂²	C₂×#67290	1	1	yes	characteristic
321	C₂²	C₂×#67290	1	1	yes	characteristic
322	C₂²	#41×D₂₃	1	1	yes	characteristic
323	C₂²	#41×D₂₃	1	1	yes	characteristic
324	C₂²	#41×D₂₃	1	1	yes	characteristic
325	C₂²	#41×D₂₃	1	1	yes	characteristic
326	C₂²	#41×D₂₃	1	1	yes	characteristic
327	C₄	—	2	2	no	—
328	C₄	—	4	1	no	—
329	C₄	—	4	1	no	—
330	C₄	—	4	1	no	—
331	C₄	—	4	1	no	—
332	C₄	—	4	1	no	—
333	C₄	—	23	4	no	—
334	C₄	—	46	2	no	—
335	C₄	—	46	2	no	—
336	C₄	—	92	1	no	—
337	C₄	—	92	1	no	—
338	C₄	—	92	1	no	—
339	C₄	—	92	1	no	—
340	C₄	—	92	1	no	—
341	C₂	—	4	2	no	—
342	C₂	#126×D₂₃	1	1	yes	characteristic
343	C₂	#13092944	1	1	yes	characteristic
344	C₂	#13092944	1	1	yes	characteristic
345	C₂	#13092944	1	1	yes	characteristic
346	C₂	#13092946	1	1	yes	characteristic
347	C₂	#13092948	1	1	yes	characteristic
348	C₂	#13092950	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₂₃
commutator, socle, cosocle	C₂³×C₂₃	C₂⁴
Fitting	C₂₃×#397	C₂
prime radical	C₂⁴×C₂₃	C₂³
hypercenter	#397	D₂₃
hypocenter	C₂₃	#2401
LCS	G ▷ C₂³×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂³ ◁ #397

Complements

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

N	H ≅ G/N	type
1. C₂₃⋊#401	C₂	semidirect
2. #13092359	C₂	semidirect
3. #13092358	C₂	semidirect
4. #13092355	C₂	semidirect
8. C₄⋊C₄×Dic₂₃	C₂	semidirect
9. #13092268	C₂	semidirect
10. #13092268	C₂	semidirect
14. #13092262	C₂	semidirect
225. C₂₃	#2401	semidirect

Maximal subgroups

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

subgroup	index
C₂₃⋊#401	2
#13092359	2
#13092358	2
#13092355	2
#13092345	2
#13092340	2
C₄⋊C₄×Dic₂₃	2
#13092268	2
C₂²⋊C₄×Dic₂₃	2
#13092262	2
C₂₃×#397	2
#2401	23