Subgroups — #15683200

Order 2944.

376 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13092973	C₂	1	1	yes	maximal, characteristic
2	C₄×Q₈×D₂₃	C₂	1	1	yes	maximal, characteristic
3	#13092725	C₂	1	1	yes	maximal, characteristic, complemented
4	#13092724	C₂	1	1	yes	maximal, characteristic, complemented
5	#13092721	C₂	1	1	yes	maximal, characteristic, complemented
6	#13092711	C₂	1	1	yes	maximal, characteristic, complemented
7	#13092710	C₂	1	1	yes	maximal, characteristic, complemented
8	#13092701	C₂	1	1	yes	maximal, characteristic, complemented
9	#13092700	C₂	1	1	yes	maximal, characteristic, complemented
10	#13092458	C₂	1	1	yes	maximal, characteristic
11	#13092456	C₂	1	1	yes	maximal, characteristic
12	#13092439	C₂	1	1	yes	maximal, characteristic
13	#13092424	C₂	1	1	yes	maximal, characteristic
14	#13092424	C₂	1	1	yes	maximal, characteristic
15	C₂₃×#479	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×Q₈×D₂₃	C₂²	1	1	yes	characteristic
17	C₂₃×#125	C₂²	1	1	yes	characteristic
18	C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
19	Q₈×Dic₂₃	C₂²	1	1	yes	characteristic
20	C₂×#67262	C₂²	1	1	yes	characteristic
21	C₂×#67262	C₂²	1	1	yes	characteristic
22	#10625067	—	2	1	no	—
23	#10625067	—	2	1	no	—
24	#10625063	C₂²	1	1	yes	characteristic
25	#10625063	C₂²	1	1	yes	characteristic
26	C₄⋊C₄×D₂₃	—	2	1	no	—
27	C₄⋊C₄×D₂₃	C₂²	1	1	yes	characteristic
28	#10625059	C₂²	1	1	yes	characteristic
29	#10625059	C₂²	1	1	yes	characteristic
30	#10625058	—	2	1	no	—
31	#10625058	—	2	1	no	—
32	#10625058	—	2	1	no	—
33	#10625058	C₂²	1	1	yes	characteristic
34	C₄²×D₂₃	—	2	1	no	—
35	C₄²×D₂₃	C₂²	1	1	yes	characteristic
36	#10625041	C₂²	1	1	yes	characteristic
37	C₄×Dic₄₆	—	2	1	no	—
38	C₄×Dic₄₆	C₂²	1	1	yes	characteristic
39	C₂₃×#107	C₂²	1	1	yes	characteristic
40	C₂₃×#106	C₂²	1	1	yes	characteristic
41	C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
42	C₂₃×#103	C₂²	1	1	yes	characteristic
43	C₂₃×#103	C₂²	1	1	yes	characteristic
44	#10625017	C₂²	1	1	yes	characteristic
45	#10625017	C₂²	1	1	yes	characteristic
46	#10625000	C₂²	1	1	yes	characteristic
47	#10624999	C₂²	1	1	yes	characteristic
48	#10624998	C₂²	1	1	yes	characteristic
49	#10624998	C₂²	1	1	yes	characteristic
50	Dic₂₃⋊C₈	C₂²	1	1	yes	characteristic
51	Dic₂₃⋊C₈	C₂²	1	1	yes	characteristic
52	#10624992	C₂²	1	1	yes	characteristic
53	#10624992	C₂²	1	1	yes	characteristic
54	#10624990	C₂²	1	1	yes	characteristic
55	#10624990	C₂²	1	1	yes	characteristic
56	#10624989	C₂²	1	1	yes	characteristic
57	#10624989	C₂²	1	1	yes	characteristic
58	C₄⋊(C₂₃⋊C₈)	C₂²	1	1	yes	characteristic
59	C₂×Q₈×C₂₃	C₂³	1	1	yes	characteristic
60	Q₈×D₂₃	—	2	2	no	—
61	Q₈×D₂₃	Q₈	1	4	yes	complemented
62	C₂×C₄×D₂₃	—	2	1	no	—
63	C₂×C₄×D₂₃	—	2	1	no	—
64	C₂×C₄×D₂₃	D₄	1	1	yes	characteristic
65	C₂×C₄×D₂₃	D₄	1	1	yes	characteristic
66	C₂×C₄×D₂₃	C₂³	1	1	yes	characteristic
67	C₂×Dic₄₆	—	2	1	no	—
68	C₂×Dic₄₆	—	2	1	no	—
69	C₂×Dic₄₆	—	2	1	no	—
70	C₂×Dic₄₆	C₂³	1	1	yes	characteristic
71	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
72	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
73	C₄⋊C₄×C₂₃	—	2	1	no	—
74	C₄⋊C₄×C₂₃	—	2	1	no	—
75	C₄⋊C₄×C₂₃	—	2	1	no	—
76	C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
77	C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
78	C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
79	C₄²×C₂₃	—	2	1	no	—
80	C₄²×C₂₃	C₂³	1	1	yes	characteristic
81	(C₂×D₂₃)⋊C₄	—	2	1	no	—
82	(C₂×D₂₃)⋊C₄	—	2	1	no	—
83	C₄⋊Dic₂₃	—	2	1	no	—
84	C₄⋊Dic₂₃	C₂³	1	1	yes	characteristic
85	C₄⋊Dic₂₃	C₂³	1	1	yes	characteristic
86	C₄⋊Dic₂₃	C₂³	1	1	yes	characteristic
87	Dic₂₃⋊C₄	—	2	1	no	—
88	Dic₂₃⋊C₄	—	2	1	no	—
89	Dic₂₃⋊C₄	—	2	1	no	—
90	Dic₂₃⋊C₄	—	2	1	no	—
91	Dic₂₃⋊C₄	—	2	1	no	—
92	Dic₂₃⋊C₄	—	2	1	no	—
93	Dic₂₃⋊C₄	—	2	1	no	—
94	C₄×Dic₂₃	—	2	1	no	—
95	C₄×Dic₂₃	—	2	1	no	—
96	C₄×Dic₂₃	—	2	1	no	—
97	C₄×Dic₂₃	—	2	1	no	—
98	C₄×Dic₂₃	—	2	1	no	—
99	C₄×Dic₂₃	D₄	1	1	yes	characteristic
100	C₄×Dic₂₃	D₄	1	1	yes	characteristic
101	C₄×Dic₂₃	C₂³	1	1	yes	characteristic
102	C₂×C₂₃⋊C₈	C₂³	1	1	yes	characteristic
103	C₂×C₂₃⋊C₈	C₂³	1	1	yes	characteristic
104	#67262	—	2	2	no	—
105	#67262	—	2	2	no	—
106	C₂²×D₂₃	C₂×D₄	1	1	yes	characteristic
107	Q₈×C₂₃	—	2	1	no	—
108	Q₈×C₂₃	C₂×Q₈	1	2	yes	—
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	—	2	1	no	—
113	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
114	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
115	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
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	C₂×D₄	1	1	yes	characteristic
121	C₂×Dic₂₃	C₂×D₄	1	1	yes	characteristic
122	C₂×Dic₂₃	C₂×D₄	1	1	yes	characteristic
123	C₄×D₂₃	—	2	4	no	—
124	C₄×D₂₃	—	2	2	no	—
125	C₄×D₂₃	—	4	1	no	—
126	C₄×D₂₃	—	4	2	no	—
127	C₄×D₂₃	C₂×Q₈	1	2	yes	—
128	C₄×D₂₃	#41	1	2	yes	—
129	Dic₄₆	—	2	1	no	—
130	Dic₄₆	—	2	4	no	—
131	Dic₄₆	—	4	2	no	—
132	Dic₄₆	—	4	2	no	—
133	Dic₄₆	C₂×Q₈	1	2	yes	—
134	C₈×C₂₃	—	2	1	no	—
135	C₈×C₂₃	—	2	1	no	—
136	C₂₃⋊C₈	—	2	1	no	—
137	C₂₃⋊C₈	—	2	1	no	—
138	#3078	—	23	1	no	maximal, Sylow
139	C₂²×C₂₃	C₂²×D₄	1	1	yes	characteristic
140	C₂×D₂₃	—	2	1	no	—
141	C₂×D₂₃	—	2	1	no	—
142	C₂×D₂₃	C₂²⋊Q₈	1	2	yes	—
143	C₄×C₂₃	—	2	2	no	—
144	C₄×C₂₃	—	2	1	no	—
145	C₄×C₂₃	—	2	1	no	—
146	C₄×C₂₃	—	4	1	no	—
147	C₄×C₂₃	—	4	1	no	—
148	C₄×C₂₃	—	4	1	no	—
149	C₄×C₂₃	C₂²×Q₈	1	1	yes	characteristic
150	C₄×C₂₃	C₂×#41	1	1	yes	characteristic
151	Dic₂₃	—	2	2	no	—
152	Dic₂₃	—	2	1	no	—
153	Dic₂₃	—	4	1	no	—
154	Dic₂₃	—	4	1	no	—
155	Dic₂₃	—	4	1	no	—
156	Dic₂₃	C₂²⋊Q₈	1	2	yes	—
157	Dic₂₃	#137	1	2	yes	—
158	#533	—	23	1	no	—
159	C₂×C₄×Q₈	—	23	1	no	—
160	#479	—	23	1	no	—
161	#479	—	23	1	no	—
162	#479	—	23	1	no	—
163	#479	D₂₃	1	1	yes	characteristic, complemented
164	#477	—	23	1	no	—
165	#477	—	23	1	no	—
166	#475	—	23	1	no	—
167	#475	—	23	1	no	—
168	#428	—	23	1	no	—
169	#428	—	23	1	no	—
170	#423	—	23	1	no	—
171	#419	—	23	1	no	—
172	#419	—	23	1	no	—
173	C₂×C₂₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
174	C₂×C₂₃	C₂×#137	1	1	yes	characteristic
175	C₂×C₂₃	#575	1	1	yes	characteristic
176	D₂₃	—	2	2	no	—
177	C₂²×Q₈	—	23	1	no	—
178	C₂×M₁₆	—	23	1	no	—
179	C₂×M₁₆	—	23	1	no	—
180	C₄⋊Q₈	—	23	1	no	—
181	C₄⋊Q₈	—	23	1	no	—
182	#125	—	23	1	no	—
183	#125	C₂×D₂₃	1	1	yes	characteristic
184	C₂²⋊Q₈	—	46	1	no	—
185	C₂²⋊Q₈	—	46	1	no	—
186	C₄×Q₈	—	23	1	no	—
187	C₄×Q₈	—	23	1	no	—
188	C₄×Q₈	—	23	1	no	—
189	C₄×Q₈	—	46	1	no	—
190	C₄×Q₈	—	46	1	no	—
191	C₄×Q₈	—	46	1	no	—
192	C₄×Q₈	—	46	1	no	—
193	C₄×Q₈	C₂×D₂₃	1	1	yes	characteristic
194	#117	—	23	1	no	—
195	#117	—	23	1	no	—
196	C₂×C₄⋊C₄	—	23	1	no	—
197	C₂×C₄⋊C₄	—	46	1	no	—
198	C₂×C₄²	—	23	1	no	—
199	C₂×C₄²	—	46	1	no	—
200	#107	—	23	1	no	—
201	#107	—	23	1	no	—
202	#107	—	23	1	no	—
203	#107	C₂×D₂₃	1	1	yes	characteristic
204	#106	—	23	1	no	—
205	#106	—	23	1	no	—
206	#106	—	23	1	no	—
207	#106	C₂×D₂₃	1	1	yes	characteristic
208	C₄⋊C₈	—	23	1	no	—
209	C₄⋊C₈	—	23	1	no	—
210	C₄⋊C₈	—	23	1	no	—
211	C₄⋊C₈	C₂×D₂₃	1	1	yes	characteristic
212	#103	—	23	1	no	—
213	#103	—	23	1	no	—
214	#103	—	23	1	no	—
215	#103	—	23	1	no	—
216	#103	—	23	1	no	—
217	#103	—	23	1	no	—
218	#103	C₂×D₂₃	1	1	yes	characteristic
219	#103	C₂×D₂₃	1	1	yes	characteristic
220	C₂₃	#3078	1	1	yes	characteristic, complemented, Sylow
221	C₂×Q₈	—	23	4	no	—
222	C₂×Q₈	—	23	1	no	—
223	C₂×Q₈	—	46	1	no	—
224	C₂×Q₈	—	46	1	no	—
225	C₂×Q₈	—	46	1	no	—
226	C₂×Q₈	—	46	2	no	—
227	C₂×Q₈	C₂²×D₂₃	1	1	yes	characteristic
228	C₂²×C₄	—	23	1	no	—
229	C₂²×C₄	—	23	1	no	—
230	C₂²×C₄	—	23	1	no	—
231	C₂²×C₄	—	46	1	no	—
232	C₂²×C₄	—	46	1	no	—
233	M₁₆	—	46	2	no	—
234	M₁₆	—	46	2	no	—
235	C₂×C₈	—	23	1	no	—
236	C₂×C₈	—	23	1	no	—
237	C₂×C₈	C₂²×D₂₃	1	1	yes	characteristic
238	C₂×C₈	C₂²×D₂₃	1	1	yes	characteristic
239	C₄⋊C₄	—	2	1	no	—
240	C₄⋊C₄	—	2	1	no	—
241	C₄⋊C₄	—	2	1	no	—
242	C₄⋊C₄	—	23	1	no	—
243	C₄⋊C₄	—	23	1	no	—
244	C₄⋊C₄	—	23	1	no	—
245	C₄⋊C₄	—	46	1	no	—
246	C₄⋊C₄	—	46	1	no	—
247	C₄⋊C₄	—	46	1	no	—
248	C₄⋊C₄	—	46	1	no	—
249	C₄⋊C₄	—	46	1	no	—
250	C₄⋊C₄	—	46	1	no	—
251	C₄⋊C₄	—	46	1	no	—
252	C₄⋊C₄	—	46	1	no	—
253	C₄⋊C₄	C₂²×D₂₃	1	1	yes	characteristic
254	C₄⋊C₄	C₂²×D₂₃	1	1	yes	characteristic
255	C₄⋊C₄	C₂²×D₂₃	1	1	yes	characteristic
256	C₂²⋊C₄	—	46	1	no	—
257	C₂²⋊C₄	—	46	1	no	—
258	C₄²	—	2	1	no	—
259	C₄²	—	23	1	no	—
260	C₄²	—	23	1	no	—
261	C₄²	—	23	1	no	—
262	C₄²	—	46	1	no	—
263	C₄²	—	46	1	no	—
264	C₄²	—	46	1	no	—
265	C₄²	—	46	1	no	—
266	C₄²	—	46	1	no	—
267	C₄²	C₂²×D₂₃	1	1	yes	characteristic
268	C₂³	—	23	1	no	—
269	Q₈	—	2	1	no	—
270	Q₈	—	23	2	no	—
271	Q₈	—	46	4	no	—
272	Q₈	—	46	1	no	—
273	Q₈	—	92	2	no	—
274	Q₈	—	92	2	no	—
275	Q₈	Q₈×D₂₃	1	2	yes	—
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	—	23	1	no	—
281	C₂×C₄	—	23	1	no	—
282	C₂×C₄	—	23	2	no	—
283	C₂×C₄	—	23	2	no	—
284	C₂×C₄	—	23	1	no	—
285	C₂×C₄	—	46	2	no	—
286	C₂×C₄	—	46	4	no	—
287	C₂×C₄	—	46	1	no	—
288	C₂×C₄	—	46	1	no	—
289	C₂×C₄	—	46	1	no	—
290	C₂×C₄	—	46	1	no	—
291	C₂×C₄	—	92	1	no	—
292	C₂×C₄	—	92	2	no	—
293	C₂×C₄	D₄×D₂₃	1	1	yes	characteristic
294	C₂×C₄	D₄×D₂₃	1	1	yes	characteristic
295	C₂×C₄	C₂³×D₂₃	1	1	yes	characteristic
296	C₈	—	2	1	no	—
297	C₈	—	2	1	no	—
298	C₈	—	46	1	no	—
299	C₈	—	46	1	no	—
300	C₂²	—	23	2	no	—
301	C₂²	—	46	1	no	—
302	C₂²	—	46	1	no	—
303	C₂²	C₂×D₄×D₂₃	1	1	yes	characteristic
304	C₄	—	2	2	no	—
305	C₄	—	2	1	no	—
306	C₄	—	2	1	no	—
307	C₄	—	4	1	no	—
308	C₄	—	4	1	no	—
309	C₄	—	4	1	no	—
310	C₄	—	23	2	no	—
311	C₄	—	23	2	no	—
312	C₄	—	46	2	no	—
313	C₄	—	46	1	no	—
314	C₄	—	92	1	no	—
315	C₄	—	92	1	no	—
316	C₄	—	92	1	no	—
317	C₄	C₂×Q₈×D₂₃	1	1	yes	characteristic
318	C₄	#41×D₂₃	1	1	yes	characteristic
319	C₂	—	46	2	no	—
320	C₂	C₂²⋊Q₈×D₂₃	1	1	yes	characteristic
321	C₂	#137×D₂₃	1	1	yes	characteristic
322	C₂	#13093214	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₂₃
commutator, cosocle	C₂×C₄×C₂₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₂₃
Fitting	C₂₃×#479	C₂
socle	C₂²×C₂₃	C₂²×D₄
prime radical	C₂²×D₂₃	C₂×D₄
hypercenter	#479	D₂₃
hypocenter	C₂₃	#3078
LCS	G ▷ C₂×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #479

Complements

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

N	H ≅ G/N	type
3. #13092725	C₂	semidirect
4. #13092724	C₂	semidirect
5. #13092721	C₂	semidirect
6. #13092711	C₂	semidirect
7. #13092710	C₂	semidirect
8. #13092701	C₂	semidirect
9. #13092700	C₂	semidirect
15. C₂₃×#479	C₂	semidirect
61. Q₈×D₂₃	Q₈	semidirect
163. #479	D₂₃	semidirect
220. C₂₃	#3078	semidirect

Maximal subgroups

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

subgroup	index
#13092973	2
C₄×Q₈×D₂₃	2
#13092725	2
#13092724	2
#13092721	2
#13092711	2
#13092710	2
#13092701	2
#13092700	2
#13092458	2
#13092456	2
#13092439	2
#13092424	2
C₂₃×#479	2
#3078	23