Subgroups — #15683238

Order 2944.

362 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13093047	C₂	1	1	yes	maximal, characteristic
2	#13092893	C₂	1	1	yes	maximal, characteristic
3	#13092726	C₂	1	1	yes	maximal, characteristic, complemented
4	#13092723	C₂	1	1	yes	maximal, characteristic, complemented
5	#13092721	C₂	1	1	yes	maximal, characteristic, complemented
6	#13092713	C₂	1	1	yes	maximal, characteristic, complemented
7	#13092710	C₂	1	1	yes	maximal, characteristic, complemented
8	#13092702	C₂	1	1	yes	maximal, characteristic, complemented
9	#13092701	C₂	1	1	yes	maximal, characteristic, complemented
10	#13092456	C₂	1	1	yes	maximal, characteristic
11	#13092454	C₂	1	1	yes	maximal, characteristic
12	#13092442	C₂	1	1	yes	maximal, characteristic
13	#13092424	C₂	1	1	yes	maximal, characteristic
14	#103×D₂₃	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×C₈×D₂₃	C₂²	1	1	yes	characteristic
22	#10625068	—	2	1	no	—
23	#10625067	—	2	1	no	—
24	#10625063	—	2	1	no	—
25	#10625063	C₂²	1	1	yes	characteristic
26	#10625063	C₂²	1	1	yes	characteristic
27	C₄⋊C₄×D₂₃	C₂²	1	1	yes	characteristic
28	#10625061	C₂²	1	1	yes	characteristic
29	#10625060	—	2	1	no	—
30	#10625059	—	2	1	no	—
31	#10625059	C₂²	1	1	yes	characteristic
32	#10625058	—	2	1	no	—
33	#10625058	C₂²	1	1	yes	characteristic
34	#10625043	—	2	1	no	—
35	#10625043	C₂²	1	1	yes	characteristic
36	#10625040	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	#10625000	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	#10624990	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	C₂³	1	1	yes	characteristic
65	C₂×Dic₄₆	—	2	1	no	—
66	C₂×Dic₄₆	—	2	1	no	—
67	C₂×Dic₄₆	—	2	1	no	—
68	C₂×Dic₄₆	C₂³	1	1	yes	characteristic
69	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
70	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
71	C₄⋊C₄×C₂₃	—	2	1	no	—
72	C₄⋊C₄×C₂₃	—	2	1	no	—
73	C₄⋊C₄×C₂₃	—	2	1	no	—
74	C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
75	C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
76	C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
77	C₄²×C₂₃	—	2	1	no	—
78	C₄²×C₂₃	C₂³	1	1	yes	characteristic
79	(C₂×D₂₃)⋊C₄	—	2	1	no	—
80	(C₂×D₂₃)⋊C₄	—	2	1	no	—
81	(C₂×D₂₃)⋊C₄	D₄	1	1	yes	characteristic
82	(C₂×D₂₃)⋊C₄	D₄	1	1	yes	characteristic
83	C₄⋊Dic₂₃	—	2	1	no	—
84	C₄⋊Dic₂₃	—	2	1	no	—
85	C₄⋊Dic₂₃	—	2	1	no	—
86	C₄⋊Dic₂₃	C₂³	1	1	yes	characteristic
87	C₄⋊Dic₂₃	C₂³	1	1	yes	characteristic
88	C₄⋊Dic₂₃	C₂³	1	1	yes	characteristic
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	Dic₂₃⋊C₄	D₄	1	1	yes	characteristic
95	Dic₂₃⋊C₄	D₄	1	1	yes	characteristic
96	C₄×Dic₂₃	—	2	1	no	—
97	C₄×Dic₂₃	—	2	1	no	—
98	C₄×Dic₂₃	—	2	1	no	—
99	C₄×Dic₂₃	—	2	1	no	—
100	C₄×Dic₂₃	—	2	1	no	—
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	C₈×D₂₃	—	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	—	8	1	no	—
125	C₄×D₂₃	C₂×Q₈	1	2	yes	—
126	C₄×D₂₃	#41	1	2	yes	—
127	Dic₄₆	—	2	1	no	—
128	Dic₄₆	—	2	4	no	—
129	Dic₄₆	—	8	1	no	—
130	Dic₄₆	—	8	1	no	—
131	Dic₄₆	C₂×Q₈	1	2	yes	—
132	C₈×C₂₃	—	2	1	no	—
133	C₈×C₂₃	—	2	1	no	—
134	C₂₃⋊C₈	—	2	1	no	—
135	C₂₃⋊C₈	—	2	1	no	—
136	#3082	—	23	1	no	maximal, Sylow
137	C₂²×C₂₃	C₂²×D₄	1	1	yes	characteristic
138	C₂×D₂₃	—	2	1	no	—
139	C₂×D₂₃	—	2	1	no	—
140	C₂×D₂₃	C₂²⋊Q₈	1	2	yes	—
141	C₄×C₂₃	—	2	2	no	—
142	C₄×C₂₃	—	2	1	no	—
143	C₄×C₂₃	—	2	1	no	—
144	C₄×C₂₃	—	4	1	no	—
145	C₄×C₂₃	—	4	1	no	—
146	C₄×C₂₃	—	4	1	no	—
147	C₄×C₂₃	C₂²×Q₈	1	1	yes	characteristic
148	C₄×C₂₃	C₂×#41	1	1	yes	characteristic
149	Dic₂₃	—	2	2	no	—
150	Dic₂₃	—	2	1	no	—
151	Dic₂₃	—	2	1	no	—
152	Dic₂₃	—	4	1	no	—
153	Dic₂₃	—	4	1	no	—
154	Dic₂₃	—	4	1	no	—
155	Dic₂₃	C₂²⋊Q₈	1	2	yes	—
156	#544	—	23	1	no	—
157	#519	—	23	1	no	—
158	#479	—	23	1	no	—
159	#479	—	23	1	no	—
160	#479	—	23	1	no	—
161	#479	D₂₃	1	1	yes	characteristic, complemented
162	#477	—	23	1	no	—
163	#477	—	23	1	no	—
164	#475	—	23	1	no	—
165	#475	—	23	1	no	—
166	#428	—	23	1	no	—
167	#427	—	23	1	no	—
168	#424	—	23	1	no	—
169	#419	—	23	1	no	—
170	C₂×#103	—	23	1	no	—
171	C₂×C₂₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
172	C₂×C₂₃	#577	1	1	yes	characteristic
173	C₂×C₂₃	#578	1	1	yes	characteristic
174	D₂₃	—	2	2	no	—
175	C₂²×Q₈	—	23	1	no	—
176	C₂×M₁₆	—	23	1	no	—
177	C₂²×C₈	—	23	1	no	—
178	C₄⋊Q₈	—	23	1	no	—
179	C₄⋊Q₈	—	23	1	no	—
180	C₄⋊Q₈	—	46	1	no	—
181	#125	—	23	1	no	—
182	#125	—	46	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	C₂×D₂₃	1	1	yes	characteristic
192	#117	—	23	1	no	—
193	#117	—	23	1	no	—
194	#117	—	23	1	no	—
195	#117	—	46	1	no	—
196	#117	—	46	1	no	—
197	C₂×C₄⋊C₄	—	23	1	no	—
198	#107	—	23	1	no	—
199	#107	—	23	1	no	—
200	#107	—	23	1	no	—
201	#107	C₂×D₂₃	1	1	yes	characteristic
202	#106	—	23	1	no	—
203	#106	—	23	1	no	—
204	#106	—	23	1	no	—
205	#106	C₂×D₂₃	1	1	yes	characteristic
206	C₄⋊C₈	—	23	1	no	—
207	C₄⋊C₈	—	23	1	no	—
208	C₄⋊C₈	—	23	1	no	—
209	C₄⋊C₈	C₂×D₂₃	1	1	yes	characteristic
210	#103	—	23	1	no	—
211	#103	—	23	1	no	—
212	#103	—	23	1	no	—
213	#103	—	23	1	no	—
214	#103	—	23	1	no	—
215	#103	—	23	1	no	—
216	#103	C₂×D₂₃	1	1	yes	characteristic
217	#103	C₂×D₂₃	1	1	yes	characteristic
218	C₂₃	#3082	1	1	yes	characteristic, complemented, Sylow
219	C₂×Q₈	—	23	4	no	—
220	C₂×Q₈	—	23	1	no	—
221	C₂×Q₈	—	46	1	no	—
222	C₂×Q₈	—	46	1	no	—
223	C₂×Q₈	—	46	2	no	—
224	C₂×Q₈	—	46	1	no	—
225	C₂×Q₈	C₂²×D₂₃	1	1	yes	characteristic
226	C₂²×C₄	—	23	1	no	—
227	C₂²×C₄	—	46	1	no	—
228	C₂²×C₄	—	46	1	no	—
229	M₁₆	—	46	2	no	—
230	C₂×C₈	—	23	1	no	—
231	C₂×C₈	—	23	1	no	—
232	C₂×C₈	—	46	2	no	—
233	C₂×C₈	C₂²×D₂₃	1	1	yes	characteristic
234	C₂×C₈	C₂²×D₂₃	1	1	yes	characteristic
235	C₄⋊C₄	—	2	1	no	—
236	C₄⋊C₄	—	2	1	no	—
237	C₄⋊C₄	—	2	1	no	—
238	C₄⋊C₄	—	23	1	no	—
239	C₄⋊C₄	—	23	1	no	—
240	C₄⋊C₄	—	23	1	no	—
241	C₄⋊C₄	—	23	1	no	—
242	C₄⋊C₄	—	23	1	no	—
243	C₄⋊C₄	—	46	1	no	—
244	C₄⋊C₄	—	46	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	C₂²×D₂₃	1	1	yes	characteristic
252	C₄⋊C₄	C₂²×D₂₃	1	1	yes	characteristic
253	C₄⋊C₄	C₂²×D₂₃	1	1	yes	characteristic
254	C₂²⋊C₄	—	23	1	no	—
255	C₂²⋊C₄	—	23	1	no	—
256	C₂²⋊C₄	—	46	1	no	—
257	C₂²⋊C₄	—	46	1	no	—
258	C₄²	—	2	1	no	—
259	C₄²	—	23	1	no	—
260	C₄²	—	46	1	no	—
261	C₄²	—	46	1	no	—
262	C₄²	—	46	1	no	—
263	C₄²	—	46	1	no	—
264	C₄²	—	46	1	no	—
265	C₄²	C₂²×D₂₃	1	1	yes	characteristic
266	C₂³	—	23	1	no	—
267	Q₈	—	2	1	no	—
268	Q₈	—	23	2	no	—
269	Q₈	—	46	4	no	—
270	Q₈	—	46	1	no	—
271	Q₈	—	184	1	no	—
272	Q₈	—	184	1	no	—
273	Q₈	Q₈×D₂₃	1	2	yes	—
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	—	23	1	no	—
279	C₂×C₄	—	23	2	no	—
280	C₂×C₄	—	23	2	no	—
281	C₂×C₄	—	23	1	no	—
282	C₂×C₄	—	23	1	no	—
283	C₂×C₄	—	46	4	no	—
284	C₂×C₄	—	46	1	no	—
285	C₂×C₄	—	46	1	no	—
286	C₂×C₄	—	46	1	no	—
287	C₂×C₄	—	46	1	no	—
288	C₂×C₄	—	184	1	no	—
289	C₂×C₄	D₄×D₂₃	1	1	yes	characteristic
290	C₂×C₄	D₄×D₂₃	1	1	yes	characteristic
291	C₂×C₄	C₂³×D₂₃	1	1	yes	characteristic
292	C₈	—	2	1	no	—
293	C₈	—	2	1	no	—
294	C₈	—	46	1	no	—
295	C₈	—	46	1	no	—
296	C₂²	—	23	2	no	—
297	C₂²	—	46	1	no	—
298	C₂²	—	46	1	no	—
299	C₂²	C₂×D₄×D₂₃	1	1	yes	characteristic
300	C₄	—	2	2	no	—
301	C₄	—	2	1	no	—
302	C₄	—	2	1	no	—
303	C₄	—	4	1	no	—
304	C₄	—	4	1	no	—
305	C₄	—	4	1	no	—
306	C₄	—	23	2	no	—
307	C₄	—	46	2	no	—
308	C₄	—	46	1	no	—
309	C₄	—	46	1	no	—
310	C₄	—	92	1	no	—
311	C₄	—	92	1	no	—
312	C₄	—	92	1	no	—
313	C₄	C₂×Q₈×D₂₃	1	1	yes	characteristic
314	C₄	#41×D₂₃	1	1	yes	characteristic
315	C₂	—	46	2	no	—
316	C₂	C₂²⋊Q₈×D₂₃	1	1	yes	characteristic
317	C₂	#13093226	1	1	yes	characteristic
318	C₂	#13093231	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	#3082
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. #13092726	C₂	semidirect
4. #13092723	C₂	semidirect
5. #13092721	C₂	semidirect
6. #13092713	C₂	semidirect
7. #13092710	C₂	semidirect
8. #13092702	C₂	semidirect
9. #13092701	C₂	semidirect
15. C₂₃×#479	C₂	semidirect
61. Q₈×D₂₃	Q₈	semidirect
161. #479	D₂₃	semidirect
218. C₂₃	#3082	semidirect

Maximal subgroups

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

subgroup	index
#13093047	2
#13092893	2
#13092726	2
#13092723	2
#13092721	2
#13092713	2
#13092710	2
#13092702	2
#13092701	2
#13092456	2
#13092454	2
#13092442	2
#13092424	2
#103×D₂₃	2
C₂₃×#479	2
#3082	23