Subgroups — #15683621

Order 2944.

370 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13093034	C₂	1	1	yes	maximal, characteristic
2	#13092894	C₂	1	1	yes	maximal, characteristic, complemented
3	#13092758	C₂	1	1	yes	maximal, characteristic, complemented
4	#13092711	C₂	1	1	yes	maximal, characteristic, complemented
5	#13092701	C₂	1	1	yes	maximal, characteristic, complemented
6	#13092638	C₂	1	1	yes	maximal, characteristic, complemented
7	#13092638	C₂	1	1	yes	maximal, characteristic, complemented
8	#13092575	C₂	1	1	yes	maximal, characteristic, complemented
9	#13092555	C₂	1	1	yes	maximal, characteristic
10	#13092523	C₂	1	1	yes	maximal, characteristic, complemented
11	#13092514	C₂	1	1	yes	maximal, characteristic, complemented
12	#13092511	C₂	1	1	yes	maximal, characteristic, complemented
13	#13092508	C₂	1	1	yes	maximal, characteristic, complemented
14	#13092475	C₂	1	1	yes	maximal, characteristic, complemented
15	C₂₃×#482	C₂	1	1	yes	maximal, characteristic
16	C₂²×Dic₄₆	C₂²	1	1	yes	characteristic
17	C₂₃×#121	C₂²	1	1	yes	characteristic
18	C₂₃×#117	C₂²	1	1	yes	characteristic
19	#10625107	—	2	1	no	—
20	#10625107	C₂²	1	1	yes	characteristic
21	#10625106	—	2	1	no	—
22	D₄×Dic₂₃	C₂²	1	1	yes	characteristic
23	C₂×#67273	C₂²	1	1	yes	characteristic
24	C₂×#67273	C₂²	1	1	yes	characteristic
25	#10625097	C₂²	1	1	yes	characteristic
26	#10625095	C₂²	1	1	yes	characteristic
27	C₂×Dic₉₂	C₂²	1	1	yes	characteristic
28	C₂×Dic₉₂	C₂²	1	1	yes	characteristic
29	#10625059	C₂²	1	1	yes	characteristic
30	#10625058	C₂²	1	1	yes	characteristic
31	#10625058	C₂²	1	1	yes	characteristic
32	#10625058	C₂²	1	1	yes	characteristic
33	#10625058	C₂²	1	1	yes	characteristic
34	#10625058	C₂²	1	1	yes	characteristic
35	#10625050	—	2	1	no	—
36	#10625049	—	2	1	no	—
37	#10625048	—	2	1	no	—
38	#10625048	—	2	1	no	—
39	C₄×Dic₄₆	—	2	1	no	—
40	C₄×Dic₄₆	—	2	1	no	—
41	C₂₃×#107	C₂²	1	1	yes	characteristic
42	C₂₃×#106	C₂²	1	1	yes	characteristic
43	C₂₃×#102	C₂²	1	1	yes	characteristic
44	C₂₃×#102	C₂²	1	1	yes	characteristic
45	C₂₃×C₂²⋊C₈	C₂²	1	1	yes	characteristic
46	C₂²⋊(C₂₃⋊C₈)	C₂²	1	1	yes	characteristic
47	#10624998	C₂²	1	1	yes	characteristic
48	#10624998	C₂²	1	1	yes	characteristic
49	#10624997	C₂²	1	1	yes	characteristic
50	Dic₂₃⋊C₈	C₂²	1	1	yes	characteristic
51	Dic₂₃⋊C₈	C₂²	1	1	yes	characteristic
52	C₈×Dic₂₃	C₂²	1	1	yes	characteristic
53	#10624992	C₂²	1	1	yes	characteristic
54	#10624992	C₂²	1	1	yes	characteristic
55	#10624992	C₂²	1	1	yes	characteristic
56	#10624992	C₂²	1	1	yes	characteristic
57	#10624990	C₂²	1	1	yes	characteristic
58	#10624989	C₂²	1	1	yes	characteristic
59	C₂×D₄×C₂₃	—	2	1	no	—
60	C₂×D₄×C₂₃	C₂³	1	1	yes	characteristic
61	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
62	C₂²×Dic₂₃	—	2	1	no	—
63	C₂²×Dic₂₃	—	2	1	no	—
64	C₂×Dic₄₆	—	2	1	no	—
65	C₂×Dic₄₆	—	2	2	no	—
66	C₂×Dic₄₆	—	2	2	no	—
67	C₂×Dic₄₆	—	2	1	no	—
68	C₂×Dic₄₆	C₂³	1	1	yes	characteristic
69	C₂×Dic₄₆	C₂³	1	1	yes	characteristic
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	C₂³	1	1	yes	characteristic
74	C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
75	C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
76	C₂²⋊C₄×C₂₃	—	2	1	no	—
77	C₂²⋊C₄×C₂₃	—	2	1	no	—
78	C₄²×C₂₃	—	2	1	no	—
79	C₂²⋊Dic₂₃	—	2	1	no	—
80	C₂²⋊Dic₂₃	—	2	1	no	—
81	C₂²⋊Dic₂₃	—	2	1	no	—
82	C₂²⋊Dic₂₃	—	2	1	no	—
83	C₂²⋊Dic₂₃	D₄	1	1	yes	characteristic
84	C₂²⋊Dic₂₃	D₄	1	1	yes	characteristic
85	#67273	—	2	2	no	—
86	#67273	—	2	2	no	—
87	C₄⋊Dic₂₃	D₄	1	1	yes	characteristic
88	C₄⋊Dic₂₃	D₄	1	1	yes	characteristic
89	C₄⋊Dic₂₃	C₂³	1	1	yes	characteristic
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	—	2	1	no	—
95	Dic₂₃⋊C₄	—	2	1	no	—
96	Dic₂₃⋊C₄	—	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	—	2	1	no	—
102	C₄×Dic₂₃	C₂³	1	1	yes	characteristic
103	C₄×Dic₂₃	C₂³	1	1	yes	characteristic
104	C₂×C₂₃⋊C₈	C₂³	1	1	yes	characteristic
105	C₂×C₂₃⋊C₈	C₂³	1	1	yes	characteristic
106	Dic₉₂	—	2	2	no	—
107	Dic₉₂	—	2	2	no	—
108	C₂³×C₂₃	—	2	1	no	—
109	C₂³×C₂₃	C₂×D₄	1	1	yes	characteristic
110	D₄×C₂₃	—	2	1	no	—
111	D₄×C₂₃	—	2	1	no	—
112	D₄×C₂₃	—	8	1	no	—
113	C₂×C₄×C₂₃	—	2	1	no	—
114	C₂×C₄×C₂₃	—	2	1	no	—
115	C₂×C₄×C₂₃	—	2	1	no	—
116	C₂×C₄×C₂₃	—	2	1	no	—
117	C₂×C₄×C₂₃	—	2	1	no	—
118	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
119	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
120	C₂×Dic₂₃	—	2	4	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	—	8	1	no	—
126	C₂×Dic₂₃	C₂×D₄	1	1	yes	characteristic
127	C₂×Dic₂₃	C₂×D₄	1	1	yes	characteristic
128	C₂×Dic₂₃	C₂×D₄	1	1	yes	characteristic
129	C₂×Dic₂₃	C₂×D₄	1	1	yes	characteristic
130	Dic₄₆	—	2	4	no	—
131	Dic₄₆	—	2	1	no	—
132	Dic₄₆	—	2	1	no	—
133	Dic₄₆	—	2	1	no	—
134	Dic₄₆	—	2	1	no	—
135	Dic₄₆	—	8	1	no	—
136	Dic₄₆	#41	1	2	yes	—
137	Dic₄₆	#41	1	2	yes	—
138	C₈×C₂₃	—	2	1	no	—
139	C₈×C₂₃	—	2	1	no	—
140	C₂₃⋊C₈	—	2	1	no	—
141	C₂₃⋊C₈	—	2	1	no	—
142	#3121	—	23	1	no	maximal, Sylow
143	C₂²×C₂₃	—	2	1	no	—
144	C₂²×C₂₃	—	2	1	no	—
145	C₂²×C₂₃	—	2	1	no	—
146	C₂²×C₂₃	—	4	1	no	—
147	C₂²×C₂₃	—	4	1	no	—
148	C₂²×C₂₃	—	4	1	no	—
149	C₂²×C₂₃	C₂²×D₄	1	1	yes	characteristic
150	C₄×C₂₃	—	2	1	no	—
151	C₄×C₂₃	—	4	1	no	—
152	C₄×C₂₃	—	4	1	no	—
153	C₄×C₂₃	—	4	1	no	—
154	C₄×C₂₃	C₂×#41	1	1	yes	characteristic
155	C₄×C₂₃	C₂×#41	1	1	yes	characteristic
156	Dic₂₃	—	2	2	no	—
157	Dic₂₃	—	2	2	no	—
158	Dic₂₃	—	2	1	no	—
159	Dic₂₃	—	2	1	no	—
160	Dic₂₃	—	2	1	no	—
161	Dic₂₃	—	2	1	no	—
162	Dic₂₃	—	4	1	no	—
163	Dic₂₃	—	4	1	no	—
164	#542	—	23	1	no	—
165	#519	—	23	1	no	—
166	#485	—	23	1	no	—
167	#482	D₂₃	1	1	yes	characteristic
168	#477	—	23	1	no	—
169	#475	—	23	1	no	—
170	#464	—	23	1	no	—
171	#464	—	23	1	no	—
172	#451	—	23	1	no	—
173	#448	—	23	1	no	—
174	#441	—	23	1	no	—
175	#440	—	23	1	no	—
176	C₄×Q₁₆	—	23	1	no	—
177	C₄×SD₁₆	—	23	1	no	—
178	#433	—	23	1	no	—
179	C₂×C₂₃	—	4	1	no	—
180	C₂×C₂₃	—	8	1	no	—
181	C₂×C₂₃	#525	1	1	yes	characteristic
182	C₂×C₂₃	#577	1	1	yes	characteristic
183	C₂×C₂₃	#578	1	1	yes	characteristic
184	C₂²×Q₈	—	23	1	no	—
185	C₂×Q₁₆	—	23	1	no	—
186	C₂×Q₁₆	—	23	1	no	—
187	C₂×SD₁₆	—	23	1	no	—
188	C₂×SD₁₆	—	23	1	no	—
189	C₄⋊Q₈	—	23	1	no	—
190	#126	—	46	1	no	—
191	#124	—	23	1	no	—
192	#124	—	46	1	no	—
193	#123	—	46	1	no	—
194	C₂²⋊Q₈	—	23	1	no	—
195	C₂²⋊Q₈	—	46	1	no	—
196	#121	C₂×D₂₃	1	1	yes	characteristic
197	C₄×Q₈	—	23	1	no	—
198	C₄×Q₈	—	23	1	no	—
199	C₄×Q₈	—	23	1	no	—
200	C₄×Q₈	—	23	1	no	—
201	C₄×Q₈	—	23	1	no	—
202	C₄×Q₈	—	46	1	no	—
203	C₄×Q₈	—	46	1	no	—
204	C₄×D₄	—	23	1	no	—
205	#117	—	23	1	no	—
206	#117	—	46	1	no	—
207	#117	—	46	1	no	—
208	#117	C₂×D₂₃	1	1	yes	characteristic
209	#107	—	23	1	no	—
210	#107	C₂×D₂₃	1	1	yes	characteristic
211	#106	—	23	1	no	—
212	#106	C₂×D₂₃	1	1	yes	characteristic
213	C₄⋊C₈	—	23	1	no	—
214	C₄⋊C₈	—	23	1	no	—
215	#103	—	23	1	no	—
216	#103	—	23	1	no	—
217	#103	—	23	1	no	—
218	#103	—	23	1	no	—
219	#103	—	23	1	no	—
220	#103	—	23	1	no	—
221	#102	C₂×D₂₃	1	1	yes	characteristic
222	#102	C₂×D₂₃	1	1	yes	characteristic
223	C₂²⋊C₈	—	23	1	no	—
224	C₂²⋊C₈	C₂×D₂₃	1	1	yes	characteristic
225	#97	—	23	1	no	—
226	C₄×C₈	—	23	1	no	—
227	C₂₃	#3121	1	1	yes	characteristic, complemented, Sylow
228	C₂×Q₈	—	23	1	no	—
229	C₂×Q₈	—	23	1	no	—
230	C₂×Q₈	—	23	1	no	—
231	C₂×Q₈	—	46	1	no	—
232	C₂×Q₈	—	46	2	no	—
233	C₂×Q₈	—	46	2	no	—
234	C₂×Q₈	—	46	1	no	—
235	C₂×D₄	—	2	1	no	—
236	C₂×D₄	C₂²×D₂₃	1	1	yes	characteristic
237	C₂²×C₄	—	46	1	no	—
238	C₂²×C₄	—	46	1	no	—
239	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
240	Q₁₆	—	46	2	no	—
241	Q₁₆	—	46	2	no	—
242	SD₁₆	—	46	2	no	—
243	SD₁₆	—	46	2	no	—
244	C₂×C₈	—	23	1	no	—
245	C₂×C₈	—	23	1	no	—
246	C₂×C₈	C₂²×D₂₃	1	1	yes	characteristic
247	C₂×C₈	C₂²×D₂₃	1	1	yes	characteristic
248	C₄⋊C₄	—	23	1	no	—
249	C₄⋊C₄	—	23	1	no	—
250	C₄⋊C₄	—	23	1	no	—
251	C₄⋊C₄	—	46	1	no	—
252	C₄⋊C₄	—	46	1	no	—
253	C₄⋊C₄	—	46	1	no	—
254	C₄⋊C₄	—	46	1	no	—
255	C₄⋊C₄	—	46	1	no	—
256	C₄⋊C₄	—	46	1	no	—
257	C₄⋊C₄	—	46	1	no	—
258	C₄⋊C₄	C₂²×D₂₃	1	1	yes	characteristic
259	C₄⋊C₄	C₂²×D₂₃	1	1	yes	characteristic
260	C₄⋊C₄	C₂²×D₂₃	1	1	yes	characteristic
261	C₂²⋊C₄	—	2	1	no	—
262	C₂²⋊C₄	—	2	1	no	—
263	C₂²⋊C₄	—	23	1	no	—
264	C₂²⋊C₄	—	23	1	no	—
265	C₂²⋊C₄	—	46	1	no	—
266	C₂²⋊C₄	—	46	1	no	—
267	C₂²⋊C₄	—	46	1	no	—
268	C₂²⋊C₄	—	46	1	no	—
269	C₄²	—	2	1	no	—
270	C₄²	—	23	1	no	—
271	C₄²	—	23	1	no	—
272	C₄²	—	46	1	no	—
273	C₄²	—	46	1	no	—
274	C₄²	—	46	1	no	—
275	C₄²	—	46	1	no	—
276	C₄²	—	46	1	no	—
277	C₂³	—	2	1	no	—
278	C₂³	D₄×D₂₃	1	1	yes	characteristic
279	Q₈	—	23	2	no	—
280	Q₈	—	23	2	no	—
281	Q₈	—	46	4	no	—
282	Q₈	—	46	1	no	—
283	Q₈	—	46	1	no	—
284	Q₈	—	46	1	no	—
285	Q₈	—	46	1	no	—
286	Q₈	—	184	1	no	—
287	D₄	—	2	1	no	—
288	D₄	—	2	1	no	—
289	D₄	—	8	1	no	—
290	C₂×C₄	—	2	1	no	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	23	1	no	—
296	C₂×C₄	—	23	1	no	—
297	C₂×C₄	—	23	1	no	—
298	C₂×C₄	—	23	1	no	—
299	C₂×C₄	—	46	4	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	—	184	1	no	—
305	C₂×C₄	D₄×D₂₃	1	1	yes	characteristic
306	C₂×C₄	C₂³×D₂₃	1	1	yes	characteristic
307	C₈	—	2	1	no	—
308	C₈	—	2	1	no	—
309	C₈	—	46	1	no	—
310	C₈	—	46	1	no	—
311	C₂²	—	2	1	no	—
312	C₂²	—	2	1	no	—
313	C₂²	—	2	1	no	—
314	C₂²	—	4	1	no	—
315	C₂²	—	4	1	no	—
316	C₂²	—	4	1	no	—
317	C₂²	C₂×D₄×D₂₃	1	1	yes	characteristic
318	C₄	—	2	1	no	—
319	C₄	—	4	1	no	—
320	C₄	—	4	1	no	—
321	C₄	—	4	1	no	—
322	C₄	—	46	2	no	—
323	C₄	—	46	2	no	—
324	C₄	—	46	1	no	—
325	C₄	—	46	1	no	—
326	C₄	—	46	1	no	—
327	C₄	—	46	1	no	—
328	C₄	—	92	1	no	—
329	C₄	—	92	1	no	—
330	C₄	#41×D₂₃	1	1	yes	characteristic
331	C₄	#41×D₂₃	1	1	yes	characteristic
332	C₂	—	4	1	no	—
333	C₂	—	8	1	no	—
334	C₂	#13092925	1	1	yes	characteristic
335	C₂	#13093227	1	1	yes	characteristic
336	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×#482	C₂
socle	C₂²×C₂₃	C₂²×D₄
prime radical	C₂₃×#121	C₂²
hypercenter	#482	D₂₃
hypocenter	C₂₃	#3121
LCS	G ▷ C₂×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #482

Complements

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

N	H ≅ G/N	type
2. #13092894	C₂	semidirect
3. #13092758	C₂	semidirect
4. #13092711	C₂	semidirect
5. #13092701	C₂	semidirect
6. #13092638	C₂	semidirect
7. #13092638	C₂	semidirect
8. #13092575	C₂	semidirect
10. #13092523	C₂	semidirect
11. #13092514	C₂	semidirect
12. #13092511	C₂	semidirect
13. #13092508	C₂	semidirect
14. #13092475	C₂	semidirect
227. C₂₃	#3121	semidirect

Maximal subgroups

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

subgroup	index
#13093034	2
#13092894	2
#13092758	2
#13092711	2
#13092701	2
#13092638	2
#13092575	2
#13092555	2
#13092523	2
#13092514	2
#13092511	2
#13092508	2
#13092475	2
C₂₃×#482	2
#3121	23