Subgroups — #15683181

Order 2944.

388 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13092970	C₂	1	1	yes	maximal, characteristic, complemented
2	#41×Dic₂₃	C₂	1	1	yes	maximal, characteristic
3	#13092726	C₂	1	1	yes	maximal, characteristic, complemented
4	#13092722	C₂	1	1	yes	maximal, characteristic, complemented
5	#13092719	C₂	1	1	yes	maximal, characteristic, complemented
6	#13092715	C₂	1	1	yes	maximal, characteristic, complemented
7	#13092710	C₂	1	1	yes	maximal, characteristic, complemented
8	#13092705	C₂	1	1	yes	maximal, characteristic, complemented
9	#13092700	C₂	1	1	yes	maximal, characteristic, complemented
10	#13092695	C₂	1	1	yes	maximal, characteristic, complemented
11	#13092455	C₂	1	1	yes	maximal, characteristic, complemented
12	#13092453	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂×Dic₂₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
14	#13092413	C₂	1	1	yes	maximal, characteristic
15	C₂₃×#416	C₂	1	1	yes	maximal, characteristic
16	C₂×#41×C₂₃	C₂²	1	1	yes	characteristic
17	C₂²×C₈×C₂₃	C₂²	1	1	yes	characteristic
18	C₂₃×#117	C₂²	1	1	yes	characteristic
19	Q₈×Dic₂₃	C₂²	1	1	yes	characteristic
20	Q₈×Dic₂₃	C₂²	1	1	yes	characteristic
21	D₄×Dic₂₃	—	2	1	no	—
22	D₄×Dic₂₃	—	2	1	no	—
23	D₄×Dic₂₃	C₂²	1	1	yes	characteristic
24	D₄×Dic₂₃	C₂²	1	1	yes	characteristic
25	#10625097	—	2	1	no	—
26	#10625097	C₂²	1	1	yes	characteristic
27	#10625097	C₂²	1	1	yes	characteristic
28	C₂×C₄×Dic₂₃	—	2	1	no	—
29	C₂×C₄×Dic₂₃	C₂²	1	1	yes	characteristic
30	C₂²×C₂₃⋊C₈	C₂²	1	1	yes	characteristic
31	#10625061	C₂²	1	1	yes	characteristic
32	#10625061	C₂²	1	1	yes	characteristic
33	#10625059	C₂²	1	1	yes	characteristic
34	#10625059	C₂²	1	1	yes	characteristic
35	#10625049	—	2	1	no	—
36	#10625049	—	2	1	no	—
37	C₄×Dic₄₆	—	2	1	no	—
38	C₄×Dic₄₆	—	2	1	no	—
39	C₂₃×#103	C₂²	1	1	yes	characteristic
40	C₂₃×#103	C₂²	1	1	yes	characteristic
41	C₂₃×#102	C₂²	1	1	yes	characteristic
42	C₂₃×#102	C₂²	1	1	yes	characteristic
43	#10625017	C₂²	1	1	yes	characteristic
44	#10625017	C₂²	1	1	yes	characteristic
45	#10625014	C₂²	1	1	yes	characteristic
46	#10625014	C₂²	1	1	yes	characteristic
47	#10625000	C₂²	1	1	yes	characteristic
48	#10625000	C₂²	1	1	yes	characteristic
49	#10624999	C₂²	1	1	yes	characteristic
50	#10624999	C₂²	1	1	yes	characteristic
51	Dic₂₃⋊C₈	C₂²	1	1	yes	characteristic
52	Dic₂₃⋊C₈	C₂²	1	1	yes	characteristic
53	Dic₂₃⋊C₈	C₂²	1	1	yes	characteristic
54	Dic₂₃⋊C₈	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	#10624989	C₂²	1	1	yes	characteristic
59	#41×C₂₃	—	2	2	no	—
60	#41×C₂₃	Q₈	1	4	yes	complemented
61	C₂×Q₈×C₂₃	C₂³	1	1	yes	characteristic
62	C₂×D₄×C₂₃	—	2	1	no	—
63	C₂×D₄×C₂₃	C₂³	1	1	yes	characteristic
64	C₂²×C₄×C₂₃	—	2	1	no	—
65	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
66	C₂²×Dic₂₃	—	2	1	no	—
67	C₂²×Dic₂₃	—	2	1	no	—
68	C₂²×Dic₂₃	D₄	1	1	yes	characteristic
69	C₂²×Dic₂₃	D₄	1	1	yes	characteristic
70	C₂×Dic₄₆	—	2	1	no	—
71	C₂×Dic₄₆	—	2	1	no	—
72	C₂×C₈×C₂₃	—	2	2	no	—
73	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
74	C₂×C₈×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	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	C₄⋊Dic₂₃	—	2	1	no	—
84	C₄⋊Dic₂₃	—	2	1	no	—
85	C₄⋊Dic₂₃	—	2	1	no	—
86	C₄⋊Dic₂₃	—	2	1	no	—
87	C₄⋊Dic₂₃	C₂³	1	1	yes	characteristic
88	C₄⋊Dic₂₃	C₂³	1	1	yes	characteristic
89	C₄⋊Dic₂₃	C₂³	1	1	yes	characteristic
90	C₄⋊Dic₂₃	C₂³	1	1	yes	characteristic
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	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	C₄×Dic₂₃	C₂³	1	1	yes	characteristic
103	C₂×C₂₃⋊C₈	—	2	2	no	—
104	C₂×C₂₃⋊C₈	C₂³	1	1	yes	characteristic
105	C₂×C₂₃⋊C₈	C₂³	1	1	yes	characteristic
106	C₂³×C₂₃	—	2	1	no	—
107	C₂³×C₂₃	C₂×D₄	1	1	yes	characteristic
108	Q₈×C₂₃	—	2	1	no	—
109	Q₈×C₂₃	C₂×Q₈	1	2	yes	—
110	D₄×C₂₃	—	2	4	no	—
111	D₄×C₂₃	—	2	1	no	—
112	D₄×C₂₃	C₂×Q₈	1	2	yes	—
113	C₂×C₄×C₂₃	—	2	1	no	—
114	C₂×C₄×C₂₃	—	2	4	no	—
115	C₂×C₄×C₂₃	—	2	1	no	—
116	C₂×C₄×C₂₃	—	2	1	no	—
117	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
118	C₂×C₄×C₂₃	C₂×Q₈	1	2	yes	—
119	C₂×C₄×C₂₃	#41	1	2	yes	—
120	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
121	C₂×Dic₂₃	—	2	2	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	—	4	1	no	—
127	C₂×Dic₂₃	—	4	2	no	—
128	C₂×Dic₂₃	—	4	2	no	—
129	C₂×Dic₂₃	C₂×D₄	1	1	yes	characteristic
130	C₂×Dic₂₃	C₂×D₄	1	1	yes	characteristic
131	C₂×Dic₂₃	C₂×D₄	1	1	yes	characteristic
132	C₂×Dic₂₃	C₂×D₄	1	1	yes	characteristic
133	Dic₄₆	—	4	2	no	—
134	Dic₄₆	—	4	2	no	—
135	C₈×C₂₃	—	2	1	no	—
136	C₈×C₂₃	—	2	1	no	—
137	C₂₃⋊C₈	—	2	1	no	—
138	C₂₃⋊C₈	—	2	1	no	—
139	#3076	—	23	1	no	maximal, Sylow
140	C₂²×C₂₃	—	2	1	no	—
141	C₂²×C₂₃	—	2	1	no	—
142	C₂²×C₂₃	—	2	2	no	—
143	C₂²×C₂₃	—	4	1	no	—
144	C₂²×C₂₃	—	4	1	no	—
145	C₂²×C₂₃	C₂²⋊Q₈	1	2	yes	—
146	C₂²×C₂₃	C₂²×D₄	1	1	yes	characteristic
147	C₄×C₂₃	—	2	2	no	—
148	C₄×C₂₃	—	4	1	no	—
149	C₄×C₂₃	—	4	1	no	—
150	C₄×C₂₃	C₂²⋊Q₈	1	2	yes	—
151	C₄×C₂₃	C₂²×Q₈	1	1	yes	characteristic
152	C₄×C₂₃	C₂×#41	1	1	yes	characteristic
153	Dic₂₃	—	2	1	no	—
154	Dic₂₃	—	2	1	no	—
155	Dic₂₃	—	2	1	no	—
156	Dic₂₃	—	4	1	no	—
157	Dic₂₃	—	4	1	no	—
158	Dic₂₃	—	4	1	no	—
159	Dic₂₃	—	4	1	no	—
160	Dic₂₃	#135	1	2	yes	—
161	#533	—	23	1	no	—
162	C₄×#41	—	23	1	no	—
163	#479	—	23	1	no	—
164	#479	—	23	1	no	—
165	#478	—	23	1	no	—
166	#478	—	23	1	no	—
167	#477	—	23	1	no	—
168	#476	—	23	1	no	—
169	#475	—	23	1	no	—
170	#474	—	23	1	no	—
171	#427	—	23	1	no	—
172	#427	—	23	1	no	—
173	C₂×C₄⋊C₈	—	23	1	no	—
174	#416	—	23	1	no	—
175	#416	D₂₃	1	1	yes	characteristic
176	C₂×C₂₃	—	2	2	no	—
177	C₂×C₂₃	—	4	2	no	—
178	C₂×C₂₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
179	C₂×C₂₃	C₂×#135	1	1	yes	characteristic
180	C₂×C₂₃	#575	1	1	yes	characteristic
181	C₂×#41	C₂×D₂₃	1	1	yes	characteristic
182	C₂²×C₈	—	23	1	no	—
183	C₂²×C₈	C₂×D₂₃	1	1	yes	characteristic
184	C₄⋊Q₈	—	23	1	no	—
185	C₄⋊Q₈	—	23	1	no	—
186	#125	—	23	1	no	—
187	#125	—	23	1	no	—
188	C₂²⋊Q₈	—	46	1	no	—
189	C₂²⋊Q₈	—	46	1	no	—
190	C₄×Q₈	—	23	1	no	—
191	C₄×Q₈	—	23	1	no	—
192	C₄×Q₈	—	46	1	no	—
193	C₄×Q₈	—	46	1	no	—
194	C₄×D₄	—	23	1	no	—
195	C₄×D₄	—	23	1	no	—
196	C₄×D₄	—	46	1	no	—
197	C₄×D₄	—	46	1	no	—
198	#117	—	23	1	no	—
199	#117	—	23	1	no	—
200	#117	—	46	1	no	—
201	#117	C₂×D₂₃	1	1	yes	characteristic
202	C₂×C₄²	—	23	1	no	—
203	C₂×C₄²	—	46	1	no	—
204	#107	—	23	1	no	—
205	#107	—	23	1	no	—
206	#107	—	23	1	no	—
207	#107	—	23	1	no	—
208	#106	—	23	1	no	—
209	#106	—	23	1	no	—
210	#106	—	23	1	no	—
211	#106	—	23	1	no	—
212	C₄⋊C₈	—	23	1	no	—
213	C₄⋊C₈	—	23	1	no	—
214	C₄⋊C₈	—	23	1	no	—
215	C₄⋊C₈	—	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	#102	—	23	1	no	—
221	#102	—	23	1	no	—
222	#102	C₂×D₂₃	1	1	yes	characteristic
223	#102	C₂×D₂₃	1	1	yes	characteristic
224	C₂₃	#3076	1	1	yes	characteristic, complemented, Sylow
225	#41	—	2	2	no	—
226	#41	Dic₄₆	1	4	yes	complemented
227	C₂×Q₈	—	46	1	no	—
228	C₂×Q₈	—	46	1	no	—
229	C₂×Q₈	C₂²×D₂₃	1	1	yes	characteristic
230	C₂×D₄	—	2	1	no	—
231	C₂×D₄	C₂²×D₂₃	1	1	yes	characteristic
232	C₂²×C₄	—	2	1	no	—
233	C₂²×C₄	—	23	1	no	—
234	C₂²×C₄	—	23	1	no	—
235	C₂²×C₄	—	46	1	no	—
236	C₂²×C₄	—	46	1	no	—
237	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
238	C₂×C₈	—	2	2	no	—
239	C₂×C₈	—	23	1	no	—
240	C₂×C₈	—	23	1	no	—
241	C₂×C₈	—	46	2	no	—
242	C₂×C₈	C₂²×D₂₃	1	1	yes	characteristic
243	C₂×C₈	C₂²×D₂₃	1	1	yes	characteristic
244	C₄⋊C₄	—	23	1	no	—
245	C₄⋊C₄	—	23	1	no	—
246	C₄⋊C₄	—	23	1	no	—
247	C₄⋊C₄	—	23	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	—	46	1	no	—
254	C₄⋊C₄	—	46	1	no	—
255	C₄⋊C₄	—	46	1	no	—
256	C₄⋊C₄	C₂²×D₂₃	1	1	yes	characteristic
257	C₄⋊C₄	C₂²×D₂₃	1	1	yes	characteristic
258	C₂²⋊C₄	—	2	1	no	—
259	C₂²⋊C₄	—	46	1	no	—
260	C₂²⋊C₄	—	46	1	no	—
261	C₂²⋊C₄	—	46	1	no	—
262	C₂²⋊C₄	—	46	1	no	—
263	C₄²	—	2	1	no	—
264	C₄²	—	23	1	no	—
265	C₄²	—	23	1	no	—
266	C₄²	—	23	1	no	—
267	C₄²	—	23	1	no	—
268	C₄²	—	46	1	no	—
269	C₄²	—	46	1	no	—
270	C₄²	—	46	1	no	—
271	C₄²	—	46	1	no	—
272	C₂³	—	2	1	no	—
273	C₂³	D₄×D₂₃	1	1	yes	characteristic
274	Q₈	—	2	1	no	—
275	Q₈	—	92	2	no	—
276	Q₈	—	92	2	no	—
277	Q₈	C₂×Dic₄₆	1	2	yes	—
278	D₄	—	2	4	no	—
279	D₄	—	2	1	no	—
280	D₄	C₂×Dic₄₆	1	2	yes	—
281	C₂×C₄	—	2	1	no	—
282	C₂×C₄	—	2	4	no	—
283	C₂×C₄	—	2	1	no	—
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	23	1	no	—
286	C₂×C₄	—	23	1	no	—
287	C₂×C₄	—	23	1	no	—
288	C₂×C₄	—	23	1	no	—
289	C₂×C₄	—	46	2	no	—
290	C₂×C₄	—	46	1	no	—
291	C₂×C₄	—	46	1	no	—
292	C₂×C₄	—	46	1	no	—
293	C₂×C₄	—	46	1	no	—
294	C₂×C₄	—	92	2	no	—
295	C₂×C₄	—	92	1	no	—
296	C₂×C₄	—	92	2	no	—
297	C₂×C₄	C₂×Dic₄₆	1	2	yes	—
298	C₂×C₄	D₄×D₂₃	1	1	yes	characteristic
299	C₂×C₄	#67290	1	2	yes	—
300	C₂×C₄	C₂³×D₂₃	1	1	yes	characteristic
301	C₈	—	2	1	no	—
302	C₈	—	2	1	no	—
303	C₈	—	46	1	no	—
304	C₈	—	46	1	no	—
305	C₂²	—	2	1	no	—
306	C₂²	—	2	1	no	—
307	C₂²	—	2	2	no	—
308	C₂²	—	4	1	no	—
309	C₂²	—	4	1	no	—
310	C₂²	#10625049	1	2	yes	—
311	C₂²	C₂×D₄×D₂₃	1	1	yes	characteristic
312	C₄	—	2	2	no	—
313	C₄	—	4	1	no	—
314	C₄	—	4	1	no	—
315	C₄	—	23	2	no	—
316	C₄	—	46	1	no	—
317	C₄	—	46	1	no	—
318	C₄	—	46	1	no	—
319	C₄	—	92	1	no	—
320	C₄	—	92	1	no	—
321	C₄	—	92	1	no	—
322	C₄	—	92	1	no	—
323	C₄	#10625049	1	2	yes	—
324	C₄	C₂²×Dic₄₆	1	1	yes	characteristic
325	C₄	C₂×#67290	1	1	yes	characteristic
326	C₂	—	2	2	no	—
327	C₂	—	4	2	no	—
328	C₂	C₂×#10625049	1	1	yes	characteristic
329	C₂	#135×D₂₃	1	1	yes	characteristic
330	C₂	#13093214	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₄×D₂₃
commutator, cosocle	C₂×C₄×C₂₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₂₃
Fitting	C₂₃×#416	C₂
socle	C₂²×C₂₃	C₂²×D₄
prime radical	C₂×#41×C₂₃	C₂²
hypercenter	#416	D₂₃
hypocenter	C₂₃	#3076
LCS	G ▷ C₂×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #416

Complements

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

N	H ≅ G/N	type
1. #13092970	C₂	semidirect
3. #13092726	C₂	semidirect
4. #13092722	C₂	semidirect
5. #13092719	C₂	semidirect
6. #13092715	C₂	semidirect
7. #13092710	C₂	semidirect
8. #13092705	C₂	semidirect
9. #13092700	C₂	semidirect
10. #13092695	C₂	semidirect
11. #13092455	C₂	semidirect
12. #13092453	C₂	semidirect
13. C₂×Dic₂₃⋊C₈	C₂	semidirect
60. #41×C₂₃	Q₈	semidirect
224. C₂₃	#3076	semidirect
226. #41	Dic₄₆	semidirect

Maximal subgroups

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

subgroup	index
#13092970	2
#41×Dic₂₃	2
#13092726	2
#13092722	2
#13092719	2
#13092715	2
#13092710	2
#13092705	2
#13092700	2
#13092695	2
#13092455	2
#13092453	2
C₂×Dic₂₃⋊C₈	2
#13092413	2
C₂₃×#416	2
#3076	23