Subgroups — #15683306

Order 2944.

420 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#10625097	C₂	1	1	yes	maximal, characteristic, complemented
2	#13092755	C₂	1	2	yes	maximal, complemented
3	#13092750	C₂	1	1	yes	maximal, characteristic, complemented
4	#13092745	C₂	1	1	yes	maximal, characteristic, complemented
5	#13092738	C₂	1	2	yes	maximal, complemented
6	#13092733	C₂	1	1	yes	maximal, characteristic, complemented
7	#13092728	C₂	1	1	yes	maximal, characteristic, complemented
8	#13092453	C₂	1	2	yes	maximal, complemented
9	#13092413	C₂	1	2	yes	maximal, complemented
10	C₂×C₂²⋊(C₂₃⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂₃×#525	C₂	1	1	yes	maximal, characteristic
12	C₂×#41×C₂₃	C₂²	1	1	yes	characteristic
13	C₂³×C₄×C₂₃	C₂²	1	1	yes	characteristic
14	C₂₃×#123	—	2	1	no	—
15	C₂₃×C₂²⋊Q₈	C₂²	1	1	yes	characteristic
16	C₂₃×C₂²⋊Q₈	C₂²	1	1	yes	characteristic
17	C₂₃×#121	C₂²	1	1	yes	characteristic
18	C₂₃×#121	C₂²	1	1	yes	characteristic
19	C₂₃×#120	—	2	1	no	—
20	C₄×D₄×C₂₃	—	2	1	no	—
21	C₄×D₄×C₂₃	—	2	1	no	—
22	C₂₃×#117	C₂²	1	1	yes	characteristic
23	C₂×C₂²⋊Dic₂₃	—	2	1	no	—
24	#10625097	—	2	1	no	—
25	#10625097	—	2	1	no	—
26	#10625097	C₂²	1	2	yes	—
27	#10625097	C₂²	1	2	yes	—
28	C₂×C₄⋊Dic₂₃	C₂²	1	1	yes	characteristic
29	C₂×C₄⋊Dic₂₃	C₂²	1	1	yes	characteristic
30	C₂×C₄×Dic₂₃	—	2	1	no	—
31	C₂²×C₂₃⋊C₈	C₂²	1	2	yes	—
32	#10625017	C₂²	1	2	yes	complemented
33	#10625017	C₂²	1	2	yes	complemented
34	#10625014	C₂²	1	2	yes	complemented
35	#10625014	C₂²	1	2	yes	complemented
36	C₂²⋊(C₂₃⋊C₈)	C₂²	1	1	yes	characteristic
37	C₂²⋊(C₂₃⋊C₈)	C₂²	1	1	yes	characteristic
38	C₂²⋊(C₂₃⋊C₈)	C₂²	1	2	yes	—
39	#10624990	C₂²	1	2	yes	complemented
40	#10624990	C₂²	1	2	yes	complemented
41	#10624989	C₂²	1	2	yes	complemented
42	#10624989	C₂²	1	2	yes	complemented
43	C₂⁴×C₂₃	D₄	1	1	yes	characteristic
44	#41×C₂₃	—	2	4	no	—
45	C₂×Q₈×C₂₃	C₂³	1	1	yes	characteristic
46	C₂×D₄×C₂₃	—	2	1	no	—
47	C₂×D₄×C₂₃	—	2	1	no	—
48	C₂×D₄×C₂₃	—	2	1	no	—
49	C₂×D₄×C₂₃	C₂³	1	1	yes	characteristic
50	C₂²×C₄×C₂₃	—	2	4	no	—
51	C₂²×C₄×C₂₃	—	2	1	no	—
52	C₂²×C₄×C₂₃	D₄	1	1	yes	characteristic
53	C₂²×C₄×C₂₃	D₄	1	1	yes	characteristic
54	C₂²×C₄×C₂₃	D₄	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	1	no	—
59	C₂²×Dic₂₃	—	2	1	no	—
60	C₄⋊C₄×C₂₃	—	2	1	no	—
61	C₄⋊C₄×C₂₃	—	2	1	no	—
62	C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
63	C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
64	C₂²⋊C₄×C₂₃	—	2	1	no	—
65	C₂²⋊C₄×C₂₃	—	2	1	no	—
66	C₂²⋊C₄×C₂₃	—	2	1	no	—
67	C₂²⋊C₄×C₂₃	—	2	1	no	—
68	C₂²⋊C₄×C₂₃	—	2	1	no	—
69	C₄²×C₂₃	—	2	1	no	—
70	C₂²⋊Dic₂₃	—	2	2	no	—
71	C₂²⋊Dic₂₃	—	2	2	no	—
72	C₄⋊Dic₂₃	—	2	1	no	—
73	C₄⋊Dic₂₃	—	2	1	no	—
74	C₄⋊Dic₂₃	C₂³	1	2	yes	—
75	C₄⋊Dic₂₃	C₂³	1	2	yes	—
76	C₄×Dic₂₃	—	2	2	no	—
77	C₄×Dic₂₃	—	2	2	no	—
78	C₂×C₂₃⋊C₈	—	2	4	no	—
79	C₂×C₂₃⋊C₈	C₂³	1	2	yes	—
80	C₂×C₂₃⋊C₈	C₂³	1	2	yes	—
81	C₂³×C₂₃	—	2	2	no	—
82	C₂³×C₂₃	—	2	2	no	—
83	C₂³×C₂₃	—	2	1	no	—
84	C₂³×C₂₃	—	4	1	no	—
85	C₂³×C₂₃	C₂×D₄	1	1	yes	characteristic
86	C₂³×C₂₃	C₂×D₄	1	1	yes	characteristic
87	C₂³×C₂₃	C₂×D₄	1	1	yes	characteristic
88	Q₈×C₂₃	—	2	2	no	—
89	D₄×C₂₃	—	2	2	no	—
90	D₄×C₂₃	—	4	2	no	—
91	D₄×C₂₃	—	4	2	no	—
92	D₄×C₂₃	—	4	2	no	—
93	C₂×C₄×C₂₃	—	2	2	no	—
94	C₂×C₄×C₂₃	—	2	2	no	—
95	C₂×C₄×C₂₃	—	2	2	no	—
96	C₂×C₄×C₂₃	—	2	2	no	—
97	C₂×C₄×C₂₃	—	2	2	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	—	4	2	no	—
102	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
103	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
104	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
105	C₂×C₄×C₂₃	#41	1	4	yes	—
106	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
107	C₂×Dic₂₃	—	2	2	no	—
108	C₂×Dic₂₃	—	2	2	no	—
109	C₂×Dic₂₃	—	4	2	no	—
110	C₂×Dic₂₃	—	4	2	no	—
111	C₂₃⋊C₈	—	2	2	no	—
112	C₂₃⋊C₈	—	2	2	no	—
113	#3091	—	23	1	no	maximal, Sylow
114	C₂²×C₂₃	—	2	1	no	—
115	C₂²×C₂₃	—	2	1	no	—
116	C₂²×C₂₃	—	2	1	no	—
117	C₂²×C₂₃	—	2	1	no	—
118	C₂²×C₂₃	—	2	1	no	—
119	C₂²×C₂₃	—	2	1	no	—
120	C₂²×C₂₃	—	2	1	no	—
121	C₂²×C₂₃	—	4	4	no	—
122	C₂²×C₂₃	—	4	1	no	—
123	C₂²×C₂₃	—	4	1	no	—
124	C₂²×C₂₃	—	4	1	no	—
125	C₂²×C₂₃	#123	1	2	yes	—
126	C₂²×C₂₃	#135	1	2	yes	—
127	C₂²×C₂₃	C₂²×D₄	1	1	yes	characteristic
128	C₄×C₂₃	—	2	1	no	—
129	C₄×C₂₃	—	2	1	no	—
130	C₄×C₂₃	—	4	1	no	—
131	C₄×C₂₃	—	4	1	no	—
132	C₄×C₂₃	—	4	1	no	—
133	C₄×C₂₃	#123	1	2	yes	—
134	C₄×C₂₃	C₂×#41	1	2	yes	—
135	Dic₂₃	—	4	2	no	—
136	Dic₂₃	—	4	2	no	—
137	#525	D₂₃	1	1	yes	characteristic
138	C₂×#117	—	23	1	no	—
139	#485	—	23	2	no	—
140	#484	—	23	1	no	—
141	#483	—	23	1	no	—
142	#482	—	23	2	no	—
143	#481	—	23	1	no	—
144	#480	—	23	1	no	—
145	#427	—	23	2	no	—
146	#416	—	23	2	no	—
147	C₂×C₂²⋊C₈	—	23	1	no	—
148	C₂×C₂₃	—	2	2	no	—
149	C₂×C₂₃	—	2	2	no	—
150	C₂×C₂₃	—	4	1	no	—
151	C₂×C₂₃	—	8	1	no	—
152	C₂×C₂₃	C₂×#123	1	1	yes	characteristic
153	C₂×C₂₃	C₂×#135	1	1	yes	characteristic
154	C₂×C₂₃	#575	1	1	yes	characteristic
155	C₂×#41	C₂×D₂₃	1	1	yes	characteristic
156	C₂³×C₄	C₂×D₂₃	1	1	yes	characteristic
157	C₂²×C₈	—	23	2	no	—
158	#123	—	2	1	no	—
159	C₂²⋊Q₈	C₂×D₂₃	1	1	yes	characteristic
160	C₂²⋊Q₈	C₂×D₂₃	1	1	yes	characteristic
161	#121	C₂×D₂₃	1	1	yes	characteristic
162	#121	C₂×D₂₃	1	1	yes	characteristic
163	#120	—	2	1	no	—
164	C₄×D₄	—	2	1	no	—
165	C₄×D₄	—	2	1	no	—
166	#117	—	23	2	no	—
167	#117	—	23	2	no	—
168	#117	—	46	1	no	—
169	#117	—	46	1	no	—
170	#117	C₂×D₂₃	1	1	yes	characteristic
171	C₂×C₄⋊C₄	—	23	1	no	—
172	C₂×C₄⋊C₄	—	23	1	no	—
173	C₂×C₂²⋊C₄	—	46	1	no	—
174	C₂×C₄²	—	46	1	no	—
175	#107	—	23	2	no	—
176	#107	—	23	2	no	—
177	#106	—	23	2	no	—
178	#106	—	23	2	no	—
179	#103	—	23	2	no	—
180	#103	—	23	2	no	—
181	#102	—	23	2	no	—
182	#102	—	23	2	no	—
183	C₂²⋊C₈	—	23	1	no	—
184	C₂²⋊C₈	—	23	1	no	—
185	C₂²⋊C₈	—	23	2	no	—
186	C₂₃	#3091	1	1	yes	characteristic, complemented, Sylow
187	C₂⁴	C₂²⋊D₂₃	1	1	yes	characteristic
188	#41	—	2	4	no	—
189	C₂×Q₈	C₂²×D₂₃	1	1	yes	characteristic
190	C₂×D₄	—	2	1	no	—
191	C₂×D₄	—	2	1	no	—
192	C₂×D₄	—	2	1	no	—
193	C₂×D₄	C₂²×D₂₃	1	1	yes	characteristic
194	C₂²×C₄	—	2	4	no	—
195	C₂²×C₄	—	2	1	no	—
196	C₂²×C₄	—	46	1	no	—
197	C₂²×C₄	—	46	1	no	—
198	C₂²×C₄	C₂²⋊D₂₃	1	1	yes	characteristic
199	C₂²×C₄	C₂²⋊D₂₃	1	1	yes	characteristic
200	C₂²×C₄	C₂²⋊D₂₃	1	1	yes	characteristic
201	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
202	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
203	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
204	C₂×C₈	—	23	2	no	—
205	C₂×C₈	—	23	2	no	—
206	C₂×C₈	—	46	4	no	—
207	C₄⋊C₄	—	2	1	no	—
208	C₄⋊C₄	—	2	1	no	—
209	C₄⋊C₄	—	23	2	no	—
210	C₄⋊C₄	—	23	2	no	—
211	C₄⋊C₄	—	46	1	no	—
212	C₄⋊C₄	—	46	1	no	—
213	C₄⋊C₄	C₂²×D₂₃	1	1	yes	characteristic
214	C₄⋊C₄	C₂²×D₂₃	1	1	yes	characteristic
215	C₂²⋊C₄	—	2	1	no	—
216	C₂²⋊C₄	—	2	1	no	—
217	C₂²⋊C₄	—	2	1	no	—
218	C₂²⋊C₄	—	2	1	no	—
219	C₂²⋊C₄	—	2	1	no	—
220	C₂²⋊C₄	—	46	2	no	—
221	C₂²⋊C₄	—	46	2	no	—
222	C₄²	—	2	1	no	—
223	C₄²	—	46	2	no	—
224	C₄²	—	46	2	no	—
225	C₂³	—	2	2	no	—
226	C₂³	—	2	2	no	—
227	C₂³	—	2	1	no	—
228	C₂³	—	4	1	no	—
229	C₂³	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
230	C₂³	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
231	C₂³	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
232	Q₈	—	2	2	no	—
233	D₄	—	2	2	no	—
234	D₄	—	4	2	no	—
235	D₄	—	4	2	no	—
236	D₄	—	4	2	no	—
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	2	2	no	—
242	C₂×C₄	—	2	1	no	—
243	C₂×C₄	—	2	1	no	—
244	C₂×C₄	—	2	1	no	—
245	C₂×C₄	—	4	2	no	—
246	C₂×C₄	—	46	2	no	—
247	C₂×C₄	—	46	2	no	—
248	C₂×C₄	—	92	2	no	—
249	C₂×C₄	—	92	2	no	—
250	C₂×C₄	#67290	1	4	yes	—
251	C₂×C₄	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
252	C₂×C₄	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
253	C₂×C₄	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
254	C₂×C₄	C₂³×D₂₃	1	1	yes	characteristic
255	C₈	—	46	2	no	—
256	C₈	—	46	2	no	—
257	C₂²	—	2	1	no	—
258	C₂²	—	2	1	no	—
259	C₂²	—	2	1	no	—
260	C₂²	—	2	1	no	—
261	C₂²	—	2	1	no	—
262	C₂²	—	2	1	no	—
263	C₂²	—	2	1	no	—
264	C₂²	—	4	4	no	—
265	C₂²	—	4	1	no	—
266	C₂²	—	4	1	no	—
267	C₂²	—	4	1	no	—
268	C₂²	#10625106	1	2	yes	—
269	C₂²	#10625121	1	2	yes	—
270	C₂²	C₂²×C₂²⋊D₂₃	1	1	yes	characteristic
271	C₄	—	2	1	no	—
272	C₄	—	2	1	no	—
273	C₄	—	4	1	no	—
274	C₄	—	4	1	no	—
275	C₄	—	4	1	no	—
276	C₄	—	92	2	no	—
277	C₄	—	92	2	no	—
278	C₄	#10625106	1	2	yes	—
279	C₄	C₂×#67290	1	2	yes	—
280	C₂	—	2	2	no	—
281	C₂	—	2	2	no	—
282	C₂	—	4	1	no	—
283	C₂	—	8	1	no	—
284	C₂	C₂×#10625106	1	1	yes	characteristic
285	C₂	C₂×#10625121	1	1	yes	characteristic
286	C₂	#13093215	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₂₃
commutator, cosocle	C₂×C₄×C₂₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₂₃
Fitting, prime radical	C₂₃×#525	C₂
socle	C₂²×C₂₃	C₂²×D₄
hypercenter	#525	D₂₃
hypocenter	C₂₃	#3091
LCS	G ▷ C₂×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ #525

Complements

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

N	H ≅ G/N	type
1. C₂×#10625097	C₂	semidirect
2. #13092755	C₂	semidirect
3. #13092750	C₂	semidirect
4. #13092745	C₂	semidirect
5. #13092738	C₂	semidirect
6. #13092733	C₂	semidirect
7. #13092728	C₂	semidirect
8. #13092453	C₂	semidirect
9. #13092413	C₂	semidirect
10. C₂×C₂²⋊(C₂₃⋊C₈)	C₂	semidirect
32. #10625017	C₂²	semidirect
33. #10625017	C₂²	semidirect
34. #10625014	C₂²	semidirect
35. #10625014	C₂²	semidirect
39. #10624990	C₂²	semidirect
40. #10624990	C₂²	semidirect
41. #10624989	C₂²	semidirect
42. #10624989	C₂²	semidirect
186. C₂₃	#3091	semidirect

Maximal subgroups

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

subgroup	index
C₂×#10625097	2
#13092755	2
#13092750	2
#13092745	2
#13092738	2
#13092733	2
#13092728	2
#13092453	2
#13092413	2
C₂×C₂²⋊(C₂₃⋊C₈)	2
C₂₃×#525	2
#3091	23