Subgroups — #15683185

Order 2944.

388 conjugacy classes of subgroups.

The subgroup lattice has 270 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13092974	C₂	1	1	yes	maximal, characteristic, complemented
2	#13092724	C₂	1	2	yes	maximal, complemented
3	#13092717	C₂	1	2	yes	maximal, complemented
4	#13092712	C₂	1	1	yes	maximal, characteristic, complemented
5	#13092707	C₂	1	1	yes	maximal, characteristic, complemented
6	#13092702	C₂	1	1	yes	maximal, characteristic, complemented
7	#13092697	C₂	1	1	yes	maximal, characteristic, complemented
8	#13092453	C₂	1	2	yes	maximal, complemented
9	C₂×C₄⋊(C₂₃⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
10	#13092413	C₂	1	2	yes	maximal
11	C₄×#41×C₂₃	C₂	1	1	yes	maximal, characteristic
12	C₂×#41×C₂₃	C₂²	1	1	yes	characteristic
13	C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
14	C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
15	C₄×D₄×C₂₃	—	2	1	no	—
16	C₄×D₄×C₂₃	—	2	1	no	—
17	C₄×D₄×C₂₃	C₂²	1	1	yes	characteristic
18	C₄×D₄×C₂₃	C₂²	1	1	yes	characteristic
19	C₂₃×#117	—	2	1	no	—
20	C₂₃×#117	C₂²	1	1	yes	characteristic
21	C₂×C₄²×C₂₃	—	2	1	no	—
22	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
23	#10625097	C₂²	1	2	yes	—
24	#10625095	—	2	1	no	—
25	#10625095	—	2	1	no	—
26	C₂²×C₂₃⋊C₈	C₂²	1	2	yes	—
27	#10625041	C₂²	1	2	yes	—
28	#10625040	C₂²	1	1	yes	characteristic
29	#10625040	C₂²	1	1	yes	characteristic
30	C₄×Dic₄₆	—	2	1	no	—
31	C₄×Dic₄₆	—	2	1	no	—
32	#10625017	C₂²	1	2	yes	—
33	#10625017	C₂²	1	2	yes	—
34	#10625014	C₂²	1	2	yes	—
35	#10625014	C₂²	1	2	yes	—
36	#10624990	C₂²	1	2	yes	—
37	#10624990	C₂²	1	2	yes	—
38	#10624989	C₂²	1	2	yes	—
39	#10624989	C₂²	1	2	yes	—
40	C₄⋊(C₂₃⋊C₈)	C₂²	1	1	yes	characteristic
41	C₄⋊(C₂₃⋊C₈)	C₂²	1	1	yes	characteristic
42	C₄⋊(C₂₃⋊C₈)	C₂²	1	2	yes	—
43	#41×C₂₃	—	2	2	no	—
44	#41×C₂₃	Q₈	1	4	yes	complemented
45	C₂×Q₈×C₂₃	C₂³	1	1	yes	characteristic
46	C₂×D₄×C₂₃	—	2	1	no	—
47	C₂×D₄×C₂₃	C₂³	1	1	yes	characteristic
48	C₂²×C₄×C₂₃	—	2	1	no	—
49	C₂²×C₄×C₂₃	—	2	1	no	—
50	C₂²×C₄×C₂₃	—	2	1	no	—
51	C₂²×C₄×C₂₃	D₄	1	1	yes	characteristic
52	C₂²×C₄×C₂₃	D₄	1	1	yes	characteristic
53	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
54	C₂×Dic₄₆	—	2	1	no	—
55	C₂×Dic₄₆	—	2	1	no	—
56	C₄⋊C₄×C₂₃	—	2	1	no	—
57	C₄⋊C₄×C₂₃	—	2	1	no	—
58	C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
59	C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
60	C₂²⋊C₄×C₂₃	—	2	1	no	—
61	C₂²⋊C₄×C₂₃	—	2	1	no	—
62	C₂²⋊C₄×C₂₃	—	2	1	no	—
63	C₄²×C₂₃	—	2	1	no	—
64	C₄²×C₂₃	—	2	1	no	—
65	C₄²×C₂₃	—	2	1	no	—
66	C₄²×C₂₃	D₄	1	1	yes	characteristic
67	C₄²×C₂₃	D₄	1	1	yes	characteristic
68	C₄²×C₂₃	C₂³	1	1	yes	characteristic
69	C₄²×C₂₃	C₂³	1	1	yes	characteristic
70	C₂²⋊Dic₂₃	—	2	2	no	—
71	C₄⋊Dic₂₃	—	2	1	no	—
72	C₄⋊Dic₂₃	—	2	1	no	—
73	C₄⋊Dic₂₃	C₂³	1	2	yes	—
74	C₄⋊Dic₂₃	C₂³	1	2	yes	—
75	Dic₂₃⋊C₄	—	2	2	no	—
76	Dic₂₃⋊C₄	—	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	1	no	—
82	C₂³×C₂₃	C₂×D₄	1	1	yes	characteristic
83	Q₈×C₂₃	—	2	1	no	—
84	Q₈×C₂₃	C₂×Q₈	1	2	yes	—
85	D₄×C₂₃	—	2	4	no	—
86	D₄×C₂₃	—	2	1	no	—
87	D₄×C₂₃	C₂×Q₈	1	2	yes	—
88	C₂×C₄×C₂₃	—	2	1	no	—
89	C₂×C₄×C₂₃	—	2	4	no	—
90	C₂×C₄×C₂₃	—	2	2	no	—
91	C₂×C₄×C₂₃	—	2	1	no	—
92	C₂×C₄×C₂₃	—	2	1	no	—
93	C₂×C₄×C₂₃	—	4	1	no	—
94	C₂×C₄×C₂₃	—	4	2	no	—
95	C₂×C₄×C₂₃	—	4	2	no	—
96	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
97	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
98	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
99	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
100	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
101	C₂×C₄×C₂₃	C₂×Q₈	1	2	yes	—
102	C₂×C₄×C₂₃	#41	1	2	yes	—
103	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
104	C₂×Dic₂₃	—	2	2	no	—
105	C₂×Dic₂₃	—	2	2	no	—
106	Dic₄₆	—	4	2	no	—
107	Dic₄₆	—	4	2	no	—
108	C₂₃⋊C₈	—	2	2	no	—
109	C₂₃⋊C₈	—	2	2	no	—
110	#3076	—	23	1	no	maximal, Sylow
111	C₂²×C₂₃	—	2	1	no	—
112	C₂²×C₂₃	—	2	1	no	—
113	C₂²×C₂₃	—	2	2	no	—
114	C₂²×C₂₃	—	4	1	no	—
115	C₂²×C₂₃	—	4	1	no	—
116	C₂²×C₂₃	C₂²⋊Q₈	1	2	yes	—
117	C₂²×C₂₃	C₂²×D₄	1	1	yes	characteristic
118	C₄×C₂₃	—	2	2	no	—
119	C₄×C₂₃	—	2	1	no	—
120	C₄×C₂₃	—	2	1	no	—
121	C₄×C₂₃	—	2	1	no	—
122	C₄×C₂₃	—	4	1	no	—
123	C₄×C₂₃	—	4	1	no	—
124	C₄×C₂₃	C₂²⋊Q₈	1	2	yes	—
125	C₄×C₂₃	#135	1	2	yes	—
126	C₄×C₂₃	C₂²×Q₈	1	1	yes	characteristic
127	C₄×C₂₃	C₂×#41	1	1	yes	characteristic
128	Dic₂₃	—	4	2	no	—
129	Dic₂₃	—	4	2	no	—
130	#533	—	23	1	no	—
131	C₄×#41	D₂₃	1	1	yes	characteristic
132	#479	—	23	2	no	—
133	#478	—	23	2	no	—
134	#477	—	23	1	no	—
135	#476	—	23	1	no	—
136	#475	—	23	1	no	—
137	#474	—	23	1	no	—
138	#427	—	23	2	no	—
139	C₂×C₄⋊C₈	—	23	1	no	—
140	#416	—	23	2	no	—
141	C₂×C₂₃	—	2	2	no	—
142	C₂×C₂₃	—	4	2	no	—
143	C₂×C₂₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
144	C₂×C₂₃	C₂×#135	1	1	yes	characteristic
145	C₂×C₂₃	#575	1	1	yes	characteristic
146	C₂×#41	C₂×D₂₃	1	1	yes	characteristic
147	C₂²×C₈	—	23	2	no	—
148	C₄⋊Q₈	—	23	1	no	—
149	C₄⋊Q₈	—	23	1	no	—
150	#125	—	23	2	no	—
151	C₂²⋊Q₈	—	46	1	no	—
152	C₂²⋊Q₈	—	46	1	no	—
153	C₄×Q₈	—	46	1	no	—
154	C₄×Q₈	—	46	1	no	—
155	C₄×Q₈	C₂×D₂₃	1	1	yes	characteristic
156	C₄×Q₈	C₂×D₂₃	1	1	yes	characteristic
157	C₄×D₄	—	2	1	no	—
158	C₄×D₄	—	2	1	no	—
159	C₄×D₄	C₂×D₂₃	1	1	yes	characteristic
160	C₄×D₄	C₂×D₂₃	1	1	yes	characteristic
161	#117	—	2	1	no	—
162	#117	—	23	2	no	—
163	#117	C₂×D₂₃	1	1	yes	characteristic
164	C₂×C₄²	—	2	1	no	—
165	C₂×C₄²	C₂×D₂₃	1	1	yes	characteristic
166	#107	—	23	2	no	—
167	#107	—	23	2	no	—
168	#106	—	23	2	no	—
169	#106	—	23	2	no	—
170	C₄⋊C₈	—	23	1	no	—
171	C₄⋊C₈	—	23	1	no	—
172	C₄⋊C₈	—	23	2	no	—
173	#103	—	23	2	no	—
174	#103	—	23	2	no	—
175	#102	—	23	2	no	—
176	#102	—	23	2	no	—
177	C₂₃	#3076	1	1	yes	characteristic, complemented, Sylow
178	#41	—	2	2	no	—
179	#41	Dic₄₆	1	4	yes	complemented
180	C₂×Q₈	—	46	1	no	—
181	C₂×Q₈	—	46	1	no	—
182	C₂×Q₈	C₂²×D₂₃	1	1	yes	characteristic
183	C₂×D₄	—	2	1	no	—
184	C₂×D₄	C₂²×D₂₃	1	1	yes	characteristic
185	C₂²×C₄	—	2	1	no	—
186	C₂²×C₄	—	2	1	no	—
187	C₂²×C₄	—	2	1	no	—
188	C₂²×C₄	C₂²⋊D₂₃	1	1	yes	characteristic
189	C₂²×C₄	C₂²⋊D₂₃	1	1	yes	characteristic
190	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
191	C₂×C₈	—	23	2	no	—
192	C₂×C₈	—	23	2	no	—
193	C₂×C₈	—	46	4	no	—
194	C₄⋊C₄	—	2	1	no	—
195	C₄⋊C₄	—	2	1	no	—
196	C₄⋊C₄	—	23	2	no	—
197	C₄⋊C₄	—	23	2	no	—
198	C₄⋊C₄	—	46	2	no	—
199	C₄⋊C₄	—	46	2	no	—
200	C₄⋊C₄	—	46	1	no	—
201	C₄⋊C₄	—	46	1	no	—
202	C₄⋊C₄	C₂²×D₂₃	1	1	yes	characteristic
203	C₄⋊C₄	C₂²×D₂₃	1	1	yes	characteristic
204	C₂²⋊C₄	—	2	1	no	—
205	C₂²⋊C₄	—	2	1	no	—
206	C₂²⋊C₄	—	2	1	no	—
207	C₂²⋊C₄	—	46	2	no	—
208	C₄²	—	2	1	no	—
209	C₄²	—	2	1	no	—
210	C₄²	—	2	1	no	—
211	C₄²	—	46	2	no	—
212	C₄²	C₂²⋊D₂₃	1	1	yes	characteristic
213	C₄²	C₂²⋊D₂₃	1	1	yes	characteristic
214	C₄²	C₂²×D₂₃	1	1	yes	characteristic
215	C₄²	C₂²×D₂₃	1	1	yes	characteristic
216	C₂³	—	2	1	no	—
217	C₂³	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
218	Q₈	—	2	1	no	—
219	Q₈	—	92	2	no	—
220	Q₈	—	92	2	no	—
221	Q₈	C₂×Dic₄₆	1	2	yes	—
222	D₄	—	2	4	no	—
223	D₄	—	2	1	no	—
224	D₄	C₂×Dic₄₆	1	2	yes	—
225	C₂×C₄	—	2	1	no	—
226	C₂×C₄	—	2	4	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	2	1	no	—
229	C₂×C₄	—	2	1	no	—
230	C₂×C₄	—	4	1	no	—
231	C₂×C₄	—	4	2	no	—
232	C₂×C₄	—	4	2	no	—
233	C₂×C₄	—	46	2	no	—
234	C₂×C₄	—	46	2	no	—
235	C₂×C₄	C₂×Dic₄₆	1	2	yes	—
236	C₂×C₄	#67288	1	2	yes	—
237	C₂×C₄	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
238	C₂×C₄	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
239	C₂×C₄	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
240	C₂×C₄	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
241	C₂×C₄	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
242	C₂×C₄	C₂³×D₂₃	1	1	yes	characteristic
243	C₈	—	46	2	no	—
244	C₈	—	46	2	no	—
245	C₂²	—	2	1	no	—
246	C₂²	—	2	1	no	—
247	C₂²	—	2	2	no	—
248	C₂²	—	4	1	no	—
249	C₂²	—	4	1	no	—
250	C₂²	#10625095	1	2	yes	—
251	C₂²	C₂²×C₂²⋊D₂₃	1	1	yes	characteristic
252	C₄	—	2	2	no	—
253	C₄	—	2	1	no	—
254	C₄	—	2	1	no	—
255	C₄	—	2	1	no	—
256	C₄	—	4	1	no	—
257	C₄	—	4	1	no	—
258	C₄	—	92	2	no	—
259	C₄	—	92	2	no	—
260	C₄	#10625095	1	2	yes	—
261	C₄	#10625121	1	2	yes	—
262	C₄	C₂²×Dic₄₆	1	1	yes	characteristic
263	C₄	C₂×#67288	1	1	yes	characteristic
264	C₂	—	2	2	no	—
265	C₂	—	4	2	no	—
266	C₂	C₂×#10625095	1	1	yes	characteristic
267	C₂	C₂×#10625121	1	1	yes	characteristic
268	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	C₄×#41×C₂₃	C₂
socle	C₂²×C₂₃	C₂²×D₄
prime radical	C₂×#41×C₂₃	C₂²
hypercenter	C₄×#41	D₂₃
hypocenter	C₂₃	#3076
LCS	G ▷ C₂×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄×#41

Complements

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

N	H ≅ G/N	type
1. #13092974	C₂	semidirect
2. #13092724	C₂	semidirect
3. #13092717	C₂	semidirect
4. #13092712	C₂	semidirect
5. #13092707	C₂	semidirect
6. #13092702	C₂	semidirect
7. #13092697	C₂	semidirect
8. #13092453	C₂	semidirect
9. C₂×C₄⋊(C₂₃⋊C₈)	C₂	semidirect
44. #41×C₂₃	Q₈	semidirect
177. C₂₃	#3076	semidirect
179. #41	Dic₄₆	semidirect

Maximal subgroups

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

subgroup	index
#13092974	2
#13092724	2
#13092717	2
#13092712	2
#13092707	2
#13092702	2
#13092697	2
#13092453	2
C₂×C₄⋊(C₂₃⋊C₈)	2
#13092413	2
C₄×#41×C₂₃	2
#3076	23