Subgroups — #15682846

Order 2944.

496 conjugacy classes of subgroups.

The subgroup lattice has 302 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#10625121	C₂	1	2	yes	maximal, complemented
2	C₄×#67288	C₂	1	1	yes	maximal, characteristic, complemented
3	#13092636	C₂	1	2	yes	maximal, complemented
4	#13092629	C₂	1	2	yes	maximal, complemented
5	#13092624	C₂	1	1	yes	maximal, characteristic, complemented
6	#13092619	C₂	1	1	yes	maximal, characteristic, complemented
7	#13092614	C₂	1	1	yes	maximal, characteristic, complemented
8	#13092609	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₄⋊(C₂₃⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
10	#13092413	C₂	1	2	yes	maximal, complemented
11	C₂₃×#532	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#41×C₂₃	C₂²	1	2	yes	—
13	C₂×#67288	C₂²	1	1	yes	characteristic
14	C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
15	C₂₃×#127	C₂²	1	1	yes	characteristic
16	C₂₃×#124	C₂²	1	2	yes	—
17	C₂₃×#121	—	2	1	no	—
18	C₂₃×#121	—	2	1	no	—
19	C₄×D₄×C₂₃	—	2	1	no	—
20	C₄×D₄×C₂₃	—	2	1	no	—
21	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
22	#10625121	—	2	8	no	—
23	C₂×#67275	C₂²	1	2	yes	—
24	C₂×#67274	C₂²	1	2	yes	—
25	C₂×#67273	C₂²	1	2	yes	—
26	C₂×#67272	C₂²	1	2	yes	—
27	C₄×C₂²⋊D₂₃	—	2	1	no	—
28	C₄×C₂²⋊D₂₃	—	2	1	no	—
29	#10625097	C₂²	1	1	yes	characteristic
30	C₂²×C₂₃⋊C₈	C₂²	1	2	yes	—
31	C₄×D₉₂	C₂²	1	1	yes	characteristic
32	C₄×D₉₂	C₂²	1	1	yes	characteristic
33	#10625043	—	2	1	no	—
34	C₄²×D₂₃	—	2	1	no	—
35	C₄×Dic₄₆	C₂²	1	1	yes	characteristic
36	C₄×Dic₄₆	C₂²	1	1	yes	characteristic
37	#10625017	C₂²	1	2	yes	—
38	#10625017	C₂²	1	2	yes	—
39	#10625014	C₂²	1	2	yes	—
40	#10625014	C₂²	1	2	yes	—
41	C₄⋊(C₂₃⋊C₈)	C₂²	1	1	yes	characteristic
42	C₄⋊(C₂₃⋊C₈)	C₂²	1	1	yes	characteristic
43	C₄⋊(C₂₃⋊C₈)	C₂²	1	2	yes	—
44	#41×C₂₃	—	2	4	no	—
45	#41×C₂₃	—	2	4	no	—
46	C₂×Q₈×C₂₃	C₂³	1	2	yes	—
47	C₂×D₄×C₂₃	—	2	2	no	—
48	C₂×D₄×C₂₃	—	2	1	no	—
49	C₂×D₄×C₂₃	—	2	1	no	—
50	C₂×D₄×C₂₃	C₂³	1	2	yes	—
51	C₂²×C₄×C₂₃	—	2	2	no	—
52	C₂²×C₄×C₂₃	D₄	1	1	yes	characteristic
53	C₂²×C₄×C₂₃	D₄	1	1	yes	characteristic
54	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
55	C₂×C₂²⋊D₂₃	—	2	1	no	—
56	#67288	—	2	2	no	—
57	#67288	D₄	1	4	yes	complemented
58	C₂×D₉₂	C₂³	1	1	yes	characteristic
59	C₂×C₄×D₂₃	—	2	1	no	—
60	C₂×C₄×D₂₃	—	2	1	no	—
61	C₂×C₄×D₂₃	—	2	1	no	—
62	C₂×Dic₄₆	C₂³	1	1	yes	characteristic
63	C₄⋊C₄×C₂₃	—	2	1	no	—
64	C₄⋊C₄×C₂₃	—	2	1	no	—
65	C₂²⋊C₄×C₂₃	—	2	2	no	—
66	C₂²⋊C₄×C₂₃	—	2	2	no	—
67	C₄²×C₂₃	D₄	1	1	yes	characteristic
68	C₄²×C₂₃	D₄	1	1	yes	characteristic
69	C₄²×C₂₃	C₂³	1	1	yes	characteristic
70	C₄²×C₂₃	C₂³	1	1	yes	characteristic
71	C₂²⋊Dic₂₃	—	2	1	no	—
72	#67275	—	2	4	no	—
73	#67274	—	2	4	no	—
74	#67273	—	2	4	no	—
75	#67272	—	2	4	no	—
76	(C₂×D₂₃)⋊C₄	—	2	1	no	—
77	(C₂×D₂₃)⋊C₄	—	2	1	no	—
78	C₄⋊Dic₂₃	C₂³	1	1	yes	characteristic
79	C₄⋊Dic₂₃	C₂³	1	1	yes	characteristic
80	Dic₂₃⋊C₄	—	2	1	no	—
81	Dic₂₃⋊C₄	—	2	1	no	—
82	C₄×Dic₂₃	—	2	1	no	—
83	C₄×Dic₂₃	—	2	1	no	—
84	C₄×Dic₂₃	—	2	1	no	—
85	C₂×C₂₃⋊C₈	—	2	4	no	—
86	C₂×C₂₃⋊C₈	C₂³	1	2	yes	—
87	C₂×C₂₃⋊C₈	C₂³	1	2	yes	—
88	C₂³×C₂₃	—	2	2	no	—
89	C₂³×C₂₃	C₂×D₄	1	1	yes	characteristic
90	C₂²×D₂₃	—	2	1	no	—
91	Q₈×C₂₃	—	2	2	no	—
92	Q₈×C₂₃	—	2	2	no	—
93	D₄×C₂₃	—	2	2	no	—
94	D₄×C₂₃	—	2	2	no	—
95	D₄×C₂₃	—	4	2	no	—
96	D₄×C₂₃	—	4	4	no	—
97	D₄×C₂₃	—	4	2	no	—
98	C₂×C₄×C₂₃	—	2	2	no	—
99	C₂×C₄×C₂₃	—	2	2	no	—
100	C₂×C₄×C₂₃	—	4	1	no	—
101	C₂×C₄×C₂₃	—	4	4	no	—
102	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
103	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
104	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
105	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
106	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
107	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
108	C₂×C₄×C₂₃	#41	1	2	yes	—
109	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
110	C₂²⋊D₂₃	—	2	4	no	—
111	C₂×Dic₂₃	—	2	1	no	—
112	C₂×Dic₂₃	—	2	1	no	—
113	C₂×Dic₂₃	—	2	1	no	—
114	D₉₂	—	2	1	no	—
115	D₉₂	C₂×D₄	1	2	yes	—
116	C₄×D₂₃	—	2	4	no	—
117	C₄×D₂₃	—	4	2	no	—
118	C₄×D₂₃	—	4	2	no	—
119	Dic₄₆	—	2	1	no	—
120	Dic₄₆	C₂×D₄	1	2	yes	—
121	C₂₃⋊C₈	—	2	2	no	—
122	C₂₃⋊C₈	—	2	2	no	—
123	#3035	—	23	1	no	maximal, Sylow
124	C₂²×C₂₃	—	2	1	no	—
125	C₂²×C₂₃	—	2	1	no	—
126	C₂²×C₂₃	—	4	2	no	—
127	C₂²×C₂₃	—	4	2	no	—
128	C₂²×C₂₃	—	4	2	no	—
129	C₂²×C₂₃	#121	1	2	yes	—
130	C₂²×C₂₃	C₂²×D₄	1	1	yes	characteristic
131	C₂×D₂₃	—	2	2	no	—
132	C₂×D₂₃	—	4	1	no	—
133	C₂×D₂₃	—	4	1	no	—
134	C₄×C₂₃	—	2	1	no	—
135	C₄×C₂₃	—	2	1	no	—
136	C₄×C₂₃	—	2	1	no	—
137	C₄×C₂₃	—	4	2	no	—
138	C₄×C₂₃	#121	1	2	yes	—
139	C₄×C₂₃	#135	1	2	yes	complemented
140	C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
141	C₄×C₂₃	C₂×#41	1	1	yes	characteristic
142	Dic₂₃	—	2	2	no	—
143	Dic₂₃	—	4	1	no	—
144	Dic₂₃	—	4	1	no	—
145	C₂×#135	—	23	2	no	—
146	#532	D₂₃	1	1	yes	characteristic, complemented
147	C₄×#41	—	23	1	no	—
148	#464	—	23	2	no	—
149	#463	—	23	2	no	—
150	#462	—	23	1	no	—
151	#461	—	23	1	no	—
152	#460	—	23	1	no	—
153	#459	—	23	1	no	—
154	C₂×C₄⋊C₈	—	23	1	no	—
155	#416	—	23	2	no	—
156	C₂×C₂₃	—	2	2	no	—
157	C₂×C₂₃	—	8	2	no	—
158	C₂×C₂₃	C₂×#121	1	1	yes	characteristic
159	C₂×C₂₃	C₂×#135	1	1	yes	characteristic
160	C₂×C₂₃	#575	1	1	yes	characteristic
161	D₂₃	—	4	2	no	—
162	C₂×#41	—	23	1	no	—
163	C₂×#41	C₂×D₂₃	1	2	yes	—
164	#135	—	46	8	no	—
165	C₂×Q₁₆	—	23	2	no	—
166	C₂×SD₁₆	—	23	2	no	—
167	C₂×SD₁₆	—	23	2	no	—
168	C₂×D₈	—	23	2	no	—
169	C₂²×C₈	—	23	2	no	—
170	C₄⋊Q₈	C₂×D₂₃	1	1	yes	characteristic
171	#127	C₂×D₂₃	1	1	yes	characteristic
172	#124	C₂×D₂₃	1	2	yes	—
173	#121	—	2	1	no	—
174	#121	—	2	1	no	—
175	C₄×Q₈	—	23	1	no	—
176	C₄×Q₈	—	23	1	no	—
177	C₄×D₄	—	2	1	no	—
178	C₄×D₄	—	2	1	no	—
179	C₄×D₄	—	23	1	no	—
180	C₄×D₄	—	23	1	no	—
181	C₄×D₄	—	46	1	no	—
182	C₄×D₄	—	46	1	no	—
183	#117	—	23	1	no	—
184	#117	—	46	1	no	—
185	C₂×C₄²	—	46	1	no	—
186	C₂×C₄²	C₂×D₂₃	1	1	yes	characteristic
187	C₄⋊C₈	—	23	1	no	—
188	C₄⋊C₈	—	23	1	no	—
189	C₄⋊C₈	—	23	2	no	—
190	#103	—	23	2	no	—
191	#103	—	23	2	no	—
192	#102	—	23	2	no	—
193	#102	—	23	2	no	—
194	C₂₃	#3035	1	1	yes	characteristic, complemented, Sylow
195	#41	—	2	4	no	—
196	#41	—	2	4	no	—
197	#41	—	23	4	no	—
198	#41	—	46	2	no	—
199	C₂×Q₈	—	23	1	no	—
200	C₂×Q₈	C₂²×D₂₃	1	2	yes	—
201	C₂×D₄	—	2	2	no	—
202	C₂×D₄	—	2	1	no	—
203	C₂×D₄	—	2	1	no	—
204	C₂×D₄	—	23	1	no	—
205	C₂×D₄	—	46	1	no	—
206	C₂×D₄	C₂²×D₂₃	1	2	yes	—
207	C₂²×C₄	—	2	2	no	—
208	C₂²×C₄	—	46	1	no	—
209	C₂²×C₄	—	46	1	no	—
210	C₂²×C₄	—	46	1	no	—
211	C₂²×C₄	C₂²⋊D₂₃	1	1	yes	characteristic
212	C₂²×C₄	C₂²⋊D₂₃	1	1	yes	characteristic
213	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
214	Q₁₆	—	46	4	no	—
215	SD₁₆	—	46	4	no	—
216	SD₁₆	—	46	4	no	—
217	D₈	—	46	4	no	—
218	C₂×C₈	—	23	2	no	—
219	C₂×C₈	—	23	2	no	—
220	C₂×C₈	—	46	4	no	—
221	C₄⋊C₄	—	2	1	no	—
222	C₄⋊C₄	—	2	1	no	—
223	C₄⋊C₄	—	23	1	no	—
224	C₄⋊C₄	—	23	1	no	—
225	C₄⋊C₄	—	46	1	no	—
226	C₄⋊C₄	—	46	1	no	—
227	C₂²⋊C₄	—	2	2	no	—
228	C₂²⋊C₄	—	2	2	no	—
229	C₂²⋊C₄	—	46	1	no	—
230	C₂²⋊C₄	—	46	1	no	—
231	C₂²⋊C₄	—	46	1	no	—
232	C₄²	—	46	1	no	—
233	C₄²	—	46	1	no	—
234	C₄²	—	46	1	no	—
235	C₄²	C₂²⋊D₂₃	1	1	yes	characteristic
236	C₄²	C₂²⋊D₂₃	1	1	yes	characteristic
237	C₄²	C₂²×D₂₃	1	1	yes	characteristic
238	C₄²	C₂²×D₂₃	1	1	yes	characteristic
239	C₂³	—	2	2	no	—
240	C₂³	—	46	1	no	—
241	C₂³	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
242	Q₈	—	2	2	no	—
243	Q₈	—	2	2	no	—
244	Q₈	—	23	2	no	—
245	Q₈	—	46	1	no	—
246	D₄	—	2	2	no	—
247	D₄	—	2	2	no	—
248	D₄	—	4	2	no	—
249	D₄	—	4	4	no	—
250	D₄	—	4	2	no	—
251	D₄	—	23	2	no	—
252	D₄	—	46	4	no	—
253	D₄	—	46	1	no	—
254	C₂×C₄	—	2	2	no	—
255	C₂×C₄	—	2	2	no	—
256	C₂×C₄	—	4	1	no	—
257	C₂×C₄	—	4	4	no	—
258	C₂×C₄	—	46	1	no	—
259	C₂×C₄	—	46	4	no	—
260	C₂×C₄	—	46	1	no	—
261	C₂×C₄	—	46	1	no	—
262	C₂×C₄	—	92	2	no	—
263	C₂×C₄	—	92	2	no	—
264	C₂×C₄	D₄×D₂₃	1	2	yes	—
265	C₂×C₄	#67290	1	2	yes	—
266	C₂×C₄	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
267	C₂×C₄	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
268	C₂×C₄	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
269	C₂×C₄	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
270	C₂×C₄	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
271	C₂×C₄	C₂³×D₂₃	1	1	yes	characteristic
272	C₈	—	46	2	no	—
273	C₈	—	46	2	no	—
274	C₂²	—	2	1	no	—
275	C₂²	—	2	1	no	—
276	C₂²	—	4	2	no	—
277	C₂²	—	4	2	no	—
278	C₂²	—	4	2	no	—
279	C₂²	—	46	2	no	—
280	C₂²	—	92	1	no	—
281	C₂²	—	92	1	no	—
282	C₂²	#10625109	1	2	yes	—
283	C₂²	C₂²×C₂²⋊D₂₃	1	1	yes	characteristic
284	C₄	—	2	1	no	—
285	C₄	—	2	1	no	—
286	C₄	—	2	1	no	—
287	C₄	—	4	2	no	—
288	C₄	—	46	2	no	—
289	C₄	—	92	1	no	—
290	C₄	—	92	1	no	—
291	C₄	#10625109	1	2	yes	—
292	C₄	#10625121	1	2	yes	complemented
293	C₄	C₂×D₄×D₂₃	1	1	yes	characteristic
294	C₄	C₂×#67290	1	1	yes	characteristic
295	C₂	—	2	2	no	—
296	C₂	—	8	2	no	—
297	C₂	—	92	2	no	—
298	C₂	C₂×#10625109	1	1	yes	characteristic
299	C₂	C₂×#10625121	1	1	yes	characteristic
300	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	C₂₃×#532	C₂
socle	C₂²×C₂₃	C₂²×D₄
hypercenter	#532	D₂₃
hypocenter	C₂₃	#3035
LCS	G ▷ C₂×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #532

Complements

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

N	H ≅ G/N	type
1. C₂×#10625121	C₂	semidirect
2. C₄×#67288	C₂	semidirect
3. #13092636	C₂	semidirect
4. #13092629	C₂	semidirect
5. #13092624	C₂	semidirect
6. #13092619	C₂	semidirect
7. #13092614	C₂	semidirect
8. #13092609	C₂	semidirect
9. C₂×C₄⋊(C₂₃⋊C₈)	C₂	semidirect
10. #13092413	C₂	semidirect
11. C₂₃×#532	C₂	semidirect
57. #67288	D₄	semidirect
139. C₄×C₂₃	#135	semidirect
146. #532	D₂₃	semidirect
194. C₂₃	#3035	semidirect
292. C₄	#10625121	semidirect

Maximal subgroups

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

subgroup	index
C₂×#10625121	2
C₄×#67288	2
#13092636	2
#13092629	2
#13092624	2
#13092619	2
#13092614	2
#13092609	2
C₂×C₄⋊(C₂₃⋊C₈)	2
#13092413	2
C₂₃×#532	2
#3035	23