Subgroups — #28404379

Order 14464.

964 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	Q₈×Dic₂₂₆	C₂	1	2	yes	maximal, complemented
2	#20299328	C₂	1	2	yes	maximal, complemented
3	#20299326	C₂	1	4	yes	maximal, complemented
4	#20299326	C₂	1	2	yes	maximal, complemented
5	#20299325	C₂	1	4	yes	maximal, complemented
6	#20299323	C₂	1	2	yes	maximal, complemented
7	#20299227	C₂	1	4	yes	maximal
8	#20299226	C₂	1	2	yes	maximal
9	#20299155	C₂	1	1	yes	maximal, characteristic
10	#20299150	C₂	1	4	yes	maximal
11	#20299075	C₂	1	2	yes	maximal
12	C₂×C₄×Dic₂₂₆	C₂	1	1	yes	maximal, characteristic
13	#533×C₁₁₃	C₂	1	1	yes	maximal, characteristic
14	C₂²×Dic₂₂₆	C₂²	1	1	yes	characteristic
15	#16813622	C₂²	1	1	yes	characteristic
16	#16813622	C₂²	1	4	yes	—
17	#16813622	C₂²	1	1	yes	characteristic
18	#16813621	C₂²	1	4	yes	—
19	#16813621	C₂²	1	8	yes	—
20	#16813621	C₂²	1	4	yes	—
21	#16813620	C₂²	1	4	yes	—
22	#16813620	C₂²	1	8	yes	—
23	#16813614	C₂²	1	2	yes	—
24	#16813614	C₂²	1	4	yes	—
25	#16813613	C₂²	1	4	yes	—
26	#16813612	C₂²	1	8	yes	—
27	#16813612	C₂²	1	8	yes	—
28	#16813612	C₂²	1	8	yes	—
29	#16813601	C₂²	1	2	yes	—
30	#16813601	C₂²	1	2	yes	—
31	#16813599	C₂²	1	8	yes	—
32	Q₈×Dic₁₁₃	C₂²	1	4	yes	—
33	C₄×Dic₂₂₆	C₂²	1	2	yes	—
34	C₄×Dic₂₂₆	C₂²	1	2	yes	—
35	C₄×Dic₂₂₆	C₂²	1	2	yes	—
36	C₄×Dic₂₂₆	C₂²	1	2	yes	—
37	C₄×Dic₂₂₆	C₂²	1	2	yes	—
38	C₄×Dic₂₂₆	C₂²	1	2	yes	—
39	C₄×Dic₂₂₆	C₂²	1	4	yes	—
40	C₄×Dic₂₂₆	C₂²	1	4	yes	—
41	#16813587	C₂²	1	4	yes	—
42	#16813587	C₂²	1	8	yes	—
43	#16813587	C₂²	1	4	yes	—
44	#16813587	C₂²	1	4	yes	—
45	#16813581	C₂²	1	2	yes	—
46	#16813578	C₂²	1	8	yes	—
47	C₂×C₄⋊Dic₁₁₃	C₂²	1	1	yes	characteristic
48	C₂×#14268626	C₂²	1	2	yes	—
49	C₂×C₄×Dic₁₁₃	C₂²	1	2	yes	—
50	C₄⋊Q₈×C₁₁₃	C₂²	1	1	yes	characteristic
51	C₄⋊Q₈×C₁₁₃	C₂²	1	1	yes	characteristic
52	#125×C₁₁₃	C₂²	1	2	yes	—
53	C₂²⋊Q₈×C₁₁₃	C₂²	1	2	yes	—
54	C₂²⋊Q₈×C₁₁₃	C₂²	1	2	yes	—
55	C₄×Q₈×C₁₁₃	C₂²	1	2	yes	—
56	C₄×Q₈×C₁₁₃	C₂²	1	2	yes	—
57	#117×C₁₁₃	C₂²	1	2	yes	—
58	C₂×C₄²×C₁₁₃	C₂²	1	1	yes	characteristic
59	C₂×Dic₂₂₆	Q₈	1	8	yes	complemented
60	C₂×Dic₂₂₆	C₂³	1	2	yes	—
61	C₂×Dic₂₂₆	C₂³	1	8	yes	—
62	C₂×Dic₂₂₆	C₂³	1	2	yes	—
63	C₂×Dic₂₂₆	C₂³	1	2	yes	—
64	C₂×Dic₂₂₆	C₂³	1	2	yes	—
65	C₂²×Dic₁₁₃	C₂³	1	2	yes	—
66	C₂²×Dic₁₁₃	C₂³	1	2	yes	—
67	C₄⋊Dic₁₁₃	C₂³	1	2	yes	—
68	C₄⋊Dic₁₁₃	C₂³	1	2	yes	—
69	C₄⋊Dic₁₁₃	C₂³	1	2	yes	—
70	C₄⋊Dic₁₁₃	C₂³	1	2	yes	—
71	C₄⋊Dic₁₁₃	C₂³	1	2	yes	—
72	C₄⋊Dic₁₁₃	C₂³	1	2	yes	—
73	C₄⋊Dic₁₁₃	C₂³	1	8	yes	—
74	#14268626	C₂³	1	4	yes	—
75	#14268626	C₂³	1	4	yes	—
76	#14268626	C₂³	1	8	yes	—
77	#14268626	C₂³	1	8	yes	—
78	#14268626	C₂³	1	4	yes	—
79	#14268626	C₂³	1	8	yes	—
80	#14268626	C₂³	1	8	yes	—
81	#14268626	C₂³	1	4	yes	—
82	C₂²⋊Dic₁₁₃	C₂³	1	4	yes	—
83	C₄×Dic₁₁₃	C₂³	1	4	yes	—
84	C₄×Dic₁₁₃	C₂³	1	4	yes	—
85	C₄×Dic₁₁₃	C₂³	1	8	yes	—
86	C₄×Dic₁₁₃	C₂³	1	8	yes	—
87	C₄×Dic₁₁₃	C₂³	1	4	yes	—
88	C₂×Q₈×C₁₁₃	C₂³	1	2	yes	—
89	C₂×Q₈×C₁₁₃	C₂³	1	2	yes	—
90	C₂²×C₄×C₁₁₃	C₂³	1	1	yes	characteristic
91	C₂²×C₄×C₁₁₃	C₂³	1	1	yes	characteristic
92	C₂²×C₄×C₁₁₃	C₂³	1	1	yes	characteristic
93	C₄⋊C₄×C₁₁₃	C₂³	1	4	yes	—
94	C₄⋊C₄×C₁₁₃	C₂³	1	2	yes	—
95	C₄⋊C₄×C₁₁₃	C₂³	1	2	yes	—
96	C₄⋊C₄×C₁₁₃	C₂³	1	2	yes	—
97	C₄⋊C₄×C₁₁₃	C₂³	1	2	yes	—
98	C₄⋊C₄×C₁₁₃	C₂³	1	4	yes	—
99	C₂²⋊C₄×C₁₁₃	C₂³	1	4	yes	—
100	C₄²×C₁₁₃	C₂³	1	1	yes	characteristic
101	C₄²×C₁₁₃	C₂³	1	1	yes	characteristic
102	C₄²×C₁₁₃	C₂³	1	1	yes	characteristic
103	C₄²×C₁₁₃	C₂³	1	4	yes	—
104	C₄²×C₁₁₃	C₂³	1	1	yes	characteristic
105	C₂³×C₁₁₃	C₂⁴	1	1	yes	characteristic
106	Q₈×C₁₁₃	—	2	4	no	—
107	Q₈×C₁₁₃	—	4	2	no	—
108	C₂×C₄×C₁₁₃	—	2	2	no	—
109	C₂×C₄×C₁₁₃	—	4	1	no	—
110	C₂×C₄×C₁₁₃	C₂×Q₈	1	4	yes	—
111	C₂×C₄×C₁₁₃	C₂⁴	1	1	yes	characteristic
112	C₂×C₄×C₁₁₃	C₂⁴	1	1	yes	characteristic
113	C₂×C₄×C₁₁₃	C₂⁴	1	1	yes	characteristic
114	C₂×C₄×C₁₁₃	C₂⁴	1	4	yes	—
115	C₂×C₄×C₁₁₃	C₂⁴	1	4	yes	—
116	C₂×C₄×C₁₁₃	C₂⁴	1	1	yes	characteristic
117	C₂×C₄×C₁₁₃	C₂⁴	1	1	yes	characteristic
118	C₂×C₄×C₁₁₃	C₂⁴	1	1	yes	characteristic
119	C₂×Dic₁₁₃	—	4	2	no	—
120	C₂×Dic₁₁₃	C₂×Q₈	1	8	yes	—
121	C₂×Dic₁₁₃	C₂⁴	1	4	yes	—
122	C₂×Dic₁₁₃	C₂⁴	1	4	yes	—
123	C₂×Dic₁₁₃	C₂⁴	1	4	yes	—
124	C₂×Dic₁₁₃	C₂⁴	1	4	yes	—
125	Dic₂₂₆	—	2	4	no	—
126	Dic₂₂₆	—	4	2	no	—
127	Dic₂₂₆	—	4	8	no	—
128	Dic₂₂₆	C₂×Q₈	1	8	yes	—
129	Dic₂₂₆	C₂×Q₈	1	8	yes	—
130	C₂²×C₁₁₃	—	2	1	no	—
131	C₂²×C₁₁₃	—	2	1	no	—
132	C₂²×C₁₁₃	C₂²×Q₈	1	2	yes	—
133	C₂²×C₁₁₃	C₂⁵	1	1	yes	characteristic
134	C₄×C₁₁₃	—	2	1	no	—
135	C₄×C₁₁₃	—	2	4	no	—
136	C₄×C₁₁₃	—	2	1	no	—
137	C₄×C₁₁₃	—	2	4	no	—
138	C₄×C₁₁₃	—	2	1	no	—
139	C₄×C₁₁₃	C₂²×Q₈	1	2	yes	—
140	C₄×C₁₁₃	C₂²×Q₈	1	2	yes	—
141	C₄×C₁₁₃	2₋¹⁺⁴	1	2	yes	—
142	Dic₁₁₃	—	2	4	no	—
143	Dic₁₁₃	—	2	4	no	—
144	Dic₁₁₃	—	2	4	no	—
145	Dic₁₁₃	C₂²×Q₈	1	8	yes	—
146	C₂×C₁₁₃	—	2	2	no	—
147	C₂×C₁₁₃	C₂³×Q₈	1	1	yes	characteristic
148	C₂×C₁₁₃	C₂×2₋¹⁺⁴	1	1	yes	characteristic
149	C₂×C₁₁₃	#585	1	1	yes	characteristic
150	#3502	—	113	1	no	maximal, Sylow
151	C₁₁₃	#3502	1	1	yes	characteristic, complemented, Sylow
152	Q₈²	—	113	2	no	—
153	Q₈²	—	113	2	no	—
154	#557	—	113	2	no	—
155	#557	—	113	4	no	—
156	#557	—	113	4	no	—
157	#557	—	113	2	no	—
158	#544	—	113	4	no	—
159	#544	—	113	2	no	—
160	#533	—	113	4	no	—
161	#533	—	113	1	no	—
162	#533	D₁₁₃	1	1	yes	characteristic
163	#519	—	113	2	no	—
164	C₂×C₄×Q₈	—	113	1	no	—
165	C₂²×Q₈	—	113	1	no	—
166	C₄⋊Q₈	—	113	4	no	—
167	C₄⋊Q₈	—	113	8	no	—
168	C₄⋊Q₈	—	113	4	no	—
169	C₄⋊Q₈	—	113	8	no	—
170	C₄⋊Q₈	—	113	4	no	—
171	C₄⋊Q₈	—	113	1	no	—
172	C₄⋊Q₈	—	113	1	no	—
173	C₄⋊Q₈	—	113	4	no	—
174	C₄⋊Q₈	C₂×D₁₁₃	1	1	yes	characteristic
175	C₄⋊Q₈	C₂×D₁₁₃	1	1	yes	characteristic
176	#125	—	113	8	no	—
177	#125	—	113	8	no	—
178	#125	—	113	8	no	—
179	#125	—	113	4	no	—
180	#125	—	113	2	no	—
181	#125	—	113	4	no	—
182	#125	C₂×D₁₁₃	1	2	yes	—
183	C₂²⋊Q₈	—	113	2	no	—
184	C₂²⋊Q₈	—	113	2	no	—
185	C₂²⋊Q₈	—	113	8	no	—
186	C₂²⋊Q₈	C₂×D₁₁₃	1	2	yes	—
187	C₂²⋊Q₈	C₂×D₁₁₃	1	2	yes	—
188	C₄×Q₈	—	113	4	no	—
189	C₄×Q₈	—	113	4	no	—
190	C₄×Q₈	—	113	4	no	—
191	C₄×Q₈	—	113	8	no	—
192	C₄×Q₈	—	113	4	no	—
193	C₄×Q₈	—	113	2	no	—
194	C₄×Q₈	—	113	2	no	—
195	C₄×Q₈	—	113	2	no	—
196	C₄×Q₈	—	113	2	no	—
197	C₄×Q₈	—	113	4	no	—
198	C₄×Q₈	—	113	4	no	—
199	C₄×Q₈	—	113	2	no	—
200	C₄×Q₈	—	113	2	no	—
201	C₄×Q₈	C₂×D₁₁₃	1	2	yes	—
202	C₄×Q₈	C₂×D₁₁₃	1	2	yes	—
203	#117	—	113	2	no	—
204	#117	—	113	8	no	—
205	#117	C₂×D₁₁₃	1	2	yes	—
206	C₂×C₄⋊C₄	—	113	2	no	—
207	C₂×C₄⋊C₄	—	113	1	no	—
208	C₂×C₄²	—	113	2	no	—
209	C₂×C₄²	C₂×D₁₁₃	1	1	yes	characteristic
210	C₂×Q₈	—	113	8	no	—
211	C₂×Q₈	—	113	2	no	—
212	C₂×Q₈	—	113	2	no	—
213	C₂×Q₈	—	113	2	no	—
214	C₂×Q₈	—	113	8	no	—
215	C₂×Q₈	—	113	2	no	—
216	C₂×Q₈	C₂²×D₁₁₃	1	2	yes	—
217	C₂×Q₈	C₂²×D₁₁₃	1	2	yes	—
218	C₂²×C₄	—	113	2	no	—
219	C₂²×C₄	—	113	2	no	—
220	C₂²×C₄	C₂²×D₁₁₃	1	1	yes	characteristic
221	C₂²×C₄	C₂²×D₁₁₃	1	1	yes	characteristic
222	C₂²×C₄	C₂²×D₁₁₃	1	1	yes	characteristic
223	C₄⋊C₄	—	113	4	no	—
224	C₄⋊C₄	—	113	4	no	—
225	C₄⋊C₄	—	113	2	no	—
226	C₄⋊C₄	—	113	2	no	—
227	C₄⋊C₄	—	113	8	no	—
228	C₄⋊C₄	—	113	2	no	—
229	C₄⋊C₄	—	113	8	no	—
230	C₄⋊C₄	—	113	2	no	—
231	C₄⋊C₄	—	113	4	no	—
232	C₄⋊C₄	—	113	2	no	—
233	C₄⋊C₄	—	113	8	no	—
234	C₄⋊C₄	—	113	2	no	—
235	C₄⋊C₄	—	113	4	no	—
236	C₄⋊C₄	—	113	8	no	—
237	C₄⋊C₄	—	113	8	no	—
238	C₄⋊C₄	C₂²×D₁₁₃	1	2	yes	—
239	C₄⋊C₄	C₂²×D₁₁₃	1	2	yes	—
240	C₄⋊C₄	C₂²×D₁₁₃	1	4	yes	—
241	C₄⋊C₄	C₂²×D₁₁₃	1	2	yes	—
242	C₄⋊C₄	C₂²×D₁₁₃	1	2	yes	—
243	C₄⋊C₄	C₂²×D₁₁₃	1	4	yes	—
244	C₂²⋊C₄	—	113	4	no	—
245	C₂²⋊C₄	C₂²×D₁₁₃	1	4	yes	—
246	C₄²	—	113	4	no	—
247	C₄²	—	113	4	no	—
248	C₄²	—	113	8	no	—
249	C₄²	—	113	8	no	—
250	C₄²	—	113	4	no	—
251	C₄²	C₂²×D₁₁₃	1	1	yes	characteristic
252	C₄²	C₂²×D₁₁₃	1	1	yes	characteristic
253	C₄²	C₂²×D₁₁₃	1	1	yes	characteristic
254	C₄²	C₂²×D₁₁₃	1	1	yes	characteristic
255	C₄²	C₂²×D₁₁₃	1	4	yes	—
256	C₂³	C₂³×D₁₁₃	1	1	yes	characteristic
257	Q₈	—	2	4	no	—
258	Q₈	—	4	2	no	—
259	Q₈	—	113	8	no	—
260	Q₈	—	113	8	no	—
261	Q₈	—	452	2	no	—
262	Q₈	—	452	8	no	—
263	Q₈	—	226	4	no	—
264	C₂×C₄	—	2	2	no	—
265	C₂×C₄	—	4	1	no	—
266	C₂×C₄	—	113	8	no	—
267	C₂×C₄	—	113	4	no	—
268	C₂×C₄	—	113	4	no	—
269	C₂×C₄	—	113	4	no	—
270	C₂×C₄	—	113	4	no	—
271	C₂×C₄	—	452	2	no	—
272	C₂×C₄	Q₈×D₁₁₃	1	4	yes	—
273	C₂×C₄	C₂³×D₁₁₃	1	1	yes	characteristic
274	C₂×C₄	C₂³×D₁₁₃	1	4	yes	—
275	C₂×C₄	C₂³×D₁₁₃	1	4	yes	—
276	C₂×C₄	C₂³×D₁₁₃	1	1	yes	characteristic
277	C₂×C₄	C₂³×D₁₁₃	1	1	yes	characteristic
278	C₂×C₄	C₂³×D₁₁₃	1	1	yes	characteristic
279	C₂×C₄	C₂³×D₁₁₃	1	1	yes	characteristic
280	C₂×C₄	C₂³×D₁₁₃	1	1	yes	characteristic
281	C₂²	—	2	1	no	—
282	C₂²	—	2	1	no	—
283	C₂²	C₂×Q₈×D₁₁₃	1	2	yes	—
284	C₂²	C₂⁴×D₁₁₃	1	1	yes	characteristic
285	C₄	—	2	1	no	—
286	C₄	—	2	4	no	—
287	C₄	—	2	1	no	—
288	C₄	—	2	1	no	—
289	C₄	—	2	4	no	—
290	C₄	—	113	8	no	—
291	C₄	—	226	4	no	—
292	C₄	—	226	4	no	—
293	C₄	—	226	4	no	—
294	C₄	C₂×Q₈×D₁₁₃	1	2	yes	—
295	C₄	C₂×Q₈×D₁₁₃	1	2	yes	—
296	C₄	#16813669	1	2	yes	—
297	C₂	—	2	2	no	—
298	C₂	C₂²×Q₈×D₁₁₃	1	1	yes	characteristic
299	C₂	C₂×#16813669	1	1	yes	characteristic
300	C₂	#20299434	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₁₁₃
commutator, socle, cosocle	C₂²×C₁₁₃	C₂⁵
Fitting	#533×C₁₁₃	C₂
prime radical	C₂³×C₁₁₃	C₂⁴
hypercenter	#533	D₁₁₃
hypocenter	C₁₁₃	#3502
LCS	G ▷ C₂²×C₁₁₃ ▷ C₁₁₃
UCS	C₁ ◁ C₂² ◁ #533

Complements

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

N	H ≅ G/N	type
1. Q₈×Dic₂₂₆	C₂	semidirect
2. #20299328	C₂	semidirect
3. #20299326	C₂	semidirect
4. #20299326	C₂	semidirect
5. #20299325	C₂	semidirect
6. #20299323	C₂	semidirect
59. C₂×Dic₂₂₆	Q₈	semidirect
151. C₁₁₃	#3502	semidirect

Maximal subgroups

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

subgroup	index
Q₈×Dic₂₂₆	2
#20299328	2
#20299326	2
#20299325	2
#20299323	2
#20299227	2
#20299226	2
#20299155	2
#20299150	2
#20299075	2
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