Subgroups — #12256995

Order 1280.

472 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#3383	C₂	1	1	yes	maximal, characteristic, complemented
2	#10611643	C₂	1	1	yes	maximal, characteristic
3	#10602180	C₂	1	1	yes	maximal, characteristic
4	#10602175	C₂	1	1	yes	maximal, characteristic, complemented
5	#10602150	C₂	1	2	yes	maximal, complemented
6	#10602144	C₂	1	2	yes	maximal
7	Q₈×#672	C₂	1	1	yes	maximal, characteristic, complemented
8	#10601709	C₂	1	1	yes	maximal, characteristic, complemented
9	#10601708	C₂	1	2	yes	maximal
10	C₄×#3961	C₂	1	2	yes	maximal, complemented
11	C₄×#3960	C₂	1	1	yes	maximal, characteristic
12	C₅×#557	—	2	1	no	—
13	#65918	—	2	1	no	—
14	C₅×#500	C₂²	1	1	yes	characteristic
15	C₅×#499	C₂²	1	2	yes	—
16	C₅×#479	—	2	2	no	—
17	C₅×#475	—	2	1	no	—
18	C₅×C₈×Q₈	C₂²	1	1	yes	characteristic
19	C₄×C₅×Q₁₆	C₂²	1	1	yes	characteristic
20	C₄×C₅×Q₁₆	C₂²	1	2	yes	—
21	C₂×#3961	C₂²	1	2	yes	—
22	C₂×#3960	C₂²	1	1	yes	characteristic
23	#65134	—	2	2	no	—
24	#65130	—	2	1	no	—
25	#64988	C₂²	1	1	yes	characteristic
26	C₄×Dic₂₀	C₂²	1	2	yes	—
27	#64985	C₂²	1	2	yes	—
28	C₄×D₄₀	C₂²	1	1	yes	characteristic
29	#64784	C₂²	1	1	yes	characteristic
30	#64712	C₂²	1	2	yes	—
31	#64712	C₂²	1	2	yes	—
32	#64712	C₂²	1	2	yes	—
33	#64711	C₂²	1	1	yes	characteristic
34	#64711	C₂²	1	2	yes	—
35	#64710	C₂²	1	1	yes	characteristic
36	#64709	C₂²	1	2	yes	—
37	C₄⋊#672	C₂²	1	2	yes	—
38	C₄⋊#672	C₂²	1	1	yes	characteristic
39	C₄×#672	C₂²	1	2	yes	—
40	C₄×#672	C₂²	1	1	yes	characteristic
41	#33931	—	5	1	no	maximal, Sylow
42	C₂×C₅×Q₁₆	C₂³	1	1	yes	characteristic
43	C₅×C₄⋊Q₈	—	2	1	no	—
44	C₅×C₄⋊Q₈	—	4	1	no	—
45	C₅×#125	—	2	2	no	—
46	C₅×#125	—	4	2	no	—
47	C₄×C₅×Q₈	—	2	1	no	—
48	C₄×C₅×Q₈	—	2	2	no	—
49	C₄×C₅×Q₈	—	4	1	no	—
50	C₄×C₅×Q₈	D₄	1	1	yes	characteristic
51	#4092	—	4	1	no	—
52	C₂×Dic₂₀	—	2	1	no	—
53	C₂×Dic₂₀	C₂³	1	2	yes	—
54	C₂×D₄₀	C₂³	1	1	yes	characteristic
55	C₂×#677	—	2	2	no	—
56	#4044	—	4	2	no	—
57	#4038	—	4	1	no	—
58	#4021	—	2	2	no	—
59	C₄×D₂₀	—	2	1	no	—
60	#4015	—	2	1	no	—
61	C₄×Dic₁₀	—	2	2	no	—
62	C₅×#107	—	2	1	no	—
63	C₅×#107	C₂³	1	1	yes	characteristic
64	C₅×#107	C₂³	1	2	yes	—
65	C₅×#106	—	2	2	no	—
66	C₅×C₄⋊C₈	D₄	1	1	yes	characteristic
67	C₅×C₄⋊C₈	D₄	1	2	yes	—
68	C₅×#103	—	2	2	no	—
69	C₅×#103	—	2	1	no	—
70	C₄×C₅×C₈	C₂³	1	2	yes	—
71	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
72	#3961	—	2	4	no	—
73	#3960	—	2	2	no	—
74	#3953	—	2	1	no	—
75	#3950	C₂³	1	1	yes	characteristic
76	#3948	—	2	2	no	—
77	C₂×#672	C₂³	1	1	yes	characteristic
78	C₂×#672	C₂³	1	1	yes	characteristic
79	C₂×#672	C₂³	1	2	yes	—
80	#3393	—	5	1	no	—
81	#3383	D₅	1	1	yes	characteristic, complemented
82	#2242	—	5	2	no	—
83	#2241	—	5	1	no	—
84	#2229	—	5	2	no	—
85	#2224	—	5	1	no	—
86	#2211	—	5	2	no	—
87	#2209	—	5	1	no	—
88	Q₈×C₁₆	—	5	1	no	—
89	C₄×Q₃₂	—	5	2	no	—
90	C₄×SD₃₂	—	5	1	no	—
91	C₂×C₅×Q₈	—	2	1	no	—
92	C₂×C₅×Q₈	—	4	1	no	—
93	C₂×C₅×Q₈	D₈	1	1	yes	characteristic
94	C₂×D₂₀	—	2	1	no	—
95	C₂×C₄×D₅	—	4	1	no	—
96	C₂×Dic₁₀	—	2	2	no	—
97	C₅×Q₁₆	—	2	1	no	—
98	C₅×Q₁₆	#41	1	2	yes	—
99	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
100	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
101	C₂×C₅×C₈	C₂⁴	1	1	yes	characteristic
102	C₅×C₄⋊C₄	—	2	1	no	—
103	C₅×C₄⋊C₄	—	2	2	no	—
104	C₅×C₄⋊C₄	—	4	1	no	—
105	C₅×C₄⋊C₄	—	4	2	no	—
106	C₅×C₄⋊C₄	—	4	2	no	—
107	C₅×C₄⋊C₄	—	4	2	no	—
108	C₅×C₄⋊C₄	—	4	1	no	—
109	C₅×C₄⋊C₄	D₈	1	1	yes	characteristic
110	C₅×C₄⋊C₄	D₈	1	2	yes	—
111	C₄²×C₅	—	4	2	no	—
112	C₄²×C₅	—	4	1	no	—
113	C₄²×C₅	C₂×D₄	1	2	yes	—
114	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
115	#685	—	4	1	no	—
116	#685	—	4	2	no	—
117	#685	—	4	2	no	—
118	C₄⋊Dic₅	—	2	1	no	—
119	C₄⋊Dic₅	—	4	1	no	—
120	C₄⋊Dic₅	—	4	1	no	—
121	#683	—	4	2	no	—
122	C₄×Dic₅	—	4	2	no	—
123	Dic₂₀	—	2	2	no	—
124	Dic₂₀	—	2	2	no	—
125	Dic₂₀	—	8	1	no	—
126	D₄₀	—	2	1	no	—
127	D₄₀	—	2	1	no	—
128	#677	—	8	2	no	—
129	#672	—	2	1	no	—
130	#672	—	2	2	no	—
131	#672	—	2	1	no	—
132	#557	—	2	1	no	—
133	#556	—	10	1	no	—
134	C₂×Q₃₂	—	5	2	no	—
135	C₂×SD₃₂	—	5	1	no	—
136	#500	C₂×D₅	1	1	yes	characteristic
137	#499	C₂×D₅	1	2	yes	—
138	#495	—	5	2	no	—
139	#494	—	5	1	no	—
140	#479	—	2	2	no	—
141	#475	—	2	1	no	—
142	#464	—	10	2	no	—
143	#461	—	10	1	no	—
144	C₈×Q₈	C₂×D₅	1	1	yes	characteristic
145	C₄×Q₁₆	—	5	2	no	—
146	C₄×Q₁₆	C₂×D₅	1	1	yes	characteristic
147	C₄×Q₁₆	C₂×D₅	1	2	yes	—
148	C₄×D₈	—	5	1	no	—
149	#367	—	5	1	no	—
150	#366	—	5	2	no	—
151	C₄⋊C₁₆	—	5	1	no	—
152	C₄⋊C₁₆	—	5	2	no	—
153	#358	—	5	1	no	—
154	#358	—	5	2	no	—
155	#358	—	5	2	no	—
156	#358	—	5	2	no	—
157	#357	—	5	1	no	—
158	#357	—	5	2	no	—
159	C₄×C₁₆	—	5	1	no	—
160	C₄×C₁₆	—	5	2	no	—
161	C₂²×D₅	—	4	1	no	—
162	C₅×Q₈	—	2	2	no	—
163	C₅×Q₈	—	2	2	no	—
164	C₅×Q₈	—	4	1	no	—
165	C₅×Q₈	—	8	2	no	—
166	C₂×C₄×C₅	—	4	1	no	—
167	C₂×C₄×C₅	—	4	1	no	—
168	C₂×C₄×C₅	—	4	2	no	—
169	C₂×C₄×C₅	C₂×D₈	1	2	yes	—
170	C₂×C₄×C₅	C₂×D₈	1	2	yes	—
171	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
172	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
173	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
174	C₂×Dic₅	—	4	1	no	—
175	C₂×Dic₅	—	4	2	no	—
176	D₂₀	—	4	1	no	—
177	D₂₀	—	4	1	no	—
178	C₄×D₅	—	8	2	no	—
179	Dic₁₀	—	4	2	no	—
180	Dic₁₀	—	4	2	no	—
181	C₅×C₈	—	2	2	no	—
182	C₅×C₈	—	2	1	no	—
183	C₅×C₈	C₂×#41	1	1	yes	characteristic
184	C₅×C₈	2₋¹⁺⁴	1	1	yes	characteristic
185	C₂×Q₁₆	—	5	2	no	—
186	C₂×Q₁₆	—	10	1	no	—
187	C₂×Q₁₆	C₂²×D₅	1	1	yes	characteristic
188	C₂×SD₁₆	—	10	2	no	—
189	C₂×D₈	—	5	1	no	—
190	C₄⋊Q₈	—	2	1	no	—
191	C₄⋊Q₈	—	4	1	no	—
192	C₄⋊Q₈	—	10	1	no	—
193	#126	—	20	2	no	—
194	#125	—	2	2	no	—
195	#125	—	4	2	no	—
196	#124	—	10	2	no	—
197	C₂²⋊Q₈	—	20	1	no	—
198	C₄×Q₈	—	2	1	no	—
199	C₄×Q₈	—	2	2	no	—
200	C₄×Q₈	—	4	1	no	—
201	C₄×Q₈	—	10	2	no	—
202	C₄×Q₈	C₂²⋊D₅	1	1	yes	characteristic
203	C₄×D₄	—	10	1	no	—
204	#117	—	20	1	no	—
205	Q₃₂	—	10	4	no	—
206	SD₃₂	—	10	2	no	—
207	C₂×C₁₆	—	5	1	no	—
208	C₂×C₁₆	—	5	2	no	—
209	C₂×C₁₆	—	5	1	no	—
210	#107	—	2	1	no	—
211	#107	—	5	1	no	—
212	#107	C₂²×D₅	1	2	yes	—
213	#107	C₂²×D₅	1	1	yes	characteristic
214	#106	—	2	2	no	—
215	C₄⋊C₈	C₂²⋊D₅	1	1	yes	characteristic
216	C₄⋊C₈	C₂²⋊D₅	1	2	yes	—
217	#103	—	2	2	no	—
218	#103	—	2	1	no	—
219	#103	—	10	2	no	—
220	#102	—	10	1	no	—
221	C₄×C₈	C₂²×D₅	1	1	yes	characteristic
222	C₄×C₈	C₂²×D₅	1	2	yes	—
223	C₂²×C₅	C₂²×D₈	1	1	yes	characteristic
224	C₂×D₅	—	8	1	no	—
225	C₂×D₅	—	8	1	no	—
226	C₂×D₅	—	8	1	no	—
227	C₄×C₅	—	2	2	no	—
228	C₄×C₅	—	2	2	no	—
229	C₄×C₅	—	2	1	no	—
230	C₄×C₅	—	4	2	no	—
231	C₄×C₅	—	8	2	no	—
232	C₄×C₅	—	8	1	no	—
233	C₄×C₅	#508	1	2	yes	—
234	C₄×C₅	#548	1	1	yes	characteristic
235	C₄×C₅	#577	1	1	yes	characteristic
236	Dic₅	—	8	2	no	—
237	Dic₅	—	8	1	no	—
238	C₂×Q₈	—	2	1	no	—
239	C₂×Q₈	—	4	1	no	—
240	C₂×Q₈	—	10	2	no	—
241	C₂×Q₈	#686	1	1	yes	characteristic
242	C₂×D₄	—	10	1	no	—
243	C₂²×C₄	—	20	1	no	—
244	Q₁₆	—	2	1	no	—
245	Q₁₆	—	10	2	no	—
246	Q₁₆	—	10	2	no	—
247	Q₁₆	—	40	1	no	—
248	Q₁₆	#713	1	2	yes	—
249	SD₁₆	—	40	2	no	—
250	D₈	—	10	1	no	—
251	D₈	—	10	1	no	—
252	C₂×C₈	C₂×C₂²⋊D₅	1	1	yes	characteristic
253	C₂×C₈	C₂×C₂²⋊D₅	1	2	yes	—
254	C₂×C₈	C₂³×D₅	1	1	yes	characteristic
255	C₄⋊C₄	—	2	1	no	—
256	C₄⋊C₄	—	2	2	no	—
257	C₄⋊C₄	—	4	1	no	—
258	C₄⋊C₄	—	4	2	no	—
259	C₄⋊C₄	—	4	2	no	—
260	C₄⋊C₄	—	4	2	no	—
261	C₄⋊C₄	—	4	1	no	—
262	C₄⋊C₄	—	10	1	no	—
263	C₄⋊C₄	—	20	1	no	—
264	C₄⋊C₄	—	20	2	no	—
265	C₄⋊C₄	—	20	1	no	—
266	C₄⋊C₄	#686	1	1	yes	characteristic
267	C₄⋊C₄	#686	1	2	yes	—
268	C₂²⋊C₄	—	20	2	no	—
269	C₂²⋊C₄	—	20	1	no	—
270	C₂²⋊C₄	—	20	2	no	—
271	C₄²	—	4	2	no	—
272	C₄²	—	4	1	no	—
273	C₄²	—	20	2	no	—
274	C₄²	C₂×C₂²⋊D₅	1	1	yes	characteristic
275	C₄²	C₂×C₂²⋊D₅	1	2	yes	—
276	C₁₆	—	10	1	no	—
277	C₁₆	—	10	2	no	—
278	C₁₆	—	10	1	no	—
279	C₂×C₅	#3358	1	1	yes	characteristic
280	C₂×C₅	C₂×#508	1	1	yes	characteristic
281	C₂×C₅	#3417	1	1	yes	characteristic
282	D₅	—	16	1	no	—
283	C₂³	—	20	1	no	—
284	Q₈	—	2	2	no	—
285	Q₈	—	2	2	no	—
286	Q₈	—	4	1	no	—
287	Q₈	—	8	2	no	—
288	Q₈	—	20	2	no	—
289	Q₈	—	20	2	no	—
290	D₄	—	20	1	no	—
291	D₄	—	20	1	no	—
292	C₂×C₄	—	4	1	no	—
293	C₂×C₄	—	4	1	no	—
294	C₂×C₄	—	4	2	no	—
295	C₂×C₄	—	20	1	no	—
296	C₂×C₄	—	20	2	no	—
297	C₂×C₄	—	40	2	no	—
298	C₂×C₄	C₂×#686	1	2	yes	—
299	C₂×C₄	C₂×#686	1	1	yes	characteristic
300	C₂×C₄	C₂×#686	1	2	yes	—
301	C₂×C₄	C₂×#686	1	1	yes	characteristic
302	C₂×C₄	C₂²×C₂²⋊D₅	1	1	yes	characteristic
303	C₈	—	2	2	no	—
304	C₈	—	2	1	no	—
305	C₈	C₂×#713	1	1	yes	characteristic
306	C₈	#4147	1	1	yes	characteristic
307	C₅	#33931	1	1	yes	characteristic, complemented, Sylow
308	C₂²	—	40	1	no	—
309	C₂²	—	40	1	no	—
310	C₂²	—	40	1	no	—
311	C₂²	C₂²×#686	1	1	yes	characteristic
312	C₄	—	2	2	no	—
313	C₄	—	2	2	no	—
314	C₄	—	2	1	no	—
315	C₄	—	4	2	no	—
316	C₄	—	8	2	no	—
317	C₄	—	8	1	no	—
318	C₄	—	40	1	no	—
319	C₄	—	40	2	no	—
320	C₄	#65483	1	2	yes	—
321	C₄	#66151	1	1	yes	characteristic
322	C₄	#66171	1	1	yes	characteristic
323	C₂	—	80	1	no	—
324	C₂	#10615293	1	1	yes	characteristic
325	C₂	C₂×#65483	1	1	yes	characteristic
326	C₂	#10615748	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×#686
commutator, cosocle	C₂×C₅×C₈	C₂⁴
Frattini	C₂×C₈	C₂³×D₅
Fitting	C₅×#3383	C₂
socle	C₂²×C₅	C₂²×D₈
prime radical	C₂×D₄₀	C₂³
hypercenter	#3383	D₅
hypocenter	C₅	#33931
LCS	G ▷ C₂×C₅×C₈ ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ C₈×Q₈ ◁ #3383

Complements

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

N	H ≅ G/N	type
1. C₅×#3383	C₂	semidirect
4. #10602175	C₂	semidirect
5. #10602150	C₂	semidirect
7. Q₈×#672	C₂	semidirect
8. #10601709	C₂	semidirect
10. C₄×#3961	C₂	semidirect
81. #3383	D₅	semidirect
307. C₅	#33931	semidirect

Maximal subgroups

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

subgroup	index
C₅×#3383	2
#10611643	2
#10602180	2
#10602175	2
#10602150	2
#10602144	2
Q₈×#672	2
#10601709	2
#10601708	2
C₄×#3961	2
C₄×#3960	2
#33931	5