Subgroups — #31912189

Order 18176.

262 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22590615	C₂	1	4	yes	maximal
2	#22587958	C₂	1	2	yes	maximal
3	C₈×C₁₆×C₇₁	C₂	1	1	yes	maximal, characteristic
4	#17986404	—	2	2	no	—
5	#17986404	C₂²	1	4	yes	—
6	#17986379	—	2	2	no	—
7	#17985825	—	2	8	no	—
8	C₄×C₁₆×C₇₁	C₂²	1	2	yes	—
9	C₈²×C₇₁	C₂²	1	1	yes	characteristic
10	C₂×Dic₂₈₄	—	4	4	no	—
11	#14556128	—	2	4	no	—
12	#14556062	—	2	8	no	—
13	#14556062	—	4	4	no	—
14	C₂×C₁₆×C₇₁	D₄	1	4	yes	—
15	C₄×C₈×C₇₁	D₄	1	1	yes	characteristic
16	C₄×C₈×C₇₁	D₄	1	1	yes	characteristic
17	C₄×C₈×C₇₁	C₂³	1	1	yes	characteristic
18	C₂×Dic₁₄₂	—	4	4	no	—
19	Dic₂₈₄	—	8	8	no	—
20	C₄⋊Dic₇₁	—	4	4	no	—
21	C₄⋊Dic₇₁	—	4	4	no	—
22	C₂×C₈×C₇₁	D₈	1	2	yes	—
23	C₂×C₈×C₇₁	D₈	1	2	yes	—
24	C₂×C₈×C₇₁	C₂×D₄	1	2	yes	—
25	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
26	C₁₆×C₇₁	Q₁₆	1	8	yes	complemented
27	C₂×C₄×C₇₁	#127	1	1	yes	characteristic
28	C₂×C₄×C₇₁	C₂×D₈	1	1	yes	characteristic
29	C₂×C₄×C₇₁	C₂×D₈	1	1	yes	characteristic
30	C₂×Dic₇₁	—	8	4	no	—
31	Dic₁₄₂	—	8	8	no	—
32	C₈×C₇₁	D₁₆	1	4	yes	—
33	C₈×C₇₁	Q₃₂	1	4	yes	—
34	C₈×C₇₁	C₂×Q₁₆	1	4	yes	—
35	C₂²×C₇₁	#493	1	1	yes	characteristic
36	C₄×C₇₁	#494	1	2	yes	—
37	C₄×C₇₁	C₂×D₁₆	1	2	yes	—
38	C₄×C₇₁	C₂×Q₃₂	1	2	yes	—
39	Dic₇₁	—	16	4	no	—
40	#12521	—	71	1	no	maximal, Sylow
41	C₂×C₇₁	#1713	1	1	yes	characteristic
42	C₂×C₇₁	#2246	1	1	yes	characteristic
43	C₂×C₇₁	#2247	1	1	yes	characteristic
44	#2252	—	71	4	no	—
45	#1713	—	71	2	no	—
46	C₈×C₁₆	D₇₁	1	1	yes	characteristic
47	C₇₁	#12521	1	1	yes	characteristic, complemented, Sylow
48	#500	—	71	4	no	—
49	#500	—	142	2	no	—
50	#494	—	142	2	no	—
51	#366	—	142	8	no	—
52	C₄×C₁₆	C₂×D₇₁	1	2	yes	—
53	C₈²	C₂×D₇₁	1	1	yes	characteristic
54	C₂×Q₁₆	—	284	4	no	—
55	C₄⋊Q₈	—	142	4	no	—
56	C₂×C₁₆	D₂₈₄	1	4	yes	—
57	#107	—	284	4	no	—
58	#107	—	142	8	no	—
59	C₄×C₈	D₂₈₄	1	1	yes	characteristic
60	C₄×C₈	D₂₈₄	1	1	yes	characteristic
61	C₄×C₈	C₂²×D₇₁	1	1	yes	characteristic
62	C₂×Q₈	—	284	4	no	—
63	Q₁₆	—	568	8	no	—
64	C₂×C₈	D₅₆₈	1	2	yes	—
65	C₂×C₈	D₅₆₈	1	2	yes	—
66	C₂×C₈	C₂×D₂₈₄	1	2	yes	—
67	C₄⋊C₄	—	284	4	no	—
68	C₄⋊C₄	—	284	4	no	—
69	C₄²	C₂×D₂₈₄	1	1	yes	characteristic
70	C₁₆	Dic₂₈₄	1	8	yes	complemented
71	Q₈	—	568	8	no	—
72	C₂×C₄	—	568	4	no	—
73	C₂×C₄	#14556125	1	1	yes	characteristic
74	C₂×C₄	C₂×D₅₆₈	1	1	yes	characteristic
75	C₂×C₄	C₂×D₅₆₈	1	1	yes	characteristic
76	C₈	D₁₁₃₆	1	4	yes	—
77	C₈	Dic₅₆₈	1	4	yes	—
78	C₈	C₂×Dic₂₈₄	1	4	yes	—
79	C₂²	#17986376	1	1	yes	characteristic
80	C₄	—	1136	4	no	—
81	C₄	#17986379	1	2	yes	—
82	C₄	C₂×D₁₁₃₆	1	2	yes	—
83	C₄	C₂×Dic₅₆₈	1	2	yes	—
84	C₂	#22587958	1	1	yes	characteristic
85	C₂	#22590590	1	1	yes	characteristic
86	C₂	#22590593	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#17986376
commutator, cosocle	C₄×C₈×C₇₁	C₂³
Frattini	C₄×C₈	C₂²×D₇₁
Fitting	C₈×C₁₆×C₇₁	C₂
socle, prime radical	C₂²×C₇₁	#493
hypercenter	C₈×C₁₆	D₇₁
hypocenter	C₇₁	#12521
LCS	G ▷ C₄×C₈×C₇₁ ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ C₈×C₁₆

Complements

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

N	H ≅ G/N	type
26. C₁₆×C₇₁	Q₁₆	semidirect
47. C₇₁	#12521	semidirect
70. C₁₆	Dic₂₈₄	semidirect

Maximal subgroups

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

subgroup	index
#22590615	2
#22587958	2
C₈×C₁₆×C₇₁	2
#12521	71