Subgroups — #31891233

Order 18176.

486 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22588500	C₂	1	6	yes	maximal
2	C₂×C₈²×C₇₁	C₂	1	1	yes	maximal, characteristic
3	C₂×#14556062	—	2	12	no	—
4	#17985869	C₂²	1	4	yes	—
5	C₂×C₄×C₈×C₇₁	C₂²	1	3	yes	—
6	C₈²×C₇₁	C₄	1	4	yes	complemented
7	C₂×C₄⋊Dic₇₁	—	2	12	no	—
8	#14556062	—	4	24	no	—
9	C₂²×C₈×C₇₁	D₄	1	6	yes	—
10	C₂×C₄²×C₇₁	C₂³	1	1	yes	characteristic
11	C₄×C₈×C₇₁	C₂×C₄	1	6	yes	—
12	C₄×C₈×C₇₁	Q₈	1	6	yes	—
13	C₂²×Dic₇₁	—	4	4	no	—
14	C₄⋊Dic₇₁	—	4	24	no	—
15	C₂²×C₄×C₇₁	C₂×D₄	1	3	yes	—
16	C₂×C₈×C₇₁	C₄⋊C₄	1	12	yes	—
17	C₂×C₈×C₇₁	D₈	1	12	yes	—
18	C₂×C₈×C₇₁	Q₁₆	1	12	yes	—
19	C₄²×C₇₁	C₂²×C₄	1	1	yes	characteristic
20	C₄²×C₇₁	C₂×Q₈	1	3	yes	—
21	C₂³×C₇₁	#127	1	1	yes	characteristic
22	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	3	yes	—
23	C₂×C₄×C₇₁	C₄⋊Q₈	1	3	yes	—
24	C₂×C₄×C₇₁	C₂×D₈	1	6	yes	—
25	C₂×C₄×C₇₁	C₂×Q₁₆	1	6	yes	—
26	C₂×Dic₇₁	—	8	12	no	—
27	C₈×C₇₁	#107	1	24	yes	complemented
28	C₂²×C₇₁	#384	1	1	yes	characteristic
29	C₂²×C₇₁	#493	1	3	yes	—
30	C₂²×C₇₁	#494	1	3	yes	—
31	C₄×C₇₁	C₂×#107	1	6	yes	—
32	C₄×C₇₁	#500	1	6	yes	—
33	Dic₇₁	—	16	4	no	—
34	#8818	—	71	1	no	maximal, Sylow
35	C₂×C₇₁	#1712	1	1	yes	characteristic
36	C₂×C₇₁	#1713	1	3	yes	—
37	C₂×C₇₁	#1845	1	3	yes	—
38	#1845	—	71	6	no	—
39	C₂×C₈²	D₇₁	1	1	yes	characteristic
40	C₇₁	#8818	1	1	yes	characteristic, complemented, Sylow
41	C₂×#107	—	142	12	no	—
42	C₂×C₄×C₈	C₂×D₇₁	1	3	yes	—
43	#384	—	71	4	no	—
44	C₈²	Dic₇₁	1	4	yes	complemented
45	C₂²×C₈	D₂₈₄	1	6	yes	—
46	C₂×C₄⋊C₄	—	142	12	no	—
47	C₂×C₄²	C₂²×D₇₁	1	1	yes	characteristic
48	#107	—	284	24	no	—
49	C₄×C₈	Dic₁₄₂	1	6	yes	—
50	C₄×C₈	C₂×Dic₇₁	1	6	yes	—
51	C₂²×C₄	—	284	4	no	—
52	C₂²×C₄	C₂×D₂₈₄	1	3	yes	—
53	C₂×C₈	C₄⋊Dic₇₁	1	12	yes	—
54	C₂×C₈	D₅₆₈	1	12	yes	—
55	C₂×C₈	Dic₂₈₄	1	12	yes	—
56	C₄⋊C₄	—	284	24	no	—
57	C₄²	C₂²×Dic₇₁	1	1	yes	characteristic
58	C₄²	C₂×Dic₁₄₂	1	3	yes	—
59	C₂³	#14556125	1	1	yes	characteristic
60	C₂×C₄	—	568	12	no	—
61	C₂×C₄	C₂×C₄⋊Dic₇₁	1	3	yes	—
62	C₂×C₄	#14556128	1	3	yes	—
63	C₂×C₄	C₂×D₅₆₈	1	6	yes	—
64	C₂×C₄	C₂×Dic₂₈₄	1	6	yes	—
65	C₈	#14556062	1	24	yes	complemented
66	C₂²	#17985869	1	1	yes	characteristic
67	C₂²	#17986376	1	3	yes	—
68	C₂²	#17986379	1	3	yes	—
69	C₄	—	1136	4	no	—
70	C₄	C₂×#14556062	1	6	yes	—
71	C₄	#17986404	1	6	yes	—
72	C₂	#22587955	1	1	yes	characteristic
73	C₂	#22587958	1	3	yes	—
74	C₂	#22588500	1	3	yes	—

	subgroup	G/H
center	C₂³	#14556125
commutator	C₄²×C₇₁	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₁
Fitting	C₂×C₈²×C₇₁	C₂
socle, prime radical	C₂³×C₇₁	#127
cosocle	C₂×C₄²×C₇₁	C₂³
hypercenter	C₂×C₈²	D₇₁
hypocenter	C₇₁	#8818
LCS	G ▷ C₄²×C₇₁ ▷ C₂²×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₈²

Complements

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

N	H ≅ G/N	type
6. C₈²×C₇₁	C₄	semidirect
27. C₈×C₇₁	#107	semidirect
40. C₇₁	#8818	semidirect
44. C₈²	Dic₇₁	semidirect
65. C₈	#14556062	semidirect

Maximal subgroups

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

subgroup	index
#22588500	2
C₂×C₈²×C₇₁	2
#8818	71