Subgroups — #31884192

Order 18176.

102 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22586410	C₂	1	2	yes	maximal
2	C₈×C₁₆×C₇₁	C₂	1	1	yes	maximal, characteristic
3	C₄⋊(C₇₁⋊C₁₆)	—	2	2	no	—
4	C₄×C₁₆×C₇₁	C₄	1	1	yes	characteristic
5	C₄×C₁₆×C₇₁	C₄	1	1	yes	characteristic
6	C₈²×C₇₁	C₂²	1	1	yes	characteristic
7	C₂×C₇₁⋊C₁₆	—	4	2	no	—
8	C₂×C₁₆×C₇₁	—	2	1	no	—
9	C₂×C₁₆×C₇₁	C₈	1	1	yes	characteristic
10	C₂×C₁₆×C₇₁	C₈	1	1	yes	characteristic
11	C₄×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
12	C₄×C₈×C₇₁	D₄	1	1	yes	characteristic
13	C₄×C₈×C₇₁	Q₈	1	1	yes	characteristic
14	C₇₁⋊C₁₆	—	8	2	no	—
15	C₂×C₈×C₇₁	—	2	1	no	—
16	C₂×C₈×C₇₁	C₂×C₈	1	1	yes	characteristic
17	C₂×C₈×C₇₁	M₁₆	1	1	yes	characteristic
18	C₂×C₈×C₇₁	D₈	1	1	yes	characteristic
19	C₂×C₈×C₇₁	Q₁₆	1	1	yes	characteristic
20	C₄²×C₇₁	C₄⋊C₄	1	1	yes	characteristic
21	C₁₆×C₇₁	—	2	2	no	—
22	C₁₆×C₇₁	—	2	1	no	—
23	C₁₆×C₇₁	C₁₆	1	2	yes	complemented
24	C₂×C₄×C₇₁	C₄⋊C₈	1	1	yes	characteristic
25	C₂×C₄×C₇₁	#107	1	1	yes	characteristic
26	C₂×C₄×C₇₁	#108	1	1	yes	characteristic
27	C₈×C₇₁	—	2	2	no	—
28	C₈×C₇₁	—	2	1	no	—
29	C₈×C₇₁	—	2	1	no	—
30	C₈×C₇₁	C₂×C₁₆	1	1	yes	characteristic
31	C₈×C₇₁	M₃₂	1	1	yes	characteristic
32	C₈×C₇₁	D₁₆	1	1	yes	characteristic
33	C₈×C₇₁	Q₃₂	1	1	yes	characteristic
34	C₂²×C₇₁	#335	1	1	yes	characteristic
35	C₄×C₇₁	—	2	1	no	—
36	C₄×C₇₁	C₄⋊C₁₆	1	1	yes	characteristic
37	C₄×C₇₁	#364	1	1	yes	characteristic
38	C₄×C₇₁	#366	1	1	yes	characteristic
39	C₄×C₇₁	#368	1	1	yes	characteristic
40	#7100	—	71	1	no	maximal, Sylow
41	C₂×C₇₁	#1370	1	1	yes	characteristic
42	C₂×C₇₁	#1371	1	1	yes	characteristic
43	C₂×C₇₁	#1373	1	1	yes	characteristic
44	#1370	—	71	2	no	—
45	C₈×C₁₆	D₇₁	1	1	yes	characteristic
46	C₇₁	#7100	1	1	yes	characteristic, complemented, Sylow
47	C₄⋊C₁₆	—	142	2	no	—
48	C₄×C₁₆	Dic₇₁	1	1	yes	characteristic
49	C₄×C₁₆	Dic₇₁	1	1	yes	characteristic
50	C₈²	C₂×D₇₁	1	1	yes	characteristic
51	C₂×C₁₆	—	2	1	no	—
52	C₂×C₁₆	—	284	2	no	—
53	C₂×C₁₆	C₇₁⋊C₈	1	1	yes	characteristic
54	C₂×C₁₆	C₇₁⋊C₈	1	1	yes	characteristic
55	C₄×C₈	Dic₁₄₂	1	1	yes	characteristic
56	C₄×C₈	D₂₈₄	1	1	yes	characteristic
57	C₄×C₈	C₂×Dic₇₁	1	1	yes	characteristic
58	C₂×C₈	—	2	1	no	—
59	C₂×C₈	C₂×C₇₁⋊C₈	1	1	yes	characteristic
60	C₂×C₈	#11765849	1	1	yes	characteristic
61	C₂×C₈	D₅₆₈	1	1	yes	characteristic
62	C₂×C₈	Dic₂₈₄	1	1	yes	characteristic
63	C₄²	C₄⋊Dic₇₁	1	1	yes	characteristic
64	C₁₆	—	2	2	no	—
65	C₁₆	—	2	1	no	—
66	C₁₆	—	568	2	no	—
67	C₁₆	C₇₁⋊C₁₆	1	2	yes	complemented
68	C₂×C₄	C₄⋊(C₇₁⋊C₈)	1	1	yes	characteristic
69	C₂×C₄	#14556062	1	1	yes	characteristic
70	C₂×C₄	#14556064	1	1	yes	characteristic
71	C₈	—	2	2	no	—
72	C₈	—	2	1	no	—
73	C₈	—	2	1	no	—
74	C₈	C₂×C₇₁⋊C₁₆	1	1	yes	characteristic
75	C₈	#14556068	1	1	yes	characteristic
76	C₈	D₁₁₃₆	1	1	yes	characteristic
77	C₈	Dic₅₆₈	1	1	yes	characteristic
78	C₂²	#17985752	1	1	yes	characteristic
79	C₄	—	2	1	no	—
80	C₄	C₄⋊(C₇₁⋊C₁₆)	1	1	yes	characteristic
81	C₄	#17985820	1	1	yes	characteristic
82	C₄	#17985825	1	1	yes	characteristic
83	C₄	#17985829	1	1	yes	characteristic
84	C₂	#22586410	1	1	yes	characteristic
85	C₂	#22586412	1	1	yes	characteristic
86	C₂	#22586416	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	D₅₆₈
commutator	C₈×C₇₁	C₂×C₁₆
Frattini	C₈²	C₂×D₇₁
Fitting	C₈×C₁₆×C₇₁	C₂
socle, prime radical	C₂²×C₇₁	#335
cosocle	C₈²×C₇₁	C₂²
hypercenter	C₈×C₁₆	D₇₁
hypocenter	C₇₁	#7100
LCS	G ▷ C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₈ ◁ C₄×C₈ ◁ C₈² ◁ C₈×C₁₆

Complements

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

N	H ≅ G/N	type
23. C₁₆×C₇₁	C₁₆	semidirect
46. C₇₁	#7100	semidirect
67. C₁₆	C₇₁⋊C₁₆	semidirect

Maximal subgroups

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

subgroup	index
#22586410	2
C₈×C₁₆×C₇₁	2
#7100	71