Subgroups — C₂×#22586551 #31918489

Order 18176.

134 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22586551	C₂	1	4	yes	maximal, direct product factor
2	C₂×C₇₁⋊C₆₄	C₂	1	2	yes	maximal, complemented
3	C₂²×C₃₂×C₇₁	C₂	1	1	yes	maximal, characteristic
4	C₇₁⋊C₆₄	C₂²	1	4	yes	complemented
5	C₂²×C₁₆×C₇₁	C₄	1	1	yes	characteristic
6	C₂×C₃₂×C₇₁	C₄	1	1	yes	characteristic
7	C₂×C₃₂×C₇₁	C₄	1	2	yes	—
8	C₂×C₃₂×C₇₁	C₂²	1	2	yes	—
9	C₂×C₃₂×C₇₁	C₂²	1	1	yes	characteristic
10	C₂²×C₈×C₇₁	C₈	1	1	yes	characteristic
11	C₂×C₁₆×C₇₁	C₈	1	1	yes	characteristic
12	C₂×C₁₆×C₇₁	C₈	1	2	yes	—
13	C₂×C₁₆×C₇₁	C₂×C₄	1	2	yes	—
14	C₂×C₁₆×C₇₁	C₂×C₄	1	1	yes	characteristic
15	C₃₂×C₇₁	C₂×C₄	1	2	yes	—
16	C₃₂×C₇₁	C₂×C₄	1	1	yes	characteristic
17	C₃₂×C₇₁	C₂³	1	1	yes	characteristic
18	C₂²×C₄×C₇₁	C₁₆	1	1	yes	characteristic
19	C₂×C₈×C₇₁	C₁₆	1	2	yes	—
20	C₂×C₈×C₇₁	C₁₆	1	1	yes	characteristic
21	C₂×C₈×C₇₁	C₂×C₈	1	2	yes	—
22	C₂×C₈×C₇₁	C₂×C₈	1	1	yes	characteristic
23	C₁₆×C₇₁	C₂×C₈	1	2	yes	—
24	C₁₆×C₇₁	C₂×C₈	1	1	yes	characteristic
25	C₁₆×C₇₁	C₂²×C₄	1	1	yes	characteristic
26	C₂³×C₇₁	C₃₂	1	1	yes	characteristic
27	C₂×C₄×C₇₁	C₃₂	1	2	yes	—
28	C₂×C₄×C₇₁	C₃₂	1	1	yes	characteristic
29	C₂×C₄×C₇₁	C₂×C₁₆	1	2	yes	—
30	C₂×C₄×C₇₁	C₂×C₁₆	1	1	yes	characteristic
31	C₈×C₇₁	C₂×C₁₆	1	2	yes	—
32	C₈×C₇₁	C₂×C₁₆	1	1	yes	characteristic
33	C₈×C₇₁	C₂²×C₈	1	1	yes	characteristic
34	C₂²×C₇₁	—	2	2	no	—
35	C₂²×C₇₁	C₂×C₃₂	1	2	yes	—
36	C₂²×C₇₁	C₂×C₃₂	1	1	yes	characteristic
37	C₄×C₇₁	C₂×C₃₂	1	1	yes	characteristic
38	C₄×C₇₁	C₂×C₃₂	1	2	yes	—
39	C₄×C₇₁	C₂²×C₁₆	1	1	yes	characteristic
40	C₂×M₁₂₈	—	71	1	no	maximal, Sylow
41	C₂×C₇₁	—	2	2	no	—
42	C₂×C₇₁	M₁₂₈	1	2	yes	complemented
43	C₂×C₇₁	C₂²×C₃₂	1	1	yes	characteristic
44	C₂²×C₃₂	D₇₁	1	1	yes	characteristic
45	M₁₂₈	—	71	4	no	—
46	C₂×C₆₄	—	71	2	no	—
47	C₇₁	C₂×M₁₂₈	1	1	yes	characteristic, complemented, Sylow
48	C₂²×C₁₆	Dic₇₁	1	1	yes	characteristic
49	C₂×C₃₂	Dic₇₁	1	1	yes	characteristic
50	C₂×C₃₂	Dic₇₁	1	2	yes	—
51	C₂×C₃₂	C₂×D₇₁	1	2	yes	—
52	C₂×C₃₂	C₂×D₇₁	1	1	yes	characteristic
53	C₆₄	—	71	4	no	—
54	C₂²×C₈	C₇₁⋊C₈	1	1	yes	characteristic
55	C₂×C₁₆	C₇₁⋊C₈	1	1	yes	characteristic
56	C₂×C₁₆	C₇₁⋊C₈	1	2	yes	—
57	C₂×C₁₆	C₂×Dic₇₁	1	2	yes	—
58	C₂×C₁₆	C₂×Dic₇₁	1	1	yes	characteristic
59	C₃₂	C₂×Dic₇₁	1	2	yes	—
60	C₃₂	C₂×Dic₇₁	1	1	yes	characteristic
61	C₃₂	C₂²×D₇₁	1	1	yes	characteristic
62	C₂²×C₄	C₇₁⋊C₁₆	1	1	yes	characteristic
63	C₂×C₈	C₇₁⋊C₁₆	1	2	yes	—
64	C₂×C₈	C₇₁⋊C₁₆	1	1	yes	characteristic
65	C₂×C₈	C₂×C₇₁⋊C₈	1	2	yes	—
66	C₂×C₈	C₂×C₇₁⋊C₈	1	1	yes	characteristic
67	C₁₆	C₂×C₇₁⋊C₈	1	2	yes	—
68	C₁₆	C₂×C₇₁⋊C₈	1	1	yes	characteristic
69	C₁₆	C₂²×Dic₇₁	1	1	yes	characteristic
70	C₂³	C₇₁⋊C₃₂	1	1	yes	characteristic
71	C₂×C₄	C₇₁⋊C₃₂	1	2	yes	—
72	C₂×C₄	C₇₁⋊C₃₂	1	1	yes	characteristic
73	C₂×C₄	C₂×C₇₁⋊C₁₆	1	2	yes	—
74	C₂×C₄	C₂×C₇₁⋊C₁₆	1	1	yes	characteristic
75	C₈	C₂×C₇₁⋊C₁₆	1	2	yes	—
76	C₈	C₂×C₇₁⋊C₁₆	1	1	yes	characteristic
77	C₈	C₂²×C₇₁⋊C₈	1	1	yes	characteristic
78	C₂²	—	2	2	no	—
79	C₂²	C₂×C₇₁⋊C₃₂	1	2	yes	—
80	C₂²	C₂×C₇₁⋊C₃₂	1	1	yes	characteristic
81	C₄	C₂×C₇₁⋊C₃₂	1	1	yes	characteristic
82	C₄	C₂×C₇₁⋊C₃₂	1	2	yes	—
83	C₄	C₂²×C₇₁⋊C₁₆	1	1	yes	characteristic
84	C₂	—	2	2	no	—
85	C₂	#22586551	1	2	yes	direct product factor
86	C₂	C₂²×C₇₁⋊C₃₂	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃₂	C₂×D₇₁
commutator	C₂×C₇₁	C₂²×C₃₂
Frattini	C₃₂	C₂²×D₇₁
Fitting	C₂²×C₃₂×C₇₁	C₂
socle	C₂²×C₇₁	C₂×C₃₂
prime radical	C₂³×C₇₁	C₃₂
cosocle	C₃₂×C₇₁	C₂³
hypercenter	C₂²×C₃₂	D₇₁
hypocenter	C₇₁	C₂×M₁₂₈
LCS	G ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₃₂ ◁ C₂²×C₃₂

Complements

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

N	H ≅ G/N	type
1. #22586551	C₂	direct (N × H)
2. C₂×C₇₁⋊C₆₄	C₂	semidirect
4. C₇₁⋊C₆₄	C₂²	semidirect
42. C₂×C₇₁	M₁₂₈	semidirect
47. C₇₁	C₂×M₁₂₈	semidirect
85. C₂	#22586551	direct (N × H)

Maximal subgroups

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

subgroup	index
#22586551	2
C₂×C₇₁⋊C₆₄	2
C₂²×C₃₂×C₇₁	2
C₂×M₁₂₈	71

Subgroups — C2×#22586551 #31918489

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22586551 #31918489