Subgroups — C₂×#22586550 #31918488

Order 18176.

134 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₂×D₇₁	C₂	1	1	yes	maximal, characteristic
2	#22586550	C₂	1	4	yes	maximal, direct product factor
3	C₂×C₇₁⋊C₆₄	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₆₄×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₁₆×D₇₁	C₄	1	1	yes	characteristic
6	C₂×C₇₁⋊C₃₂	C₄	1	1	yes	characteristic
7	C₃₂×D₇₁	C₄	1	2	yes	—
8	C₃₂×D₇₁	C₂²	1	2	yes	—
9	C₇₁⋊C₆₄	C₂²	1	2	yes	complemented
10	C₂×C₃₂×C₇₁	C₂²	1	1	yes	characteristic
11	C₆₄×C₇₁	C₂²	1	2	yes	complemented
12	C₂×C₈×D₇₁	C₈	1	1	yes	characteristic
13	C₂×C₇₁⋊C₁₆	C₈	1	1	yes	characteristic
14	C₁₆×D₇₁	C₈	1	2	yes	—
15	C₁₆×D₇₁	C₂×C₄	1	2	yes	—
16	C₇₁⋊C₃₂	C₂×C₄	1	2	yes	—
17	C₂×C₁₆×C₇₁	C₂×C₄	1	1	yes	characteristic
18	C₃₂×C₇₁	C₂×C₄	1	1	yes	characteristic
19	C₃₂×C₇₁	C₂³	1	1	yes	characteristic
20	C₂×C₄×D₇₁	C₁₆	1	1	yes	characteristic
21	C₂×C₇₁⋊C₈	C₁₆	1	1	yes	characteristic
22	C₈×D₇₁	C₁₆	1	2	yes	—
23	C₈×D₇₁	C₂×C₈	1	2	yes	—
24	C₇₁⋊C₁₆	C₂×C₈	1	2	yes	—
25	C₂×C₈×C₇₁	C₂×C₈	1	1	yes	characteristic
26	C₁₆×C₇₁	C₂×C₈	1	1	yes	characteristic
27	C₁₆×C₇₁	C₂²×C₄	1	1	yes	characteristic
28	C₂²×D₇₁	C₃₂	1	1	yes	characteristic
29	C₂×C₄×C₇₁	C₂×C₁₆	1	1	yes	characteristic
30	C₂×Dic₇₁	C₃₂	1	1	yes	characteristic
31	C₄×D₇₁	C₃₂	1	2	yes	—
32	C₄×D₇₁	C₂×C₁₆	1	2	yes	—
33	C₈×C₇₁	C₂×C₁₆	1	1	yes	characteristic
34	C₈×C₇₁	C₂²×C₈	1	1	yes	characteristic
35	C₇₁⋊C₈	C₂×C₁₆	1	2	yes	—
36	C₂²×C₇₁	C₂×C₃₂	1	1	yes	characteristic
37	C₂×D₇₁	—	2	2	no	—
38	C₂×D₇₁	C₂×C₃₂	1	2	yes	—
39	C₄×C₇₁	C₂×C₃₂	1	1	yes	characteristic
40	C₄×C₇₁	C₂²×C₁₆	1	1	yes	characteristic
41	Dic₇₁	C₂×C₃₂	1	2	yes	—
42	C₂×M₁₂₈	—	71	1	no	maximal, Sylow
43	C₂×C₇₁	M₁₂₈	1	2	yes	complemented
44	C₂×C₇₁	C₂²×C₃₂	1	1	yes	characteristic
45	D₇₁	—	2	2	no	—
46	C₂²×C₃₂	—	71	1	no	—
47	M₁₂₈	—	71	4	no	—
48	C₂×C₆₄	—	71	1	no	—
49	C₂×C₆₄	D₇₁	1	1	yes	characteristic, complemented
50	C₇₁	C₂×M₁₂₈	1	1	yes	characteristic, complemented, Sylow
51	C₂²×C₁₆	—	71	1	no	—
52	C₂×C₃₂	—	71	1	no	—
53	C₂×C₃₂	—	71	2	no	—
54	C₂×C₃₂	—	71	2	no	—
55	C₂×C₃₂	C₂×D₇₁	1	1	yes	characteristic
56	C₆₄	—	71	2	no	—
57	C₆₄	C₂×D₇₁	1	2	yes	complemented
58	C₂²×C₈	—	71	1	no	—
59	C₂×C₁₆	—	71	1	no	—
60	C₂×C₁₆	—	71	2	no	—
61	C₂×C₁₆	—	71	2	no	—
62	C₂×C₁₆	C₄×D₇₁	1	1	yes	characteristic
63	C₃₂	—	71	2	no	—
64	C₃₂	C₄×D₇₁	1	1	yes	characteristic
65	C₃₂	C₂²×D₇₁	1	1	yes	characteristic
66	C₂²×C₄	—	71	1	no	—
67	C₂×C₈	—	71	1	no	—
68	C₂×C₈	—	71	2	no	—
69	C₂×C₈	—	71	2	no	—
70	C₂×C₈	C₈×D₇₁	1	1	yes	characteristic
71	C₁₆	—	71	2	no	—
72	C₁₆	C₈×D₇₁	1	1	yes	characteristic
73	C₁₆	C₂×C₄×D₇₁	1	1	yes	characteristic
74	C₂³	—	71	1	no	—
75	C₂×C₄	—	71	1	no	—
76	C₂×C₄	—	71	2	no	—
77	C₂×C₄	—	71	2	no	—
78	C₂×C₄	C₁₆×D₇₁	1	1	yes	characteristic
79	C₈	—	71	2	no	—
80	C₈	C₁₆×D₇₁	1	1	yes	characteristic
81	C₈	C₂×C₈×D₇₁	1	1	yes	characteristic
82	C₂²	—	71	2	no	—
83	C₂²	—	142	2	no	—
84	C₂²	C₃₂×D₇₁	1	1	yes	characteristic
85	C₄	—	71	2	no	—
86	C₄	C₃₂×D₇₁	1	1	yes	characteristic
87	C₄	C₂×C₁₆×D₇₁	1	1	yes	characteristic
88	C₂	—	142	2	no	—
89	C₂	#22586550	1	2	yes	direct product factor
90	C₂	C₂×C₃₂×D₇₁	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃₂	C₂×D₇₁
commutator	C₂×C₇₁	C₂²×C₃₂
Frattini	C₃₂	C₂²×D₇₁
Fitting	C₂×C₆₄×C₇₁	C₂
socle	C₂²×C₇₁	C₂×C₃₂
prime radical	C₂²×D₇₁	C₃₂
cosocle	C₃₂×C₇₁	C₂³
hypercenter	C₂×C₆₄	D₇₁
hypocenter	C₇₁	C₂×M₁₂₈
LCS	G ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₃₂ ◁ C₂×C₆₄

Complements

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

N	H ≅ G/N	type
2. #22586550	C₂	direct (N × H)
3. C₂×C₇₁⋊C₆₄	C₂	semidirect
4. C₂×C₆₄×C₇₁	C₂	semidirect
9. C₇₁⋊C₆₄	C₂²	semidirect
11. C₆₄×C₇₁	C₂²	semidirect
43. C₂×C₇₁	M₁₂₈	semidirect
49. C₂×C₆₄	D₇₁	semidirect
50. C₇₁	C₂×M₁₂₈	semidirect
57. C₆₄	C₂×D₇₁	semidirect
89. C₂	#22586550	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃₂×D₇₁	2
#22586550	2
C₂×C₇₁⋊C₆₄	2
C₂×C₆₄×C₇₁	2
C₂×M₁₂₈	71

Subgroups — C2×#22586550 #31918488

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22586550 #31918488