Group #31994772 — subgroups

Subgroups — C₂×C₄²×C₇₁⋊C₈ #31994772

Order 18176.

2046 conjugacy classes of subgroups; only the 1659 normal subgroups are shown.

normal subgroups only

This group has 2046 conjugacy classes of subgroups — too many to draw. Only its 1659 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×C₇₁⋊C₈	C₂	1	6	yes	maximal
2	C₄²×C₇₁⋊C₈	C₂	1	8	yes	maximal, direct product factor
3	C₂×C₄³×C₇₁	C₂	1	1	yes	maximal, characteristic
4	C₂³×C₇₁⋊C₈	C₂²	1	4	yes	—
5	C₂×C₄×C₇₁⋊C₈	C₄	1	48	yes	direct product factor
6	C₂×C₄×C₇₁⋊C₈	C₂²	1	24	yes	—
7	C₂²×C₄²×C₇₁	C₄	1	4	yes	—
8	C₂²×C₄²×C₇₁	C₂²	1	3	yes	—
9	C₄³×C₇₁	C₄	1	4	yes	—
10	C₄³×C₇₁	C₂²	1	4	yes	—
11	C₂²×C₇₁⋊C₈	C₂×C₄	1	48	yes	—
12	C₂²×C₇₁⋊C₈	C₂³	1	8	yes	—
13	C₄×C₇₁⋊C₈	C₂×C₄	1	96	yes	direct product factor
14	C₂³×C₄×C₇₁	C₂×C₄	1	6	yes	—
15	C₂³×C₄×C₇₁	C₂³	1	1	yes	characteristic
16	C₂×C₄²×C₇₁	C₈	1	32	yes	complemented
17	C₂×C₄²×C₇₁	C₂×C₄	1	6	yes	—
18	C₂×C₄²×C₇₁	C₂×C₄	1	24	yes	—
19	C₂×C₄²×C₇₁	C₂×C₄	1	16	yes	—
20	C₂×C₄²×C₇₁	C₂³	1	6	yes	—
21	C₂×C₇₁⋊C₈	C₄²	1	64	yes	direct product factor
22	C₂×C₇₁⋊C₈	C₂²×C₄	1	48	yes	—
23	C₂⁴×C₇₁	C₂²×C₄	1	1	yes	characteristic
24	C₂²×C₄×C₇₁	C₄²	1	24	yes	—
25	C₂²×C₄×C₇₁	C₂×C₈	1	48	yes	—
26	C₂²×C₄×C₇₁	C₂²×C₄	1	12	yes	—
27	C₂²×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
28	C₂²×C₄×C₇₁	C₂²×C₄	1	12	yes	—
29	C₂²×C₄×C₇₁	C₂⁴	1	1	yes	characteristic
30	C₄²×C₇₁	C₄²	1	24	yes	—
31	C₄²×C₇₁	C₂×C₈	1	64	yes	complemented
32	C₄²×C₇₁	C₂²×C₄	1	24	yes	—
33	C₂³×C₇₁	C₂×C₄²	1	6	yes	—
34	C₂³×C₇₁	C₂²×C₈	1	8	yes	—
35	C₂³×C₇₁	C₂³×C₄	1	1	yes	characteristic
36	C₂×C₄×C₇₁	C₄×C₈	1	96	yes	complemented
37	C₂×C₄×C₇₁	C₂×C₄²	1	16	yes	—
38	C₂×C₄×C₇₁	C₂×C₄²	1	6	yes	—
39	C₂×C₄×C₇₁	C₂×C₄²	1	24	yes	—
40	C₂×C₄×C₇₁	C₂²×C₈	1	48	yes	—
41	C₂×C₄×C₇₁	C₂³×C₄	1	6	yes	—
42	C₇₁⋊C₈	C₂×C₄²	1	32	yes	direct product factor
43	C₂²×C₇₁	C₄³	1	4	yes	—
44	C₂²×C₇₁	C₂×C₄×C₈	1	24	yes	—
45	C₂²×C₇₁	C₂²×C₄²	1	3	yes	—
46	C₂²×C₇₁	C₂³×C₈	1	4	yes	—
47	C₄×C₇₁	C₄³	1	4	yes	—
48	C₄×C₇₁	C₂×C₄×C₈	1	48	yes	complemented
49	C₄×C₇₁	C₂²×C₄²	1	4	yes	—
50	C₂×C₄²×C₈	—	71	1	no	maximal, Sylow
51	C₂×C₇₁	C₄²×C₈	1	8	yes	complemented
52	C₂×C₇₁	C₂×C₄³	1	1	yes	characteristic
53	C₂×C₇₁	C₂²×C₄×C₈	1	6	yes	—
54	C₂²×C₄×C₈	—	71	6	no	—
55	C₂×C₄³	D₇₁	1	1	yes	characteristic
56	C₄²×C₈	—	71	8	no	—
57	C₇₁	C₂×C₄²×C₈	1	1	yes	characteristic, complemented, Sylow
58	C₂³×C₈	—	71	4	no	—
59	C₂²×C₄²	Dic₇₁	1	4	yes	—
60	C₂²×C₄²	C₂×D₇₁	1	3	yes	—
61	C₂×C₄×C₈	—	71	48	no	—
62	C₂×C₄×C₈	—	71	24	no	—
63	C₄³	Dic₇₁	1	4	yes	—
64	C₄³	C₂×D₇₁	1	4	yes	—
65	C₂³×C₄	C₂×Dic₇₁	1	6	yes	—
66	C₂³×C₄	C₂²×D₇₁	1	1	yes	characteristic
67	C₂²×C₈	—	71	48	no	—
68	C₂²×C₈	—	71	8	no	—
69	C₂×C₄²	C₇₁⋊C₈	1	32	yes	direct product factor
70	C₂×C₄²	C₄×D₇₁	1	24	yes	—
71	C₂×C₄²	C₂×Dic₇₁	1	6	yes	—
72	C₂×C₄²	C₂×Dic₇₁	1	16	yes	—
73	C₂×C₄²	C₂²×D₇₁	1	6	yes	—
74	C₄×C₈	—	71	96	no	—
75	C₂⁴	C₂²×Dic₇₁	1	1	yes	characteristic
76	C₂²×C₄	C₄×Dic₇₁	1	24	yes	—
77	C₂²×C₄	C₂×C₇₁⋊C₈	1	48	yes	—
78	C₂²×C₄	C₂×C₄×D₇₁	1	12	yes	—
79	C₂²×C₄	C₂²×Dic₇₁	1	1	yes	characteristic
80	C₂²×C₄	C₂²×Dic₇₁	1	12	yes	—
81	C₂²×C₄	C₂³×D₇₁	1	1	yes	characteristic
82	C₂×C₈	—	71	48	no	—
83	C₂×C₈	—	71	64	no	—
84	C₄²	C₄×Dic₇₁	1	24	yes	—
85	C₄²	C₂×C₇₁⋊C₈	1	64	yes	direct product factor
86	C₄²	C₂×C₄×D₇₁	1	24	yes	—
87	C₂³	C₂×C₄×Dic₇₁	1	6	yes	—
88	C₂³	C₂²×C₇₁⋊C₈	1	8	yes	—
89	C₂³	C₂³×Dic₇₁	1	1	yes	characteristic
90	C₂×C₄	C₄×C₇₁⋊C₈	1	96	yes	direct product factor
91	C₂×C₄	C₄²×D₇₁	1	16	yes	—
92	C₂×C₄	C₂×C₄×Dic₇₁	1	6	yes	—
93	C₂×C₄	C₂×C₄×Dic₇₁	1	24	yes	—
94	C₂×C₄	C₂²×C₇₁⋊C₈	1	48	yes	—
95	C₂×C₄	C₂²×C₄×D₇₁	1	6	yes	—
96	C₈	—	71	32	no	—
97	C₂²	C₄²×Dic₇₁	1	4	yes	—
98	C₂²	C₂×C₄×C₇₁⋊C₈	1	24	yes	—
99	C₂²	C₂²×C₄×Dic₇₁	1	3	yes	—
100	C₂²	C₂³×C₇₁⋊C₈	1	4	yes	—
101	C₄	C₄²×Dic₇₁	1	4	yes	—
102	C₄	C₂×C₄×C₇₁⋊C₈	1	48	yes	direct product factor
103	C₄	C₂×C₄²×D₇₁	1	4	yes	—
104	C₂	C₄²×C₇₁⋊C₈	1	8	yes	direct product factor
105	C₂	C₂×C₄²×Dic₇₁	1	1	yes	characteristic
106	C₂	C₂²×C₄×C₇₁⋊C₈	1	6	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×C₄×C₇₁⋊C₈

C₂

yes

maximal

C₄²×C₇₁⋊C₈

C₂

yes

maximal, direct product factor

C₂×C₄³×C₇₁

C₂

yes

maximal, characteristic

C₂³×C₇₁⋊C₈

C₂²

yes

—

C₂×C₄×C₇₁⋊C₈

C₄

yes

direct product factor

C₂×C₄×C₇₁⋊C₈

C₂²

yes

—

C₂²×C₄²×C₇₁

C₄

yes

—

C₂²×C₄²×C₇₁

C₂²

yes

—

C₄³×C₇₁

C₄

yes

—

C₄³×C₇₁

C₂²

yes

—

C₂²×C₇₁⋊C₈

C₂×C₄

yes

—

C₂²×C₇₁⋊C₈

C₂³

yes

—

C₄×C₇₁⋊C₈

C₂×C₄

yes

direct product factor

C₂³×C₄×C₇₁

C₂×C₄

yes

—

C₂³×C₄×C₇₁

C₂³

yes

characteristic

C₂×C₄²×C₇₁

C₈

yes

complemented

C₂×C₄²×C₇₁

C₂×C₄

yes

—

C₂×C₄²×C₇₁

C₂×C₄

yes

—

C₂×C₄²×C₇₁

C₂×C₄

yes

—

C₂×C₄²×C₇₁

C₂³

yes

—

C₂×C₇₁⋊C₈

C₄²

yes

direct product factor

C₂×C₇₁⋊C₈

C₂²×C₄

yes

—

C₂⁴×C₇₁

C₂²×C₄

yes

characteristic

C₂²×C₄×C₇₁

C₄²

yes

—

C₂²×C₄×C₇₁

C₂×C₈

yes

—

C₂²×C₄×C₇₁

C₂²×C₄

yes

—

C₂²×C₄×C₇₁

C₂²×C₄

yes

characteristic

C₂²×C₄×C₇₁

C₂²×C₄

yes

—

C₂²×C₄×C₇₁

C₂⁴

yes

characteristic

C₄²×C₇₁

C₄²

yes

—

C₄²×C₇₁

C₂×C₈

yes

complemented

C₄²×C₇₁

C₂²×C₄

yes

—

C₂³×C₇₁

C₂×C₄²

yes

—

C₂³×C₇₁

C₂²×C₈

yes

—

C₂³×C₇₁

C₂³×C₄

yes

characteristic

C₂×C₄×C₇₁

C₄×C₈

yes

complemented

C₂×C₄×C₇₁

C₂×C₄²

yes

—

C₂×C₄×C₇₁

C₂×C₄²

yes

—

C₂×C₄×C₇₁

C₂×C₄²

yes

—

C₂×C₄×C₇₁

C₂²×C₈

yes

—

C₂×C₄×C₇₁

C₂³×C₄

yes

—

C₇₁⋊C₈

C₂×C₄²

yes

direct product factor

C₂²×C₇₁

C₄³

yes

—

C₂²×C₇₁

C₂×C₄×C₈

yes

—

C₂²×C₇₁

C₂²×C₄²

yes

—

C₂²×C₇₁

C₂³×C₈

yes

—

C₄×C₇₁

C₄³

yes

—

C₄×C₇₁

C₂×C₄×C₈

yes

complemented

C₄×C₇₁

C₂²×C₄²

yes

—

C₂×C₄²×C₈

—

maximal, Sylow

C₂×C₇₁

C₄²×C₈

yes

complemented

C₂×C₇₁

C₂×C₄³

yes

characteristic

C₂×C₇₁

C₂²×C₄×C₈

yes

—

C₂²×C₄×C₈

—

C₂×C₄³

D₇₁

yes

characteristic

C₄²×C₈

—

C₇₁

C₂×C₄²×C₈

yes

characteristic, complemented, Sylow

C₂³×C₈

—

C₂²×C₄²

Dic₇₁

yes

—

C₂²×C₄²

C₂×D₇₁

yes

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₄³

Dic₇₁

yes

—

C₄³

C₂×D₇₁

yes

—

C₂³×C₄

C₂×Dic₇₁

yes

—

C₂³×C₄

C₂²×D₇₁

yes

characteristic

C₂²×C₈

—

C₂²×C₈

—

C₂×C₄²

C₇₁⋊C₈

yes

direct product factor

C₂×C₄²

C₄×D₇₁

yes

—

C₂×C₄²

C₂×Dic₇₁

yes

—

C₂×C₄²

C₂×Dic₇₁

yes

—

C₂×C₄²

C₂²×D₇₁

yes

—

C₄×C₈

—

C₂⁴

C₂²×Dic₇₁

yes

characteristic

C₂²×C₄

C₄×Dic₇₁

yes

—

C₂²×C₄

C₂×C₇₁⋊C₈

yes

—

C₂²×C₄

C₂×C₄×D₇₁

yes

—

C₂²×C₄

C₂²×Dic₇₁

yes

characteristic

C₂²×C₄

C₂²×Dic₇₁

yes

—

C₂²×C₄

C₂³×D₇₁

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₄²

C₄×Dic₇₁

yes

—

C₄²

C₂×C₇₁⋊C₈

yes

direct product factor

C₄²

C₂×C₄×D₇₁

yes

—

C₂³

C₂×C₄×Dic₇₁

yes

—

C₂³

C₂²×C₇₁⋊C₈

yes

—

C₂³

C₂³×Dic₇₁

yes

characteristic

C₂×C₄

C₄×C₇₁⋊C₈

yes

direct product factor

C₂×C₄

C₄²×D₇₁

yes

—

C₂×C₄

C₂×C₄×Dic₇₁

yes

—

C₂×C₄

C₂×C₄×Dic₇₁

yes

—

C₂×C₄

C₂²×C₇₁⋊C₈

yes

—

C₂×C₄

C₂²×C₄×D₇₁

yes

—

C₈

—

C₂²

C₄²×Dic₇₁

yes

—

C₂²

C₂×C₄×C₇₁⋊C₈

yes

—

C₂²

C₂²×C₄×Dic₇₁

yes

—

100

C₂²

C₂³×C₇₁⋊C₈

yes

—

101

C₄

C₄²×Dic₇₁

yes

—

102

C₄

C₂×C₄×C₇₁⋊C₈

yes

direct product factor

103

C₄

C₂×C₄²×D₇₁

yes

—

104

C₂

C₄²×C₇₁⋊C₈

yes

direct product factor

105

C₂

C₂×C₄²×Dic₇₁

yes

characteristic

106

C₂

C₂²×C₄×C₇₁⋊C₈

yes

—

	subgroup	G/H
center, hypercenter	C₂×C₄³	D₇₁
commutator, hypocenter	C₇₁	C₂×C₄²×C₈
Frattini	C₂²×C₄	C₂³×D₇₁
Fitting	C₂×C₄³×C₇₁	C₂
socle, prime radical	C₂⁴×C₇₁	C₂²×C₄
cosocle	C₂²×C₄×C₇₁	C₂⁴
LCS	G ▷ C₇₁
UCS	C₁ ◁ C₂×C₄³

subgroup

G/H

center, hypercenter

C₂×C₄³

D₇₁

commutator, hypocenter

C₇₁

C₂×C₄²×C₈

Frattini

C₂²×C₄

C₂³×D₇₁

Fitting

C₂×C₄³×C₇₁

C₂

socle, prime radical

C₂⁴×C₇₁

C₂²×C₄

cosocle

C₂²×C₄×C₇₁

C₂⁴

LCS

G ▷ C₇₁

UCS

C₁ ◁ C₂×C₄³

N	H ≅ G/N	type
2. C₄²×C₇₁⋊C₈	C₂	direct (N × H)
5. C₂×C₄×C₇₁⋊C₈	C₄	direct (N × H)
13. C₄×C₇₁⋊C₈	C₂×C₄	direct (N × H)
16. C₂×C₄²×C₇₁	C₈	semidirect
21. C₂×C₇₁⋊C₈	C₄²	direct (N × H)
31. C₄²×C₇₁	C₂×C₈	semidirect
36. C₂×C₄×C₇₁	C₄×C₈	semidirect
42. C₇₁⋊C₈	C₂×C₄²	direct (N × H)
48. C₄×C₇₁	C₂×C₄×C₈	semidirect
51. C₂×C₇₁	C₄²×C₈	semidirect
57. C₇₁	C₂×C₄²×C₈	semidirect
69. C₂×C₄²	C₇₁⋊C₈	direct (N × H)
85. C₄²	C₂×C₇₁⋊C₈	direct (N × H)
90. C₂×C₄	C₄×C₇₁⋊C₈	direct (N × H)
102. C₄	C₂×C₄×C₇₁⋊C₈	direct (N × H)
104. C₂	C₄²×C₇₁⋊C₈	direct (N × H)

H ≅ G/N

type

2. C₄²×C₇₁⋊C₈

C₂

direct (N × H)

5. C₂×C₄×C₇₁⋊C₈

C₄

direct (N × H)

13. C₄×C₇₁⋊C₈

C₂×C₄

direct (N × H)

16. C₂×C₄²×C₇₁

C₈

semidirect

21. C₂×C₇₁⋊C₈

C₄²

direct (N × H)

31. C₄²×C₇₁

C₂×C₈

semidirect

36. C₂×C₄×C₇₁

C₄×C₈

semidirect

42. C₇₁⋊C₈

C₂×C₄²

direct (N × H)

48. C₄×C₇₁

C₂×C₄×C₈

semidirect

51. C₂×C₇₁

C₄²×C₈

semidirect

57. C₇₁

C₂×C₄²×C₈

semidirect

69. C₂×C₄²

C₇₁⋊C₈

direct (N × H)

85. C₄²

C₂×C₇₁⋊C₈

direct (N × H)

90. C₂×C₄

C₄×C₇₁⋊C₈

direct (N × H)

102. C₄

C₂×C₄×C₇₁⋊C₈

direct (N × H)

104. C₂

C₄²×C₇₁⋊C₈

direct (N × H)

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

subgroup	index
C₂²×C₄×C₇₁⋊C₈	2
C₄²×C₇₁⋊C₈	2
C₂×C₄³×C₇₁	2
C₂×C₄²×C₈	71

subgroup

index

C₂²×C₄×C₇₁⋊C₈

C₄²×C₇₁⋊C₈

C₂×C₄³×C₇₁

C₂×C₄²×C₈

Subgroups — C₂×C₄²×C₇₁⋊C₈ #31994772

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×C42×C71⋊C8 #31994772

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄²×C₇₁⋊C₈ #31994772