Group #31963017 — subgroups

Subgroups — C₂×C₈²×D₇₁ #31963017

Order 18176.

1342 conjugacy classes of subgroups; only the 811 normal subgroups are shown.

normal subgroups only

This group has 1342 conjugacy classes of subgroups — too many to draw. Only its 811 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	C₂×C₄×C₈×D₇₁	C₂	1	3	yes	maximal
2	C₂×C₈×C₇₁⋊C₈	C₂	1	3	yes	maximal, complemented
3	C₈²×D₇₁	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₈²×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×C₈×D₇₁	C₄	1	6	yes	—
6	C₂×C₄²×D₇₁	C₂²	1	1	yes	characteristic
7	C₂×C₄×C₇₁⋊C₈	C₂²	1	3	yes	—
8	C₂×C₈×Dic₇₁	C₄	1	6	yes	—
9	C₄×C₈×D₇₁	C₄	1	12	yes	—
10	C₄×C₈×D₇₁	C₂²	1	12	yes	—
11	C₈×C₇₁⋊C₈	C₂²	1	12	yes	complemented
12	C₂×C₄×C₈×C₇₁	C₂²	1	3	yes	—
13	C₈²×C₇₁	C₂²	1	4	yes	complemented
14	C₂²×C₄×D₇₁	C₂×C₄	1	3	yes	—
15	C₂²×C₇₁⋊C₈	C₂×C₄	1	6	yes	—
16	C₂×C₈×D₇₁	C₈	1	48	yes	direct product factor
17	C₂×C₈×D₇₁	C₂×C₄	1	24	yes	—
18	C₂×C₄×Dic₇₁	C₂×C₄	1	3	yes	—
19	C₄²×D₇₁	C₂×C₄	1	6	yes	—
20	C₄²×D₇₁	C₂³	1	2	yes	—
21	C₄×C₇₁⋊C₈	C₂×C₄	1	6	yes	—
22	C₄×C₇₁⋊C₈	C₂³	1	6	yes	—
23	C₈×Dic₇₁	C₂×C₄	1	24	yes	—
24	C₂²×C₈×C₇₁	C₂×C₄	1	6	yes	—
25	C₂×C₄²×C₇₁	C₂³	1	1	yes	characteristic
26	C₄×C₈×C₇₁	C₂×C₄	1	6	yes	—
27	C₄×C₈×C₇₁	C₂³	1	6	yes	—
28	C₂³×D₇₁	C₄²	1	1	yes	characteristic
29	C₂²×Dic₇₁	C₄²	1	3	yes	—
30	C₂×C₄×D₇₁	C₄²	1	6	yes	—
31	C₂×C₄×D₇₁	C₂×C₈	1	24	yes	—
32	C₂×C₄×D₇₁	C₂²×C₄	1	6	yes	—
33	C₂×C₇₁⋊C₈	C₂×C₈	1	24	yes	complemented
34	C₂×C₇₁⋊C₈	C₂²×C₄	1	12	yes	—
35	C₈×D₇₁	C₂×C₈	1	96	yes	direct product factor
36	C₄×Dic₇₁	C₄²	1	6	yes	—
37	C₄×Dic₇₁	C₂²×C₄	1	6	yes	—
38	C₂²×C₄×C₇₁	C₂²×C₄	1	3	yes	—
39	C₂×C₈×C₇₁	C₂×C₈	1	24	yes	complemented
40	C₂×C₈×C₇₁	C₂²×C₄	1	12	yes	—
41	C₄²×C₇₁	C₂²×C₄	1	3	yes	—
42	C₄²×C₇₁	C₂⁴	1	1	yes	characteristic
43	C₂³×C₇₁	C₂×C₄²	1	1	yes	characteristic
44	C₂²×D₇₁	C₄×C₈	1	12	yes	—
45	C₂²×D₇₁	C₂×C₄²	1	2	yes	—
46	C₂×C₄×C₇₁	C₂×C₄²	1	3	yes	—
47	C₂×C₄×C₇₁	C₂²×C₈	1	12	yes	—
48	C₂×C₄×C₇₁	C₂³×C₄	1	3	yes	—
49	C₂×Dic₇₁	C₄×C₈	1	12	yes	—
50	C₂×Dic₇₁	C₂×C₄²	1	6	yes	—
51	C₄×D₇₁	C₄×C₈	1	24	yes	—
52	C₄×D₇₁	C₂²×C₈	1	24	yes	—
53	C₈×C₇₁	C₂²×C₈	1	24	yes	complemented
54	C₇₁⋊C₈	C₂²×C₈	1	24	yes	complemented
55	C₂²×C₇₁	C₂×C₄×C₈	1	6	yes	—
56	C₂²×C₇₁	C₂²×C₄²	1	1	yes	characteristic
57	C₂×D₇₁	C₈²	1	16	yes	direct product factor
58	C₂×D₇₁	C₂×C₄×C₈	1	12	yes	—
59	C₄×C₇₁	C₂×C₄×C₈	1	6	yes	—
60	C₄×C₇₁	C₂³×C₈	1	6	yes	—
61	Dic₇₁	C₂×C₄×C₈	1	12	yes	—
62	C₂²×C₈²	—	71	1	no	maximal, Sylow
63	C₂×C₇₁	C₂×C₈²	1	4	yes	complemented
64	C₂×C₇₁	C₂²×C₄×C₈	1	3	yes	—
65	D₇₁	C₂×C₈²	1	8	yes	direct product factor
66	C₂²×C₄×C₈	—	71	3	no	—
67	C₂×C₈²	—	71	3	no	—
68	C₂×C₈²	—	71	8	no	—
69	C₂×C₈²	D₇₁	1	1	yes	characteristic, direct product factor
70	C₇₁	C₂²×C₈²	1	1	yes	characteristic, complemented, Sylow
71	C₂³×C₈	—	71	6	no	—
72	C₂²×C₄²	—	71	1	no	—
73	C₂×C₄×C₈	—	71	6	no	—
74	C₂×C₄×C₈	—	71	3	no	—
75	C₂×C₄×C₈	—	71	12	no	—
76	C₂×C₄×C₈	—	71	12	no	—
77	C₂×C₄×C₈	C₂×D₇₁	1	3	yes	—
78	C₈²	—	71	12	no	—
79	C₈²	C₂×D₇₁	1	4	yes	direct product factor
80	C₂³×C₄	—	71	3	no	—
81	C₂²×C₈	—	71	6	no	—
82	C₂²×C₈	—	71	48	no	—
83	C₂²×C₈	—	71	24	no	—
84	C₂²×C₈	C₄×D₇₁	1	6	yes	—
85	C₂×C₄²	—	71	3	no	—
86	C₂×C₄²	—	71	6	no	—
87	C₂×C₄²	—	71	2	no	—
88	C₂×C₄²	C₂²×D₇₁	1	1	yes	characteristic
89	C₄×C₈	—	71	24	no	—
90	C₄×C₈	—	71	6	no	—
91	C₄×C₈	—	71	6	no	—
92	C₄×C₈	C₄×D₇₁	1	6	yes	—
93	C₄×C₈	C₂²×D₇₁	1	6	yes	—
94	C₂⁴	—	71	1	no	—
95	C₂²×C₄	—	71	3	no	—
96	C₂²×C₄	—	71	6	no	—
97	C₂²×C₄	—	71	6	no	—
98	C₂²×C₄	—	71	24	no	—
99	C₂²×C₄	C₂×C₄×D₇₁	1	3	yes	—
100	C₂×C₈	—	71	12	no	—
101	C₂×C₈	—	71	24	no	—
102	C₂×C₈	—	71	96	no	—
103	C₂×C₈	C₈×D₇₁	1	24	yes	direct product factor
104	C₂×C₈	C₂×C₄×D₇₁	1	12	yes	—
105	C₄²	—	71	6	no	—
106	C₄²	—	71	6	no	—
107	C₄²	C₂×C₄×D₇₁	1	3	yes	—
108	C₄²	C₂³×D₇₁	1	1	yes	characteristic
109	C₂³	—	71	2	no	—
110	C₂³	—	71	12	no	—
111	C₂³	C₄²×D₇₁	1	1	yes	characteristic
112	C₂×C₄	—	71	6	no	—
113	C₂×C₄	—	71	12	no	—
114	C₂×C₄	—	71	24	no	—
115	C₂×C₄	—	71	24	no	—
116	C₂×C₄	C₄²×D₇₁	1	3	yes	—
117	C₂×C₄	C₂×C₈×D₇₁	1	12	yes	—
118	C₂×C₄	C₂²×C₄×D₇₁	1	3	yes	—
119	C₈	—	71	24	no	—
120	C₈	C₂×C₈×D₇₁	1	24	yes	direct product factor
121	C₂²	—	71	12	no	—
122	C₂²	—	71	16	no	—
123	C₂²	C₄×C₈×D₇₁	1	6	yes	—
124	C₂²	C₂×C₄²×D₇₁	1	1	yes	characteristic
125	C₄	—	71	12	no	—
126	C₄	C₄×C₈×D₇₁	1	6	yes	—
127	C₄	C₂²×C₈×D₇₁	1	6	yes	—
128	C₂	—	71	8	no	—
129	C₂	C₈²×D₇₁	1	4	yes	direct product factor
130	C₂	C₂×C₄×C₈×D₇₁	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄×C₈×D₇₁

C₂

yes

maximal

C₂×C₈×C₇₁⋊C₈

C₂

yes

maximal, complemented

C₈²×D₇₁

C₂

yes

maximal, direct product factor

C₂×C₈²×C₇₁

C₂

yes

maximal, characteristic, complemented

C₂²×C₈×D₇₁

C₄

yes

—

C₂×C₄²×D₇₁

C₂²

yes

characteristic

C₂×C₄×C₇₁⋊C₈

C₂²

yes

—

C₂×C₈×Dic₇₁

C₄

yes

—

C₄×C₈×D₇₁

C₄

yes

—

C₄×C₈×D₇₁

C₂²

yes

—

C₈×C₇₁⋊C₈

C₂²

yes

complemented

C₂×C₄×C₈×C₇₁

C₂²

yes

—

C₈²×C₇₁

C₂²

yes

complemented

C₂²×C₄×D₇₁

C₂×C₄

yes

—

C₂²×C₇₁⋊C₈

C₂×C₄

yes

—

C₂×C₈×D₇₁

C₈

yes

direct product factor

C₂×C₈×D₇₁

C₂×C₄

yes

—

C₂×C₄×Dic₇₁

C₂×C₄

yes

—

C₄²×D₇₁

C₂×C₄

yes

—

C₄²×D₇₁

C₂³

yes

—

C₄×C₇₁⋊C₈

C₂×C₄

yes

—

C₄×C₇₁⋊C₈

C₂³

yes

—

C₈×Dic₇₁

C₂×C₄

yes

—

C₂²×C₈×C₇₁

C₂×C₄

yes

—

C₂×C₄²×C₇₁

C₂³

yes

characteristic

C₄×C₈×C₇₁

C₂×C₄

yes

—

C₄×C₈×C₇₁

C₂³

yes

—

C₂³×D₇₁

C₄²

yes

characteristic

C₂²×Dic₇₁

C₄²

yes

—

C₂×C₄×D₇₁

C₄²

yes

—

C₂×C₄×D₇₁

C₂×C₈

yes

—

C₂×C₄×D₇₁

C₂²×C₄

yes

—

C₂×C₇₁⋊C₈

C₂×C₈

yes

complemented

C₂×C₇₁⋊C₈

C₂²×C₄

yes

—

C₈×D₇₁

C₂×C₈

yes

direct product factor

C₄×Dic₇₁

C₄²

yes

—

C₄×Dic₇₁

C₂²×C₄

yes

—

C₂²×C₄×C₇₁

C₂²×C₄

yes

—

C₂×C₈×C₇₁

C₂×C₈

yes

complemented

C₂×C₈×C₇₁

C₂²×C₄

yes

—

C₄²×C₇₁

C₂²×C₄

yes

—

C₄²×C₇₁

C₂⁴

yes

characteristic

C₂³×C₇₁

C₂×C₄²

yes

characteristic

C₂²×D₇₁

C₄×C₈

yes

—

C₂²×D₇₁

C₂×C₄²

yes

—

C₂×C₄×C₇₁

C₂×C₄²

yes

—

C₂×C₄×C₇₁

C₂²×C₈

yes

—

C₂×C₄×C₇₁

C₂³×C₄

yes

—

C₂×Dic₇₁

C₄×C₈

yes

—

C₂×Dic₇₁

C₂×C₄²

yes

—

C₄×D₇₁

C₄×C₈

yes

—

C₄×D₇₁

C₂²×C₈

yes

—

C₈×C₇₁

C₂²×C₈

yes

complemented

C₇₁⋊C₈

C₂²×C₈

yes

complemented

C₂²×C₇₁

C₂×C₄×C₈

yes

—

C₂²×C₇₁

C₂²×C₄²

yes

characteristic

C₂×D₇₁

C₈²

yes

direct product factor

C₂×D₇₁

C₂×C₄×C₈

yes

—

C₄×C₇₁

C₂×C₄×C₈

yes

—

C₄×C₇₁

C₂³×C₈

yes

—

Dic₇₁

C₂×C₄×C₈

yes

—

C₂²×C₈²

—

maximal, Sylow

C₂×C₇₁

C₂×C₈²

yes

complemented

C₂×C₇₁

C₂²×C₄×C₈

yes

—

D₇₁

C₂×C₈²

yes

direct product factor

C₂²×C₄×C₈

—

C₂×C₈²

—

C₂×C₈²

—

C₂×C₈²

D₇₁

yes

characteristic, direct product factor

C₇₁

C₂²×C₈²

yes

characteristic, complemented, Sylow

C₂³×C₈

—

C₂²×C₄²

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₂×C₄×C₈

C₂×D₇₁

yes

—

C₈²

—

C₈²

C₂×D₇₁

yes

direct product factor

C₂³×C₄

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

C₄×D₇₁

yes

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

C₂²×D₇₁

yes

characteristic

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₄×D₇₁

yes

—

C₄×C₈

C₂²×D₇₁

yes

—

C₂⁴

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×C₄×D₇₁

yes

—

100

C₂×C₈

—

101

C₂×C₈

—

102

C₂×C₈

—

103

C₂×C₈

C₈×D₇₁

yes

direct product factor

104

C₂×C₈

C₂×C₄×D₇₁

yes

—

105

C₄²

—

106

C₄²

—

107

C₄²

C₂×C₄×D₇₁

yes

—

108

C₄²

C₂³×D₇₁

yes

characteristic

109

C₂³

—

110

C₂³

—

111

C₂³

C₄²×D₇₁

yes

characteristic

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

C₄²×D₇₁

yes

—

117

C₂×C₄

C₂×C₈×D₇₁

yes

—

118

C₂×C₄

C₂²×C₄×D₇₁

yes

—

119

C₈

—

120

C₈

C₂×C₈×D₇₁

yes

direct product factor

121

C₂²

—

122

C₂²

—

123

C₂²

C₄×C₈×D₇₁

yes

—

124

C₂²

C₂×C₄²×D₇₁

yes

characteristic

125

C₄

—

126

C₄

C₄×C₈×D₇₁

yes

—

127

C₄

C₂²×C₈×D₇₁

yes

—

128

C₂

—

129

C₂

C₈²×D₇₁

yes

direct product factor

130

C₂

C₂×C₄×C₈×D₇₁

yes

—

	subgroup	G/H
center, hypercenter	C₂×C₈²	D₇₁
commutator, hypocenter	C₇₁	C₂²×C₈²
Frattini	C₄²	C₂³×D₇₁
Fitting	C₂×C₈²×C₇₁	C₂
socle	C₂³×C₇₁	C₂×C₄²
prime radical	C₂³×D₇₁	C₄²
cosocle	C₄²×C₇₁	C₂⁴
LCS	G ▷ C₇₁
UCS	C₁ ◁ C₂×C₈²

subgroup

G/H

center, hypercenter

C₂×C₈²

D₇₁

commutator, hypocenter

C₇₁

C₂²×C₈²

Frattini

C₄²

C₂³×D₇₁

Fitting

C₂×C₈²×C₇₁

C₂

socle

C₂³×C₇₁

C₂×C₄²

prime radical

C₂³×D₇₁

C₄²

cosocle

C₄²×C₇₁

C₂⁴

LCS

G ▷ C₇₁

UCS

C₁ ◁ C₂×C₈²

N	H ≅ G/N	type
2. C₂×C₈×C₇₁⋊C₈	C₂	semidirect
3. C₈²×D₇₁	C₂	direct (N × H)
4. C₂×C₈²×C₇₁	C₂	semidirect
11. C₈×C₇₁⋊C₈	C₂²	semidirect
13. C₈²×C₇₁	C₂²	semidirect
16. C₂×C₈×D₇₁	C₈	direct (N × H)
33. C₂×C₇₁⋊C₈	C₂×C₈	semidirect
35. C₈×D₇₁	C₂×C₈	direct (N × H)
39. C₂×C₈×C₇₁	C₂×C₈	semidirect
53. C₈×C₇₁	C₂²×C₈	semidirect
54. C₇₁⋊C₈	C₂²×C₈	semidirect
57. C₂×D₇₁	C₈²	direct (N × H)
63. C₂×C₇₁	C₂×C₈²	semidirect
65. D₇₁	C₂×C₈²	direct (N × H)
69. C₂×C₈²	D₇₁	direct (N × H)
70. C₇₁	C₂²×C₈²	semidirect
79. C₈²	C₂×D₇₁	direct (N × H)
103. C₂×C₈	C₈×D₇₁	direct (N × H)
120. C₈	C₂×C₈×D₇₁	direct (N × H)
129. C₂	C₈²×D₇₁	direct (N × H)

H ≅ G/N

type

2. C₂×C₈×C₇₁⋊C₈

C₂

semidirect

3. C₈²×D₇₁

C₂

direct (N × H)

4. C₂×C₈²×C₇₁

C₂

semidirect

11. C₈×C₇₁⋊C₈

C₂²

semidirect

13. C₈²×C₇₁

C₂²

semidirect

16. C₂×C₈×D₇₁

C₈

direct (N × H)

33. C₂×C₇₁⋊C₈

C₂×C₈

semidirect

35. C₈×D₇₁

C₂×C₈

direct (N × H)

39. C₂×C₈×C₇₁

C₂×C₈

semidirect

53. C₈×C₇₁

C₂²×C₈

semidirect

54. C₇₁⋊C₈

C₂²×C₈

semidirect

57. C₂×D₇₁

C₈²

direct (N × H)

63. C₂×C₇₁

C₂×C₈²

semidirect

65. D₇₁

C₂×C₈²

direct (N × H)

69. C₂×C₈²

D₇₁

direct (N × H)

70. C₇₁

C₂²×C₈²

semidirect

79. C₈²

C₂×D₇₁

direct (N × H)

103. C₂×C₈

C₈×D₇₁

direct (N × H)

120. C₈

C₂×C₈×D₇₁

direct (N × H)

129. C₂

C₈²×D₇₁

direct (N × H)

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

subgroup	index
C₂×C₄×C₈×D₇₁	2
C₂×C₈×C₇₁⋊C₈	2
C₈²×D₇₁	2
C₂×C₈²×C₇₁	2
C₂²×C₈²	71

subgroup

index

C₂×C₄×C₈×D₇₁

C₂×C₈×C₇₁⋊C₈

C₈²×D₇₁

C₂×C₈²×C₇₁

C₂²×C₈²

Subgroups — C₂×C₈²×D₇₁ #31963017

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×C82×D71 #31963017

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₈²×D₇₁ #31963017