Group #31917621 — subgroups

Subgroups — C₂×C₃₂×Dic₇₁ #31917621

Order 18176.

270 conjugacy classes of subgroups; only the 195 normal subgroups are shown.

normal subgroups only

This group has 270 conjugacy classes of subgroups — too many to draw. Only its 195 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×C₇₁⋊C₃₂	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₆×Dic₇₁	C₂	1	1	yes	maximal, characteristic
3	C₃₂×Dic₇₁	C₂	1	4	yes	maximal, direct product factor
4	C₂²×C₃₂×C₇₁	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₇₁⋊C₁₆	C₄	1	1	yes	characteristic
6	C₂×C₈×Dic₇₁	C₄	1	1	yes	characteristic
7	C₂×C₇₁⋊C₃₂	C₄	1	4	yes	complemented
8	C₂×C₇₁⋊C₃₂	C₂²	1	2	yes	—
9	C₁₆×Dic₇₁	C₄	1	2	yes	—
10	C₁₆×Dic₇₁	C₂²	1	2	yes	—
11	C₂²×C₁₆×C₇₁	C₂²	1	1	yes	characteristic
12	C₂×C₃₂×C₇₁	C₄	1	4	yes	complemented
13	C₂×C₃₂×C₇₁	C₂²	1	2	yes	—
14	C₂²×C₇₁⋊C₈	C₈	1	1	yes	characteristic
15	C₂×C₄×Dic₇₁	C₈	1	1	yes	characteristic
16	C₂×C₇₁⋊C₁₆	C₈	1	4	yes	—
17	C₂×C₇₁⋊C₁₆	C₂×C₄	1	2	yes	—
18	C₈×Dic₇₁	C₈	1	2	yes	—
19	C₈×Dic₇₁	C₂×C₄	1	2	yes	—
20	C₇₁⋊C₃₂	C₂×C₄	1	4	yes	complemented
21	C₂²×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
22	C₂×C₁₆×C₇₁	C₂×C₄	1	1	yes	characteristic
23	C₂×C₁₆×C₇₁	C₂×C₄	1	2	yes	—
24	C₂×C₁₆×C₇₁	C₂×C₄	1	2	yes	—
25	C₂×C₁₆×C₇₁	C₂³	1	1	yes	characteristic
26	C₃₂×C₇₁	C₂×C₄	1	4	yes	complemented
27	C₂²×Dic₇₁	C₁₆	1	2	yes	—
28	C₂×C₇₁⋊C₈	C₁₆	1	4	yes	—
29	C₂×C₇₁⋊C₈	C₂×C₈	1	2	yes	—
30	C₄×Dic₇₁	C₁₆	1	2	yes	—
31	C₄×Dic₇₁	C₂×C₈	1	2	yes	—
32	C₇₁⋊C₁₆	C₂×C₈	1	4	yes	—
33	C₂²×C₄×C₇₁	C₂×C₈	1	1	yes	characteristic
34	C₂×C₈×C₇₁	C₄²	1	2	yes	—
35	C₂×C₈×C₇₁	C₂×C₈	1	1	yes	characteristic
36	C₂×C₈×C₇₁	C₂×C₈	1	2	yes	—
37	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
38	C₁₆×C₇₁	C₄²	1	2	yes	—
39	C₁₆×C₇₁	C₂²×C₄	1	1	yes	characteristic
40	C₁₆×C₇₁	C₂²×C₄	1	1	yes	characteristic
41	C₂³×C₇₁	C₂×C₁₆	1	1	yes	characteristic
42	C₂×C₄×C₇₁	C₄×C₈	1	2	yes	—
43	C₂×C₄×C₇₁	C₂×C₁₆	1	1	yes	characteristic
44	C₂×C₄×C₇₁	C₂×C₁₆	1	2	yes	—
45	C₂×C₄×C₇₁	C₂²×C₈	1	1	yes	characteristic
46	C₂×Dic₇₁	C₃₂	1	8	yes	direct product factor
47	C₂×Dic₇₁	C₂×C₁₆	1	4	yes	—
48	C₈×C₇₁	C₄×C₈	1	2	yes	—
49	C₈×C₇₁	C₂×C₄²	1	1	yes	characteristic
50	C₈×C₇₁	C₂²×C₈	1	1	yes	characteristic
51	C₇₁⋊C₈	C₂×C₁₆	1	4	yes	—
52	C₂²×C₇₁	C₄×C₁₆	1	2	yes	—
53	C₂²×C₇₁	C₂×C₃₂	1	4	yes	—
54	C₂²×C₇₁	C₂²×C₁₆	1	1	yes	characteristic
55	C₄×C₇₁	C₄×C₁₆	1	2	yes	—
56	C₄×C₇₁	C₂×C₄×C₈	1	1	yes	characteristic
57	C₄×C₇₁	C₂²×C₁₆	1	1	yes	characteristic
58	Dic₇₁	C₂×C₃₂	1	8	yes	direct product factor
59	C₂×C₄×C₃₂	—	71	1	no	maximal, Sylow
60	C₂×C₇₁	C₄×C₃₂	1	4	yes	complemented
61	C₂×C₇₁	C₂×C₄×C₁₆	1	1	yes	characteristic
62	C₂×C₇₁	C₂²×C₃₂	1	2	yes	—
63	C₂²×C₃₂	—	71	1	no	—
64	C₂²×C₃₂	D₇₁	1	1	yes	characteristic
65	C₂×C₄×C₁₆	—	71	1	no	—
66	C₄×C₃₂	—	71	4	no	—
67	C₇₁	C₂×C₄×C₃₂	1	1	yes	characteristic, complemented, Sylow
68	C₂²×C₁₆	—	71	1	no	—
69	C₂²×C₁₆	C₂×D₇₁	1	1	yes	characteristic
70	C₂×C₄×C₈	—	71	1	no	—
71	C₂×C₃₂	—	71	4	no	—
72	C₂×C₃₂	—	71	2	no	—
73	C₂×C₃₂	Dic₇₁	1	4	yes	direct product factor
74	C₂×C₃₂	C₂×D₇₁	1	2	yes	—
75	C₄×C₁₆	—	71	2	no	—
76	C₄×C₁₆	—	71	2	no	—
77	C₂²×C₈	—	71	1	no	—
78	C₂²×C₈	C₄×D₇₁	1	1	yes	characteristic
79	C₂×C₄²	—	71	1	no	—
80	C₂×C₁₆	—	71	4	no	—
81	C₂×C₁₆	—	71	2	no	—
82	C₂×C₁₆	C₄×D₇₁	1	1	yes	characteristic
83	C₂×C₁₆	C₄×D₇₁	1	2	yes	—
84	C₂×C₁₆	C₂×Dic₇₁	1	2	yes	—
85	C₂×C₁₆	C₂²×D₇₁	1	1	yes	characteristic
86	C₄×C₈	—	71	2	no	—
87	C₄×C₈	—	71	2	no	—
88	C₃₂	—	71	4	no	—
89	C₃₂	C₂×Dic₇₁	1	4	yes	direct product factor
90	C₂²×C₄	—	71	2	no	—
91	C₂²×C₄	C₈×D₇₁	1	1	yes	characteristic
92	C₂×C₈	—	71	4	no	—
93	C₂×C₈	—	71	2	no	—
94	C₂×C₈	C₄×Dic₇₁	1	2	yes	—
95	C₂×C₈	C₈×D₇₁	1	1	yes	characteristic
96	C₂×C₈	C₈×D₇₁	1	2	yes	—
97	C₂×C₈	C₂×C₄×D₇₁	1	1	yes	characteristic
98	C₄²	—	71	2	no	—
99	C₄²	—	71	2	no	—
100	C₁₆	—	71	4	no	—
101	C₁₆	C₄×Dic₇₁	1	2	yes	—
102	C₁₆	C₂×C₄×D₇₁	1	1	yes	characteristic
103	C₁₆	C₂²×Dic₇₁	1	1	yes	characteristic
104	C₂³	C₁₆×D₇₁	1	1	yes	characteristic
105	C₂×C₄	—	71	8	no	—
106	C₂×C₄	—	71	4	no	—
107	C₂×C₄	C₈×Dic₇₁	1	2	yes	—
108	C₂×C₄	C₁₆×D₇₁	1	2	yes	—
109	C₂×C₄	C₁₆×D₇₁	1	1	yes	characteristic
110	C₂×C₄	C₂×C₈×D₇₁	1	1	yes	characteristic
111	C₈	—	71	4	no	—
112	C₈	C₈×Dic₇₁	1	2	yes	—
113	C₈	C₂×C₄×Dic₇₁	1	1	yes	characteristic
114	C₈	C₂×C₈×D₇₁	1	1	yes	characteristic
115	C₂²	C₁₆×Dic₇₁	1	2	yes	—
116	C₂²	C₃₂×D₇₁	1	4	yes	—
117	C₂²	C₂×C₁₆×D₇₁	1	1	yes	characteristic
118	C₄	—	71	8	no	—
119	C₄	C₁₆×Dic₇₁	1	2	yes	—
120	C₄	C₂×C₈×Dic₇₁	1	1	yes	characteristic
121	C₄	C₂×C₁₆×D₇₁	1	1	yes	characteristic
122	C₂	C₃₂×Dic₇₁	1	4	yes	direct product factor
123	C₂	C₂×C₁₆×Dic₇₁	1	1	yes	characteristic
124	C₂	C₂×C₃₂×D₇₁	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×C₇₁⋊C₃₂

C₂

yes

maximal, characteristic

C₂×C₁₆×Dic₇₁

C₂

yes

maximal, characteristic

C₃₂×Dic₇₁

C₂

yes

maximal, direct product factor

C₂²×C₃₂×C₇₁

C₂

yes

maximal, characteristic

C₂²×C₇₁⋊C₁₆

C₄

yes

characteristic

C₂×C₈×Dic₇₁

C₄

yes

characteristic

C₂×C₇₁⋊C₃₂

C₄

yes

complemented

C₂×C₇₁⋊C₃₂

C₂²

yes

—

C₁₆×Dic₇₁

C₄

yes

—

C₁₆×Dic₇₁

C₂²

yes

—

C₂²×C₁₆×C₇₁

C₂²

yes

characteristic

C₂×C₃₂×C₇₁

C₄

yes

complemented

C₂×C₃₂×C₇₁

C₂²

yes

—

C₂²×C₇₁⋊C₈

C₈

yes

characteristic

C₂×C₄×Dic₇₁

C₈

yes

characteristic

C₂×C₇₁⋊C₁₆

C₈

yes

—

C₂×C₇₁⋊C₁₆

C₂×C₄

yes

—

C₈×Dic₇₁

C₈

yes

—

C₈×Dic₇₁

C₂×C₄

yes

—

C₇₁⋊C₃₂

C₂×C₄

yes

complemented

C₂²×C₈×C₇₁

C₂×C₄

yes

characteristic

C₂×C₁₆×C₇₁

C₂×C₄

yes

characteristic

C₂×C₁₆×C₇₁

C₂×C₄

yes

—

C₂×C₁₆×C₇₁

C₂×C₄

yes

—

C₂×C₁₆×C₇₁

C₂³

yes

characteristic

C₃₂×C₇₁

C₂×C₄

yes

complemented

C₂²×Dic₇₁

C₁₆

yes

—

C₂×C₇₁⋊C₈

C₁₆

yes

—

C₂×C₇₁⋊C₈

C₂×C₈

yes

—

C₄×Dic₇₁

C₁₆

yes

—

C₄×Dic₇₁

C₂×C₈

yes

—

C₇₁⋊C₁₆

C₂×C₈

yes

—

C₂²×C₄×C₇₁

C₂×C₈

yes

characteristic

C₂×C₈×C₇₁

C₄²

yes

—

C₂×C₈×C₇₁

C₂×C₈

yes

characteristic

C₂×C₈×C₇₁

C₂×C₈

yes

—

C₂×C₈×C₇₁

C₂²×C₄

yes

characteristic

C₁₆×C₇₁

C₄²

yes

—

C₁₆×C₇₁

C₂²×C₄

yes

characteristic

C₁₆×C₇₁

C₂²×C₄

yes

characteristic

C₂³×C₇₁

C₂×C₁₆

yes

characteristic

C₂×C₄×C₇₁

C₄×C₈

yes

—

C₂×C₄×C₇₁

C₂×C₁₆

yes

characteristic

C₂×C₄×C₇₁

C₂×C₁₆

yes

—

C₂×C₄×C₇₁

C₂²×C₈

yes

characteristic

C₂×Dic₇₁

C₃₂

yes

direct product factor

C₂×Dic₇₁

C₂×C₁₆

yes

—

C₈×C₇₁

C₄×C₈

yes

—

C₈×C₇₁

C₂×C₄²

yes

characteristic

C₈×C₇₁

C₂²×C₈

yes

characteristic

C₇₁⋊C₈

C₂×C₁₆

yes

—

C₂²×C₇₁

C₄×C₁₆

yes

—

C₂²×C₇₁

C₂×C₃₂

yes

—

C₂²×C₇₁

C₂²×C₁₆

yes

characteristic

C₄×C₇₁

C₄×C₁₆

yes

—

C₄×C₇₁

C₂×C₄×C₈

yes

characteristic

C₄×C₇₁

C₂²×C₁₆

yes

characteristic

Dic₇₁

C₂×C₃₂

yes

direct product factor

C₂×C₄×C₃₂

—

maximal, Sylow

C₂×C₇₁

C₄×C₃₂

yes

complemented

C₂×C₇₁

C₂×C₄×C₁₆

yes

characteristic

C₂×C₇₁

C₂²×C₃₂

yes

—

C₂²×C₃₂

—

C₂²×C₃₂

D₇₁

yes

characteristic

C₂×C₄×C₁₆

—

C₄×C₃₂

—

C₇₁

C₂×C₄×C₃₂

yes

characteristic, complemented, Sylow

C₂²×C₁₆

—

C₂²×C₁₆

C₂×D₇₁

yes

characteristic

C₂×C₄×C₈

—

C₂×C₃₂

—

C₂×C₃₂

—

C₂×C₃₂

Dic₇₁

yes

direct product factor

C₂×C₃₂

C₂×D₇₁

yes

—

C₄×C₁₆

—

C₄×C₁₆

—

C₂²×C₈

—

C₂²×C₈

C₄×D₇₁

yes

characteristic

C₂×C₄²

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₄×D₇₁

yes

characteristic

C₂×C₁₆

C₄×D₇₁

yes

—

C₂×C₁₆

C₂×Dic₇₁

yes

—

C₂×C₁₆

C₂²×D₇₁

yes

characteristic

C₄×C₈

—

C₄×C₈

—

C₃₂

—

C₃₂

C₂×Dic₇₁

yes

direct product factor

C₂²×C₄

—

C₂²×C₄

C₈×D₇₁

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₄×Dic₇₁

yes

—

C₂×C₈

C₈×D₇₁

yes

characteristic

C₂×C₈

C₈×D₇₁

yes

—

C₂×C₈

C₂×C₄×D₇₁

yes

characteristic

C₄²

—

C₄²

—

100

C₁₆

—

101

C₁₆

C₄×Dic₇₁

yes

—

102

C₁₆

C₂×C₄×D₇₁

yes

characteristic

103

C₁₆

C₂²×Dic₇₁

yes

characteristic

104

C₂³

C₁₆×D₇₁

yes

characteristic

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

C₈×Dic₇₁

yes

—

108

C₂×C₄

C₁₆×D₇₁

yes

—

109

C₂×C₄

C₁₆×D₇₁

yes

characteristic

110

C₂×C₄

C₂×C₈×D₇₁

yes

characteristic

111

C₈

—

112

C₈

C₈×Dic₇₁

yes

—

113

C₈

C₂×C₄×Dic₇₁

yes

characteristic

114

C₈

C₂×C₈×D₇₁

yes

characteristic

115

C₂²

C₁₆×Dic₇₁

yes

—

116

C₂²

C₃₂×D₇₁

yes

—

117

C₂²

C₂×C₁₆×D₇₁

yes

characteristic

118

C₄

—

119

C₄

C₁₆×Dic₇₁

yes

—

120

C₄

C₂×C₈×Dic₇₁

yes

characteristic

121

C₄

C₂×C₁₆×D₇₁

yes

characteristic

122

C₂

C₃₂×Dic₇₁

yes

direct product factor

123

C₂

C₂×C₁₆×Dic₇₁

yes

characteristic

124

C₂

C₂×C₃₂×D₇₁

yes

—

	subgroup	G/H
center, hypercenter	C₂²×C₃₂	D₇₁
commutator, hypocenter	C₇₁	C₂×C₄×C₃₂
Frattini	C₂×C₁₆	C₂²×D₇₁
Fitting	C₂²×C₃₂×C₇₁	C₂
socle, prime radical	C₂³×C₇₁	C₂×C₁₆
cosocle	C₂×C₁₆×C₇₁	C₂³
LCS	G ▷ C₇₁
UCS	C₁ ◁ C₂²×C₃₂

subgroup

G/H

center, hypercenter

C₂²×C₃₂

D₇₁

commutator, hypocenter

C₇₁

C₂×C₄×C₃₂

Frattini

C₂×C₁₆

C₂²×D₇₁

Fitting

C₂²×C₃₂×C₇₁

C₂

socle, prime radical

C₂³×C₇₁

C₂×C₁₆

cosocle

C₂×C₁₆×C₇₁

C₂³

LCS

G ▷ C₇₁

UCS

C₁ ◁ C₂²×C₃₂

N	H ≅ G/N	type
3. C₃₂×Dic₇₁	C₂	direct (N × H)
7. C₂×C₇₁⋊C₃₂	C₄	semidirect
12. C₂×C₃₂×C₇₁	C₄	semidirect
20. C₇₁⋊C₃₂	C₂×C₄	semidirect
26. C₃₂×C₇₁	C₂×C₄	semidirect
46. C₂×Dic₇₁	C₃₂	direct (N × H)
58. Dic₇₁	C₂×C₃₂	direct (N × H)
60. C₂×C₇₁	C₄×C₃₂	semidirect
67. C₇₁	C₂×C₄×C₃₂	semidirect
73. C₂×C₃₂	Dic₇₁	direct (N × H)
89. C₃₂	C₂×Dic₇₁	direct (N × H)
122. C₂	C₃₂×Dic₇₁	direct (N × H)

H ≅ G/N

type

3. C₃₂×Dic₇₁

C₂

direct (N × H)

7. C₂×C₇₁⋊C₃₂

C₄

semidirect

12. C₂×C₃₂×C₇₁

C₄

semidirect

20. C₇₁⋊C₃₂

C₂×C₄

semidirect

26. C₃₂×C₇₁

C₂×C₄

semidirect

46. C₂×Dic₇₁

C₃₂

direct (N × H)

58. Dic₇₁

C₂×C₃₂

direct (N × H)

60. C₂×C₇₁

C₄×C₃₂

semidirect

67. C₇₁

C₂×C₄×C₃₂

semidirect

73. C₂×C₃₂

Dic₇₁

direct (N × H)

89. C₃₂

C₂×Dic₇₁

direct (N × H)

122. C₂

C₃₂×Dic₇₁

direct (N × H)

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

subgroup	index
C₂²×C₇₁⋊C₃₂	2
C₂×C₁₆×Dic₇₁	2
C₃₂×Dic₇₁	2
C₂²×C₃₂×C₇₁	2
C₂×C₄×C₃₂	71

subgroup

index

C₂²×C₇₁⋊C₃₂

C₂×C₁₆×Dic₇₁

C₃₂×Dic₇₁

C₂²×C₃₂×C₇₁

C₂×C₄×C₃₂

Subgroups — C₂×C₃₂×Dic₇₁ #31917621

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×C32×Dic71 #31917621

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃₂×Dic₇₁ #31917621