Group #31918021 — subgroups

Subgroups — #31918021

Order 18176.

184 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 184 conjugacy classes of subgroups — too many to draw. Only its 105 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	#22590634	C₂	1	2	yes	maximal, complemented
2	#22586538	C₂	1	1	yes	maximal, characteristic, complemented
3	#22586482	C₂	1	2	yes	maximal, complemented
4	#22586474	C₂	1	1	yes	maximal, characteristic, complemented
5	C₇₁×#2170	C₂	1	1	yes	maximal, characteristic
6	#17985833	—	2	2	no	—
7	#17985833	C₂²	1	2	yes	—
8	C₂×C₇₁⋊C₃₂	—	2	2	no	—
9	C₂×C₇₁⋊C₃₂	C₂²	1	1	yes	characteristic
10	C₂×C₇₁⋊C₃₂	C₂²	1	1	yes	characteristic
11	#504×C₇₁	C₂²	1	2	yes	—
12	#443×C₇₁	C₄	1	1	yes	characteristic
13	#364×C₇₁	C₄	1	1	yes	characteristic
14	#350×C₇₁	C₄	1	2	yes	—
15	C₄×C₁₆×C₇₁	C₂²	1	1	yes	characteristic
16	C₇₁⋊C₃₂	—	2	2	no	—
17	C₇₁⋊C₃₂	—	2	1	no	—
18	C₇₁⋊C₃₂	D₄	1	2	yes	—
19	#135×C₇₁	C₈	1	1	yes	characteristic
20	#131×C₇₁	C₂×C₄	1	2	yes	—
21	M₃₂×C₇₁	—	2	2	no	—
22	M₃₂×C₇₁	C₂×C₄	1	2	yes	—
23	C₂×C₁₆×C₇₁	—	2	2	no	—
24	C₂×C₁₆×C₇₁	C₂×C₄	1	1	yes	characteristic
25	C₂×C₁₆×C₇₁	C₂³	1	1	yes	characteristic
26	#108×C₇₁	C₈	1	1	yes	characteristic
27	#104×C₇₁	C₈	1	2	yes	—
28	C₄×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
29	#41×C₇₁	C₂×C₈	1	2	yes	—
30	Q₁₆×C₇₁	C₁₆	1	1	yes	characteristic
31	SD₁₆×C₇₁	C₁₆	1	2	yes	—
32	D₈×C₇₁	C₁₆	1	1	yes	characteristic
33	M₁₆×C₇₁	—	2	2	no	—
34	M₁₆×C₇₁	C₂×C₈	1	2	yes	—
35	C₂×C₈×C₇₁	—	2	2	no	—
36	C₂×C₈×C₇₁	C₂×C₈	1	1	yes	characteristic
37	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
38	C₄²×C₇₁	C₂×C₈	1	1	yes	characteristic
39	C₁₆×C₇₁	—	2	2	no	—
40	C₁₆×C₇₁	—	2	1	no	—
41	C₁₆×C₇₁	C₂×D₄	1	1	yes	characteristic
42	C₁₆×C₇₁	#41	1	1	yes	characteristic
43	Q₈×C₇₁	C₂×C₁₆	1	2	yes	—
44	D₄×C₇₁	—	2	2	no	—
45	D₄×C₇₁	C₂×C₁₆	1	2	yes	—
46	C₂×C₄×C₇₁	—	2	2	no	—
47	C₂×C₄×C₇₁	—	2	1	no	—
48	C₂×C₄×C₇₁	C₂²×C₈	1	1	yes	characteristic
49	C₈×C₇₁	—	2	2	no	—
50	C₈×C₇₁	C₂×C₁₆	1	1	yes	characteristic
51	C₈×C₇₁	C₂×C₁₆	1	1	yes	characteristic
52	C₈×C₇₁	C₄×D₄	1	1	yes	characteristic
53	C₈×C₇₁	#131	1	1	yes	characteristic
54	C₂²×C₇₁	—	2	2	no	—
55	C₂²×C₇₁	#504	1	1	yes	characteristic
56	C₄×C₇₁	—	2	2	no	—
57	C₄×C₇₁	—	4	1	no	—
58	C₄×C₇₁	C₈×D₄	1	1	yes	characteristic
59	C₄×C₇₁	C₂²×C₁₆	1	1	yes	characteristic
60	#13648	—	71	1	no	maximal, Sylow
61	C₂×C₇₁	—	2	1	no	—
62	C₂×C₇₁	—	4	2	no	—
63	C₂×C₇₁	D₄×C₁₆	1	1	yes	characteristic
64	#2258	—	71	2	no	—
65	#2170	D₇₁	1	1	yes	characteristic
66	#1422	—	71	1	no	—
67	#1401	—	71	2	no	—
68	#1397	—	71	1	no	—
69	C₇₁	#13648	1	1	yes	characteristic, complemented, Sylow
70	#504	C₂×D₇₁	1	2	yes	—
71	#443	Dic₇₁	1	1	yes	characteristic
72	M₆₄	—	71	2	no	—
73	M₆₄	—	142	2	no	—
74	C₂×C₃₂	—	71	1	no	—
75	C₂×C₃₂	—	71	1	no	—
76	C₂×C₃₂	—	142	2	no	—
77	#364	Dic₇₁	1	1	yes	characteristic
78	#350	Dic₇₁	1	2	yes	—
79	C₄×C₁₆	C₂×D₇₁	1	1	yes	characteristic
80	#135	C₇₁⋊C₈	1	1	yes	characteristic
81	#131	C₂×Dic₇₁	1	2	yes	—
82	M₃₂	—	2	2	no	—
83	M₃₂	C₂×Dic₇₁	1	2	yes	—
84	C₂×C₁₆	—	2	2	no	—
85	C₂×C₁₆	C₂×Dic₇₁	1	1	yes	characteristic
86	C₂×C₁₆	C₂²×D₇₁	1	1	yes	characteristic
87	#108	C₇₁⋊C₈	1	1	yes	characteristic
88	#104	C₇₁⋊C₈	1	2	yes	—
89	C₄×C₈	C₂×Dic₇₁	1	1	yes	characteristic
90	C₃₂	—	71	2	no	—
91	C₃₂	—	142	2	no	—
92	C₃₂	—	142	1	no	—
93	#41	C₂×C₇₁⋊C₈	1	2	yes	—
94	Q₁₆	C₇₁⋊C₁₆	1	1	yes	characteristic
95	SD₁₆	C₇₁⋊C₁₆	1	2	yes	—
96	D₈	C₇₁⋊C₁₆	1	1	yes	characteristic
97	M₁₆	—	2	2	no	—
98	M₁₆	C₂×C₇₁⋊C₈	1	2	yes	—
99	C₂×C₈	—	2	2	no	—
100	C₂×C₈	C₂×C₇₁⋊C₈	1	1	yes	characteristic
101	C₂×C₈	C₂²×Dic₇₁	1	1	yes	characteristic
102	C₄²	C₂×C₇₁⋊C₈	1	1	yes	characteristic
103	C₁₆	—	2	2	no	—
104	C₁₆	—	2	1	no	—
105	C₁₆	D₄×D₇₁	1	1	yes	characteristic
106	C₁₆	#11765864	1	1	yes	characteristic
107	Q₈	C₂×C₇₁⋊C₁₆	1	2	yes	—
108	D₄	—	2	2	no	—
109	D₄	C₂×C₇₁⋊C₁₆	1	2	yes	—
110	C₂×C₄	—	2	2	no	—
111	C₂×C₄	—	2	1	no	—
112	C₂×C₄	C₂²×C₇₁⋊C₈	1	1	yes	characteristic
113	C₈	—	2	2	no	—
114	C₈	C₂×C₇₁⋊C₁₆	1	1	yes	characteristic
115	C₈	C₂×C₇₁⋊C₁₆	1	1	yes	characteristic
116	C₈	D₄×Dic₇₁	1	1	yes	characteristic
117	C₈	#14556135	1	1	yes	characteristic
118	C₂²	—	2	2	no	—
119	C₂²	#17986416	1	1	yes	characteristic
120	C₄	—	2	2	no	—
121	C₄	—	4	1	no	—
122	C₄	D₄×C₇₁⋊C₈	1	1	yes	characteristic
123	C₄	C₂²×C₇₁⋊C₁₆	1	1	yes	characteristic
124	C₂	—	2	1	no	—
125	C₂	—	4	2	no	—
126	C₂	D₄×C₇₁⋊C₁₆	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22590634

C₂

yes

maximal, complemented

#22586538

C₂

yes

maximal, characteristic, complemented

#22586482

C₂

yes

maximal, complemented

#22586474

C₂

yes

maximal, characteristic, complemented

C₇₁×#2170

C₂

yes

maximal, characteristic

#17985833

—

#17985833

C₂²

yes

—

C₂×C₇₁⋊C₃₂

—

C₂×C₇₁⋊C₃₂

C₂²

yes

characteristic

C₂×C₇₁⋊C₃₂

C₂²

yes

characteristic

#504×C₇₁

C₂²

yes

—

#443×C₇₁

C₄

yes

characteristic

#364×C₇₁

C₄

yes

characteristic

#350×C₇₁

C₄

yes

—

C₄×C₁₆×C₇₁

C₂²

yes

characteristic

C₇₁⋊C₃₂

—

C₇₁⋊C₃₂

—

C₇₁⋊C₃₂

D₄

yes

—

#135×C₇₁

C₈

yes

characteristic

#131×C₇₁

C₂×C₄

yes

—

M₃₂×C₇₁

—

M₃₂×C₇₁

C₂×C₄

yes

—

C₂×C₁₆×C₇₁

—

C₂×C₁₆×C₇₁

C₂×C₄

yes

characteristic

C₂×C₁₆×C₇₁

C₂³

yes

characteristic

#108×C₇₁

C₈

yes

characteristic

#104×C₇₁

C₈

yes

—

C₄×C₈×C₇₁

C₂×C₄

yes

characteristic

#41×C₇₁

C₂×C₈

yes

—

Q₁₆×C₇₁

C₁₆

yes

characteristic

SD₁₆×C₇₁

C₁₆

yes

—

D₈×C₇₁

C₁₆

yes

characteristic

M₁₆×C₇₁

—

M₁₆×C₇₁

C₂×C₈

yes

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

C₂×C₈

yes

characteristic

C₂×C₈×C₇₁

C₂²×C₄

yes

characteristic

C₄²×C₇₁

C₂×C₈

yes

characteristic

C₁₆×C₇₁

—

C₁₆×C₇₁

—

C₁₆×C₇₁

C₂×D₄

yes

characteristic

C₁₆×C₇₁

#41

yes

characteristic

Q₈×C₇₁

C₂×C₁₆

yes

—

D₄×C₇₁

—

D₄×C₇₁

C₂×C₁₆

yes

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

C₂²×C₈

yes

characteristic

C₈×C₇₁

—

C₈×C₇₁

C₂×C₁₆

yes

characteristic

C₈×C₇₁

C₂×C₁₆

yes

characteristic

C₈×C₇₁

C₄×D₄

yes

characteristic

C₈×C₇₁

#131

yes

characteristic

C₂²×C₇₁

—

C₂²×C₇₁

#504

yes

characteristic

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

C₈×D₄

yes

characteristic

C₄×C₇₁

C₂²×C₁₆

yes

characteristic

#13648

—

maximal, Sylow

C₂×C₇₁

—

C₂×C₇₁

—

C₂×C₇₁

D₄×C₁₆

yes

characteristic

#2258

—

#2170

D₇₁

yes

—

—

—

C₇₁

#13648

yes

characteristic, complemented, Sylow

#504

C₂×D₇₁

yes

—

#443

Dic₇₁

yes

characteristic

M₆₄

—

M₆₄

—

142

—

C₂×C₃₂

—

C₂×C₃₂

—

C₂×C₃₂

—

142

—

#364

Dic₇₁

yes

characteristic

#350

Dic₇₁

yes

—

C₄×C₁₆

C₂×D₇₁

yes

characteristic

#135

C₇₁⋊C₈

yes

characteristic

#131

C₂×Dic₇₁

yes

—

M₃₂

—

M₃₂

C₂×Dic₇₁

yes

—

C₂×C₁₆

—

C₂×C₁₆

C₂×Dic₇₁

yes

characteristic

C₂×C₁₆

C₂²×D₇₁

yes

characteristic

#108

C₇₁⋊C₈

yes

characteristic

#104

C₇₁⋊C₈

yes

—

C₄×C₈

C₂×Dic₇₁

yes

characteristic

C₃₂

—

C₃₂

—

142

—

C₃₂

—

142

—

#41

C₂×C₇₁⋊C₈

yes

—

Q₁₆

C₇₁⋊C₁₆

yes

characteristic

SD₁₆

C₇₁⋊C₁₆

yes

—

D₈

C₇₁⋊C₁₆

yes

characteristic

M₁₆

—

M₁₆

C₂×C₇₁⋊C₈

yes

—

C₂×C₈

—

100

C₂×C₈

C₂×C₇₁⋊C₈

yes

characteristic

101

C₂×C₈

C₂²×Dic₇₁

yes

characteristic

102

C₄²

C₂×C₇₁⋊C₈

yes

characteristic

103

C₁₆

—

104

C₁₆

—

105

C₁₆

D₄×D₇₁

yes

characteristic

106

C₁₆

#11765864

yes

characteristic

107

Q₈

C₂×C₇₁⋊C₁₆

yes

—

108

D₄

—

109

D₄

C₂×C₇₁⋊C₁₆

yes

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

C₂²×C₇₁⋊C₈

yes

characteristic

113

C₈

—

114

C₈

C₂×C₇₁⋊C₁₆

yes

characteristic

115

C₈

C₂×C₇₁⋊C₁₆

yes

characteristic

116

C₈

D₄×Dic₇₁

yes

characteristic

117

C₈

#14556135

yes

characteristic

118

C₂²

—

119

C₂²

#17986416

yes

characteristic

120

C₄

—

121

C₄

—

122

C₄

D₄×C₇₁⋊C₈

yes

characteristic

123

C₄

C₂²×C₇₁⋊C₁₆

yes

characteristic

124

C₂

—

125

C₂

—

126

C₂

D₄×C₇₁⋊C₁₆

yes

characteristic

	subgroup	G/H
center	C₁₆	D₄×D₇₁
commutator	C₄×C₇₁	C₂²×C₁₆
Frattini	C₂×C₁₆	C₂²×D₇₁
Fitting	C₇₁×#2170	C₂
socle	C₂×C₇₁	D₄×C₁₆
prime radical	#135×C₇₁	C₈
cosocle	C₂×C₁₆×C₇₁	C₂³
hypercenter	#2170	D₇₁
hypocenter	C₇₁	#13648
LCS	G ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₁₆ ◁ C₂×C₁₆ ◁ #2170

subgroup

G/H

center

C₁₆

D₄×D₇₁

commutator

C₄×C₇₁

C₂²×C₁₆

Frattini

C₂×C₁₆

C₂²×D₇₁

Fitting

C₇₁×#2170

C₂

socle

C₂×C₇₁

D₄×C₁₆

prime radical

#135×C₇₁

C₈

cosocle

C₂×C₁₆×C₇₁

C₂³

D₇₁

C₇₁

G ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₁₆ ◁ C₂×C₁₆ ◁ #2170

N	H ≅ G/N	type
1. #22590634	C₂	semidirect
2. #22586538	C₂	semidirect
3. #22586482	C₂	semidirect
4. #22586474	C₂	semidirect
69. C₇₁	#13648	semidirect

H ≅ G/N

type

1. #22590634

C₂

semidirect

2. #22586538

C₂

semidirect

3. #22586482

C₂

semidirect

4. #22586474

C₂

semidirect

69. C₇₁

#13648

semidirect

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

subgroup	index
#22590634	2
#22586538	2
#22586482	2
#22586474	2
C₇₁×#2170	2
#13648	71

subgroup

index

C₇₁×#2170

Subgroups — #31918021

Conjugacy classes of subgroups

Complements

Maximal subgroups