Group #31884854 — subgroups

Subgroups — #31884854

Order 18176.

126 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 126 conjugacy classes of subgroups — too many to draw. Only its 45 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	#22590095	C₂	1	1	yes	maximal, characteristic
2	#22586480	C₂	1	1	yes	maximal, characteristic
3	C₇₁×#1400	C₂	1	1	yes	maximal, characteristic
4	C₂×#14556068	C₄	1	1	yes	characteristic
5	C₂×C₈×Dic₇₁	C₄	1	1	yes	characteristic
6	#17985833	—	2	1	no	—
7	Dic₇₁⋊C₁₆	—	2	1	no	—
8	Dic₇₁⋊C₁₆	—	2	1	no	—
9	C₂×M₃₂×C₇₁	C₂²	1	1	yes	characteristic
10	M₆₄×C₇₁	—	2	1	no	—
11	C₂²×C₇₁⋊C₈	C₈	1	1	yes	characteristic
12	C₂×C₄×Dic₇₁	C₈	1	1	yes	characteristic
13	#14556068	—	4	1	no	—
14	C₂×C₇₁⋊C₁₆	—	2	1	no	—
15	C₈×Dic₇₁	—	2	1	no	—
16	C₈×Dic₇₁	—	2	1	no	—
17	C₇₁⋊C₃₂	—	4	1	no	—
18	C₂²×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
19	M₃₂×C₇₁	—	4	1	no	—
20	C₂×C₁₆×C₇₁	D₄	1	1	yes	characteristic
21	C₂×C₁₆×C₇₁	D₄	1	1	yes	characteristic
22	C₃₂×C₇₁	—	4	1	no	—
23	C₂²×Dic₇₁	—	2	1	no	—
24	C₂×C₇₁⋊C₈	—	2	1	no	—
25	C₂×C₇₁⋊C₈	—	4	1	no	—
26	C₄×Dic₇₁	—	2	1	no	—
27	C₄×Dic₇₁	C₁₆	1	1	yes	characteristic
28	C₄×Dic₇₁	C₁₆	1	1	yes	characteristic
29	C₇₁⋊C₁₆	—	4	1	no	—
30	C₂²×C₄×C₇₁	C₂×C₈	1	1	yes	characteristic
31	C₂×C₈×C₇₁	—	2	1	no	—
32	C₂×C₈×C₇₁	—	2	1	no	—
33	C₂×C₈×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
34	C₂×C₈×C₇₁	M₁₆	1	1	yes	characteristic
35	C₁₆×C₇₁	—	2	1	no	—
36	C₁₆×C₇₁	—	2	1	no	—
37	C₂³×C₇₁	M₃₂	1	1	yes	characteristic
38	C₂×C₄×C₇₁	—	2	1	no	—
39	C₂×C₄×C₇₁	—	2	1	no	—
40	C₂×C₄×C₇₁	C₂²⋊C₈	1	1	yes	characteristic
41	C₂×C₄×C₇₁	C₂×C₁₆	1	1	yes	characteristic
42	C₂×Dic₇₁	—	2	1	no	—
43	C₂×Dic₇₁	—	2	1	no	—
44	C₂×Dic₇₁	—	4	2	no	—
45	C₈×C₇₁	—	2	1	no	—
46	C₈×C₇₁	#99	1	1	yes	characteristic
47	C₈×C₇₁	#101	1	1	yes	characteristic
48	C₇₁⋊C₈	—	4	1	no	—
49	C₂²×C₇₁	—	2	1	no	—
50	C₂²×C₇₁	—	4	1	no	—
51	C₂²×C₇₁	C₂²⋊C₁₆	1	1	yes	characteristic
52	C₄×C₇₁	—	2	1	no	—
53	C₄×C₇₁	#324	1	1	yes	characteristic
54	C₄×C₇₁	#349	1	1	yes	characteristic
55	Dic₇₁	—	2	2	no	—
56	Dic₇₁	—	4	1	no	—
57	#7364	—	71	1	no	maximal, Sylow
58	C₂×C₇₁	—	2	1	no	—
59	C₂×C₇₁	—	4	1	no	—
60	C₂×C₇₁	#1322	1	1	yes	characteristic
61	#2150	—	71	1	no	—
62	#1400	—	71	1	no	—
63	#1400	D₇₁	1	1	yes	characteristic
64	C₇₁	#7364	1	1	yes	characteristic, complemented, Sylow
65	C₂×M₃₂	—	71	1	no	—
66	C₂×M₃₂	C₂×D₇₁	1	1	yes	characteristic
67	C₂×C₄×C₈	—	71	1	no	—
68	M₆₄	—	2	1	no	—
69	M₆₄	—	142	1	no	—
70	C₄⋊C₁₆	—	142	1	no	—
71	C₄⋊C₁₆	—	142	1	no	—
72	C₂²×C₈	—	71	1	no	—
73	C₂²×C₈	C₄×D₇₁	1	1	yes	characteristic
74	C₂×C₄²	—	71	1	no	—
75	M₃₂	—	4	1	no	—
76	M₃₂	—	284	1	no	—
77	C₂×C₁₆	—	142	1	no	—
78	C₂×C₁₆	D₂₈₄	1	1	yes	characteristic
79	C₂×C₁₆	C₂²⋊D₇₁	1	1	yes	characteristic
80	C₄×C₈	—	142	1	no	—
81	C₄×C₈	—	142	1	no	—
82	C₃₂	—	4	1	no	—
83	C₃₂	—	284	1	no	—
84	C₂²×C₄	—	142	1	no	—
85	C₂²×C₄	C₈×D₇₁	1	1	yes	characteristic
86	C₂×C₈	—	2	1	no	—
87	C₂×C₈	—	2	1	no	—
88	C₂×C₈	—	284	1	no	—
89	C₂×C₈	—	142	1	no	—
90	C₂×C₈	(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
91	C₂×C₈	#11765848	1	1	yes	characteristic
92	C₄²	—	71	1	no	—
93	C₄²	—	71	1	no	—
94	C₄²	—	142	1	no	—
95	C₁₆	—	2	1	no	—
96	C₁₆	—	2	1	no	—
97	C₁₆	—	284	1	no	—
98	C₂³	#14556067	1	1	yes	characteristic
99	C₂×C₄	—	2	1	no	—
100	C₂×C₄	—	2	1	no	—
101	C₂×C₄	—	284	2	no	—
102	C₂×C₄	—	142	1	no	—
103	C₂×C₄	—	142	1	no	—
104	C₂×C₄	(C₂×D₇₁)⋊C₈	1	1	yes	characteristic
105	C₂×C₄	C₁₆×D₇₁	1	1	yes	characteristic
106	C₈	—	2	1	no	—
107	C₈	—	284	1	no	—
108	C₈	#14556042	1	1	yes	characteristic
109	C₈	#14556046	1	1	yes	characteristic
110	C₂²	—	2	1	no	—
111	C₂²	—	4	1	no	—
112	C₂²	(C₂×D₇₁)⋊C₁₆	1	1	yes	characteristic
113	C₄	—	2	1	no	—
114	C₄	—	284	1	no	—
115	C₄	—	142	2	no	—
116	C₄	#17985724	1	1	yes	characteristic
117	C₄	#17985780	1	1	yes	characteristic
118	C₂	—	2	1	no	—
119	C₂	—	4	1	no	—
120	C₂	#22586279	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22590095

C₂

yes

maximal, characteristic

#22586480

C₂

yes

maximal, characteristic

C₇₁×#1400

C₂

yes

maximal, characteristic

C₂×#14556068

C₄

yes

characteristic

C₂×C₈×Dic₇₁

C₄

yes

characteristic

#17985833

—

Dic₇₁⋊C₁₆

—

Dic₇₁⋊C₁₆

—

C₂×M₃₂×C₇₁

C₂²

yes

characteristic

M₆₄×C₇₁

—

C₂²×C₇₁⋊C₈

C₈

yes

characteristic

C₂×C₄×Dic₇₁

C₈

yes

characteristic

#14556068

—

C₂×C₇₁⋊C₁₆

—

C₈×Dic₇₁

—

C₈×Dic₇₁

—

C₇₁⋊C₃₂

—

C₂²×C₈×C₇₁

C₂×C₄

yes

characteristic

M₃₂×C₇₁

—

C₂×C₁₆×C₇₁

D₄

yes

characteristic

C₂×C₁₆×C₇₁

D₄

yes

characteristic

C₃₂×C₇₁

—

C₂²×Dic₇₁

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

C₄×Dic₇₁

—

C₄×Dic₇₁

C₁₆

yes

characteristic

C₄×Dic₇₁

C₁₆

yes

characteristic

C₇₁⋊C₁₆

—

C₂²×C₄×C₇₁

C₂×C₈

yes

characteristic

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₇₁

M₁₆

yes

characteristic

C₁₆×C₇₁

—

C₁₆×C₇₁

—

C₂³×C₇₁

M₃₂

yes

characteristic

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

C₂²⋊C₈

yes

characteristic

C₂×C₄×C₇₁

C₂×C₁₆

yes

characteristic

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₈×C₇₁

—

C₈×C₇₁

yes

C₈×C₇₁

yes

C₇₁⋊C₈

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

C₂²⋊C₁₆

yes

characteristic

C₄×C₇₁

—

C₄×C₇₁

yes

C₄×C₇₁

yes

Dic₇₁

—

Dic₇₁

—

#7364

—

maximal, Sylow

C₂×C₇₁

—

C₂×C₇₁

—

C₂×C₇₁

yes

—

—

D₇₁

yes

characteristic

C₇₁

#7364

yes

characteristic, complemented, Sylow

C₂×M₃₂

—

C₂×M₃₂

C₂×D₇₁

yes

characteristic

C₂×C₄×C₈

—

M₆₄

—

M₆₄

—

142

—

C₄⋊C₁₆

—

142

—

C₄⋊C₁₆

—

142

—

C₂²×C₈

—

C₂²×C₈

C₄×D₇₁

yes

characteristic

C₂×C₄²

—

M₃₂

—

M₃₂

—

284

—

C₂×C₁₆

—

142

—

C₂×C₁₆

D₂₈₄

yes

characteristic

C₂×C₁₆

C₂²⋊D₇₁

yes

characteristic

C₄×C₈

—

142

—

C₄×C₈

—

142

—

C₃₂

—

C₃₂

—

284

—

C₂²×C₄

—

142

—

C₂²×C₄

C₈×D₇₁

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

284

—

C₂×C₈

—

142

—

C₂×C₈

(C₂×D₇₁)⋊C₄

yes

characteristic

C₂×C₈

#11765848

yes

characteristic

C₄²

—

C₄²

—

C₄²

—

142

—

C₁₆

—

C₁₆

—

C₁₆

—

284

—

C₂³

#14556067

yes

characteristic

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

284

—

102

C₂×C₄

—

142

—

103

C₂×C₄

—

142

—

104

C₂×C₄

(C₂×D₇₁)⋊C₈

yes

characteristic

105

C₂×C₄

C₁₆×D₇₁

yes

characteristic

106

C₈

—

107

C₈

—

284

—

108

C₈

#14556042

yes

characteristic

109

C₈

#14556046

yes

characteristic

110

C₂²

—

111

C₂²

—

112

C₂²

(C₂×D₇₁)⋊C₁₆

yes

characteristic

113

C₄

—

114

C₄

—

284

—

115

C₄

—

142

—

116

C₄

#17985724

yes

characteristic

117

C₄

#17985780

yes

characteristic

118

C₂

—

119

C₂

—

120

C₂

#22586279

yes

characteristic

	subgroup	G/H
center	C₈	#14556042
commutator	C₂×C₄×C₇₁	C₂×C₁₆
Frattini	C₂×M₃₂	C₂×D₇₁
Fitting	C₇₁×#1400	C₂
socle	C₂×C₇₁	#1322
prime radical	C₂³×C₇₁	M₃₂
cosocle	C₂×M₃₂×C₇₁	C₂²
hypercenter	#1400	D₇₁
hypocenter	C₇₁	#7364
LCS	G ▷ C₂×C₄×C₇₁ ▷ C₂²×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₈ ◁ C₂×C₈ ◁ C₂×M₃₂ ◁ #1400

subgroup

G/H

center

C₈

#14556042

commutator

C₂×C₄×C₇₁

C₂×C₁₆

Frattini

C₂×M₃₂

C₂×D₇₁

Fitting

C₇₁×#1400

C₂

socle

C₂×C₇₁

#1322

prime radical

C₂³×C₇₁

M₃₂

cosocle

C₂×M₃₂×C₇₁

C₂²

D₇₁

C₇₁

G ▷ C₂×C₄×C₇₁ ▷ C₂²×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₈ ◁ C₂×C₈ ◁ C₂×M₃₂ ◁ #1400

N	H ≅ G/N	type
64. C₇₁	#7364	semidirect

H ≅ G/N

type

64. C₇₁

#7364

semidirect

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

subgroup	index
#22590095	2
#22586480	2
C₇₁×#1400	2
#7364	71

subgroup

index

#22590095

#22586480

C₇₁×#1400

#7364

Subgroups — #31884854

Conjugacy classes of subgroups

Complements

Maximal subgroups