Group #31884429 — subgroups

Subgroups — #31884429

Order 18176.

210 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 210 conjugacy classes of subgroups — too many to draw. Only its 95 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	#22590143	C₂	1	2	yes	maximal
2	C₂×#322×C₇₁	C₂	1	1	yes	maximal, characteristic
3	C₄⋊(C₇₁⋊C₁₆)	—	2	2	no	—
4	C₂²⋊(C₇₁⋊C₁₆)	C₄	1	4	yes	—
5	#17985775	—	2	2	no	—
6	C₂×C₄×C₈×C₇₁	C₄	1	1	yes	characteristic
7	C₂×C₄×C₈×C₇₁	C₄	1	1	yes	characteristic
8	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
9	#322×C₇₁	—	2	2	no	—
10	C₂×C₇₁⋊C₁₆	—	2	4	no	—
11	C₂²×C₈×C₇₁	—	2	1	no	—
12	C₂²×C₈×C₇₁	—	2	1	no	—
13	C₂²×C₈×C₇₁	C₂×C₄	1	2	yes	—
14	C₂×C₄²×C₇₁	C₂×C₄	1	1	yes	characteristic
15	C₄×C₈×C₇₁	—	2	1	no	—
16	C₄×C₈×C₇₁	—	2	1	no	—
17	C₄×C₈×C₇₁	C₈	1	2	yes	—
18	C₄×C₈×C₇₁	C₈	1	2	yes	—
19	C₄×C₈×C₇₁	D₄	1	2	yes	—
20	C₄×C₈×C₇₁	D₄	1	1	yes	characteristic
21	C₄×C₈×C₇₁	Q₈	1	1	yes	characteristic
22	C₇₁⋊C₁₆	—	4	4	no	—
23	C₂²×C₄×C₇₁	C₄²	1	1	yes	characteristic
24	C₂²×C₄×C₇₁	M₁₆	1	1	yes	characteristic
25	C₂²×C₄×C₇₁	M₁₆	1	1	yes	characteristic
26	C₂×C₈×C₇₁	—	2	4	no	—
27	C₂×C₈×C₇₁	—	2	2	no	—
28	C₂×C₈×C₇₁	—	2	2	no	—
29	C₂×C₈×C₇₁	—	4	2	no	—
30	C₂×C₈×C₇₁	—	4	2	no	—
31	C₂×C₈×C₇₁	C₂²⋊C₄	1	2	yes	—
32	C₂×C₈×C₇₁	C₄⋊C₄	1	2	yes	—
33	C₄²×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
34	C₄²×C₇₁	C₄⋊C₄	1	1	yes	characteristic
35	C₄²×C₇₁	C₂×C₈	1	1	yes	characteristic
36	C₄²×C₇₁	C₂×C₈	1	1	yes	characteristic
37	C₂³×C₇₁	#97	1	1	yes	characteristic
38	C₂×C₄×C₇₁	—	2	2	no	—
39	C₂×C₄×C₇₁	—	2	2	no	—
40	C₂×C₄×C₇₁	—	2	2	no	—
41	C₂×C₄×C₇₁	#95	1	1	yes	characteristic
42	C₂×C₄×C₇₁	C₄×C₈	1	1	yes	characteristic
43	C₂×C₄×C₇₁	C₂²⋊C₈	1	1	yes	characteristic
44	C₂×C₄×C₇₁	C₂²⋊C₈	1	1	yes	characteristic
45	C₂×C₄×C₇₁	C₄⋊C₈	1	1	yes	characteristic
46	C₂×C₄×C₇₁	C₄⋊C₈	1	1	yes	characteristic
47	C₈×C₇₁	—	2	4	no	—
48	C₈×C₇₁	—	2	2	no	—
49	C₈×C₇₁	—	2	2	no	—
50	C₈×C₇₁	—	4	2	no	—
51	C₂²×C₇₁	—	2	1	no	—
52	C₂²×C₇₁	—	2	1	no	—
53	C₂²×C₇₁	#336	1	1	yes	characteristic
54	C₂²×C₇₁	#347	1	2	yes	—
55	C₄×C₇₁	—	2	1	no	—
56	C₄×C₇₁	—	2	1	no	—
57	C₄×C₇₁	—	2	1	no	—
58	C₄×C₇₁	—	2	1	no	—
59	C₄×C₇₁	#344	1	2	yes	—
60	C₄×C₇₁	#364	1	2	yes	—
61	#7196	—	71	1	no	maximal, Sylow
62	C₂×C₇₁	—	2	2	no	—
63	C₂×C₇₁	#1281	1	1	yes	characteristic
64	C₂×C₇₁	#1375	1	1	yes	characteristic
65	C₂×C₇₁	#1376	1	1	yes	characteristic
66	#2163	—	71	2	no	—
67	C₂×#322	D₇₁	1	1	yes	characteristic
68	C₇₁	#7196	1	1	yes	characteristic, complemented, Sylow
69	C₂×C₄×C₈	Dic₇₁	1	1	yes	characteristic
70	C₂×C₄×C₈	Dic₇₁	1	1	yes	characteristic
71	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
72	C₄⋊C₁₆	—	142	2	no	—
73	C₂²⋊C₁₆	—	71	4	no	—
74	#346	—	142	2	no	—
75	#322	—	2	2	no	—
76	C₂²×C₈	—	2	1	no	—
77	C₂²×C₈	—	2	1	no	—
78	C₂²×C₈	C₄×D₇₁	1	2	yes	—
79	C₂×C₄²	C₂×Dic₇₁	1	1	yes	characteristic
80	C₂×C₁₆	—	142	4	no	—
81	C₄×C₈	—	2	1	no	—
82	C₄×C₈	—	2	1	no	—
83	C₄×C₈	C₇₁⋊C₈	1	2	yes	—
84	C₄×C₈	C₇₁⋊C₈	1	2	yes	—
85	C₄×C₈	Dic₁₄₂	1	1	yes	characteristic
86	C₄×C₈	D₂₈₄	1	1	yes	characteristic
87	C₄×C₈	C₂²⋊D₇₁	1	2	yes	—
88	C₂²×C₄	C₄×Dic₇₁	1	1	yes	characteristic
89	C₂²×C₄	#11765849	1	1	yes	characteristic
90	C₂²×C₄	#11765849	1	1	yes	characteristic
91	C₂×C₈	—	2	4	no	—
92	C₂×C₈	—	2	2	no	—
93	C₂×C₈	—	2	2	no	—
94	C₂×C₈	—	4	2	no	—
95	C₂×C₈	—	4	2	no	—
96	C₂×C₈	(C₂×D₇₁)⋊C₄	1	2	yes	—
97	C₂×C₈	Dic₇₁⋊C₄	1	2	yes	—
98	C₄²	C₂²⋊Dic₇₁	1	1	yes	characteristic
99	C₄²	C₄⋊Dic₇₁	1	1	yes	characteristic
100	C₄²	C₂×C₇₁⋊C₈	1	1	yes	characteristic
101	C₄²	C₂×C₇₁⋊C₈	1	1	yes	characteristic
102	C₁₆	—	284	4	no	—
103	C₂³	#14556039	1	1	yes	characteristic
104	C₂×C₄	—	2	2	no	—
105	C₂×C₄	—	2	2	no	—
106	C₂×C₄	—	2	2	no	—
107	C₂×C₄	C₇₁⋊#95	1	1	yes	characteristic
108	C₂×C₄	C₄×C₇₁⋊C₈	1	1	yes	characteristic
109	C₂×C₄	C₂²⋊(C₇₁⋊C₈)	1	1	yes	characteristic
110	C₂×C₄	C₂²⋊(C₇₁⋊C₈)	1	1	yes	characteristic
111	C₂×C₄	C₄⋊(C₇₁⋊C₈)	1	1	yes	characteristic
112	C₂×C₄	C₄⋊(C₇₁⋊C₈)	1	1	yes	characteristic
113	C₈	—	2	4	no	—
114	C₈	—	2	2	no	—
115	C₈	—	2	2	no	—
116	C₈	—	4	2	no	—
117	C₂²	—	2	1	no	—
118	C₂²	—	2	1	no	—
119	C₂²	#17985754	1	1	yes	characteristic
120	C₂²	#17985777	1	2	yes	—
121	C₄	—	2	1	no	—
122	C₄	—	2	1	no	—
123	C₄	—	2	1	no	—
124	C₄	—	2	1	no	—
125	C₄	#17985770	1	2	yes	—
126	C₄	#17985820	1	2	yes	—
127	C₂	—	2	2	no	—
128	C₂	#22586185	1	1	yes	characteristic
129	C₂	#22586420	1	1	yes	characteristic
130	C₂	#22586422	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22590143

C₂

yes

maximal

C₂×#322×C₇₁

C₂

yes

maximal, characteristic

C₄⋊(C₇₁⋊C₁₆)

—

C₂²⋊(C₇₁⋊C₁₆)

C₄

yes

—

#17985775

—

C₂×C₄×C₈×C₇₁

C₄

yes

characteristic

C₂×C₄×C₈×C₇₁

C₄

yes

characteristic

C₂×C₄×C₈×C₇₁

C₂²

yes

characteristic

#322×C₇₁

—

C₂×C₇₁⋊C₁₆

—

C₂²×C₈×C₇₁

—

C₂²×C₈×C₇₁

—

C₂²×C₈×C₇₁

C₂×C₄

yes

—

C₂×C₄²×C₇₁

C₂×C₄

yes

characteristic

C₄×C₈×C₇₁

—

C₄×C₈×C₇₁

—

C₄×C₈×C₇₁

C₈

yes

—

C₄×C₈×C₇₁

C₈

yes

—

C₄×C₈×C₇₁

D₄

yes

—

C₄×C₈×C₇₁

D₄

yes

characteristic

C₄×C₈×C₇₁

Q₈

yes

characteristic

C₇₁⋊C₁₆

—

C₂²×C₄×C₇₁

C₄²

yes

characteristic

C₂²×C₄×C₇₁

M₁₆

yes

characteristic

C₂²×C₄×C₇₁

M₁₆

yes

characteristic

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

C₂²⋊C₄

yes

—

C₂×C₈×C₇₁

C₄⋊C₄

yes

—

C₄²×C₇₁

C₂²⋊C₄

yes

characteristic

C₄²×C₇₁

C₄⋊C₄

yes

characteristic

C₄²×C₇₁

C₂×C₈

yes

characteristic

C₄²×C₇₁

C₂×C₈

yes

characteristic

C₂³×C₇₁

#97

yes

characteristic

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

#95

yes

characteristic

C₂×C₄×C₇₁

C₄×C₈

yes

characteristic

C₂×C₄×C₇₁

C₂²⋊C₈

yes

characteristic

C₂×C₄×C₇₁

C₂²⋊C₈

yes

characteristic

C₂×C₄×C₇₁

C₄⋊C₈

yes

characteristic

C₂×C₄×C₇₁

C₄⋊C₈

yes

characteristic

C₈×C₇₁

—

C₈×C₇₁

—

C₈×C₇₁

—

C₈×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

#336

yes

characteristic

C₂²×C₇₁

#347

yes

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

#344

yes

—

C₄×C₇₁

#364

yes

—

#7196

—

maximal, Sylow

C₂×C₇₁

—

C₂×C₇₁

yes

C₂×C₇₁

yes

C₂×C₇₁

yes

—

D₇₁

yes

characteristic

C₇₁

#7196

yes

characteristic, complemented, Sylow

C₂×C₄×C₈

Dic₇₁

yes

characteristic

C₂×C₄×C₈

Dic₇₁

yes

characteristic

C₂×C₄×C₈

C₂×D₇₁

yes

characteristic

C₄⋊C₁₆

—

142

—

C₂²⋊C₁₆

—

#346

—

142

—

#322

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

C₄×D₇₁

yes

—

C₂×C₄²

C₂×Dic₇₁

yes

characteristic

C₂×C₁₆

—

142

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₇₁⋊C₈

yes

—

C₄×C₈

C₇₁⋊C₈

yes

—

C₄×C₈

Dic₁₄₂

yes

characteristic

C₄×C₈

D₂₈₄

yes

characteristic

C₄×C₈

C₂²⋊D₇₁

yes

—

C₂²×C₄

C₄×Dic₇₁

yes

C₂²×C₄

yes

C₂²×C₄

yes

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

(C₂×D₇₁)⋊C₄

yes

—

C₂×C₈

Dic₇₁⋊C₄

yes

—

C₄²

C₂²⋊Dic₇₁

yes

characteristic

C₄²

C₄⋊Dic₇₁

yes

characteristic

100

C₄²

C₂×C₇₁⋊C₈

yes

characteristic

101

C₄²

C₂×C₇₁⋊C₈

yes

characteristic

102

C₁₆

—

284

—

103

C₂³

#14556039

yes

characteristic

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

C₇₁⋊#95

yes

characteristic

108

C₂×C₄

C₄×C₇₁⋊C₈

yes

characteristic

109

C₂×C₄

C₂²⋊(C₇₁⋊C₈)

yes

characteristic

110

C₂×C₄

C₂²⋊(C₇₁⋊C₈)

yes

characteristic

111

C₂×C₄

C₄⋊(C₇₁⋊C₈)

yes

characteristic

112

C₂×C₄

C₄⋊(C₇₁⋊C₈)

yes

characteristic

113

C₈

—

114

C₈

—

115

C₈

—

116

C₈

—

117

C₂²

—

118

C₂²

—

119

C₂²

#17985754

yes

characteristic

120

C₂²

#17985777

yes

—

121

C₄

—

122

C₄

—

123

C₄

—

124

C₄

—

125

C₄

#17985770

yes

—

126

C₄

#17985820

yes

—

127

C₂

—

128

C₂

#22586185

yes

characteristic

129

C₂

#22586420

yes

characteristic

130

C₂

#22586422

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₇₁⋊#95
commutator	C₂×C₄×C₇₁	C₄×C₈
Frattini	C₂×C₄×C₈	C₂×D₇₁
Fitting	C₂×#322×C₇₁	C₂
socle	C₂²×C₇₁	#336
prime radical	C₂³×C₇₁	#97
cosocle	C₂×C₄×C₈×C₇₁	C₂²
hypercenter	C₂×#322	D₇₁
hypocenter	C₇₁	#7196
LCS	G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#322

subgroup

G/H

center

C₂×C₄

C₇₁⋊#95

commutator

C₂×C₄×C₇₁

C₄×C₈

Frattini

C₂×C₄×C₈

C₂×D₇₁

Fitting

C₂×#322×C₇₁

C₂

C₂²×C₇₁

C₂³×C₇₁

C₂×C₄×C₈×C₇₁

C₂²

D₇₁

C₇₁

G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#322

N	H ≅ G/N	type
68. C₇₁	#7196	semidirect

H ≅ G/N

type

68. C₇₁

#7196

semidirect

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

subgroup	index
#22590143	2
C₂×#322×C₇₁	2
#7196	71

subgroup

index

#22590143

C₂×#322×C₇₁

#7196

Subgroups — #31884429

Conjugacy classes of subgroups

Complements

Maximal subgroups