Group #31918323 — subgroups

Subgroups — #31918323

Order 18176.

180 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 180 conjugacy classes of subgroups — too many to draw. Only its 53 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×Dic₁₁₃₆	C₂	1	1	yes	maximal, characteristic, complemented
2	#22590633	C₂	1	1	yes	maximal, characteristic
3	#22590466	C₂	1	2	yes	maximal
4	#22586532	C₂	1	2	yes	maximal, complemented
5	C₇₁×#1425	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×Dic₅₆₈	C₂²	1	2	yes	—
7	#17986415	C₂²	1	1	yes	characteristic
8	#17986275	—	2	2	no	—
9	Dic₁₁₃₆	—	2	1	no	—
10	Dic₁₁₃₆	—	2	1	no	—
11	#17985833	C₂²	1	1	yes	characteristic
12	#17985832	—	2	1	no	—
13	C₃₂×D₇₁	—	2	1	no	—
14	#17985814	—	2	2	no	—
15	C₂×C₃₂×C₇₁	C₂²	1	1	yes	characteristic
16	#368×C₇₁	C₂²	1	2	yes	—
17	#14556161	—	4	2	no	—
18	C₂×Dic₂₈₄	—	2	2	no	—
19	#14556134	D₄	1	1	yes	characteristic
20	Dic₅₆₈	—	2	2	no	—
21	Dic₅₆₈	—	2	2	no	—
22	#14556068	D₄	1	1	yes	characteristic
23	#14556067	—	2	1	no	—
24	C₁₆×D₇₁	—	2	1	no	—
25	#14556046	—	4	2	no	—
26	C₇₁⋊C₃₂	—	2	1	no	—
27	C₂×C₁₆×C₇₁	C₂³	1	1	yes	characteristic
28	#108×C₇₁	—	2	2	no	—
29	C₃₂×C₇₁	D₄	1	1	yes	characteristic
30	C₃₂×C₇₁	D₄	1	1	yes	characteristic
31	#11765865	D₈	1	1	yes	characteristic
32	C₂×Dic₁₄₂	—	4	2	no	—
33	Dic₂₈₄	—	4	2	no	—
34	Dic₂₈₄	—	4	2	no	—
35	Dic₂₈₄	—	8	2	no	—
36	#11765854	—	8	2	no	—
37	#11765849	D₈	1	1	yes	characteristic
38	#11765848	—	2	1	no	—
39	C₈×D₇₁	—	2	1	no	—
40	C₇₁⋊C₁₆	—	2	1	no	—
41	M₁₆×C₇₁	—	4	2	no	—
42	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
43	C₁₆×C₇₁	C₂×D₄	1	1	yes	characteristic
44	C₁₆×C₇₁	#41	1	1	yes	characteristic
45	C₂×C₄×C₇₁	C₂×D₈	1	1	yes	characteristic
46	C₂²⋊D₇₁	—	2	1	no	—
47	C₂×Dic₇₁	—	8	2	no	—
48	D₂₈₄	D₁₆	1	1	yes	characteristic
49	C₄×D₇₁	—	2	1	no	—
50	Dic₁₄₂	—	8	2	no	—
51	Dic₁₄₂	—	8	2	no	—
52	Dic₁₄₂	D₁₆	1	1	yes	characteristic
53	C₈×C₇₁	—	8	2	no	—
54	C₈×C₇₁	#121	1	1	yes	characteristic
55	C₈×C₇₁	#135	1	1	yes	characteristic
56	C₇₁⋊C₈	—	2	1	no	—
57	C₂²×C₇₁	#508	1	1	yes	characteristic
58	C₂×D₇₁	—	2	1	no	—
59	C₄×C₇₁	#466	1	1	yes	characteristic
60	C₄×C₇₁	C₂×D₁₆	1	1	yes	characteristic
61	Dic₇₁	—	2	1	no	—
62	Dic₇₁	—	16	2	no	—
63	#13701	—	71	1	no	maximal, Sylow
64	C₂×C₇₁	—	2	1	no	—
65	C₂×C₇₁	#2215	1	1	yes	characteristic
66	D₇₁	—	4	1	no	—
67	C₂×Q₆₄	—	71	1	no	—
68	#2258	—	71	1	no	—
69	#2222	—	71	2	no	—
70	#1425	D₇₁	1	1	yes	characteristic, complemented
71	#1420	—	71	2	no	—
72	C₇₁	#13701	1	1	yes	characteristic, complemented, Sylow
73	C₂×Q₃₂	—	71	2	no	—
74	#504	—	71	1	no	—
75	#473	—	142	2	no	—
76	Q₆₄	—	142	1	no	—
77	Q₆₄	—	142	1	no	—
78	M₆₄	—	71	1	no	—
79	M₆₄	—	142	1	no	—
80	C₂×C₃₂	—	142	1	no	—
81	C₂×C₃₂	C₂×D₇₁	1	1	yes	characteristic
82	#368	C₂×D₇₁	1	2	yes	—
83	#362	—	142	2	no	—
84	#137	—	284	2	no	—
85	C₂×Q₁₆	—	142	2	no	—
86	#131	—	71	1	no	—
87	Q₃₂	—	142	2	no	—
88	Q₃₂	—	142	2	no	—
89	M₃₂	—	71	1	no	—
90	M₃₂	—	142	1	no	—
91	C₂×C₁₆	—	142	1	no	—
92	C₂×C₁₆	C₂²×D₇₁	1	1	yes	characteristic
93	#108	—	2	2	no	—
94	#101	—	284	2	no	—
95	C₃₂	—	142	1	no	—
96	C₃₂	D₂₈₄	1	1	yes	characteristic
97	C₃₂	D₂₈₄	1	1	yes	characteristic
98	#41	—	71	1	no	—
99	C₂×Q₈	—	284	2	no	—
100	Q₁₆	—	284	2	no	—
101	Q₁₆	—	284	2	no	—
102	Q₁₆	—	568	2	no	—
103	SD₁₆	—	568	2	no	—
104	M₁₆	—	4	2	no	—
105	M₁₆	—	71	1	no	—
106	M₁₆	—	142	1	no	—
107	C₂×C₈	—	142	1	no	—
108	C₂×C₈	D₄×D₇₁	1	1	yes	characteristic
109	C₁₆	—	142	1	no	—
110	C₁₆	C₂×D₂₈₄	1	1	yes	characteristic
111	C₁₆	#11765866	1	1	yes	characteristic
112	Q₈	—	568	2	no	—
113	Q₈	—	568	2	no	—
114	Q₈	—	71	1	no	—
115	D₄	—	71	1	no	—
116	D₄	—	142	1	no	—
117	C₂×C₄	—	568	2	no	—
118	C₂×C₄	—	142	1	no	—
119	C₂×C₄	D₈×D₇₁	1	1	yes	characteristic
120	C₈	—	8	2	no	—
121	C₈	—	142	1	no	—
122	C₈	#14556103	1	1	yes	characteristic
123	C₈	#14556151	1	1	yes	characteristic
124	C₂²	—	142	1	no	—
125	C₂²	#17986432	1	1	yes	characteristic
126	C₄	—	1136	2	no	—
127	C₄	—	142	1	no	—
128	C₄	#17986236	1	1	yes	characteristic
129	C₄	D₁₆×D₇₁	1	1	yes	characteristic
130	C₂	—	2	1	no	—
131	C₂	—	284	1	no	—
132	C₂	#22590427	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×Dic₁₁₃₆

C₂

yes

maximal, characteristic, complemented

#22590633

C₂

yes

maximal, characteristic

#22590466

C₂

yes

maximal

#22586532

C₂

yes

maximal, complemented

C₇₁×#1425

C₂

yes

maximal, characteristic, complemented

C₂×Dic₅₆₈

C₂²

yes

—

#17986415

C₂²

yes

characteristic

#17986275

—

Dic₁₁₃₆

—

Dic₁₁₃₆

—

#17985833

C₂²

yes

characteristic

#17985832

—

C₃₂×D₇₁

—

#17985814

—

C₂×C₃₂×C₇₁

C₂²

yes

characteristic

#368×C₇₁

C₂²

yes

—

#14556161

—

C₂×Dic₂₈₄

—

#14556134

D₄

yes

characteristic

Dic₅₆₈

—

Dic₅₆₈

—

#14556068

D₄

yes

characteristic

#14556067

—

C₁₆×D₇₁

—

#14556046

—

C₇₁⋊C₃₂

—

C₂×C₁₆×C₇₁

C₂³

yes

characteristic

#108×C₇₁

—

C₃₂×C₇₁

D₄

yes

characteristic

C₃₂×C₇₁

D₄

yes

characteristic

#11765865

D₈

yes

characteristic

C₂×Dic₁₄₂

—

Dic₂₈₄

—

Dic₂₈₄

—

Dic₂₈₄

—

#11765854

—

#11765849

D₈

yes

characteristic

#11765848

—

C₈×D₇₁

—

C₇₁⋊C₁₆

—

M₁₆×C₇₁

—

C₂×C₈×C₇₁

C₂×D₄

yes

characteristic

C₁₆×C₇₁

C₂×D₄

yes

characteristic

C₁₆×C₇₁

#41

yes

characteristic

C₂×C₄×C₇₁

C₂×D₈

yes

characteristic

C₂²⋊D₇₁

—

C₂×Dic₇₁

—

D₂₈₄

D₁₆

yes

characteristic

C₄×D₇₁

—

Dic₁₄₂

—

Dic₁₄₂

—

Dic₁₄₂

D₁₆

yes

characteristic

C₈×C₇₁

—

C₈×C₇₁

yes

C₈×C₇₁

yes

C₇₁⋊C₈

—

C₂²×C₇₁

#508

yes

characteristic

C₂×D₇₁

—

C₄×C₇₁

#466

yes

characteristic

C₄×C₇₁

C₂×D₁₆

yes

characteristic

Dic₇₁

—

Dic₇₁

—

#13701

—

maximal, Sylow

C₂×C₇₁

—

C₂×C₇₁

#2215

yes

characteristic

D₇₁

—

C₂×Q₆₄

—

#2258

—

#2222

—

#1425

D₇₁

yes

characteristic, complemented

#1420

—

C₇₁

#13701

yes

characteristic, complemented, Sylow

C₂×Q₃₂

—

#504

—

#473

—

142

—

Q₆₄

—

142

—

Q₆₄

—

142

—

M₆₄

—

M₆₄

—

142

—

C₂×C₃₂

—

142

—

C₂×C₃₂

C₂×D₇₁

yes

characteristic

#368

C₂×D₇₁

yes

—

#362

—

142

—

#137

—

284

—

C₂×Q₁₆

—

142

—

#131

—

Q₃₂

—

142

—

Q₃₂

—

142

—

M₃₂

—

M₃₂

—

142

—

C₂×C₁₆

—

142

—

C₂×C₁₆

C₂²×D₇₁

yes

characteristic

#108

—

#101

—

284

—

C₃₂

—

142

—

C₃₂

D₂₈₄

yes

characteristic

C₃₂

D₂₈₄

yes

characteristic

#41

—

C₂×Q₈

—

284

—

100

Q₁₆

—

284

—

101

Q₁₆

—

284

—

102

Q₁₆

—

568

—

103

SD₁₆

—

568

—

104

M₁₆

—

105

M₁₆

—

106

M₁₆

—

142

—

107

C₂×C₈

—

142

—

108

C₂×C₈

D₄×D₇₁

yes

characteristic

109

C₁₆

—

142

—

110

C₁₆

C₂×D₂₈₄

yes

characteristic

111

C₁₆

#11765866

yes

characteristic

112

Q₈

—

568

—

113

Q₈

—

568

—

114

Q₈

—

115

D₄

—

116

D₄

—

142

—

117

C₂×C₄

—

568

—

118

C₂×C₄

—

142

—

119

C₂×C₄

D₈×D₇₁

yes

characteristic

120

C₈

—

121

C₈

—

142

—

122

C₈

#14556103

yes

characteristic

123

C₈

#14556151

yes

characteristic

124

C₂²

—

142

—

125

C₂²

#17986432

yes

characteristic

126

C₄

—

1136

—

127

C₄

—

142

—

128

C₄

#17986236

yes

characteristic

129

C₄

D₁₆×D₇₁

yes

characteristic

130

C₂

—

131

C₂

—

284

—

132

C₂

#22590427

yes

characteristic

	subgroup	G/H
center	C₂	#22590427
commutator, cosocle	C₂×C₁₆×C₇₁	C₂³
Frattini	C₂×C₁₆	C₂²×D₇₁
Fitting	C₇₁×#1425	C₂
socle	C₂×C₇₁	#2215
prime radical	#11765865	D₈
hypercenter	#1425	D₇₁
hypocenter	C₇₁	#13701
LCS	G ▷ C₂×C₁₆×C₇₁ ▷ C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ #1425

subgroup

G/H

center

C₂

#22590427

commutator, cosocle

C₂×C₁₆×C₇₁

C₂³

Frattini

C₂×C₁₆

C₂²×D₇₁

Fitting

C₇₁×#1425

C₂

C₂×C₇₁

D₈

D₇₁

C₇₁

G ▷ C₂×C₁₆×C₇₁ ▷ C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ #1425

N	H ≅ G/N	type
1. C₂×Dic₁₁₃₆	C₂	semidirect
4. #22586532	C₂	semidirect
5. C₇₁×#1425	C₂	semidirect
70. #1425	D₇₁	semidirect
72. C₇₁	#13701	semidirect

H ≅ G/N

type

1. C₂×Dic₁₁₃₆

C₂

semidirect

4. #22586532

C₂

semidirect

5. C₇₁×#1425

C₂

semidirect

70. #1425

D₇₁

semidirect

72. C₇₁

#13701

semidirect

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

subgroup	index
C₂×Dic₁₁₃₆	2
#22590633	2
#22590466	2
#22586532	2
C₇₁×#1425	2
#13701	71

subgroup

index

C₂×Dic₁₁₃₆

C₇₁×#1425

Subgroups — #31918323

Conjugacy classes of subgroups

Complements

Maximal subgroups