Group #31909048 — subgroups

Subgroups — #31909048

Order 18176.

220 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 143 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×#17985820	C₂	1	2	yes	maximal
2	#22586416	C₂	1	4	yes	maximal, complemented
3	C₄×M₃₂×C₇₁	C₂	1	1	yes	maximal, characteristic
4	C₂×#14556068	—	2	2	no	—
5	#17985820	—	2	2	no	—
6	#17985820	C₂²	1	4	yes	—
7	C₂×M₃₂×C₇₁	C₄	1	1	yes	characteristic
8	C₂×M₃₂×C₇₁	C₄	1	1	yes	characteristic
9	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
10	#346×C₇₁	C₄	1	2	yes	—
11	C₄×C₁₆×C₇₁	C₂²	1	2	yes	—
12	#14556068	—	2	4	no	—
13	#14556068	—	4	2	no	—
14	C₂×C₇₁⋊C₁₆	—	4	2	no	—
15	C₂²×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
16	C₂²×C₈×C₇₁	D₄	1	1	yes	characteristic
17	C₂×C₄²×C₇₁	Q₈	1	1	yes	characteristic
18	M₃₂×C₇₁	—	2	2	no	—
19	M₃₂×C₇₁	C₈	1	4	yes	—
20	C₂×C₁₆×C₇₁	C₂×C₄	1	2	yes	—
21	C₂×C₁₆×C₇₁	C₂×C₄	1	2	yes	—
22	C₄×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
23	C₄×C₈×C₇₁	D₄	1	1	yes	characteristic
24	C₄×C₈×C₇₁	Q₈	1	1	yes	characteristic
25	C₄×C₈×C₇₁	C₂³	1	1	yes	characteristic
26	C₇₁⋊C₁₆	—	4	4	no	—
27	C₂²×C₄×C₇₁	—	2	1	no	—
28	C₂²×C₄×C₇₁	C₄⋊C₄	1	1	yes	characteristic
29	C₂×C₈×C₇₁	C₄⋊C₄	1	1	yes	characteristic
30	C₂×C₈×C₇₁	C₄⋊C₄	1	1	yes	characteristic
31	C₂×C₈×C₇₁	C₂×C₈	1	2	yes	—
32	C₂×C₈×C₇₁	M₁₆	1	2	yes	—
33	C₂×C₈×C₇₁	D₈	1	2	yes	—
34	C₂×C₈×C₇₁	Q₁₆	1	2	yes	—
35	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
36	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
37	C₄²×C₇₁	—	2	1	no	—
38	C₄²×C₇₁	C₄⋊C₄	1	1	yes	characteristic
39	C₄²×C₇₁	C₂×Q₈	1	1	yes	characteristic
40	C₁₆×C₇₁	—	2	2	no	—
41	C₁₆×C₇₁	C₂×C₈	1	4	yes	—
42	C₂³×C₇₁	#108	1	1	yes	characteristic
43	C₂×C₄×C₇₁	—	2	1	no	—
44	C₂×C₄×C₇₁	—	2	1	no	—
45	C₂×C₄×C₇₁	—	4	2	no	—
46	C₂×C₄×C₇₁	C₄⋊C₈	1	2	yes	—
47	C₂×C₄×C₇₁	#107	1	2	yes	—
48	C₂×C₄×C₇₁	#108	1	1	yes	characteristic
49	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
50	C₈×C₇₁	C₄⋊C₈	1	2	yes	—
51	C₈×C₇₁	#107	1	2	yes	—
52	C₈×C₇₁	C₂²×C₈	1	1	yes	characteristic
53	C₈×C₇₁	C₂×M₁₆	1	1	yes	characteristic
54	C₈×C₇₁	C₂×D₈	1	1	yes	characteristic
55	C₈×C₇₁	C₂×Q₁₆	1	1	yes	characteristic
56	C₂²×C₇₁	—	4	1	no	—
57	C₂²×C₇₁	#335	1	2	yes	—
58	C₂²×C₇₁	C₂×#108	1	1	yes	characteristic
59	C₄×C₇₁	—	2	2	no	—
60	C₄×C₇₁	—	4	1	no	—
61	C₄×C₇₁	#335	1	2	yes	—
62	C₄×C₇₁	C₂×C₄⋊C₈	1	1	yes	characteristic
63	C₄×C₇₁	C₂×#107	1	1	yes	characteristic
64	#12011	—	71	1	no	maximal, Sylow
65	C₂×C₇₁	—	2	1	no	—
66	C₂×C₇₁	—	2	2	no	—
67	C₂×C₇₁	C₂×#335	1	1	yes	characteristic
68	C₂×#364	—	71	2	no	—
69	C₄×M₃₂	D₇₁	1	1	yes	characteristic
70	#1373	—	71	4	no	—
71	C₇₁	#12011	1	1	yes	characteristic, complemented, Sylow
72	C₂×M₃₂	—	142	2	no	—
73	C₂×M₃₂	Dic₇₁	1	1	yes	characteristic
74	C₂×M₃₂	Dic₇₁	1	1	yes	characteristic
75	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
76	#364	—	71	4	no	—
77	#364	—	142	2	no	—
78	#346	Dic₇₁	1	2	yes	—
79	C₄×C₁₆	C₂×D₇₁	1	2	yes	—
80	C₂²×C₈	D₂₈₄	1	1	yes	characteristic
81	C₂²×C₈	C₂×Dic₇₁	1	1	yes	characteristic
82	C₂×C₄²	Dic₁₄₂	1	1	yes	characteristic
83	M₃₂	—	2	2	no	—
84	M₃₂	—	284	2	no	—
85	M₃₂	—	142	4	no	—
86	M₃₂	C₇₁⋊C₈	1	4	yes	—
87	C₂×C₁₆	—	284	2	no	—
88	C₂×C₁₆	C₂×Dic₇₁	1	2	yes	—
89	C₂×C₁₆	C₂×Dic₇₁	1	2	yes	—
90	C₄×C₈	Dic₁₄₂	1	1	yes	characteristic
91	C₄×C₈	D₂₈₄	1	1	yes	characteristic
92	C₄×C₈	C₂×Dic₇₁	1	1	yes	characteristic
93	C₄×C₈	C₂²×D₇₁	1	1	yes	characteristic
94	C₂²×C₄	—	2	1	no	—
95	C₂²×C₄	C₄⋊Dic₇₁	1	1	yes	characteristic
96	C₂×C₈	C₄⋊Dic₇₁	1	1	yes	characteristic
97	C₂×C₈	C₄⋊Dic₇₁	1	1	yes	characteristic
98	C₂×C₈	C₂×C₇₁⋊C₈	1	2	yes	—
99	C₂×C₈	#11765849	1	2	yes	—
100	C₂×C₈	D₅₆₈	1	2	yes	—
101	C₂×C₈	Dic₂₈₄	1	2	yes	—
102	C₂×C₈	C₂²×Dic₇₁	1	1	yes	characteristic
103	C₂×C₈	C₂×D₂₈₄	1	1	yes	characteristic
104	C₄²	—	2	1	no	—
105	C₄²	C₄⋊Dic₇₁	1	1	yes	characteristic
106	C₄²	C₂×Dic₁₄₂	1	1	yes	characteristic
107	C₁₆	—	2	2	no	—
108	C₁₆	—	284	4	no	—
109	C₁₆	C₂×C₇₁⋊C₈	1	4	yes	—
110	C₂³	#14556064	1	1	yes	characteristic
111	C₂×C₄	—	2	1	no	—
112	C₂×C₄	—	2	1	no	—
113	C₂×C₄	—	4	2	no	—
114	C₂×C₄	C₄⋊(C₇₁⋊C₈)	1	2	yes	—
115	C₂×C₄	#14556062	1	2	yes	—
116	C₂×C₄	#14556064	1	1	yes	characteristic
117	C₂×C₄	C₂×C₄⋊Dic₇₁	1	1	yes	characteristic
118	C₈	C₄⋊(C₇₁⋊C₈)	1	2	yes	—
119	C₈	#14556062	1	2	yes	—
120	C₈	C₂²×C₇₁⋊C₈	1	1	yes	characteristic
121	C₈	C₂×#11765849	1	1	yes	characteristic
122	C₈	C₂×D₅₆₈	1	1	yes	characteristic
123	C₈	C₂×Dic₂₈₄	1	1	yes	characteristic
124	C₂²	—	4	1	no	—
125	C₂²	#17985752	1	2	yes	—
126	C₂²	C₂×#14556064	1	1	yes	characteristic
127	C₄	—	2	2	no	—
128	C₄	—	4	1	no	—
129	C₄	#17985752	1	2	yes	—
130	C₄	C₂×C₄⋊(C₇₁⋊C₈)	1	1	yes	characteristic
131	C₄	C₂×#14556062	1	1	yes	characteristic
132	C₂	—	2	1	no	—
133	C₂	—	2	2	no	—
134	C₂	C₂×#17985752	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#17985820

C₂

yes

maximal

#22586416

C₂

yes

maximal, complemented

C₄×M₃₂×C₇₁

C₂

yes

maximal, characteristic

C₂×#14556068

—

#17985820

—

#17985820

C₂²

yes

—

C₂×M₃₂×C₇₁

C₄

yes

characteristic

C₂×M₃₂×C₇₁

C₄

yes

characteristic

C₂×C₄×C₈×C₇₁

C₂²

yes

characteristic

#346×C₇₁

C₄

yes

—

C₄×C₁₆×C₇₁

C₂²

yes

—

#14556068

—

#14556068

—

C₂×C₇₁⋊C₁₆

—

C₂²×C₈×C₇₁

C₂×C₄

yes

characteristic

C₂²×C₈×C₇₁

D₄

yes

characteristic

C₂×C₄²×C₇₁

Q₈

yes

characteristic

M₃₂×C₇₁

—

M₃₂×C₇₁

C₈

yes

—

C₂×C₁₆×C₇₁

C₂×C₄

yes

—

C₂×C₁₆×C₇₁

C₂×C₄

yes

—

C₄×C₈×C₇₁

C₂×C₄

yes

characteristic

C₄×C₈×C₇₁

D₄

yes

characteristic

C₄×C₈×C₇₁

Q₈

yes

characteristic

C₄×C₈×C₇₁

C₂³

yes

characteristic

C₇₁⋊C₁₆

—

C₂²×C₄×C₇₁

—

C₂²×C₄×C₇₁

C₄⋊C₄

yes

characteristic

C₂×C₈×C₇₁

C₄⋊C₄

yes

characteristic

C₂×C₈×C₇₁

C₄⋊C₄

yes

characteristic

C₂×C₈×C₇₁

C₂×C₈

yes

—

C₂×C₈×C₇₁

M₁₆

yes

—

C₂×C₈×C₇₁

D₈

yes

—

C₂×C₈×C₇₁

Q₁₆

yes

—

C₂×C₈×C₇₁

C₂²×C₄

yes

characteristic

C₂×C₈×C₇₁

C₂×D₄

yes

characteristic

C₄²×C₇₁

—

C₄²×C₇₁

C₄⋊C₄

yes

characteristic

C₄²×C₇₁

C₂×Q₈

yes

characteristic

C₁₆×C₇₁

—

C₁₆×C₇₁

C₂×C₈

yes

—

C₂³×C₇₁

#108

yes

characteristic

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

C₄⋊C₈

yes

—

C₂×C₄×C₇₁

#107

yes

—

C₂×C₄×C₇₁

#108

yes

characteristic

C₂×C₄×C₇₁

C₂×C₄⋊C₄

yes

characteristic

C₈×C₇₁

C₄⋊C₈

yes

—

C₈×C₇₁

#107

yes

—

C₈×C₇₁

C₂²×C₈

yes

characteristic

C₈×C₇₁

C₂×M₁₆

yes

characteristic

C₈×C₇₁

C₂×D₈

yes

characteristic

C₈×C₇₁

C₂×Q₁₆

yes

characteristic

C₂²×C₇₁

—

C₂²×C₇₁

#335

yes

—

C₂²×C₇₁

C₂×#108

yes

characteristic

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

#335

yes

—

C₄×C₇₁

C₂×C₄⋊C₈

yes

C₄×C₇₁

yes

—

C₂×C₇₁

—

C₂×C₇₁

—

C₂×C₇₁

C₂×#335

yes

characteristic

C₂×#364

—

C₄×M₃₂

D₇₁

yes

characteristic

#1373

—

C₇₁

#12011

yes

characteristic, complemented, Sylow

C₂×M₃₂

—

142

—

C₂×M₃₂

Dic₇₁

yes

characteristic

C₂×M₃₂

Dic₇₁

yes

characteristic

C₂×C₄×C₈

C₂×D₇₁

yes

characteristic

#364

—

#364

—

142

—

#346

Dic₇₁

yes

—

C₄×C₁₆

C₂×D₇₁

yes

—

C₂²×C₈

D₂₈₄

yes

characteristic

C₂²×C₈

C₂×Dic₇₁

yes

characteristic

C₂×C₄²

Dic₁₄₂

yes

characteristic

M₃₂

—

M₃₂

—

284

—

M₃₂

—

142

—

M₃₂

C₇₁⋊C₈

yes

—

C₂×C₁₆

—

284

—

C₂×C₁₆

C₂×Dic₇₁

yes

—

C₂×C₁₆

C₂×Dic₇₁

yes

—

C₄×C₈

Dic₁₄₂

yes

characteristic

C₄×C₈

D₂₈₄

yes

characteristic

C₄×C₈

C₂×Dic₇₁

yes

characteristic

C₄×C₈

C₂²×D₇₁

yes

characteristic

C₂²×C₄

—

C₂²×C₄

C₄⋊Dic₇₁

yes

characteristic

C₂×C₈

C₄⋊Dic₇₁

yes

characteristic

C₂×C₈

C₄⋊Dic₇₁

yes

characteristic

C₂×C₈

C₂×C₇₁⋊C₈

yes

—

C₂×C₈

#11765849

yes

—

100

C₂×C₈

D₅₆₈

yes

—

101

C₂×C₈

Dic₂₈₄

yes

—

102

C₂×C₈

C₂²×Dic₇₁

yes

characteristic

103

C₂×C₈

C₂×D₂₈₄

yes

characteristic

104

C₄²

—

105

C₄²

C₄⋊Dic₇₁

yes

characteristic

106

C₄²

C₂×Dic₁₄₂

yes

characteristic

107

C₁₆

—

108

C₁₆

—

284

—

109

C₁₆

C₂×C₇₁⋊C₈

yes

—

110

C₂³

#14556064

yes

characteristic

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

C₄⋊(C₇₁⋊C₈)

yes

—

115

C₂×C₄

#14556062

yes

—

116

C₂×C₄

#14556064

yes

characteristic

117

C₂×C₄

C₂×C₄⋊Dic₇₁

yes

characteristic

118

C₈

C₄⋊(C₇₁⋊C₈)

yes

—

119

C₈

#14556062

yes

—

120

C₈

C₂²×C₇₁⋊C₈

yes

characteristic

121

C₈

C₂×#11765849

yes

characteristic

122

C₈

C₂×D₅₆₈

yes

characteristic

123

C₈

C₂×Dic₂₈₄

yes

characteristic

124

C₂²

—

125

C₂²

#17985752

yes

—

126

C₂²

C₂×#14556064

yes

characteristic

127

C₄

—

128

C₄

—

129

C₄

#17985752

yes

—

130

C₄

C₂×C₄⋊(C₇₁⋊C₈)

yes

characteristic

131

C₄

C₂×#14556062

yes

characteristic

132

C₂

—

133

C₂

—

134

C₂

C₂×#17985752

yes

characteristic

	subgroup	G/H
center	C₈	C₂×D₅₆₈
commutator	C₈×C₇₁	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₇₁
Fitting	C₄×M₃₂×C₇₁	C₂
socle	C₂×C₇₁	C₂×#335
prime radical	C₂³×C₇₁	#108
cosocle	C₄×C₈×C₇₁	C₂³
hypercenter	C₄×M₃₂	D₇₁
hypocenter	C₇₁	#12011
LCS	G ▷ C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ C₂×C₄×C₈ ◁ C₄×M₃₂

subgroup

G/H

center

C₈

C₂×D₅₆₈

commutator

C₈×C₇₁

C₂²×C₈

Frattini

C₄×C₈

C₂²×D₇₁

Fitting

C₄×M₃₂×C₇₁

C₂

C₂×C₇₁

C₂³×C₇₁

C₄×C₈×C₇₁

C₂³

hypercenter

C₄×M₃₂

D₇₁

hypocenter

C₇₁

#12011

LCS

G ▷ C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₈ ◁ C₂²×C₈ ◁ C₂×C₄×C₈ ◁ C₄×M₃₂

N	H ≅ G/N	type
2. #22586416	C₂	semidirect
71. C₇₁	#12011	semidirect

H ≅ G/N

type

2. #22586416

C₂

semidirect

71. C₇₁

#12011

semidirect

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

subgroup	index
C₂×#17985820	2
#22586416	2
C₄×M₃₂×C₇₁	2
#12011	71

subgroup

index

C₂×#17985820

#22586416

C₄×M₃₂×C₇₁

#12011

Subgroups — #31909048

Conjugacy classes of subgroups

Complements

Maximal subgroups