Group #22589115 — subgroups

Subgroups — #22589115

Order 9088.

200 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 97 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×#14556063	C₂	1	2	yes	maximal
2	#17986023	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄⋊(C₇₁⋊C₈)	C₂	1	1	yes	maximal, characteristic
4	#17985763	C₂	1	2	yes	maximal
5	C₄×M₁₆×C₇₁	C₂	1	1	yes	maximal, characteristic
6	C₂×#11765849	C₂²	1	2	yes	—
7	C₂²×C₇₁⋊C₈	C₂²	1	2	yes	—
8	#14556063	—	2	4	no	—
9	C₄⋊(C₇₁⋊C₈)	—	2	1	no	—
10	C₄⋊(C₇₁⋊C₈)	—	2	1	no	—
11	C₄⋊(C₇₁⋊C₈)	C₄	1	2	yes	—
12	C₄⋊(C₇₁⋊C₈)	C₄	1	2	yes	—
13	C₂×M₁₆×C₇₁	C₂²	1	1	yes	characteristic
14	C₂×M₁₆×C₇₁	C₂²	1	1	yes	characteristic
15	C₂×C₄²×C₇₁	C₂²	1	1	yes	characteristic
16	#97×C₇₁	—	2	1	no	—
17	C₄×C₈×C₇₁	—	2	1	no	—
18	#11765849	—	2	2	no	—
19	#11765849	—	2	2	no	—
20	C₂×C₇₁⋊C₈	—	2	2	no	—
21	C₂×C₇₁⋊C₈	—	2	2	no	—
22	C₂×C₇₁⋊C₈	—	2	2	no	—
23	C₂×C₇₁⋊C₈	C₂×C₄	1	4	yes	—
24	C₂²×C₄×C₇₁	Q₈	1	1	yes	characteristic
25	C₂²×C₄×C₇₁	Q₈	1	1	yes	characteristic
26	C₂²×C₄×C₇₁	C₂³	1	1	yes	characteristic
27	M₁₆×C₇₁	—	2	1	no	—
28	M₁₆×C₇₁	—	2	1	no	—
29	M₁₆×C₇₁	D₄	1	2	yes	—
30	M₁₆×C₇₁	Q₈	1	2	yes	—
31	C₂×C₈×C₇₁	—	2	1	no	—
32	C₂×C₈×C₇₁	—	2	1	no	—
33	C₄²×C₇₁	C₂×C₄	1	1	yes	characteristic
34	C₄²×C₇₁	C₂×C₄	1	1	yes	characteristic
35	C₄²×C₇₁	D₄	1	1	yes	characteristic
36	C₄²×C₇₁	D₄	1	1	yes	characteristic
37	C₂³×C₇₁	C₂×Q₈	1	1	yes	characteristic
38	C₂×C₄×C₇₁	—	2	2	no	—
39	C₂×C₄×C₇₁	—	4	1	no	—
40	C₂×C₄×C₇₁	C₄⋊C₄	1	1	yes	characteristic
41	C₂×C₄×C₇₁	C₄⋊C₄	1	1	yes	characteristic
42	C₂×C₄×C₇₁	C₄⋊C₄	1	1	yes	characteristic
43	C₂×C₄×C₇₁	C₄⋊C₄	1	1	yes	characteristic
44	C₂×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
46	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
47	C₂×C₄×C₇₁	C₂×Q₈	1	1	yes	characteristic
48	C₂×C₄×C₇₁	#41	1	1	yes	characteristic
49	C₂×C₄×C₇₁	#41	1	1	yes	characteristic
50	C₈×C₇₁	—	2	2	no	—
51	C₈×C₇₁	—	2	2	no	—
52	C₇₁⋊C₈	—	2	4	no	—
53	C₇₁⋊C₈	—	4	2	no	—
54	C₂²×C₇₁	—	2	1	no	—
55	C₂²×C₇₁	—	2	1	no	—
56	C₂²×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
57	C₂²×C₇₁	#125	1	1	yes	characteristic
58	C₂²×C₇₁	C₄⋊Q₈	1	1	yes	characteristic
59	C₄×C₇₁	—	2	1	no	—
60	C₄×C₇₁	—	2	1	no	—
61	C₄×C₇₁	—	2	1	no	—
62	C₄×C₇₁	#108	1	2	yes	complemented
63	C₄×C₇₁	C₄×D₄	1	1	yes	characteristic
64	C₄×C₇₁	C₄×Q₈	1	1	yes	characteristic
65	C₄×C₇₁	#121	1	1	yes	characteristic
66	C₄×C₇₁	C₂²⋊Q₈	1	1	yes	characteristic
67	C₂×C₇₁	—	2	1	no	—
68	C₂×C₇₁	—	2	1	no	—
69	C₂×C₇₁	#389	1	1	yes	characteristic
70	C₂×C₇₁	C₂×#108	1	1	yes	characteristic
71	C₂×C₇₁	#430	1	1	yes	characteristic
72	#1950	—	71	1	no	maximal, Sylow
73	C₇₁	#1950	1	1	yes	characteristic, complemented, Sylow
74	C₂×#108	—	71	2	no	—
75	#423	—	71	1	no	—
76	C₂×C₄⋊C₈	—	71	1	no	—
77	C₄×M₁₆	D₇₁	1	1	yes	characteristic
78	#341	—	71	2	no	—
79	C₂×M₁₆	—	71	2	no	—
80	C₂×M₁₆	C₂×D₇₁	1	1	yes	characteristic
81	C₂×M₁₆	C₂×D₇₁	1	1	yes	characteristic
82	C₂²×C₈	—	71	2	no	—
83	C₂×C₄²	C₂×D₇₁	1	1	yes	characteristic
84	#108	—	142	4	no	—
85	C₄⋊C₈	—	71	2	no	—
86	C₄⋊C₈	—	71	2	no	—
87	C₄⋊C₈	—	142	1	no	—
88	C₄⋊C₈	—	142	1	no	—
89	#97	—	2	1	no	—
90	C₄×C₈	—	2	1	no	—
91	C₂²×C₄	Dic₁₄₂	1	1	yes	characteristic
92	C₂²×C₄	Dic₁₄₂	1	1	yes	characteristic
93	C₂²×C₄	C₂²×D₇₁	1	1	yes	characteristic
94	M₁₆	—	2	1	no	—
95	M₁₆	—	2	1	no	—
96	M₁₆	—	142	2	no	—
97	M₁₆	—	142	2	no	—
98	M₁₆	Dic₁₄₂	1	2	yes	—
99	M₁₆	D₂₈₄	1	2	yes	—
100	C₂×C₈	—	2	1	no	—
101	C₂×C₈	—	2	1	no	—
102	C₂×C₈	—	71	4	no	—
103	C₂×C₈	—	142	2	no	—
104	C₂×C₈	—	142	2	no	—
105	C₂×C₈	—	142	2	no	—
106	C₄²	C₄×D₇₁	1	1	yes	characteristic
107	C₄²	C₄×D₇₁	1	1	yes	characteristic
108	C₄²	C₂²⋊D₇₁	1	1	yes	characteristic
109	C₄²	C₂²⋊D₇₁	1	1	yes	characteristic
110	C₂³	C₂×Dic₁₄₂	1	1	yes	characteristic
111	C₂×C₄	—	2	2	no	—
112	C₂×C₄	—	4	1	no	—
113	C₂×C₄	Dic₇₁⋊C₄	1	1	yes	characteristic
114	C₂×C₄	Dic₇₁⋊C₄	1	1	yes	characteristic
115	C₂×C₄	Dic₇₁⋊C₄	1	1	yes	characteristic
116	C₂×C₄	Dic₇₁⋊C₄	1	1	yes	characteristic
117	C₂×C₄	C₂×C₄×D₇₁	1	1	yes	characteristic
118	C₂×C₄	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
119	C₂×C₄	C₂×D₂₈₄	1	1	yes	characteristic
120	C₂×C₄	C₂×Dic₁₄₂	1	1	yes	characteristic
121	C₂×C₄	#11765865	1	1	yes	characteristic
122	C₂×C₄	#11765865	1	1	yes	characteristic
123	C₈	—	2	2	no	—
124	C₈	—	2	2	no	—
125	C₈	—	284	2	no	—
126	C₈	—	142	4	no	—
127	C₂²	—	2	1	no	—
128	C₂²	—	2	1	no	—
129	C₂²	C₂×Dic₇₁⋊C₄	1	1	yes	characteristic
130	C₂²	#14556120	1	1	yes	characteristic
131	C₂²	#14556128	1	1	yes	characteristic
132	C₄	—	2	1	no	—
133	C₄	—	2	1	no	—
134	C₄	—	2	1	no	—
135	C₄	#14556063	1	2	yes	complemented
136	C₄	C₄×D₂₈₄	1	1	yes	characteristic
137	C₄	C₄×Dic₁₄₂	1	1	yes	characteristic
138	C₄	#14556102	1	1	yes	characteristic
139	C₄	#14556107	1	1	yes	characteristic
140	C₂	—	2	1	no	—
141	C₂	—	2	1	no	—
142	C₂	#17985897	1	1	yes	characteristic
143	C₂	C₂×#14556063	1	1	yes	characteristic
144	C₂	#17986047	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14556063

C₂

yes

maximal

#17986023

C₂

yes

maximal, characteristic

C₂×C₄⋊(C₇₁⋊C₈)

C₂

yes

maximal, characteristic

#17985763

C₂

yes

maximal

C₄×M₁₆×C₇₁

C₂

yes

maximal, characteristic

C₂×#11765849

C₂²

yes

—

C₂²×C₇₁⋊C₈

C₂²

yes

—

#14556063

—

C₄⋊(C₇₁⋊C₈)

—

C₄⋊(C₇₁⋊C₈)

—

C₄⋊(C₇₁⋊C₈)

C₄

yes

—

C₄⋊(C₇₁⋊C₈)

C₄

yes

—

C₂×M₁₆×C₇₁

C₂²

yes

characteristic

C₂×M₁₆×C₇₁

C₂²

yes

characteristic

C₂×C₄²×C₇₁

C₂²

yes

characteristic

#97×C₇₁

—

C₄×C₈×C₇₁

—

#11765849

—

#11765849

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

C₂×C₄

yes

—

C₂²×C₄×C₇₁

Q₈

yes

characteristic

C₂²×C₄×C₇₁

Q₈

yes

characteristic

C₂²×C₄×C₇₁

C₂³

yes

characteristic

M₁₆×C₇₁

—

M₁₆×C₇₁

—

M₁₆×C₇₁

D₄

yes

—

M₁₆×C₇₁

Q₈

yes

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

—

C₄²×C₇₁

C₂×C₄

yes

characteristic

C₄²×C₇₁

C₂×C₄

yes

characteristic

C₄²×C₇₁

D₄

yes

characteristic

C₄²×C₇₁

D₄

yes

characteristic

C₂³×C₇₁

C₂×Q₈

yes

characteristic

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

C₄⋊C₄

yes

characteristic

C₂×C₄×C₇₁

C₄⋊C₄

yes

characteristic

C₂×C₄×C₇₁

C₄⋊C₄

yes

characteristic

C₂×C₄×C₇₁

C₄⋊C₄

yes

characteristic

C₂×C₄×C₇₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₇₁

C₂×D₄

yes

characteristic

C₂×C₄×C₇₁

C₂×D₄

yes

characteristic

C₂×C₄×C₇₁

C₂×Q₈

yes

C₂×C₄×C₇₁

yes

C₂×C₄×C₇₁

yes

C₈×C₇₁

—

C₈×C₇₁

—

C₇₁⋊C₈

—

C₇₁⋊C₈

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₇₁

#125

yes

characteristic

C₂²×C₇₁

C₄⋊Q₈

yes

characteristic

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

#108

yes

complemented

C₄×C₇₁

C₄×D₄

yes

characteristic

C₄×C₇₁

C₄×Q₈

yes

characteristic

C₄×C₇₁

#121

yes

characteristic

C₄×C₇₁

C₂²⋊Q₈

yes

characteristic

C₂×C₇₁

—

C₂×C₇₁

—

C₂×C₇₁

yes

C₂×C₇₁

yes

C₂×C₇₁

yes

—

C₇₁

yes

characteristic, complemented, Sylow

C₂×#108

—

#423

—

C₂×C₄⋊C₈

—

C₄×M₁₆

D₇₁

yes

characteristic

#341

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₇₁

yes

characteristic

C₂×M₁₆

C₂×D₇₁

yes

characteristic

C₂²×C₈

—

C₂×C₄²

C₂×D₇₁

yes

characteristic

#108

—

142

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

142

—

C₄⋊C₈

—

142

—

#97

—

C₄×C₈

—

C₂²×C₄

Dic₁₄₂

yes

characteristic

C₂²×C₄

Dic₁₄₂

yes

characteristic

C₂²×C₄

C₂²×D₇₁

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

142

—

M₁₆

—

142

—

M₁₆

Dic₁₄₂

yes

—

M₁₆

D₂₈₄

yes

—

100

C₂×C₈

—

101

C₂×C₈

—

102

C₂×C₈

—

103

C₂×C₈

—

142

—

104

C₂×C₈

—

142

—

105

C₂×C₈

—

142

—

106

C₄²

C₄×D₇₁

yes

characteristic

107

C₄²

C₄×D₇₁

yes

characteristic

108

C₄²

C₂²⋊D₇₁

yes

characteristic

109

C₄²

C₂²⋊D₇₁

yes

characteristic

110

C₂³

C₂×Dic₁₄₂

yes

characteristic

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

Dic₇₁⋊C₄

yes

characteristic

114

C₂×C₄

Dic₇₁⋊C₄

yes

characteristic

115

C₂×C₄

Dic₇₁⋊C₄

yes

characteristic

116

C₂×C₄

Dic₇₁⋊C₄

yes

characteristic

117

C₂×C₄

C₂×C₄×D₇₁

yes

characteristic

118

C₂×C₄

C₂×C₂²⋊D₇₁

yes

characteristic

119

C₂×C₄

C₂×D₂₈₄

yes

characteristic

120

C₂×C₄

C₂×Dic₁₄₂

yes

characteristic

121

C₂×C₄

#11765865

yes

characteristic

122

C₂×C₄

#11765865

yes

characteristic

123

C₈

—

124

C₈

—

125

C₈

—

284

—

126

C₈

—

142

—

127

C₂²

—

128

C₂²

—

129

C₂²

C₂×Dic₇₁⋊C₄

yes

characteristic

130

C₂²

#14556120

yes

characteristic

131

C₂²

#14556128

yes

characteristic

132

C₄

—

133

C₄

—

134

C₄

—

135

C₄

#14556063

yes

complemented

136

C₄

C₄×D₂₈₄

yes

characteristic

137

C₄

C₄×Dic₁₄₂

yes

characteristic

138

C₄

#14556102

yes

characteristic

139

C₄

#14556107

yes

characteristic

140

C₂

—

141

C₂

—

142

C₂

#17985897

yes

characteristic

143

C₂

C₂×#14556063

yes

characteristic

144

C₂

#17986047

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₇₁
commutator	C₂×C₄×C₇₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₇₁
Fitting	C₄×M₁₆×C₇₁	C₂
socle	C₂²×C₇₁	C₂×C₄⋊C₄
hypercenter	C₄×M₁₆	D₇₁
hypocenter	C₇₁	#1950
LCS	G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄×M₁₆

subgroup

G/H

center

C₂×C₄

C₂×C₂²⋊D₇₁

commutator

C₂×C₄×C₇₁

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₇₁

Fitting

C₄×M₁₆×C₇₁

C₂

socle

C₂²×C₇₁

C₂×C₄⋊C₄

hypercenter

C₄×M₁₆

D₇₁

hypocenter

C₇₁

#1950

LCS

G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄×M₁₆

N	H ≅ G/N	type
62. C₄×C₇₁	#108	semidirect
73. C₇₁	#1950	semidirect
135. C₄	#14556063	semidirect

H ≅ G/N

type

62. C₄×C₇₁

#108

semidirect

73. C₇₁

#1950

semidirect

135. C₄

#14556063

semidirect

Subgroups — #22589115

Conjugacy classes of subgroups

Complements