Group #22588210 — subgroups

Subgroups — #22588210

Order 9088.

272 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 272 conjugacy classes of subgroups — too many to draw. Only its 149 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×#14556038	C₂	1	2	yes	maximal
2	C₂×C₄×C₇₁⋊C₈	C₂	1	1	yes	maximal, characteristic
3	C₄×C₄⋊Dic₇₁	C₂	1	1	yes	maximal, characteristic
4	#17985753	C₂	1	2	yes	maximal
5	C₂×C₄⋊C₈×C₇₁	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₇₁⋊C₈	C₂²	1	1	yes	characteristic
7	C₂²×C₇₁⋊C₈	C₂²	1	1	yes	characteristic
8	C₂×C₄⋊Dic₇₁	C₄	1	2	yes	—
9	C₂×C₄×Dic₇₁	C₂²	1	2	yes	—
10	#14556038	—	2	4	no	—
11	C₄×C₇₁⋊C₈	C₄	1	4	yes	complemented
12	C₇₁⋊#95	C₄	1	2	yes	—
13	C₂²×C₈×C₇₁	C₂²	1	2	yes	—
14	C₂×C₄²×C₇₁	C₂²	1	1	yes	characteristic
15	C₄⋊C₈×C₇₁	C₄	1	4	yes	complemented
16	C₂²×Dic₇₁	C₂×C₄	1	4	yes	—
17	C₂×C₇₁⋊C₈	—	2	1	no	—
18	C₂×C₇₁⋊C₈	—	2	1	no	—
19	C₂×C₇₁⋊C₈	C₂×C₄	1	2	yes	—
20	C₂×C₇₁⋊C₈	C₂×C₄	1	2	yes	—
21	C₂×C₇₁⋊C₈	D₄	1	2	yes	—
22	C₂×C₇₁⋊C₈	Q₈	1	2	yes	—
23	C₄⋊Dic₇₁	—	2	4	no	—
24	C₄×Dic₇₁	—	2	2	no	—
25	C₄×Dic₇₁	—	2	2	no	—
26	C₂²×C₄×C₇₁	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₄×C₇₁	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₄×C₇₁	C₂³	1	1	yes	characteristic
29	C₂×C₈×C₇₁	—	2	2	no	—
30	C₂×C₈×C₇₁	—	2	2	no	—
31	C₂×C₈×C₇₁	C₂×C₄	1	4	yes	—
32	C₄²×C₇₁	C₂×C₄	1	2	yes	—
33	C₄²×C₇₁	C₂×C₄	1	2	yes	—
34	C₂³×C₇₁	C₂²×C₄	1	1	yes	characteristic
35	C₂×C₄×C₇₁	—	2	2	no	—
36	C₂×C₄×C₇₁	C₄²	1	2	yes	—
37	C₂×C₄×C₇₁	C₄²	1	2	yes	—
38	C₂×C₄×C₇₁	M₁₆	1	4	yes	—
39	C₂×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
40	C₂×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
42	C₂×C₄×C₇₁	C₂×Q₈	1	1	yes	characteristic
43	C₂×C₄×C₇₁	#41	1	1	yes	characteristic
44	C₂×C₄×C₇₁	#41	1	1	yes	characteristic
45	C₂×Dic₇₁	—	2	4	no	—
46	C₂×Dic₇₁	—	2	4	no	—
47	C₂×Dic₇₁	—	2	4	no	—
48	C₈×C₇₁	—	2	4	no	—
49	C₇₁⋊C₈	—	2	2	no	—
50	C₇₁⋊C₈	C₄⋊C₄	1	4	yes	complemented
51	C₂²×C₇₁	C₂×C₄²	1	1	yes	characteristic
52	C₂²×C₇₁	C₄×D₄	1	1	yes	characteristic
53	C₂²×C₇₁	C₄×Q₈	1	1	yes	characteristic
54	C₂²×C₇₁	C₂×M₁₆	1	2	yes	—
55	C₂²×C₇₁	#131	1	2	yes	—
56	C₄×C₇₁	—	2	2	no	—
57	C₄×C₇₁	#97	1	4	yes	complemented
58	C₄×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
59	C₄×C₇₁	#117	1	1	yes	characteristic
60	C₄×C₇₁	C₄×D₄	1	1	yes	characteristic
61	C₄×C₇₁	C₄×Q₈	1	1	yes	characteristic
62	Dic₇₁	—	4	4	no	—
63	C₂×C₇₁	C₄×C₄⋊C₄	1	1	yes	characteristic
64	C₂×C₇₁	C₂×#97	1	1	yes	characteristic
65	C₂×C₇₁	#405	1	1	yes	characteristic
66	C₂×C₇₁	#436	1	2	yes	—
67	C₂×C₇₁	#446	1	2	yes	—
68	#1770	—	71	1	no	maximal, Sylow
69	C₇₁	#1770	1	1	yes	characteristic, complemented, Sylow
70	C₂×C₄⋊C₈	D₇₁	1	1	yes	characteristic
71	C₂×#97	—	71	2	no	—
72	C₂×C₄×C₈	—	71	1	no	—
73	C₄×C₄⋊C₄	—	71	1	no	—
74	#336	—	71	2	no	—
75	C₂²×C₈	—	71	1	no	—
76	C₂²×C₈	—	71	1	no	—
77	C₂²×C₈	C₂×D₇₁	1	2	yes	—
78	C₂×C₄⋊C₄	—	71	2	no	—
79	C₂×C₄²	—	71	2	no	—
80	C₂×C₄²	C₂×D₇₁	1	1	yes	characteristic
81	C₄⋊C₈	Dic₇₁	1	4	yes	complemented
82	#97	—	142	4	no	—
83	C₄×C₈	—	71	4	no	—
84	#95	—	71	2	no	—
85	C₂²×C₄	—	71	4	no	—
86	C₂²×C₄	C₄×D₇₁	1	1	yes	characteristic
87	C₂²×C₄	C₄×D₇₁	1	1	yes	characteristic
88	C₂²×C₄	C₂²×D₇₁	1	1	yes	characteristic
89	C₂×C₈	—	2	2	no	—
90	C₂×C₈	—	2	2	no	—
91	C₂×C₈	—	71	2	no	—
92	C₂×C₈	—	71	2	no	—
93	C₂×C₈	—	71	2	no	—
94	C₂×C₈	—	71	2	no	—
95	C₂×C₈	—	142	1	no	—
96	C₂×C₈	—	142	1	no	—
97	C₂×C₈	C₂×Dic₇₁	1	4	yes	—
98	C₄⋊C₄	—	142	4	no	—
99	C₄²	—	142	2	no	—
100	C₄²	—	142	2	no	—
101	C₄²	C₄×D₇₁	1	2	yes	—
102	C₄²	C₂×Dic₇₁	1	2	yes	—
103	C₂³	C₂×C₄×D₇₁	1	1	yes	characteristic
104	C₂×C₄	—	2	2	no	—
105	C₂×C₄	—	142	4	no	—
106	C₂×C₄	—	142	4	no	—
107	C₂×C₄	—	142	4	no	—
108	C₂×C₄	C₄×Dic₇₁	1	2	yes	—
109	C₂×C₄	C₄×Dic₇₁	1	2	yes	—
110	C₂×C₄	#11765848	1	4	yes	—
111	C₂×C₄	C₂×C₄×D₇₁	1	1	yes	characteristic
112	C₂×C₄	C₂²×Dic₇₁	1	1	yes	characteristic
113	C₂×C₄	D₄×D₇₁	1	1	yes	characteristic
114	C₂×C₄	Q₈×D₇₁	1	1	yes	characteristic
115	C₂×C₄	#11765864	1	1	yes	characteristic
116	C₂×C₄	#11765866	1	1	yes	characteristic
117	C₈	—	2	4	no	—
118	C₈	—	71	4	no	—
119	C₈	—	142	2	no	—
120	C₂²	C₂×C₄×Dic₇₁	1	1	yes	characteristic
121	C₂²	#14556090	1	1	yes	characteristic
122	C₂²	#14556093	1	1	yes	characteristic
123	C₂²	C₂×#11765848	1	2	yes	—
124	C₂²	#14556136	1	2	yes	—
125	C₄	—	2	2	no	—
126	C₄	—	284	4	no	—
127	C₄	#14556038	1	4	yes	complemented
128	C₄	C₄⋊C₄×D₇₁	1	1	yes	characteristic
129	C₄	#14556086	1	1	yes	characteristic
130	C₄	D₄×Dic₇₁	1	1	yes	characteristic
131	C₄	Q₈×Dic₇₁	1	1	yes	characteristic
132	C₂	C₄⋊C₄×Dic₇₁	1	1	yes	characteristic
133	C₂	C₂×#14556038	1	1	yes	characteristic
134	C₂	#17985957	1	1	yes	characteristic
135	C₂	#17986076	1	2	yes	—
136	C₂	#17986126	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14556038

C₂

yes

maximal

C₂×C₄×C₇₁⋊C₈

C₂

yes

maximal, characteristic

C₄×C₄⋊Dic₇₁

C₂

yes

maximal, characteristic

#17985753

C₂

yes

maximal

C₂×C₄⋊C₈×C₇₁

C₂

yes

maximal, characteristic

C₂²×C₇₁⋊C₈

C₂²

yes

characteristic

C₂²×C₇₁⋊C₈

C₂²

yes

characteristic

C₂×C₄⋊Dic₇₁

C₄

yes

—

C₂×C₄×Dic₇₁

C₂²

yes

—

#14556038

—

C₄×C₇₁⋊C₈

C₄

yes

complemented

C₇₁⋊#95

C₄

yes

—

C₂²×C₈×C₇₁

C₂²

yes

—

C₂×C₄²×C₇₁

C₂²

yes

characteristic

C₄⋊C₈×C₇₁

C₄

yes

complemented

C₂²×Dic₇₁

C₂×C₄

yes

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

C₂×C₄

yes

—

C₂×C₇₁⋊C₈

C₂×C₄

yes

—

C₂×C₇₁⋊C₈

D₄

yes

—

C₂×C₇₁⋊C₈

Q₈

yes

—

C₄⋊Dic₇₁

—

C₄×Dic₇₁

—

C₄×Dic₇₁

—

C₂²×C₄×C₇₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₇₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₇₁

C₂³

yes

characteristic

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

C₂×C₄

yes

—

C₄²×C₇₁

C₂×C₄

yes

—

C₄²×C₇₁

C₂×C₄

yes

—

C₂³×C₇₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

C₄²

yes

—

C₂×C₄×C₇₁

C₄²

yes

—

C₂×C₄×C₇₁

M₁₆

yes

—

C₂×C₄×C₇₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₇₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₇₁

C₂×D₄

yes

characteristic

C₂×C₄×C₇₁

C₂×Q₈

yes

C₂×C₄×C₇₁

yes

C₂×C₄×C₇₁

yes

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₈×C₇₁

—

C₇₁⋊C₈

—

C₇₁⋊C₈

C₄⋊C₄

yes

complemented

C₂²×C₇₁

C₂×C₄²

yes

characteristic

C₂²×C₇₁

C₄×D₄

yes

characteristic

C₂²×C₇₁

C₄×Q₈

yes

characteristic

C₂²×C₇₁

C₂×M₁₆

yes

—

C₂²×C₇₁

#131

yes

—

C₄×C₇₁

—

C₄×C₇₁

#97

yes

complemented

C₄×C₇₁

C₂×C₄⋊C₄

yes

characteristic

C₄×C₇₁

#117

yes

characteristic

C₄×C₇₁

C₄×D₄

yes

characteristic

C₄×C₇₁

C₄×Q₈

yes

characteristic

Dic₇₁

—

C₂×C₇₁

C₄×C₄⋊C₄

yes

characteristic

C₂×C₇₁

C₂×#97

yes

C₂×C₇₁

yes

C₂×C₇₁

yes

—

C₂×C₇₁

#446

yes

—

#1770

—

maximal, Sylow

C₇₁

#1770

yes

characteristic, complemented, Sylow

C₂×C₄⋊C₈

D₇₁

yes

characteristic

C₂×#97

—

C₂×C₄×C₈

—

C₄×C₄⋊C₄

—

#336

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

C₂×D₇₁

yes

—

C₂×C₄⋊C₄

—

C₂×C₄²

—

C₂×C₄²

C₂×D₇₁

yes

characteristic

C₄⋊C₈

Dic₇₁

yes

complemented

#97

—

142

—

C₄×C₈

—

#95

—

C₂²×C₄

—

C₂²×C₄

C₄×D₇₁

yes

characteristic

C₂²×C₄

C₄×D₇₁

yes

characteristic

C₂²×C₄

C₂²×D₇₁

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

142

—

C₂×C₈

—

142

—

C₂×C₈

C₂×Dic₇₁

yes

—

C₄⋊C₄

—

142

—

C₄²

—

142

—

100

C₄²

—

142

—

101

C₄²

C₄×D₇₁

yes

—

102

C₄²

C₂×Dic₇₁

yes

—

103

C₂³

C₂×C₄×D₇₁

yes

characteristic

104

C₂×C₄

—

105

C₂×C₄

—

142

—

106

C₂×C₄

—

142

—

107

C₂×C₄

—

142

—

108

C₂×C₄

C₄×Dic₇₁

yes

—

109

C₂×C₄

C₄×Dic₇₁

yes

—

110

C₂×C₄

#11765848

yes

—

111

C₂×C₄

C₂×C₄×D₇₁

yes

characteristic

112

C₂×C₄

C₂²×Dic₇₁

yes

characteristic

113

C₂×C₄

D₄×D₇₁

yes

characteristic

114

C₂×C₄

Q₈×D₇₁

yes

characteristic

115

C₂×C₄

#11765864

yes

characteristic

116

C₂×C₄

#11765866

yes

characteristic

117

C₈

—

118

C₈

—

119

C₈

—

142

—

120

C₂²

C₂×C₄×Dic₇₁

yes

characteristic

121

C₂²

#14556090

yes

characteristic

122

C₂²

#14556093

yes

characteristic

123

C₂²

C₂×#11765848

yes

—

124

C₂²

#14556136

yes

—

125

C₄

—

126

C₄

—

284

—

127

C₄

#14556038

yes

complemented

128

C₄

C₄⋊C₄×D₇₁

yes

characteristic

129

C₄

#14556086

yes

characteristic

130

C₄

D₄×Dic₇₁

yes

characteristic

131

C₄

Q₈×Dic₇₁

yes

characteristic

132

C₂

C₄⋊C₄×Dic₇₁

yes

characteristic

133

C₂

C₂×#14556038

yes

characteristic

134

C₂

#17985957

yes

characteristic

135

C₂

#17986076

yes

—

136

C₂

#17986126

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₇₁
commutator	C₂²×C₇₁	C₂×C₄²
Fitting	C₂×C₄⋊C₈×C₇₁	C₂
socle	C₂³×C₇₁	C₂²×C₄
hypercenter	C₂×C₄⋊C₈	D₇₁
hypocenter	C₇₁	#1770
LCS	G ▷ C₂²×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄⋊C₈

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×D₇₁

commutator

C₂²×C₇₁

C₂×C₄²

Fitting

C₂×C₄⋊C₈×C₇₁

C₂

socle

C₂³×C₇₁

C₂²×C₄

hypercenter

C₂×C₄⋊C₈

D₇₁

C₇₁

G ▷ C₂²×C₇₁ ▷ C₇₁

C₁ ◁ C₂²×C₄ ◁ C₂×C₄⋊C₈

N	H ≅ G/N	type
11. C₄×C₇₁⋊C₈	C₄	semidirect
15. C₄⋊C₈×C₇₁	C₄	semidirect
50. C₇₁⋊C₈	C₄⋊C₄	semidirect
57. C₄×C₇₁	#97	semidirect
69. C₇₁	#1770	semidirect
81. C₄⋊C₈	Dic₇₁	semidirect
127. C₄	#14556038	semidirect

H ≅ G/N

type

11. C₄×C₇₁⋊C₈

C₄

semidirect

15. C₄⋊C₈×C₇₁

C₄

semidirect

50. C₇₁⋊C₈

C₄⋊C₄

semidirect

57. C₄×C₇₁

#97

semidirect

69. C₇₁

#1770

semidirect

81. C₄⋊C₈

Dic₇₁

semidirect

127. C₄

#14556038

semidirect

Subgroups — #22588210

Conjugacy classes of subgroups

Complements