Group #22588491 — subgroups

Subgroups — #22588491

Order 9088.

300 conjugacy classes of subgroups; only the 153 normal subgroups are shown.

normal subgroups only

This group has 300 conjugacy classes of subgroups — too many to draw. Only its 153 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×C₄⋊(C₇₁⋊C₈)	C₂	1	1	yes	maximal, characteristic
2	C₂×Dic₇₁⋊C₈	C₂	1	4	yes	maximal
3	C₄²×Dic₇₁	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄⋊C₈×C₇₁	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₇₁⋊C₈	C₂²	1	2	yes	—
6	C₂×C₄×Dic₇₁	C₄	1	2	yes	—
7	C₂×C₄×Dic₇₁	C₄	1	1	yes	characteristic
8	C₂×C₄×Dic₇₁	C₄	1	1	yes	characteristic
9	C₂×C₄×Dic₇₁	C₂²	1	2	yes	—
10	C₄⋊(C₇₁⋊C₈)	—	2	2	no	—
11	Dic₇₁⋊C₈	—	2	8	no	—
12	C₂²×C₈×C₇₁	C₂²	1	2	yes	—
13	C₂×C₄²×C₇₁	C₂²	1	1	yes	characteristic
14	C₄⋊C₈×C₇₁	—	2	2	no	—
15	C₂²×Dic₇₁	C₂×C₄	1	2	yes	—
16	C₂²×Dic₇₁	C₂×C₄	1	2	yes	—
17	C₂×C₇₁⋊C₈	—	2	4	no	—
18	C₂×C₇₁⋊C₈	—	2	2	no	—
19	C₄×Dic₇₁	—	2	2	no	—
20	C₄×Dic₇₁	—	2	4	no	—
21	C₄×Dic₇₁	C₈	1	4	yes	complemented
22	C₄×Dic₇₁	D₄	1	4	yes	—
23	C₄×Dic₇₁	Q₈	1	4	yes	—
24	C₂²×C₄×C₇₁	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₄×C₇₁	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₄×C₇₁	C₂³	1	1	yes	characteristic
27	C₂×C₈×C₇₁	—	2	2	no	—
28	C₂×C₈×C₇₁	—	2	4	no	—
29	C₄²×C₇₁	D₄	1	2	yes	—
30	C₄²×C₇₁	Q₈	1	2	yes	—
31	C₂³×C₇₁	C₂²×C₄	1	1	yes	characteristic
32	C₂×C₄×C₇₁	—	2	2	no	—
33	C₂×C₄×C₇₁	C₄⋊C₄	1	2	yes	—
34	C₂×C₄×C₇₁	C₄⋊C₄	1	2	yes	—
35	C₂×C₄×C₇₁	C₂×C₈	1	2	yes	—
36	C₂×C₄×C₇₁	M₁₆	1	2	yes	—
37	C₂×C₄×C₇₁	C₂×D₄	1	2	yes	—
38	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
39	C₂×C₄×C₇₁	C₂×Q₈	1	2	yes	—
40	C₂×C₄×C₇₁	C₂×Q₈	1	1	yes	characteristic
41	C₂×Dic₇₁	—	2	2	no	—
42	C₂×Dic₇₁	—	2	2	no	—
43	C₂×Dic₇₁	C₄⋊C₄	1	4	yes	—
44	C₂×Dic₇₁	C₄⋊C₄	1	4	yes	—
45	C₂×Dic₇₁	C₂×C₈	1	4	yes	—
46	C₂×Dic₇₁	M₁₆	1	4	yes	—
47	C₈×C₇₁	—	4	2	no	—
48	C₇₁⋊C₈	—	4	2	no	—
49	C₂²×C₇₁	C₂×C₄⋊C₄	1	2	yes	—
50	C₂²×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
51	C₂²×C₇₁	C₂²×C₈	1	1	yes	characteristic
52	C₂²×C₇₁	C₂×M₁₆	1	2	yes	—
53	C₂²×C₇₁	C₂×M₁₆	1	1	yes	characteristic
54	C₄×C₇₁	—	2	2	no	—
55	C₄×C₇₁	C₄⋊C₈	1	4	yes	complemented
56	C₄×C₇₁	#127	1	1	yes	characteristic
57	C₄×C₇₁	C₄⋊Q₈	1	1	yes	characteristic
58	C₄×C₇₁	C₄⋊Q₈	1	2	yes	—
59	Dic₇₁	—	2	4	no	—
60	Dic₇₁	C₄⋊C₈	1	8	yes	complemented
61	C₂×C₇₁	#384	1	1	yes	characteristic
62	C₂×C₇₁	C₂×C₄⋊C₈	1	2	yes	—
63	C₂×C₇₁	C₂×C₄⋊C₈	1	1	yes	characteristic
64	C₂×C₇₁	#423	1	2	yes	—
65	C₂×C₇₁	#423	1	1	yes	characteristic
66	#1842	—	71	1	no	maximal, Sylow
67	C₇₁	#1842	1	1	yes	characteristic, complemented, Sylow
68	C₂×C₄⋊C₈	—	71	4	no	—
69	C₂×C₄⋊C₈	—	71	1	no	—
70	C₂×C₄⋊C₈	D₇₁	1	1	yes	characteristic
71	C₄³	—	71	1	no	—
72	C₂²×C₈	—	71	2	no	—
73	C₂²×C₈	C₂×D₇₁	1	2	yes	—
74	C₂×C₄²	—	71	1	no	—
75	C₂×C₄²	—	71	2	no	—
76	C₂×C₄²	—	71	2	no	—
77	C₂×C₄²	—	71	1	no	—
78	C₂×C₄²	C₂×D₇₁	1	1	yes	characteristic
79	C₄⋊C₈	—	2	2	no	—
80	C₄⋊C₈	—	142	8	no	—
81	C₄⋊C₈	—	142	2	no	—
82	C₂²×C₄	—	71	2	no	—
83	C₂²×C₄	—	71	2	no	—
84	C₂²×C₄	C₄×D₇₁	1	1	yes	characteristic
85	C₂²×C₄	C₄×D₇₁	1	1	yes	characteristic
86	C₂²×C₄	C₂²×D₇₁	1	1	yes	characteristic
87	C₂×C₈	—	2	4	no	—
88	C₂×C₈	—	2	2	no	—
89	C₂×C₈	—	142	4	no	—
90	C₂×C₈	—	142	2	no	—
91	C₄²	—	71	4	no	—
92	C₄²	—	71	4	no	—
93	C₄²	—	71	4	no	—
94	C₄²	—	142	4	no	—
95	C₄²	—	142	2	no	—
96	C₄²	Dic₁₄₂	1	2	yes	—
97	C₄²	C₂²⋊D₇₁	1	2	yes	—
98	C₂³	C₂×C₄×D₇₁	1	1	yes	characteristic
99	C₂×C₄	—	2	2	no	—
100	C₂×C₄	—	71	4	no	—
101	C₂×C₄	—	71	4	no	—
102	C₂×C₄	—	71	4	no	—
103	C₂×C₄	—	71	4	no	—
104	C₂×C₄	—	142	2	no	—
105	C₂×C₄	—	142	2	no	—
106	C₂×C₄	Dic₇₁⋊C₄	1	2	yes	—
107	C₂×C₄	Dic₇₁⋊C₄	1	2	yes	—
108	C₂×C₄	C₈×D₇₁	1	2	yes	—
109	C₂×C₄	#11765848	1	2	yes	—
110	C₂×C₄	D₄×D₇₁	1	2	yes	—
111	C₂×C₄	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
112	C₂×C₄	Q₈×D₇₁	1	2	yes	—
113	C₂×C₄	C₂×Dic₁₄₂	1	1	yes	characteristic
114	C₈	—	4	2	no	—
115	C₈	—	284	2	no	—
116	C₂²	C₄⋊C₄×D₇₁	1	2	yes	—
117	C₂²	C₂×Dic₇₁⋊C₄	1	1	yes	characteristic
118	C₂²	C₂×C₈×D₇₁	1	1	yes	characteristic
119	C₂²	M₁₆×D₇₁	1	2	yes	—
120	C₂²	C₂×#11765848	1	1	yes	characteristic
121	C₄	—	2	2	no	—
122	C₄	—	71	8	no	—
123	C₄	—	142	4	no	—
124	C₄	Dic₇₁⋊C₈	1	4	yes	complemented
125	C₄	#14556124	1	1	yes	characteristic
126	C₄	#14556126	1	1	yes	characteristic
127	C₄	#14556127	1	2	yes	—
128	C₂	#17985868	1	1	yes	characteristic
129	C₂	C₄⋊C₈×D₇₁	1	2	yes	—
130	C₂	C₂×Dic₇₁⋊C₈	1	1	yes	characteristic
131	C₂	#17986021	1	2	yes	—
132	C₂	#17986022	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊(C₇₁⋊C₈)

C₂

yes

maximal, characteristic

C₂×Dic₇₁⋊C₈

C₂

yes

maximal

C₄²×Dic₇₁

C₂

yes

maximal, characteristic

C₂×C₄⋊C₈×C₇₁

C₂

yes

maximal, characteristic

C₂²×C₇₁⋊C₈

C₂²

yes

—

C₂×C₄×Dic₇₁

C₄

yes

—

C₂×C₄×Dic₇₁

C₄

yes

characteristic

C₂×C₄×Dic₇₁

C₄

yes

characteristic

C₂×C₄×Dic₇₁

C₂²

yes

—

C₄⋊(C₇₁⋊C₈)

—

Dic₇₁⋊C₈

—

C₂²×C₈×C₇₁

C₂²

yes

—

C₂×C₄²×C₇₁

C₂²

yes

characteristic

C₄⋊C₈×C₇₁

—

C₂²×Dic₇₁

C₂×C₄

yes

—

C₂²×Dic₇₁

C₂×C₄

yes

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

C₄×Dic₇₁

—

C₄×Dic₇₁

—

C₄×Dic₇₁

C₈

yes

complemented

C₄×Dic₇₁

D₄

yes

—

C₄×Dic₇₁

Q₈

yes

—

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

—

C₄²×C₇₁

D₄

yes

—

C₄²×C₇₁

Q₈

yes

—

C₂³×C₇₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

C₄⋊C₄

yes

—

C₂×C₄×C₇₁

C₄⋊C₄

yes

—

C₂×C₄×C₇₁

C₂×C₈

yes

—

C₂×C₄×C₇₁

M₁₆

yes

—

C₂×C₄×C₇₁

C₂×D₄

yes

—

C₂×C₄×C₇₁

C₂×D₄

yes

characteristic

C₂×C₄×C₇₁

C₂×Q₈

yes

—

C₂×C₄×C₇₁

C₂×Q₈

yes

characteristic

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₂×Dic₇₁

C₄⋊C₄

yes

—

C₂×Dic₇₁

C₄⋊C₄

yes

—

C₂×Dic₇₁

C₂×C₈

yes

—

C₂×Dic₇₁

M₁₆

yes

—

C₈×C₇₁

—

C₇₁⋊C₈

—

C₂²×C₇₁

C₂×C₄⋊C₄

yes

—

C₂²×C₇₁

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₇₁

C₂²×C₈

yes

characteristic

C₂²×C₇₁

C₂×M₁₆

yes

—

C₂²×C₇₁

C₂×M₁₆

yes

characteristic

C₄×C₇₁

—

C₄×C₇₁

C₄⋊C₈

yes

complemented

C₄×C₇₁

#127

yes

characteristic

C₄×C₇₁

C₄⋊Q₈

yes

characteristic

C₄×C₇₁

C₄⋊Q₈

yes

—

Dic₇₁

—

Dic₇₁

C₄⋊C₈

yes

complemented

C₂×C₇₁

#384

yes

characteristic

C₂×C₇₁

C₂×C₄⋊C₈

yes

—

C₂×C₇₁

C₂×C₄⋊C₈

yes

characteristic

C₂×C₇₁

#423

yes

—

C₂×C₇₁

yes

—

C₇₁

yes

characteristic, complemented, Sylow

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

D₇₁

yes

characteristic

C₄³

—

C₂²×C₈

—

C₂²×C₈

C₂×D₇₁

yes

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

C₂×D₇₁

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

142

—

C₄⋊C₈

—

142

—

C₂²×C₄

—

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

—

142

—

C₂×C₈

—

142

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

142

—

C₄²

—

142

—

C₄²

Dic₁₄₂

yes

—

C₄²

C₂²⋊D₇₁

yes

—

C₂³

C₂×C₄×D₇₁

yes

characteristic

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

142

—

105

C₂×C₄

—

142

—

106

C₂×C₄

Dic₇₁⋊C₄

yes

—

107

C₂×C₄

Dic₇₁⋊C₄

yes

—

108

C₂×C₄

C₈×D₇₁

yes

—

109

C₂×C₄

#11765848

yes

—

110

C₂×C₄

D₄×D₇₁

yes

—

111

C₂×C₄

C₂×C₂²⋊D₇₁

yes

characteristic

112

C₂×C₄

Q₈×D₇₁

yes

—

113

C₂×C₄

C₂×Dic₁₄₂

yes

characteristic

114

C₈

—

115

C₈

—

284

—

116

C₂²

C₄⋊C₄×D₇₁

yes

—

117

C₂²

C₂×Dic₇₁⋊C₄

yes

characteristic

118

C₂²

C₂×C₈×D₇₁

yes

characteristic

119

C₂²

M₁₆×D₇₁

yes

—

120

C₂²

C₂×#11765848

yes

characteristic

121

C₄

—

122

C₄

—

123

C₄

—

142

—

124

C₄

Dic₇₁⋊C₈

yes

complemented

125

C₄

#14556124

yes

characteristic

126

C₄

#14556126

yes

characteristic

127

C₄

#14556127

yes

—

128

C₂

#17985868

yes

characteristic

129

C₂

C₄⋊C₈×D₇₁

yes

—

130

C₂

C₂×Dic₇₁⋊C₈

yes

characteristic

131

C₂

#17986021

yes

—

132

C₂

#17986022

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₇₁
commutator	C₂²×C₇₁	C₂²×C₈
Fitting	C₂×C₄⋊C₈×C₇₁	C₂
socle	C₂³×C₇₁	C₂²×C₄
hypercenter	C₂×C₄⋊C₈	D₇₁
hypocenter	C₇₁	#1842
LCS	G ▷ C₂²×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄⋊C₈

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×D₇₁

commutator

C₂²×C₇₁

C₂²×C₈

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
21. C₄×Dic₇₁	C₈	semidirect
55. C₄×C₇₁	C₄⋊C₈	semidirect
60. Dic₇₁	C₄⋊C₈	semidirect
67. C₇₁	#1842	semidirect
124. C₄	Dic₇₁⋊C₈	semidirect

H ≅ G/N

type

21. C₄×Dic₇₁

C₈

semidirect

55. C₄×C₇₁

C₄⋊C₈

semidirect

60. Dic₇₁

C₄⋊C₈

semidirect

67. C₇₁

#1842

semidirect

124. C₄

Dic₇₁⋊C₈

semidirect

Subgroups — #22588491

Conjugacy classes of subgroups

Complements