Group #22588232 — subgroups

Subgroups — #22588232

Order 9088.

280 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 280 conjugacy classes of subgroups — too many to draw. Only its 129 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×#14556061	C₂	1	2	yes	maximal
2	C₂×C₈×Dic₇₁	C₂	1	1	yes	maximal, characteristic
3	C₄⋊C₄×Dic₇₁	C₂	1	2	yes	maximal
4	#17985762	C₂	1	1	yes	maximal, characteristic
5	#340×C₇₁	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₇₁⋊C₈	C₂²	1	1	yes	characteristic
7	C₂×C₄⋊Dic₇₁	C₂²	1	2	yes	—
8	C₂×C₄×Dic₇₁	—	2	2	no	—
9	C₂×C₄×Dic₇₁	C₂²	1	1	yes	characteristic
10	#14556061	C₄	1	8	yes	complemented
11	C₈×Dic₇₁	C₄	1	4	yes	complemented
12	C₇₁⋊#95	—	2	2	no	—
13	C₂²×C₈×C₇₁	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊C₄×C₇₁	C₂²	1	2	yes	—
15	C₂²×Dic₇₁	—	2	2	no	—
16	C₂²×Dic₇₁	D₄	1	1	yes	characteristic
17	C₂²×Dic₇₁	D₄	1	1	yes	characteristic
18	C₂×C₇₁⋊C₈	C₂×C₄	1	2	yes	—
19	C₂×C₇₁⋊C₈	C₂×C₄	1	4	yes	—
20	C₄⋊Dic₇₁	—	2	2	no	—
21	C₄⋊Dic₇₁	C₂×C₄	1	4	yes	—
22	C₄×Dic₇₁	—	4	4	no	—
23	C₄×Dic₇₁	C₂×C₄	1	2	yes	—
24	C₄×Dic₇₁	Q₈	1	2	yes	—
25	C₂²×C₄×C₇₁	—	2	2	no	—
26	C₂²×C₄×C₇₁	C₂³	1	1	yes	characteristic
27	C₂×C₈×C₇₁	—	2	2	no	—
28	C₂×C₈×C₇₁	C₂×C₄	1	2	yes	—
29	C₄⋊C₄×C₇₁	—	2	2	no	—
30	C₄⋊C₄×C₇₁	C₂×C₄	1	4	yes	—
31	C₂³×C₇₁	C₂×D₄	1	1	yes	characteristic
32	C₂×C₄×C₇₁	—	2	4	no	—
33	C₂×C₄×C₇₁	—	4	2	no	—
34	C₂×C₄×C₇₁	—	4	2	no	—
35	C₂×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
36	C₂×C₄×C₇₁	C₂²×C₄	1	2	yes	—
37	C₂×C₄×C₇₁	C₂×Q₈	1	1	yes	characteristic
38	C₂×C₄×C₇₁	#41	1	2	yes	—
39	C₂×Dic₇₁	—	2	2	no	—
40	C₂×Dic₇₁	—	2	4	no	—
41	C₂×Dic₇₁	—	4	2	no	—
42	C₂×Dic₇₁	—	4	2	no	—
43	C₂×Dic₇₁	C₄⋊C₄	1	2	yes	—
44	C₂×Dic₇₁	C₄⋊C₄	1	2	yes	—
45	C₂×Dic₇₁	D₈	1	2	yes	—
46	C₂×Dic₇₁	Q₁₆	1	2	yes	—
47	C₈×C₇₁	—	2	2	no	—
48	C₇₁⋊C₈	C₄²	1	4	yes	complemented
49	C₂²×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
50	C₂²×C₇₁	C₄×D₄	1	2	yes	—
51	C₂²×C₇₁	C₂×D₈	1	1	yes	characteristic
52	C₂²×C₇₁	C₂×Q₁₆	1	1	yes	characteristic
53	C₂²×C₇₁	#135	1	2	yes	—
54	C₄×C₇₁	—	4	4	no	—
55	C₄×C₇₁	C₂×C₄²	1	1	yes	characteristic
56	C₄×C₇₁	#117	1	1	yes	characteristic
57	C₄×C₇₁	C₄×Q₈	1	2	yes	—
58	Dic₇₁	—	2	2	no	—
59	Dic₇₁	—	4	4	no	—
60	Dic₇₁	#107	1	4	yes	complemented
61	C₂×C₇₁	C₄×C₄⋊C₄	1	1	yes	characteristic
62	C₂×C₇₁	C₂×#107	1	1	yes	characteristic
63	C₂×C₇₁	#427	1	1	yes	characteristic
64	C₂×C₇₁	C₄×D₈	1	2	yes	—
65	C₂×C₇₁	C₄×Q₁₆	1	2	yes	—
66	C₄×#107	—	71	1	no	maximal, Sylow
67	C₇₁	C₄×#107	1	1	yes	characteristic, complemented, Sylow
68	C₂×#107	—	71	2	no	—
69	C₂×C₄×C₈	—	71	1	no	—
70	C₄×C₄⋊C₄	—	71	2	no	—
71	#340	—	71	1	no	—
72	#340	D₇₁	1	1	yes	characteristic
73	C₂²×C₈	—	71	1	no	—
74	C₂²×C₈	C₂×D₇₁	1	1	yes	characteristic
75	C₂×C₄⋊C₄	—	71	2	no	—
76	C₂×C₄⋊C₄	C₂×D₇₁	1	2	yes	—
77	C₂×C₄²	—	71	1	no	—
78	C₂×C₄²	—	142	2	no	—
79	#107	—	71	8	no	—
80	C₄×C₈	—	71	4	no	—
81	#95	—	142	2	no	—
82	C₂²×C₄	—	2	2	no	—
83	C₂²×C₄	—	71	1	no	—
84	C₂²×C₄	—	71	1	no	—
85	C₂²×C₄	—	142	2	no	—
86	C₂²×C₄	C₂²×D₇₁	1	1	yes	characteristic
87	C₂×C₈	—	2	2	no	—
88	C₂×C₈	—	71	2	no	—
89	C₂×C₈	—	71	4	no	—
90	C₂×C₈	C₄×D₇₁	1	2	yes	—
91	C₄⋊C₄	—	2	2	no	—
92	C₄⋊C₄	—	71	4	no	—
93	C₄⋊C₄	—	142	2	no	—
94	C₄⋊C₄	C₄×D₇₁	1	4	yes	—
95	C₄²	—	284	4	no	—
96	C₄²	—	71	2	no	—
97	C₄²	—	71	2	no	—
98	C₂³	D₄×D₇₁	1	1	yes	characteristic
99	C₂×C₄	—	2	4	no	—
100	C₂×C₄	—	4	2	no	—
101	C₂×C₄	—	4	2	no	—
102	C₂×C₄	—	284	2	no	—
103	C₂×C₄	—	284	2	no	—
104	C₂×C₄	—	71	2	no	—
105	C₂×C₄	—	71	2	no	—
106	C₂×C₄	—	71	2	no	—
107	C₂×C₄	—	71	2	no	—
108	C₂×C₄	—	142	4	no	—
109	C₂×C₄	—	142	2	no	—
110	C₂×C₄	C₂×C₄×D₇₁	1	1	yes	characteristic
111	C₂×C₄	C₂×C₄×D₇₁	1	2	yes	—
112	C₂×C₄	Q₈×D₇₁	1	1	yes	characteristic
113	C₂×C₄	#11765864	1	2	yes	—
114	C₈	—	2	2	no	—
115	C₈	—	71	4	no	—
116	C₂²	C₄⋊C₄×D₇₁	1	1	yes	characteristic
117	C₂²	(C₂²⋊D₇₁)⋊C₄	1	2	yes	—
118	C₂²	D₈×D₇₁	1	1	yes	characteristic
119	C₂²	Q₁₆×D₇₁	1	1	yes	characteristic
120	C₂²	#14556149	1	2	yes	—
121	C₄	—	4	4	no	—
122	C₄	—	284	4	no	—
123	C₄	—	71	4	no	—
124	C₄	—	142	2	no	—
125	C₄	C₄²×D₇₁	1	1	yes	characteristic
126	C₄	#14556084	1	1	yes	characteristic
127	C₄	#14556093	1	2	yes	—
128	C₂	#17985850	1	1	yes	characteristic
129	C₂	#107×D₇₁	1	1	yes	characteristic
130	C₂	#17986034	1	1	yes	characteristic
131	C₂	#17986079	1	2	yes	—
132	C₂	#17986091	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14556061

C₂

yes

maximal

C₂×C₈×Dic₇₁

C₂

yes

maximal, characteristic

C₄⋊C₄×Dic₇₁

C₂

yes

maximal

#17985762

C₂

yes

maximal, characteristic

#340×C₇₁

C₂

yes

maximal, characteristic

C₂²×C₇₁⋊C₈

C₂²

yes

characteristic

C₂×C₄⋊Dic₇₁

C₂²

yes

—

C₂×C₄×Dic₇₁

—

C₂×C₄×Dic₇₁

C₂²

yes

characteristic

#14556061

C₄

yes

complemented

C₈×Dic₇₁

C₄

yes

complemented

C₇₁⋊#95

—

C₂²×C₈×C₇₁

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₇₁

C₂²

yes

—

C₂²×Dic₇₁

—

C₂²×Dic₇₁

D₄

yes

characteristic

C₂²×Dic₇₁

D₄

yes

characteristic

C₂×C₇₁⋊C₈

C₂×C₄

yes

—

C₂×C₇₁⋊C₈

C₂×C₄

yes

—

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

C₂×C₄

yes

—

C₄×Dic₇₁

—

C₄×Dic₇₁

C₂×C₄

yes

—

C₄×Dic₇₁

Q₈

yes

—

C₂²×C₄×C₇₁

—

C₂²×C₄×C₇₁

C₂³

yes

characteristic

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

C₂×C₄

yes

—

C₄⋊C₄×C₇₁

—

C₄⋊C₄×C₇₁

C₂×C₄

yes

—

C₂³×C₇₁

C₂×D₄

yes

characteristic

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₇₁

C₂²×C₄

yes

—

C₂×C₄×C₇₁

C₂×Q₈

yes

characteristic

C₂×C₄×C₇₁

#41

yes

—

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₂×Dic₇₁

C₄⋊C₄

yes

—

C₂×Dic₇₁

C₄⋊C₄

yes

—

C₂×Dic₇₁

D₈

yes

—

C₂×Dic₇₁

Q₁₆

yes

—

C₈×C₇₁

—

C₇₁⋊C₈

C₄²

yes

complemented

C₂²×C₇₁

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₇₁

C₄×D₄

yes

—

C₂²×C₇₁

C₂×D₈

yes

characteristic

C₂²×C₇₁

C₂×Q₁₆

yes

characteristic

C₂²×C₇₁

#135

yes

—

C₄×C₇₁

—

C₄×C₇₁

C₂×C₄²

yes

characteristic

C₄×C₇₁

#117

yes

characteristic

C₄×C₇₁

C₄×Q₈

yes

—

Dic₇₁

—

Dic₇₁

—

Dic₇₁

#107

yes

complemented

C₂×C₇₁

C₄×C₄⋊C₄

yes

C₂×C₇₁

yes

C₂×C₇₁

yes

C₂×C₇₁

C₄×D₈

yes

—

C₂×C₇₁

C₄×Q₁₆

yes

—

C₄×#107

—

maximal, Sylow

C₇₁

C₄×#107

yes

characteristic, complemented, Sylow

C₂×#107

—

C₂×C₄×C₈

—

C₄×C₄⋊C₄

—

#340

—

#340

D₇₁

yes

characteristic

C₂²×C₈

—

C₂²×C₈

C₂×D₇₁

yes

characteristic

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₇₁

yes

—

C₂×C₄²

—

C₂×C₄²

—

142

—

#107

—

C₄×C₈

—

#95

—

142

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

142

—

C₂²×C₄

C₂²×D₇₁

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₄×D₇₁

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

142

—

C₄⋊C₄

C₄×D₇₁

yes

—

C₄²

—

284

—

C₄²

—

C₄²

—

C₂³

D₄×D₇₁

yes

characteristic

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

284

—

103

C₂×C₄

—

284

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

142

—

109

C₂×C₄

—

142

—

110

C₂×C₄

C₂×C₄×D₇₁

yes

characteristic

111

C₂×C₄

C₂×C₄×D₇₁

yes

—

112

C₂×C₄

Q₈×D₇₁

yes

characteristic

113

C₂×C₄

#11765864

yes

—

114

C₈

—

115

C₈

—

116

C₂²

C₄⋊C₄×D₇₁

yes

characteristic

117

C₂²

(C₂²⋊D₇₁)⋊C₄

yes

—

118

C₂²

D₈×D₇₁

yes

characteristic

119

C₂²

Q₁₆×D₇₁

yes

characteristic

120

C₂²

#14556149

yes

—

121

C₄

—

122

C₄

—

284

—

123

C₄

—

124

C₄

—

142

—

125

C₄

C₄²×D₇₁

yes

characteristic

126

C₄

#14556084

yes

characteristic

127

C₄

#14556093

yes

—

128

C₂

#17985850

yes

characteristic

129

C₂

#107×D₇₁

yes

characteristic

130

C₂

#17986034

yes

characteristic

131

C₂

#17986079

yes

—

132

C₂

#17986091

yes

—

	subgroup	G/H
center	C₂³	D₄×D₇₁
commutator	C₄×C₇₁	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₇₁
Fitting	#340×C₇₁	C₂
socle	C₂³×C₇₁	C₂×D₄
hypercenter	#340	D₇₁
hypocenter	C₇₁	C₄×#107
LCS	G ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ #340

subgroup

G/H

center

C₂³

D₄×D₇₁

commutator

C₄×C₇₁

C₂×C₄²

Frattini

C₂²×C₄

C₂²×D₇₁

Fitting

#340×C₇₁

C₂

socle

C₂³×C₇₁

C₂×D₄

D₇₁

C₇₁

G ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ #340

N	H ≅ G/N	type
10. #14556061	C₄	semidirect
11. C₈×Dic₇₁	C₄	semidirect
48. C₇₁⋊C₈	C₄²	semidirect
60. Dic₇₁	#107	semidirect
67. C₇₁	C₄×#107	semidirect

H ≅ G/N

type

10. #14556061

C₄

semidirect

11. C₈×Dic₇₁

C₄

semidirect

48. C₇₁⋊C₈

C₄²

semidirect

60. Dic₇₁

#107

semidirect

67. C₇₁

C₄×#107

semidirect

Subgroups — #22588232

Conjugacy classes of subgroups

Complements