Group #22601018 — subgroups

Subgroups — #22601018

Order 9088.

336 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 336 conjugacy classes of subgroups — too many to draw. Only its 141 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	#17986728	C₂	1	1	yes	maximal, characteristic
2	#17986351	C₂	1	3	yes	maximal
3	#17986291	C₂	1	3	yes	maximal
4	#17986209	C₂	1	3	yes	maximal
5	Q₈×C₇₁⋊C₈	C₂	1	1	yes	maximal, characteristic
6	C₄×#11765855	C₂	1	3	yes	maximal
7	Q₈²×C₇₁	C₂	1	1	yes	maximal, characteristic
8	C₂×#11765855	C₂²	1	3	yes	—
9	#14556128	C₂²	1	3	yes	—
10	#14556119	—	2	3	no	—
11	Q₈×Dic₇₁	—	2	1	no	—
12	C₄×Dic₁₄₂	C₂²	1	3	yes	—
13	#14556061	C₂²	1	3	yes	—
14	C₄⋊(C₇₁⋊C₈)	C₂²	1	3	yes	—
15	#14556052	C₂²	1	1	yes	characteristic
16	#14556051	C₂²	1	3	yes	—
17	#14556051	C₂²	1	6	yes	—
18	C₄×C₇₁⋊C₈	C₂²	1	3	yes	—
19	C₄⋊Q₈×C₇₁	—	2	3	no	—
20	C₄⋊Q₈×C₇₁	C₂²	1	3	yes	—
21	C₄×Q₈×C₇₁	—	2	1	no	—
22	C₄×Q₈×C₇₁	C₂²	1	3	yes	—
23	C₄×Q₈×C₇₁	C₂²	1	1	yes	characteristic
24	C₂×Dic₁₄₂	C₂³	1	3	yes	—
25	#11765855	—	2	6	no	—
26	C₂×C₇₁⋊C₈	C₂³	1	1	yes	characteristic
27	C₂×C₇₁⋊C₈	C₂³	1	3	yes	—
28	C₄⋊Dic₇₁	—	2	3	no	—
29	C₄⋊Dic₇₁	—	2	3	no	—
30	C₄⋊Dic₇₁	C₂³	1	1	yes	characteristic
31	Dic₇₁⋊C₄	—	2	3	no	—
32	C₄×Dic₇₁	—	2	3	no	—
33	C₂×Q₈×C₇₁	—	2	3	no	—
34	C₂×Q₈×C₇₁	D₄	1	1	yes	characteristic
35	C₂×Q₈×C₇₁	C₂³	1	1	yes	characteristic
36	C₄⋊C₄×C₇₁	—	2	3	no	—
37	C₄⋊C₄×C₇₁	—	2	3	no	—
38	C₄⋊C₄×C₇₁	D₄	1	3	yes	—
39	C₄⋊C₄×C₇₁	C₂³	1	3	yes	—
40	C₄²×C₇₁	—	2	3	no	—
41	C₄²×C₇₁	C₂³	1	3	yes	—
42	Q₈×C₇₁	—	2	1	no	—
43	Q₈×C₇₁	—	8	3	no	—
44	Q₈×C₇₁	Q₁₆	1	4	yes	complemented
45	Q₈×C₇₁	#41	1	2	yes	—
46	C₂×C₄×C₇₁	—	2	1	no	—
47	C₂×C₄×C₇₁	—	2	3	no	—
48	C₂×C₄×C₇₁	C₂×D₄	1	3	yes	—
49	C₂×C₄×C₇₁	C₂×D₄	1	3	yes	—
50	C₂×C₄×C₇₁	C₂⁴	1	1	yes	characteristic
51	C₂×Dic₇₁	—	2	1	no	—
52	C₂×Dic₇₁	—	2	3	no	—
53	Dic₁₄₂	—	2	3	no	—
54	Dic₁₄₂	—	2	3	no	—
55	C₇₁⋊C₈	—	2	3	no	—
56	C₇₁⋊C₈	—	2	1	no	—
57	C₂²×C₇₁	C₂²×D₄	1	1	yes	characteristic
58	C₄×C₇₁	—	2	2	no	—
59	C₄×C₇₁	—	2	3	no	—
60	C₄×C₇₁	—	4	3	no	—
61	C₄×C₇₁	C₂×Q₁₆	1	6	yes	—
62	C₄×C₇₁	C₂×#41	1	1	yes	characteristic
63	C₄×C₇₁	2₋¹⁺⁴	1	1	yes	characteristic
64	Dic₇₁	—	4	3	no	—
65	Dic₇₁	—	4	1	no	—
66	C₂×C₇₁	#548	1	1	yes	characteristic
67	C₂×C₇₁	C₂²×Q₁₆	1	1	yes	characteristic
68	C₂×C₇₁	#577	1	1	yes	characteristic
69	#3363	—	71	1	no	maximal, Sylow
70	C₇₁	#3363	1	1	yes	characteristic, complemented, Sylow
71	Q₈²	D₇₁	1	1	yes	characteristic
72	#557	—	71	1	no	—
73	#487	—	71	3	no	—
74	#477	—	71	3	no	—
75	#462	—	71	3	no	—
76	C₈×Q₈	—	71	1	no	—
77	C₄×Q₁₆	—	71	3	no	—
78	C₂×Q₁₆	—	71	3	no	—
79	C₄⋊Q₈	—	2	3	no	—
80	C₄⋊Q₈	—	71	3	no	—
81	C₄⋊Q₈	C₂×D₇₁	1	3	yes	—
82	#125	—	142	3	no	—
83	C₄×Q₈	—	2	1	no	—
84	C₄×Q₈	—	71	3	no	—
85	C₄×Q₈	—	142	1	no	—
86	C₄×Q₈	C₂×D₇₁	1	3	yes	—
87	C₄×Q₈	C₂×D₇₁	1	1	yes	characteristic
88	#107	—	71	3	no	—
89	C₄⋊C₈	—	71	3	no	—
90	#103	—	71	1	no	—
91	#103	—	71	3	no	—
92	#103	—	71	6	no	—
93	C₄×C₈	—	71	3	no	—
94	C₂×Q₈	—	2	3	no	—
95	C₂×Q₈	—	71	3	no	—
96	C₂×Q₈	C₂²⋊D₇₁	1	1	yes	characteristic
97	C₂×Q₈	C₂²×D₇₁	1	1	yes	characteristic
98	Q₁₆	—	142	6	no	—
99	C₂×C₈	—	71	1	no	—
100	C₂×C₈	—	71	3	no	—
101	C₄⋊C₄	—	2	3	no	—
102	C₄⋊C₄	—	2	3	no	—
103	C₄⋊C₄	—	71	1	no	—
104	C₄⋊C₄	—	142	3	no	—
105	C₄⋊C₄	—	142	3	no	—
106	C₄⋊C₄	—	142	3	no	—
107	C₄⋊C₄	C₂²⋊D₇₁	1	3	yes	—
108	C₄⋊C₄	C₂²×D₇₁	1	3	yes	—
109	C₄²	—	2	3	no	—
110	C₄²	—	142	3	no	—
111	C₄²	C₂²×D₇₁	1	3	yes	—
112	Q₈	—	2	1	no	—
113	Q₈	—	8	3	no	—
114	Q₈	—	142	3	no	—
115	Q₈	—	142	3	no	—
116	Q₈	#11765855	1	4	yes	complemented
117	Q₈	#11765866	1	2	yes	—
118	C₂×C₄	—	2	1	no	—
119	C₂×C₄	—	2	3	no	—
120	C₂×C₄	—	142	1	no	—
121	C₂×C₄	—	142	3	no	—
122	C₂×C₄	C₂×C₂²⋊D₇₁	1	3	yes	—
123	C₂×C₄	C₂×C₂²⋊D₇₁	1	3	yes	—
124	C₂×C₄	C₂³×D₇₁	1	1	yes	characteristic
125	C₈	—	142	3	no	—
126	C₈	—	142	1	no	—
127	C₂²	C₂²×C₂²⋊D₇₁	1	1	yes	characteristic
128	C₄	—	2	2	no	—
129	C₄	—	2	3	no	—
130	C₄	—	4	3	no	—
131	C₄	—	284	3	no	—
132	C₄	—	284	1	no	—
133	C₄	C₂×#11765855	1	6	yes	—
134	C₄	C₂×#11765866	1	1	yes	characteristic
135	C₄	#14556176	1	1	yes	characteristic
136	C₂	#17986660	1	1	yes	characteristic
137	C₂	C₂²×#11765855	1	1	yes	characteristic
138	C₂	#17986810	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17986728

C₂

yes

maximal, characteristic

#17986351

C₂

yes

maximal

#17986291

C₂

yes

maximal

#17986209

C₂

yes

maximal

Q₈×C₇₁⋊C₈

C₂

yes

maximal, characteristic

C₄×#11765855

C₂

yes

maximal

Q₈²×C₇₁

C₂

yes

maximal, characteristic

C₂×#11765855

C₂²

yes

—

#14556128

C₂²

yes

—

#14556119

—

Q₈×Dic₇₁

—

C₄×Dic₁₄₂

C₂²

yes

—

#14556061

C₂²

yes

—

C₄⋊(C₇₁⋊C₈)

C₂²

yes

—

#14556052

C₂²

yes

characteristic

#14556051

C₂²

yes

—

#14556051

C₂²

yes

—

C₄×C₇₁⋊C₈

C₂²

yes

—

C₄⋊Q₈×C₇₁

—

C₄⋊Q₈×C₇₁

C₂²

yes

—

C₄×Q₈×C₇₁

—

C₄×Q₈×C₇₁

C₂²

yes

—

C₄×Q₈×C₇₁

C₂²

yes

characteristic

C₂×Dic₁₄₂

C₂³

yes

—

#11765855

—

C₂×C₇₁⋊C₈

C₂³

yes

characteristic

C₂×C₇₁⋊C₈

C₂³

yes

—

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

C₂³

yes

characteristic

Dic₇₁⋊C₄

—

C₄×Dic₇₁

—

C₂×Q₈×C₇₁

—

C₂×Q₈×C₇₁

D₄

yes

characteristic

C₂×Q₈×C₇₁

C₂³

yes

characteristic

C₄⋊C₄×C₇₁

—

C₄⋊C₄×C₇₁

—

C₄⋊C₄×C₇₁

D₄

yes

—

C₄⋊C₄×C₇₁

C₂³

yes

—

C₄²×C₇₁

—

C₄²×C₇₁

C₂³

yes

—

Q₈×C₇₁

—

Q₈×C₇₁

—

Q₈×C₇₁

Q₁₆

yes

complemented

Q₈×C₇₁

#41

yes

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

C₂×D₄

yes

—

C₂×C₄×C₇₁

C₂×D₄

yes

—

C₂×C₄×C₇₁

C₂⁴

yes

characteristic

C₂×Dic₇₁

—

C₂×Dic₇₁

—

Dic₁₄₂

—

Dic₁₄₂

—

C₇₁⋊C₈

—

C₇₁⋊C₈

—

C₂²×C₇₁

C₂²×D₄

yes

characteristic

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

C₂×Q₁₆

yes

—

C₄×C₇₁

C₂×#41

yes

characteristic

C₄×C₇₁

2₋¹⁺⁴

yes

characteristic

Dic₇₁

—

Dic₇₁

—

C₂×C₇₁

#548

yes

characteristic

C₂×C₇₁

C₂²×Q₁₆

yes

C₂×C₇₁

yes

—

C₇₁

yes

characteristic, complemented, Sylow

Q₈²

D₇₁

yes

—

—

—

—

C₈×Q₈

—

C₄×Q₁₆

—

C₂×Q₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₇₁

yes

—

#125

—

142

—

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

—

142

—

C₄×Q₈

C₂×D₇₁

yes

—

C₄×Q₈

C₂×D₇₁

yes

characteristic

#107

—

C₄⋊C₈

—

#103

—

#103

—

#103

—

C₄×C₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂²⋊D₇₁

yes

characteristic

C₂×Q₈

C₂²×D₇₁

yes

characteristic

Q₁₆

—

142

—

C₂×C₈

—

100

C₂×C₈

—

101

C₄⋊C₄

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

142

—

105

C₄⋊C₄

—

142

—

106

C₄⋊C₄

—

142

—

107

C₄⋊C₄

C₂²⋊D₇₁

yes

—

108

C₄⋊C₄

C₂²×D₇₁

yes

—

109

C₄²

—

110

C₄²

—

142

—

111

C₄²

C₂²×D₇₁

yes

—

112

Q₈

—

113

Q₈

—

114

Q₈

—

142

—

115

Q₈

—

142

—

116

Q₈

#11765855

yes

complemented

117

Q₈

#11765866

yes

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

142

—

121

C₂×C₄

—

142

—

122

C₂×C₄

C₂×C₂²⋊D₇₁

yes

—

123

C₂×C₄

C₂×C₂²⋊D₇₁

yes

—

124

C₂×C₄

C₂³×D₇₁

yes

characteristic

125

C₈

—

142

—

126

C₈

—

142

—

127

C₂²

C₂²×C₂²⋊D₇₁

yes

characteristic

128

C₄

—

129

C₄

—

130

C₄

—

131

C₄

—

284

—

132

C₄

—

284

—

133

C₄

C₂×#11765855

yes

—

134

C₄

C₂×#11765866

yes

characteristic

135

C₄

#14556176

yes

characteristic

136

C₂

#17986660

yes

characteristic

137

C₂

C₂²×#11765855

yes

characteristic

138

C₂

#17986810

yes

characteristic

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₇₁
commutator	C₂×C₄×C₇₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₇₁
Fitting	Q₈²×C₇₁	C₂
socle	C₂²×C₇₁	C₂²×D₄
hypercenter	Q₈²	D₇₁
hypocenter	C₇₁	#3363
LCS	G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ Q₈²

subgroup

G/H

center

C₂²

C₂²×C₂²⋊D₇₁

commutator

C₂×C₄×C₇₁

C₂⁴

Frattini

C₂×C₄

C₂³×D₇₁

Fitting

Q₈²×C₇₁

C₂

socle

C₂²×C₇₁

C₂²×D₄

hypercenter

Q₈²

D₇₁

hypocenter

C₇₁

#3363

LCS

G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂² ◁ C₄×Q₈ ◁ Q₈²

N	H ≅ G/N	type
44. Q₈×C₇₁	Q₁₆	semidirect
70. C₇₁	#3363	semidirect
116. Q₈	#11765855	semidirect

H ≅ G/N

type

44. Q₈×C₇₁

Q₁₆

semidirect

70. C₇₁

#3363

semidirect

116. Q₈

#11765855

semidirect

Subgroups — #22601018

Conjugacy classes of subgroups

Complements