Group #22586955 — subgroups

Subgroups — #22586955

Order 9088.

212 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 212 conjugacy classes of subgroups — too many to draw. Only its 81 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	#17986056	C₂	1	1	yes	maximal, characteristic
2	C₂×#14556039	C₂	1	1	yes	maximal, characteristic
3	#17985741	C₂	1	4	yes	maximal, complemented
4	#533×C₇₁	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₇₁⋊C₈	—	2	1	no	—
6	C₄⋊(C₇₁⋊C₈)	—	2	1	no	—
7	C₂²⋊(C₇₁⋊C₈)	—	2	1	no	—
8	#14556039	—	2	1	no	—
9	#14556039	C₂²	1	2	yes	—
10	#14556039	C₂²	1	2	yes	—
11	C₄⋊Q₈×C₇₁	C₄	1	1	yes	characteristic
12	C₄⋊Q₈×C₇₁	C₄	1	1	yes	characteristic
13	#125×C₇₁	C₂²	1	2	yes	—
14	C₂²⋊Q₈×C₇₁	—	2	1	no	—
15	C₂²⋊Q₈×C₇₁	—	2	1	no	—
16	C₄×Q₈×C₇₁	—	2	1	no	—
17	C₄×Q₈×C₇₁	—	2	1	no	—
18	#117×C₇₁	C₄	1	2	yes	—
19	C₂×C₄²×C₇₁	C₂²	1	1	yes	characteristic
20	C₂×C₇₁⋊C₈	—	2	4	no	—
21	C₂×C₇₁⋊C₈	—	2	2	no	—
22	C₂×C₇₁⋊C₈	—	2	2	no	—
23	C₂×Q₈×C₇₁	—	2	1	no	—
24	C₂×Q₈×C₇₁	—	2	1	no	—
25	C₂²×C₄×C₇₁	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₄×C₇₁	D₄	1	1	yes	characteristic
27	C₂²×C₄×C₇₁	D₄	1	1	yes	characteristic
28	C₄⋊C₄×C₇₁	—	2	1	no	—
29	C₄⋊C₄×C₇₁	—	2	1	no	—
30	C₄⋊C₄×C₇₁	—	2	2	no	—
31	C₄⋊C₄×C₇₁	—	2	2	no	—
32	C₄⋊C₄×C₇₁	C₂×C₄	1	2	yes	—
33	C₄⋊C₄×C₇₁	C₂×C₄	1	2	yes	—
34	C₂²⋊C₄×C₇₁	—	2	2	no	—
35	C₄²×C₇₁	—	2	2	no	—
36	C₄²×C₇₁	C₂×C₄	1	1	yes	characteristic
37	C₄²×C₇₁	D₄	1	1	yes	characteristic
38	C₄²×C₇₁	D₄	1	1	yes	characteristic
39	C₄²×C₇₁	C₂³	1	1	yes	characteristic
40	C₂³×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
41	Q₈×C₇₁	—	4	2	no	—
42	Q₈×C₇₁	—	4	2	no	—
43	C₂×C₄×C₇₁	—	2	2	no	—
44	C₂×C₄×C₇₁	—	2	2	no	—
45	C₂×C₄×C₇₁	—	2	2	no	—
46	C₂×C₄×C₇₁	—	2	2	no	—
47	C₂×C₄×C₇₁	—	4	1	no	—
48	C₂×C₄×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
49	C₂×C₄×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
50	C₂×C₄×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
51	C₂×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
52	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
53	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
54	C₇₁⋊C₈	—	2	4	no	—
55	C₇₁⋊C₈	—	4	2	no	—
56	C₂²×C₇₁	—	2	1	no	—
57	C₂²×C₇₁	—	2	1	no	—
58	C₂²×C₇₁	#101	1	2	yes	—
59	C₂²×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
60	C₄×C₇₁	—	2	1	no	—
61	C₄×C₇₁	—	2	1	no	—
62	C₄×C₇₁	—	2	1	no	—
63	C₄×C₇₁	—	2	1	no	—
64	C₄×C₇₁	—	2	1	no	—
65	C₄×C₇₁	—	4	2	no	—
66	C₄×C₇₁	—	4	2	no	—
67	C₄×C₇₁	#104	1	2	yes	—
68	C₂×C₇₁	—	2	2	no	—
69	C₂×C₇₁	C₂×#101	1	1	yes	characteristic
70	C₂×C₇₁	C₂×#104	1	1	yes	characteristic
71	C₂×C₇₁	#421	1	1	yes	characteristic
72	#1531	—	71	1	no	maximal, Sylow
73	C₇₁	#1531	1	1	yes	characteristic, complemented, Sylow
74	#533	D₇₁	1	1	yes	characteristic
75	#432	—	71	1	no	—
76	C₂×#97	—	71	1	no	—
77	#330	—	71	4	no	—
78	C₂²×C₈	—	142	1	no	—
79	C₄⋊Q₈	Dic₇₁	1	1	yes	characteristic
80	C₄⋊Q₈	Dic₇₁	1	1	yes	characteristic
81	#125	C₂×D₇₁	1	2	yes	—
82	C₂²⋊Q₈	—	2	1	no	—
83	C₂²⋊Q₈	—	2	1	no	—
84	C₄×Q₈	—	2	1	no	—
85	C₄×Q₈	—	2	1	no	—
86	#117	Dic₇₁	1	2	yes	—
87	C₂×C₄²	C₂×D₇₁	1	1	yes	characteristic
88	C₄⋊C₈	—	142	1	no	—
89	C₂²⋊C₈	—	142	1	no	—
90	#97	—	71	2	no	—
91	#97	—	71	2	no	—
92	#97	—	142	1	no	—
93	C₂×Q₈	—	2	1	no	—
94	C₂×Q₈	—	2	1	no	—
95	C₂²×C₄	C₂×Dic₇₁	1	1	yes	characteristic
96	C₂²×C₄	C₂²⋊D₇₁	1	1	yes	characteristic
97	C₂²×C₄	C₂²⋊D₇₁	1	1	yes	characteristic
98	C₂×C₈	—	142	2	no	—
99	C₂×C₈	—	142	4	no	—
100	C₂×C₈	—	142	2	no	—
101	C₄⋊C₄	—	2	1	no	—
102	C₄⋊C₄	—	2	1	no	—
103	C₄⋊C₄	—	2	2	no	—
104	C₄⋊C₄	—	2	2	no	—
105	C₄⋊C₄	C₂×Dic₇₁	1	2	yes	—
106	C₄⋊C₄	C₂×Dic₇₁	1	2	yes	—
107	C₂²⋊C₄	—	2	2	no	—
108	C₄²	—	2	2	no	—
109	C₄²	C₂×Dic₇₁	1	1	yes	characteristic
110	C₄²	C₂²⋊D₇₁	1	1	yes	characteristic
111	C₄²	C₂²⋊D₇₁	1	1	yes	characteristic
112	C₄²	C₂²×D₇₁	1	1	yes	characteristic
113	C₂³	C₂²⋊Dic₇₁	1	1	yes	characteristic
114	Q₈	—	4	2	no	—
115	Q₈	—	4	2	no	—
116	C₂×C₄	—	2	2	no	—
117	C₂×C₄	—	2	2	no	—
118	C₂×C₄	—	2	2	no	—
119	C₂×C₄	—	2	2	no	—
120	C₂×C₄	—	4	1	no	—
121	C₂×C₄	C₂²⋊Dic₇₁	1	1	yes	characteristic
122	C₂×C₄	C₂²⋊Dic₇₁	1	1	yes	characteristic
123	C₂×C₄	C₂²⋊Dic₇₁	1	1	yes	characteristic
124	C₂×C₄	C₂²×Dic₇₁	1	1	yes	characteristic
125	C₂×C₄	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
126	C₂×C₄	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
127	C₈	—	284	2	no	—
128	C₈	—	142	4	no	—
129	C₂²	—	2	1	no	—
130	C₂²	—	2	1	no	—
131	C₂²	#14556047	1	2	yes	—
132	C₂²	C₂×C₂²⋊Dic₇₁	1	1	yes	characteristic
133	C₄	—	2	1	no	—
134	C₄	—	2	1	no	—
135	C₄	—	2	1	no	—
136	C₄	—	2	1	no	—
137	C₄	—	2	1	no	—
138	C₄	—	4	2	no	—
139	C₄	—	4	2	no	—
140	C₄	#14556055	1	2	yes	—
141	C₂	—	2	2	no	—
142	C₂	C₂×#14556047	1	1	yes	characteristic
143	C₂	C₂×#14556055	1	1	yes	characteristic
144	C₂	#17986016	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17986056

C₂

yes

maximal, characteristic

C₂×#14556039

C₂

yes

maximal, characteristic

#17985741

C₂

yes

maximal, complemented

#533×C₇₁

C₂

yes

maximal, characteristic

C₂²×C₇₁⋊C₈

—

C₄⋊(C₇₁⋊C₈)

—

C₂²⋊(C₇₁⋊C₈)

—

#14556039

—

#14556039

C₂²

yes

—

#14556039

C₂²

yes

—

C₄⋊Q₈×C₇₁

C₄

yes

characteristic

C₄⋊Q₈×C₇₁

C₄

yes

characteristic

#125×C₇₁

C₂²

yes

—

C₂²⋊Q₈×C₇₁

—

C₂²⋊Q₈×C₇₁

—

C₄×Q₈×C₇₁

—

C₄×Q₈×C₇₁

—

#117×C₇₁

C₄

yes

—

C₂×C₄²×C₇₁

C₂²

yes

characteristic

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

C₂×Q₈×C₇₁

—

C₂×Q₈×C₇₁

—

C₂²×C₄×C₇₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₇₁

D₄

yes

characteristic

C₂²×C₄×C₇₁

D₄

yes

characteristic

C₄⋊C₄×C₇₁

—

C₄⋊C₄×C₇₁

—

C₄⋊C₄×C₇₁

—

C₄⋊C₄×C₇₁

—

C₄⋊C₄×C₇₁

C₂×C₄

yes

—

C₄⋊C₄×C₇₁

C₂×C₄

yes

—

C₂²⋊C₄×C₇₁

—

C₄²×C₇₁

—

C₄²×C₇₁

C₂×C₄

yes

characteristic

C₄²×C₇₁

D₄

yes

characteristic

C₄²×C₇₁

D₄

yes

characteristic

C₄²×C₇₁

C₂³

yes

characteristic

C₂³×C₇₁

C₂²⋊C₄

yes

characteristic

Q₈×C₇₁

—

Q₈×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₇₁

C₂²⋊C₄

yes

characteristic

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×D₄

yes

characteristic

C₇₁⋊C₈

—

C₇₁⋊C₈

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

#101

yes

—

C₂²×C₇₁

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

#104

yes

—

C₂×C₇₁

—

C₂×C₇₁

yes

C₂×C₇₁

yes

C₂×C₇₁

yes

—

C₇₁

yes

characteristic, complemented, Sylow

#533

D₇₁

yes

characteristic

#432

—

C₂×#97

—

#330

—

C₂²×C₈

—

142

—

C₄⋊Q₈

Dic₇₁

yes

characteristic

C₄⋊Q₈

Dic₇₁

yes

characteristic

#125

C₂×D₇₁

yes

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

C₄×Q₈

—

C₄×Q₈

—

#117

Dic₇₁

yes

—

C₂×C₄²

C₂×D₇₁

yes

characteristic

C₄⋊C₈

—

142

—

C₂²⋊C₈

—

142

—

#97

—

#97

—

#97

—

142

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

C₂×Dic₇₁

yes

characteristic

C₂²×C₄

C₂²⋊D₇₁

yes

characteristic

C₂²×C₄

C₂²⋊D₇₁

yes

characteristic

C₂×C₈

—

142

—

C₂×C₈

—

142

—

100

C₂×C₈

—

142

—

101

C₄⋊C₄

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄⋊C₄

C₂×Dic₇₁

yes

—

106

C₄⋊C₄

C₂×Dic₇₁

yes

—

107

C₂²⋊C₄

—

108

C₄²

—

109

C₄²

C₂×Dic₇₁

yes

characteristic

110

C₄²

C₂²⋊D₇₁

yes

characteristic

111

C₄²

C₂²⋊D₇₁

yes

characteristic

112

C₄²

C₂²×D₇₁

yes

characteristic

113

C₂³

C₂²⋊Dic₇₁

yes

characteristic

114

Q₈

—

115

Q₈

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

C₂²⋊Dic₇₁

yes

characteristic

122

C₂×C₄

C₂²⋊Dic₇₁

yes

characteristic

123

C₂×C₄

C₂²⋊Dic₇₁

yes

characteristic

124

C₂×C₄

C₂²×Dic₇₁

yes

characteristic

125

C₂×C₄

C₂×C₂²⋊D₇₁

yes

characteristic

126

C₂×C₄

C₂×C₂²⋊D₇₁

yes

characteristic

127

C₈

—

284

—

128

C₈

—

142

—

129

C₂²

—

130

C₂²

—

131

C₂²

#14556047

yes

—

132

C₂²

C₂×C₂²⋊Dic₇₁

yes

characteristic

133

C₄

—

134

C₄

—

135

C₄

—

136

C₄

—

137

C₄

—

138

C₄

—

139

C₄

—

140

C₄

#14556055

yes

—

141

C₂

—

142

C₂

C₂×#14556047

yes

characteristic

143

C₂

C₂×#14556055

yes

characteristic

144

C₂

#17986016

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊Dic₇₁
commutator	C₂×C₄×C₇₁	C₂²×C₄
Frattini	C₄²	C₂²×D₇₁
Fitting	#533×C₇₁	C₂
socle	C₂²×C₇₁	C₂×C₂²⋊C₄
hypercenter	#533	D₇₁
hypocenter	C₇₁	#1531
LCS	G ▷ C₂×C₄×C₇₁ ▷ C₂²×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #533

subgroup

G/H

center

C₂²

C₂×C₂²⋊Dic₇₁

commutator

C₂×C₄×C₇₁

C₂²×C₄

Frattini

C₄²

C₂²×D₇₁

Fitting

#533×C₇₁

C₂

socle

C₂²×C₇₁

C₂×C₂²⋊C₄

D₇₁

C₇₁

G ▷ C₂×C₄×C₇₁ ▷ C₂²×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #533

N	H ≅ G/N	type
3. #17985741	C₂	semidirect
73. C₇₁	#1531	semidirect

H ≅ G/N

type

3. #17985741

C₂

semidirect

73. C₇₁

#1531

semidirect

Subgroups — #22586955

Conjugacy classes of subgroups

Complements