Group #22586977 — subgroups

Subgroups — #22586977

Order 9088.

194 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 194 conjugacy classes of subgroups — too many to draw. Only its 69 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	#17986556	C₂	1	1	yes	maximal, characteristic
2	#17986056	C₂	1	1	yes	maximal, characteristic
3	#17985740	C₂	1	4	yes	maximal, complemented
4	#432×C₇₁	C₂	1	1	yes	maximal, characteristic
5	#14556128	C₄	1	1	yes	characteristic
6	#14556128	C₄	1	1	yes	characteristic
7	#14556120	C₂²	1	2	yes	—
8	#14556107	—	2	1	no	—
9	#14556107	—	2	1	no	—
10	C₄×Dic₁₄₂	—	2	1	no	—
11	C₄×Dic₁₄₂	—	2	1	no	—
12	#14556087	C₄	1	2	yes	—
13	C₄⋊(C₇₁⋊C₈)	—	2	1	no	—
14	C₂²⋊(C₇₁⋊C₈)	—	2	1	no	—
15	#14556039	C₂²	1	2	yes	—
16	C₂×C₄²×C₇₁	C₂²	1	1	yes	characteristic
17	C₄⋊C₈×C₇₁	—	2	1	no	—
18	C₂²⋊C₈×C₇₁	—	2	1	no	—
19	#97×C₇₁	C₂²	1	2	yes	—
20	C₂×Dic₁₄₂	—	2	1	no	—
21	C₂×Dic₁₄₂	—	2	1	no	—
22	C₂×C₇₁⋊C₈	—	2	2	no	—
23	C₄⋊Dic₇₁	—	2	1	no	—
24	C₄⋊Dic₇₁	—	2	1	no	—
25	C₄⋊Dic₇₁	C₂×C₄	1	2	yes	—
26	C₄⋊Dic₇₁	C₂×C₄	1	2	yes	—
27	Dic₇₁⋊C₄	—	2	2	no	—
28	Dic₇₁⋊C₄	—	2	2	no	—
29	C₂²⋊Dic₇₁	—	2	2	no	—
30	C₄×Dic₇₁	—	2	2	no	—
31	C₂²×C₄×C₇₁	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₄×C₇₁	D₄	1	1	yes	characteristic
33	C₂²×C₄×C₇₁	D₄	1	1	yes	characteristic
34	C₂×C₈×C₇₁	—	2	2	no	—
35	C₄²×C₇₁	C₂×C₄	1	1	yes	characteristic
36	C₄²×C₇₁	D₄	1	1	yes	characteristic
37	C₄²×C₇₁	D₄	1	1	yes	characteristic
38	C₄²×C₇₁	C₂³	1	1	yes	characteristic
39	C₂³×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
40	C₂×C₄×C₇₁	—	2	2	no	—
41	C₂×C₄×C₇₁	—	4	1	no	—
42	C₂×C₄×C₇₁	—	4	1	no	—
43	C₂×C₄×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
44	C₂×C₄×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
45	C₂×C₄×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
46	C₂×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
47	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
48	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
49	C₂×Dic₇₁	—	2	2	no	—
50	C₂×Dic₇₁	—	2	2	no	—
51	Dic₁₄₂	—	4	2	no	—
52	Dic₁₄₂	—	4	2	no	—
53	C₈×C₇₁	—	4	2	no	—
54	C₇₁⋊C₈	—	4	2	no	—
55	C₂²×C₇₁	—	2	1	no	—
56	C₂²×C₇₁	—	2	1	no	—
57	C₂²×C₇₁	—	2	1	no	—
58	C₂²×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
59	C₄×C₇₁	—	2	1	no	—
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	#101	1	2	yes	—
65	Dic₇₁	—	4	2	no	—
66	Dic₇₁	—	4	2	no	—
67	C₂×C₇₁	—	4	1	no	—
68	C₂×C₇₁	C₂×#101	1	1	yes	characteristic
69	C₂×C₇₁	#421	1	1	yes	characteristic
70	C₂×C₇₁	#421	1	1	yes	characteristic
71	#1537	—	71	1	no	maximal, Sylow
72	C₇₁	#1537	1	1	yes	characteristic, complemented, Sylow
73	#533	—	71	1	no	—
74	#432	—	71	1	no	—
75	#432	D₇₁	1	1	yes	characteristic
76	#330	—	71	4	no	—
77	C₄⋊Q₈	—	71	1	no	—
78	C₄⋊Q₈	—	71	1	no	—
79	#125	—	71	2	no	—
80	C₂²⋊Q₈	—	142	1	no	—
81	C₂²⋊Q₈	—	142	1	no	—
82	C₄×Q₈	—	142	1	no	—
83	C₄×Q₈	—	142	1	no	—
84	#117	—	71	2	no	—
85	C₂×C₄²	C₂×D₇₁	1	1	yes	characteristic
86	C₄⋊C₈	—	2	1	no	—
87	C₄⋊C₈	—	142	1	no	—
88	C₂²⋊C₈	—	2	1	no	—
89	C₂²⋊C₈	—	142	1	no	—
90	#97	—	71	2	no	—
91	#97	C₂×D₇₁	1	2	yes	—
92	C₂×Q₈	—	142	1	no	—
93	C₂×Q₈	—	142	1	no	—
94	C₂²×C₄	C₄×D₇₁	1	1	yes	characteristic
95	C₂²×C₄	D₂₈₄	1	1	yes	characteristic
96	C₂²×C₄	C₂²⋊D₇₁	1	1	yes	characteristic
97	C₂×C₈	—	2	2	no	—
98	C₂×C₈	—	142	2	no	—
99	C₄⋊C₄	—	71	2	no	—
100	C₄⋊C₄	—	71	2	no	—
101	C₄⋊C₄	—	142	1	no	—
102	C₄⋊C₄	—	142	1	no	—
103	C₄⋊C₄	—	142	2	no	—
104	C₄⋊C₄	—	142	2	no	—
105	C₂²⋊C₄	—	142	2	no	—
106	C₄²	—	142	2	no	—
107	C₄²	C₄×D₇₁	1	1	yes	characteristic
108	C₄²	D₂₈₄	1	1	yes	characteristic
109	C₄²	C₂²⋊D₇₁	1	1	yes	characteristic
110	C₄²	C₂²×D₇₁	1	1	yes	characteristic
111	C₂³	(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
112	Q₈	—	284	2	no	—
113	Q₈	—	284	2	no	—
114	C₂×C₄	—	2	2	no	—
115	C₂×C₄	—	4	1	no	—
116	C₂×C₄	—	4	1	no	—
117	C₂×C₄	—	142	2	no	—
118	C₂×C₄	—	142	2	no	—
119	C₂×C₄	(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
120	C₂×C₄	(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
121	C₂×C₄	(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
122	C₂×C₄	C₂×C₄×D₇₁	1	1	yes	characteristic
123	C₂×C₄	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
124	C₂×C₄	C₂×D₂₈₄	1	1	yes	characteristic
125	C₈	—	4	2	no	—
126	C₈	—	284	2	no	—
127	C₂²	—	2	1	no	—
128	C₂²	—	2	1	no	—
129	C₂²	—	2	1	no	—
130	C₂²	C₂×(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
131	C₄	—	2	1	no	—
132	C₄	—	2	1	no	—
133	C₄	—	2	1	no	—
134	C₄	—	2	1	no	—
135	C₄	—	2	1	no	—
136	C₄	—	284	2	no	—
137	C₄	—	284	2	no	—
138	C₄	#14556046	1	2	yes	—
139	C₂	—	4	1	no	—
140	C₂	C₂×#14556046	1	1	yes	characteristic
141	C₂	#17986015	1	1	yes	characteristic
142	C₂	#17986017	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17986556

C₂

yes

maximal, characteristic

#17986056

C₂

yes

maximal, characteristic

#17985740

C₂

yes

maximal, complemented

#432×C₇₁

C₂

yes

maximal, characteristic

#14556128

C₄

yes

characteristic

#14556128

C₄

yes

characteristic

#14556120

C₂²

yes

—

#14556107

—

#14556107

—

C₄×Dic₁₄₂

—

C₄×Dic₁₄₂

—

#14556087

C₄

yes

—

C₄⋊(C₇₁⋊C₈)

—

C₂²⋊(C₇₁⋊C₈)

—

#14556039

C₂²

yes

—

C₂×C₄²×C₇₁

C₂²

yes

characteristic

C₄⋊C₈×C₇₁

—

C₂²⋊C₈×C₇₁

—

#97×C₇₁

C₂²

yes

—

C₂×Dic₁₄₂

—

C₂×Dic₁₄₂

—

C₂×C₇₁⋊C₈

—

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

C₂×C₄

yes

—

C₄⋊Dic₇₁

C₂×C₄

yes

—

Dic₇₁⋊C₄

—

Dic₇₁⋊C₄

—

C₂²⋊Dic₇₁

—

C₄×Dic₇₁

—

C₂²×C₄×C₇₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₇₁

D₄

yes

characteristic

C₂²×C₄×C₇₁

D₄

yes

characteristic

C₂×C₈×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

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

C₂×Dic₇₁

—

C₂×Dic₇₁

—

Dic₁₄₂

—

Dic₁₄₂

—

C₈×C₇₁

—

C₇₁⋊C₈

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

#101

yes

—

Dic₇₁

—

Dic₇₁

—

C₂×C₇₁

—

C₂×C₇₁

yes

C₂×C₇₁

yes

C₂×C₇₁

yes

—

C₇₁

yes

characteristic, complemented, Sylow

#533

—

#432

—

#432

D₇₁

yes

characteristic

#330

—

C₄⋊Q₈

—

C₄⋊Q₈

—

#125

—

C₂²⋊Q₈

—

142

—

C₂²⋊Q₈

—

142

—

C₄×Q₈

—

142

—

C₄×Q₈

—

142

—

#117

—

C₂×C₄²

C₂×D₇₁

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

142

—

C₂²⋊C₈

—

C₂²⋊C₈

—

142

—

#97

—

#97

C₂×D₇₁

yes

—

C₂×Q₈

—

142

—

C₂×Q₈

—

142

—

C₂²×C₄

C₄×D₇₁

yes

characteristic

C₂²×C₄

D₂₈₄

yes

characteristic

C₂²×C₄

C₂²⋊D₇₁

yes

characteristic

C₂×C₈

—

C₂×C₈

—

142

—

C₄⋊C₄

—

100

C₄⋊C₄

—

101

C₄⋊C₄

—

142

—

102

C₄⋊C₄

—

142

—

103

C₄⋊C₄

—

142

—

104

C₄⋊C₄

—

142

—

105

C₂²⋊C₄

—

142

—

106

C₄²

—

142

—

107

C₄²

C₄×D₇₁

yes

characteristic

108

C₄²

D₂₈₄

yes

characteristic

109

C₄²

C₂²⋊D₇₁

yes

characteristic

110

C₄²

C₂²×D₇₁

yes

characteristic

111

C₂³

(C₂×D₇₁)⋊C₄

yes

characteristic

112

Q₈

—

284

—

113

Q₈

—

284

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

142

—

118

C₂×C₄

—

142

—

119

C₂×C₄

(C₂×D₇₁)⋊C₄

yes

characteristic

120

C₂×C₄

(C₂×D₇₁)⋊C₄

yes

characteristic

121

C₂×C₄

(C₂×D₇₁)⋊C₄

yes

characteristic

122

C₂×C₄

C₂×C₄×D₇₁

yes

characteristic

123

C₂×C₄

C₂×C₂²⋊D₇₁

yes

characteristic

124

C₂×C₄

C₂×D₂₈₄

yes

characteristic

125

C₈

—

126

C₈

—

284

—

127

C₂²

—

128

C₂²

—

129

C₂²

—

130

C₂²

C₂×(C₂×D₇₁)⋊C₄

yes

characteristic

131

C₄

—

132

C₄

—

133

C₄

—

134

C₄

—

135

C₄

—

136

C₄

—

284

—

137

C₄

—

284

—

138

C₄

#14556046

yes

—

139

C₂

—

140

C₂

C₂×#14556046

yes

characteristic

141

C₂

#17986015

yes

characteristic

142

C₂

#17986017

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×(C₂×D₇₁)⋊C₄
commutator	C₂×C₄×C₇₁	C₂²×C₄
Frattini	C₄²	C₂²×D₇₁
Fitting	#432×C₇₁	C₂
socle	C₂²×C₇₁	C₂×C₂²⋊C₄
hypercenter	#432	D₇₁
hypocenter	C₇₁	#1537
LCS	G ▷ C₂×C₄×C₇₁ ▷ C₂²×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #432

subgroup

G/H

center

C₂²

C₂×(C₂×D₇₁)⋊C₄

commutator

C₂×C₄×C₇₁

C₂²×C₄

Frattini

C₄²

C₂²×D₇₁

Fitting

#432×C₇₁

C₂

socle

C₂²×C₇₁

C₂×C₂²⋊C₄

D₇₁

C₇₁

G ▷ C₂×C₄×C₇₁ ▷ C₂²×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #432

N	H ≅ G/N	type
3. #17985740	C₂	semidirect
72. C₇₁	#1537	semidirect

H ≅ G/N

type

3. #17985740

C₂

semidirect

72. C₇₁

#1537

semidirect

Subgroups — #22586977

Conjugacy classes of subgroups

Complements