Group #22587041 — subgroups

Subgroups — #22587041

Order 9088.

208 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 208 conjugacy classes of subgroups — too many to draw. Only its 77 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	C₂×C₄⋊(C₇₁⋊C₈)	C₂	1	1	yes	maximal, characteristic
3	#17985746	C₂	1	2	yes	maximal, complemented
4	#17985744	C₂	1	2	yes	maximal, complemented
5	#432×C₇₁	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₇₁⋊C₈	—	2	1	no	—
7	#14556128	C₂²	1	1	yes	characteristic
8	#14556128	C₂²	1	1	yes	characteristic
9	#14556120	C₄	1	2	yes	—
10	#14556107	—	2	1	no	—
11	#14556107	—	2	1	no	—
12	C₄×Dic₁₄₂	—	2	1	no	—
13	C₄×Dic₁₄₂	—	2	1	no	—
14	#14556087	C₄	1	2	yes	—
15	C₄⋊(C₇₁⋊C₈)	—	2	1	no	—
16	C₄⋊(C₇₁⋊C₈)	C₂²	1	2	yes	—
17	C₂×C₄²×C₇₁	C₂²	1	1	yes	characteristic
18	C₄⋊C₈×C₇₁	C₂²	1	2	yes	—
19	C₂²⋊C₈×C₇₁	—	2	1	no	—
20	#97×C₇₁	—	2	1	no	—
21	C₂×Dic₁₄₂	—	2	1	no	—
22	C₂×Dic₁₄₂	—	2	1	no	—
23	C₂×C₇₁⋊C₈	—	2	2	no	—
24	C₂×C₇₁⋊C₈	—	4	2	no	—
25	C₄⋊Dic₇₁	—	2	1	no	—
26	C₄⋊Dic₇₁	—	2	1	no	—
27	C₄⋊Dic₇₁	C₂×C₄	1	2	yes	—
28	C₄⋊Dic₇₁	C₂×C₄	1	2	yes	—
29	Dic₇₁⋊C₄	—	2	2	no	—
30	Dic₇₁⋊C₄	—	2	2	no	—
31	C₂²⋊Dic₇₁	—	2	2	no	—
32	C₄×Dic₇₁	—	2	2	no	—
33	C₂²×C₄×C₇₁	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₄×C₇₁	D₄	1	1	yes	characteristic
35	C₂²×C₄×C₇₁	D₄	1	1	yes	characteristic
36	C₂×C₈×C₇₁	—	2	2	no	—
37	C₄²×C₇₁	C₂×C₄	1	1	yes	characteristic
38	C₄²×C₇₁	D₄	1	1	yes	characteristic
39	C₄²×C₇₁	D₄	1	1	yes	characteristic
40	C₄²×C₇₁	C₂³	1	1	yes	characteristic
41	C₂³×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
42	C₂×C₄×C₇₁	—	2	2	no	—
43	C₂×C₄×C₇₁	—	2	2	no	—
44	C₂×C₄×C₇₁	—	2	1	no	—
45	C₂×C₄×C₇₁	—	2	1	no	—
46	C₂×C₄×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
47	C₂×C₄×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
48	C₂×C₄×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
49	C₂×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
50	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
51	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
52	C₂×Dic₇₁	—	2	2	no	—
53	C₂×Dic₇₁	—	2	2	no	—
54	Dic₁₄₂	—	4	2	no	—
55	Dic₁₄₂	—	4	1	no	—
56	Dic₁₄₂	—	4	1	no	—
57	C₈×C₇₁	—	4	2	no	—
58	C₇₁⋊C₈	—	4	2	no	—
59	C₂²×C₇₁	—	2	1	no	—
60	C₂²×C₇₁	—	2	1	no	—
61	C₂²×C₇₁	#101	1	2	yes	—
62	C₂²×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
63	C₄×C₇₁	—	2	1	no	—
64	C₄×C₇₁	—	2	1	no	—
65	C₄×C₇₁	—	2	1	no	—
66	C₄×C₇₁	—	2	1	no	—
67	C₄×C₇₁	#135	1	2	yes	—
68	C₄×C₇₁	AGL(1,ℤ/8)	1	1	yes	characteristic
69	C₄×C₇₁	#137	1	1	yes	characteristic
70	Dic₇₁	—	4	2	no	—
71	Dic₇₁	—	4	2	no	—
72	C₂×C₇₁	—	2	2	no	—
73	C₂×C₇₁	C₂×#101	1	1	yes	characteristic
74	C₂×C₇₁	#416	1	1	yes	characteristic
75	C₂×C₇₁	#417	1	1	yes	characteristic
76	#1552	—	71	1	no	maximal, Sylow
77	C₇₁	#1552	1	1	yes	characteristic, complemented, Sylow
78	#533	—	71	1	no	—
79	#432	D₇₁	1	1	yes	characteristic
80	C₂×C₄⋊C₈	—	71	1	no	—
81	#332	—	71	2	no	—
82	#332	—	71	2	no	—
83	C₂²×C₈	—	142	1	no	—
84	C₄⋊Q₈	—	71	1	no	—
85	C₄⋊Q₈	—	71	1	no	—
86	#125	—	71	2	no	—
87	C₂²⋊Q₈	—	142	1	no	—
88	C₂²⋊Q₈	—	142	1	no	—
89	C₄×Q₈	—	142	1	no	—
90	C₄×Q₈	—	142	1	no	—
91	#117	—	71	2	no	—
92	C₂×C₄²	C₂×D₇₁	1	1	yes	characteristic
93	C₄⋊C₈	—	71	2	no	—
94	C₄⋊C₈	—	142	1	no	—
95	C₄⋊C₈	C₂×D₇₁	1	2	yes	—
96	C₂²⋊C₈	—	2	1	no	—
97	#97	—	2	1	no	—
98	C₂×Q₈	—	142	1	no	—
99	C₂×Q₈	—	142	1	no	—
100	C₂²×C₄	C₄×D₇₁	1	1	yes	characteristic
101	C₂²×C₄	D₂₈₄	1	1	yes	characteristic
102	C₂²×C₄	C₂²⋊D₇₁	1	1	yes	characteristic
103	C₂×C₈	—	2	2	no	—
104	C₂×C₈	—	284	2	no	—
105	C₂×C₈	—	142	2	no	—
106	C₄⋊C₄	—	71	2	no	—
107	C₄⋊C₄	—	71	2	no	—
108	C₄⋊C₄	—	142	2	no	—
109	C₄⋊C₄	—	142	2	no	—
110	C₄⋊C₄	—	142	1	no	—
111	C₄⋊C₄	—	142	1	no	—
112	C₂²⋊C₄	—	142	2	no	—
113	C₄²	—	142	2	no	—
114	C₄²	C₄×D₇₁	1	1	yes	characteristic
115	C₄²	D₂₈₄	1	1	yes	characteristic
116	C₄²	C₂²⋊D₇₁	1	1	yes	characteristic
117	C₄²	C₂²×D₇₁	1	1	yes	characteristic
118	C₂³	(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
119	Q₈	—	284	1	no	—
120	Q₈	—	284	2	no	—
121	Q₈	—	284	1	no	—
122	C₂×C₄	—	2	2	no	—
123	C₂×C₄	—	2	1	no	—
124	C₂×C₄	—	2	2	no	—
125	C₂×C₄	—	2	1	no	—
126	C₂×C₄	—	142	2	no	—
127	C₂×C₄	—	142	2	no	—
128	C₂×C₄	(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
129	C₂×C₄	(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
130	C₂×C₄	(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
131	C₂×C₄	C₂×C₄×D₇₁	1	1	yes	characteristic
132	C₂×C₄	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
133	C₂×C₄	C₂×D₂₈₄	1	1	yes	characteristic
134	C₈	—	4	2	no	—
135	C₈	—	284	2	no	—
136	C₂²	—	2	1	no	—
137	C₂²	—	2	1	no	—
138	C₂²	#14556046	1	2	yes	—
139	C₂²	C₂×(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
140	C₄	—	2	1	no	—
141	C₄	—	2	1	no	—
142	C₄	—	2	1	no	—
143	C₄	—	2	1	no	—
144	C₄	—	284	2	no	—
145	C₄	—	284	2	no	—
146	C₄	#14556148	1	2	yes	—
147	C₄	#14556156	1	1	yes	characteristic
148	C₄	#14556161	1	1	yes	characteristic
149	C₂	—	2	2	no	—
150	C₂	C₂×#14556046	1	1	yes	characteristic
151	C₂	#17985993	1	1	yes	characteristic
152	C₂	#17986001	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17986556

C₂

yes

maximal, characteristic

C₂×C₄⋊(C₇₁⋊C₈)

C₂

yes

maximal, characteristic

#17985746

C₂

yes

maximal, complemented

#17985744

C₂

yes

maximal, complemented

#432×C₇₁

C₂

yes

maximal, characteristic

C₂²×C₇₁⋊C₈

—

#14556128

C₂²

yes

characteristic

#14556128

C₂²

yes

characteristic

#14556120

C₄

yes

—

#14556107

—

#14556107

—

C₄×Dic₁₄₂

—

C₄×Dic₁₄₂

—

#14556087

C₄

yes

—

C₄⋊(C₇₁⋊C₈)

—

C₄⋊(C₇₁⋊C₈)

C₂²

yes

—

C₂×C₄²×C₇₁

C₂²

yes

characteristic

C₄⋊C₈×C₇₁

C₂²

yes

—

C₂²⋊C₈×C₇₁

—

#97×C₇₁

—

C₂×Dic₁₄₂

—

C₂×Dic₁₄₂

—

C₂×C₇₁⋊C₈

—

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

—

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

—

Dic₁₄₂

—

C₈×C₇₁

—

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

#135

yes

—

C₄×C₇₁

yes

C₄×C₇₁

yes

Dic₇₁

—

Dic₇₁

—

C₂×C₇₁

—

C₂×C₇₁

yes

C₂×C₇₁

yes

C₂×C₇₁

yes

—

C₇₁

yes

characteristic, complemented, Sylow

#533

—

#432

D₇₁

yes

characteristic

C₂×C₄⋊C₈

—

#332

—

#332

—

C₂²×C₈

—

142

—

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

—

C₄⋊C₈

C₂×D₇₁

yes

—

C₂²⋊C₈

—

#97

—

C₂×Q₈

—

142

—

C₂×Q₈

—

142

—

100

C₂²×C₄

C₄×D₇₁

yes

characteristic

101

C₂²×C₄

D₂₈₄

yes

characteristic

102

C₂²×C₄

C₂²⋊D₇₁

yes

characteristic

103

C₂×C₈

—

104

C₂×C₈

—

284

—

105

C₂×C₈

—

142

—

106

C₄⋊C₄

—

107

C₄⋊C₄

—

108

C₄⋊C₄

—

142

—

109

C₄⋊C₄

—

142

—

110

C₄⋊C₄

—

142

—

111

C₄⋊C₄

—

142

—

112

C₂²⋊C₄

—

142

—

113

C₄²

—

142

—

114

C₄²

C₄×D₇₁

yes

characteristic

115

C₄²

D₂₈₄

yes

characteristic

116

C₄²

C₂²⋊D₇₁

yes

characteristic

117

C₄²

C₂²×D₇₁

yes

characteristic

118

C₂³

(C₂×D₇₁)⋊C₄

yes

characteristic

119

Q₈

—

284

—

120

Q₈

—

284

—

121

Q₈

—

284

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

142

—

127

C₂×C₄

—

142

—

128

C₂×C₄

(C₂×D₇₁)⋊C₄

yes

characteristic

129

C₂×C₄

(C₂×D₇₁)⋊C₄

yes

characteristic

130

C₂×C₄

(C₂×D₇₁)⋊C₄

yes

characteristic

131

C₂×C₄

C₂×C₄×D₇₁

yes

characteristic

132

C₂×C₄

C₂×C₂²⋊D₇₁

yes

characteristic

133

C₂×C₄

C₂×D₂₈₄

yes

characteristic

134

C₈

—

135

C₈

—

284

—

136

C₂²

—

137

C₂²

—

138

C₂²

#14556046

yes

—

139

C₂²

C₂×(C₂×D₇₁)⋊C₄

yes

characteristic

140

C₄

—

141

C₄

—

142

C₄

—

143

C₄

—

144

C₄

—

284

—

145

C₄

—

284

—

146

C₄

#14556148

yes

—

147

C₄

#14556156

yes

characteristic

148

C₄

#14556161

yes

characteristic

149

C₂

—

150

C₂

C₂×#14556046

yes

characteristic

151

C₂

#17985993

yes

characteristic

152

C₂

#17986001

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	#1552
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. #17985746	C₂	semidirect
4. #17985744	C₂	semidirect
77. C₇₁	#1552	semidirect

H ≅ G/N

type

3. #17985746

C₂

semidirect

4. #17985744

C₂

semidirect

77. C₇₁

#1552

semidirect

Subgroups — #22587041

Conjugacy classes of subgroups

Complements