Group #22597428 — subgroups

Subgroups — #22597428

Order 9088.

364 conjugacy classes of subgroups; only the 179 normal subgroups are shown.

normal subgroups only

This group has 364 conjugacy classes of subgroups — too many to draw. Only its 179 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	C₄×Q₈×D₇₁	C₂	1	1	yes	maximal, characteristic
2	#17986129	C₂	1	3	yes	maximal, complemented
3	#17986126	C₂	1	3	yes	maximal, complemented
4	Q₈×C₇₁⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
5	#17986055	C₂	1	3	yes	maximal
6	#17986050	C₂	1	3	yes	maximal
7	C₈×Q₈×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×Q₈×D₇₁	C₄	1	1	yes	characteristic
9	Q₈×Dic₇₁	C₄	1	1	yes	characteristic
10	C₄×Dic₁₄₂	C₂²	1	3	yes	—
11	#14556093	C₄	1	3	yes	—
12	C₄⋊C₄×D₇₁	C₄	1	3	yes	—
13	C₄²×D₇₁	C₂²	1	3	yes	—
14	C₄⋊(C₇₁⋊C₈)	C₂²	1	3	yes	—
15	Dic₇₁⋊C₈	C₂²	1	3	yes	—
16	Dic₇₁⋊C₈	C₂²	1	3	yes	—
17	(C₂×D₇₁)⋊C₈	C₂²	1	1	yes	characteristic
18	(C₂×D₇₁)⋊C₈	C₂²	1	3	yes	—
19	#14556038	C₂²	1	6	yes	—
20	C₄×C₇₁⋊C₈	C₂²	1	3	yes	—
21	C₄×Q₈×C₇₁	C₂²	1	1	yes	characteristic
22	C₄⋊C₈×C₇₁	C₂²	1	3	yes	—
23	C₄×C₈×C₇₁	C₂²	1	3	yes	—
24	C₂×Dic₁₄₂	C₂×C₄	1	3	yes	—
25	Q₈×D₇₁	—	2	4	no	—
26	C₂×C₄×D₇₁	C₂×C₄	1	3	yes	—
27	C₂×C₄×D₇₁	C₂×C₄	1	3	yes	—
28	C₂×C₄×D₇₁	C₂³	1	1	yes	characteristic
29	C₂×C₇₁⋊C₈	C₂³	1	1	yes	characteristic
30	C₂×C₇₁⋊C₈	C₂³	1	3	yes	—
31	C₄⋊Dic₇₁	C₂×C₄	1	3	yes	—
32	Dic₇₁⋊C₄	C₂×C₄	1	6	yes	—
33	C₄×Dic₇₁	C₂×C₄	1	3	yes	—
34	C₄×Dic₇₁	C₂×C₄	1	3	yes	—
35	C₄×Dic₇₁	C₂³	1	3	yes	—
36	C₂×Q₈×C₇₁	C₂×C₄	1	1	yes	characteristic
37	C₂×C₈×C₇₁	C₂³	1	1	yes	characteristic
38	C₂×C₈×C₇₁	C₂³	1	3	yes	—
39	C₄⋊C₄×C₇₁	C₂×C₄	1	3	yes	—
40	C₄²×C₇₁	C₂³	1	3	yes	—
41	C₂²×D₇₁	C₂²×C₄	1	1	yes	characteristic
42	Q₈×C₇₁	M₁₆	1	4	yes	—
43	C₂×C₄×C₇₁	C₂²×C₄	1	3	yes	—
44	C₂×C₄×C₇₁	C₂²×C₄	1	3	yes	—
45	C₂×C₄×C₇₁	C₂⁴	1	1	yes	characteristic
46	C₂×Dic₇₁	C₂²×C₄	1	1	yes	characteristic
47	C₂×Dic₇₁	C₂²×C₄	1	3	yes	—
48	C₂×Dic₇₁	C₂²×C₄	1	3	yes	—
49	C₄×D₇₁	—	2	6	no	—
50	C₄×D₇₁	—	2	6	no	—
51	C₄×D₇₁	—	2	1	no	—
52	C₄×D₇₁	—	2	1	no	—
53	Dic₁₄₂	—	2	6	no	—
54	C₈×C₇₁	—	2	3	no	—
55	C₈×C₇₁	—	2	1	no	—
56	C₇₁⋊C₈	—	2	1	no	—
57	C₇₁⋊C₈	—	2	3	no	—
58	C₂²×C₇₁	C₂³×C₄	1	1	yes	characteristic
59	C₂×D₇₁	—	2	1	no	—
60	C₂×D₇₁	—	2	1	no	—
61	C₂×D₇₁	#131	1	2	yes	—
62	C₄×C₇₁	—	2	3	no	—
63	C₄×C₇₁	C₂×M₁₆	1	6	yes	—
64	C₄×C₇₁	2₋¹⁺⁴	1	1	yes	characteristic
65	C₄×C₇₁	2₋¹⁺⁴	1	1	yes	characteristic
66	Dic₇₁	—	2	1	no	—
67	Dic₇₁	—	2	3	no	—
68	Dic₇₁	—	2	3	no	—
69	C₂×C₇₁	#519	1	1	yes	characteristic
70	C₂×C₇₁	C₂²×M₁₆	1	1	yes	characteristic
71	C₂×C₇₁	C₂×#131	1	1	yes	characteristic
72	D₇₁	—	2	2	no	—
73	#2984	—	71	1	no	maximal, Sylow
74	C₇₁	#2984	1	1	yes	characteristic, complemented, Sylow
75	C₂×C₄×Q₈	—	71	1	no	—
76	#446	—	71	3	no	—
77	#446	—	71	3	no	—
78	C₈×Q₈	—	71	1	no	—
79	C₈×Q₈	D₇₁	1	1	yes	characteristic, complemented
80	#432	—	71	3	no	—
81	#431	—	71	3	no	—
82	C₂²×Q₈	—	71	1	no	—
83	C₄×Q₈	—	71	1	no	—
84	C₄×Q₈	—	71	3	no	—
85	C₄×Q₈	—	71	3	no	—
86	C₄×Q₈	C₂×D₇₁	1	1	yes	characteristic
87	C₂×C₄⋊C₄	—	71	3	no	—
88	C₂×C₄²	—	71	3	no	—
89	C₄⋊C₈	—	71	3	no	—
90	C₄⋊C₈	—	71	3	no	—
91	C₄⋊C₈	—	71	3	no	—
92	C₄⋊C₈	C₂×D₇₁	1	3	yes	—
93	C₂²⋊C₈	—	71	1	no	—
94	C₂²⋊C₈	—	71	3	no	—
95	#97	—	71	6	no	—
96	C₄×C₈	—	71	3	no	—
97	C₄×C₈	C₂×D₇₁	1	3	yes	—
98	C₂×Q₈	—	71	3	no	—
99	C₂×Q₈	—	142	4	no	—
100	C₂×Q₈	C₄×D₇₁	1	1	yes	characteristic
101	C₂²×C₄	—	71	3	no	—
102	C₂²×C₄	—	71	3	no	—
103	C₂²×C₄	—	71	1	no	—
104	C₂×C₈	—	71	1	no	—
105	C₂×C₈	—	71	3	no	—
106	C₂×C₈	C₂²×D₇₁	1	1	yes	characteristic
107	C₂×C₈	C₂²×D₇₁	1	3	yes	—
108	C₄⋊C₄	—	71	6	no	—
109	C₄⋊C₄	—	71	3	no	—
110	C₄⋊C₄	C₄×D₇₁	1	3	yes	—
111	C₄²	—	71	3	no	—
112	C₄²	—	71	3	no	—
113	C₄²	—	71	3	no	—
114	C₄²	C₂²×D₇₁	1	3	yes	—
115	C₂³	—	71	1	no	—
116	Q₈	—	142	6	no	—
117	Q₈	#11765848	1	4	yes	—
118	C₂×C₄	—	71	1	no	—
119	C₂×C₄	—	71	3	no	—
120	C₂×C₄	—	71	3	no	—
121	C₂×C₄	—	142	6	no	—
122	C₂×C₄	—	142	1	no	—
123	C₂×C₄	—	142	1	no	—
124	C₂×C₄	—	142	6	no	—
125	C₂×C₄	C₂×C₄×D₇₁	1	3	yes	—
126	C₂×C₄	C₂×C₄×D₇₁	1	3	yes	—
127	C₂×C₄	C₂³×D₇₁	1	1	yes	characteristic
128	C₈	—	2	3	no	—
129	C₈	—	2	1	no	—
130	C₈	—	142	1	no	—
131	C₈	—	142	3	no	—
132	C₂²	—	71	2	no	—
133	C₂²	—	142	1	no	—
134	C₂²	—	142	1	no	—
135	C₂²	C₂²×C₄×D₇₁	1	1	yes	characteristic
136	C₄	—	2	3	no	—
137	C₄	—	142	1	no	—
138	C₄	—	142	3	no	—
139	C₄	—	142	3	no	—
140	C₄	C₂×#11765848	1	6	yes	—
141	C₄	#14556175	1	1	yes	characteristic
142	C₄	#14556176	1	1	yes	characteristic
143	C₂	—	142	2	no	—
144	C₂	#17986475	1	1	yes	characteristic
145	C₂	C₂²×#11765848	1	1	yes	characteristic
146	C₂	#131×D₇₁	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×Q₈×D₇₁

C₂

yes

maximal, characteristic

#17986129

C₂

yes

maximal, complemented

#17986126

C₂

yes

maximal, complemented

Q₈×C₇₁⋊C₈

C₂

yes

maximal, characteristic, complemented

#17986055

C₂

yes

maximal

#17986050

C₂

yes

maximal

C₈×Q₈×C₇₁

C₂

yes

maximal, characteristic, complemented

C₂×Q₈×D₇₁

C₄

yes

characteristic

Q₈×Dic₇₁

C₄

yes

characteristic

C₄×Dic₁₄₂

C₂²

yes

—

#14556093

C₄

yes

—

C₄⋊C₄×D₇₁

C₄

yes

—

C₄²×D₇₁

C₂²

yes

—

C₄⋊(C₇₁⋊C₈)

C₂²

yes

—

Dic₇₁⋊C₈

C₂²

yes

—

Dic₇₁⋊C₈

C₂²

yes

—

(C₂×D₇₁)⋊C₈

C₂²

yes

characteristic

(C₂×D₇₁)⋊C₈

C₂²

yes

—

#14556038

C₂²

yes

—

C₄×C₇₁⋊C₈

C₂²

yes

—

C₄×Q₈×C₇₁

C₂²

yes

characteristic

C₄⋊C₈×C₇₁

C₂²

yes

—

C₄×C₈×C₇₁

C₂²

yes

—

C₂×Dic₁₄₂

C₂×C₄

yes

—

Q₈×D₇₁

—

C₂×C₄×D₇₁

C₂×C₄

yes

—

C₂×C₄×D₇₁

C₂×C₄

yes

—

C₂×C₄×D₇₁

C₂³

yes

characteristic

C₂×C₇₁⋊C₈

C₂³

yes

characteristic

C₂×C₇₁⋊C₈

C₂³

yes

—

C₄⋊Dic₇₁

C₂×C₄

yes

—

Dic₇₁⋊C₄

C₂×C₄

yes

—

C₄×Dic₇₁

C₂×C₄

yes

—

C₄×Dic₇₁

C₂×C₄

yes

—

C₄×Dic₇₁

C₂³

yes

—

C₂×Q₈×C₇₁

C₂×C₄

yes

characteristic

C₂×C₈×C₇₁

C₂³

yes

characteristic

C₂×C₈×C₇₁

C₂³

yes

—

C₄⋊C₄×C₇₁

C₂×C₄

yes

—

C₄²×C₇₁

C₂³

yes

—

C₂²×D₇₁

C₂²×C₄

yes

characteristic

Q₈×C₇₁

M₁₆

yes

—

C₂×C₄×C₇₁

C₂²×C₄

yes

—

C₂×C₄×C₇₁

C₂²×C₄

yes

—

C₂×C₄×C₇₁

C₂⁴

yes

characteristic

C₂×Dic₇₁

C₂²×C₄

yes

characteristic

C₂×Dic₇₁

C₂²×C₄

yes

—

C₂×Dic₇₁

C₂²×C₄

yes

—

C₄×D₇₁

—

C₄×D₇₁

—

C₄×D₇₁

—

C₄×D₇₁

—

Dic₁₄₂

—

C₈×C₇₁

—

C₈×C₇₁

—

C₇₁⋊C₈

—

C₇₁⋊C₈

—

C₂²×C₇₁

C₂³×C₄

yes

characteristic

C₂×D₇₁

—

C₂×D₇₁

—

C₂×D₇₁

#131

yes

—

C₄×C₇₁

—

C₄×C₇₁

C₂×M₁₆

yes

—

C₄×C₇₁

2₋¹⁺⁴

yes

characteristic

C₄×C₇₁

2₋¹⁺⁴

yes

characteristic

Dic₇₁

—

Dic₇₁

—

Dic₇₁

—

C₂×C₇₁

#519

yes

characteristic

C₂×C₇₁

C₂²×M₁₆

yes

characteristic

C₂×C₇₁

C₂×#131

yes

characteristic

D₇₁

—

#2984

—

maximal, Sylow

C₇₁

#2984

yes

characteristic, complemented, Sylow

C₂×C₄×Q₈

—

#446

—

#446

—

C₈×Q₈

—

C₈×Q₈

D₇₁

yes

characteristic, complemented

#432

—

#431

—

C₂²×Q₈

—

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

C₂×D₇₁

yes

characteristic

C₂×C₄⋊C₄

—

C₂×C₄²

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

C₂×D₇₁

yes

—

C₂²⋊C₈

—

C₂²⋊C₈

—

#97

—

C₄×C₈

—

C₄×C₈

C₂×D₇₁

yes

—

C₂×Q₈

—

C₂×Q₈

—

142

—

100

C₂×Q₈

C₄×D₇₁

yes

characteristic

101

C₂²×C₄

—

102

C₂²×C₄

—

103

C₂²×C₄

—

104

C₂×C₈

—

105

C₂×C₈

—

106

C₂×C₈

C₂²×D₇₁

yes

characteristic

107

C₂×C₈

C₂²×D₇₁

yes

—

108

C₄⋊C₄

—

109

C₄⋊C₄

—

110

C₄⋊C₄

C₄×D₇₁

yes

—

111

C₄²

—

112

C₄²

—

113

C₄²

—

114

C₄²

C₂²×D₇₁

yes

—

115

C₂³

—

116

Q₈

—

142

—

117

Q₈

#11765848

yes

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

142

—

122

C₂×C₄

—

142

—

123

C₂×C₄

—

142

—

124

C₂×C₄

—

142

—

125

C₂×C₄

C₂×C₄×D₇₁

yes

—

126

C₂×C₄

C₂×C₄×D₇₁

yes

—

127

C₂×C₄

C₂³×D₇₁

yes

characteristic

128

C₈

—

129

C₈

—

130

C₈

—

142

—

131

C₈

—

142

—

132

C₂²

—

133

C₂²

—

142

—

134

C₂²

—

142

—

135

C₂²

C₂²×C₄×D₇₁

yes

characteristic

136

C₄

—

137

C₄

—

142

—

138

C₄

—

142

—

139

C₄

—

142

—

140

C₄

C₂×#11765848

yes

—

141

C₄

#14556175

yes

characteristic

142

C₄

#14556176

yes

characteristic

143

C₂

—

142

—

144

C₂

#17986475

yes

characteristic

145

C₂

C₂²×#11765848

yes

characteristic

146

C₂

#131×D₇₁

yes

characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₇₁
commutator, socle	C₂²×C₇₁	C₂³×C₄
Fitting	C₈×Q₈×C₇₁	C₂
hypercenter	C₈×Q₈	D₇₁
hypocenter	C₇₁	#2984
LCS	G ▷ C₂²×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₄ ◁ C₈×Q₈

subgroup

G/H

center, Frattini

C₂×C₄

C₂³×D₇₁

commutator, socle

C₂²×C₇₁

C₂³×C₄

Fitting

C₈×Q₈×C₇₁

C₂

hypercenter

C₈×Q₈

D₇₁

C₇₁

G ▷ C₂²×C₇₁ ▷ C₇₁

C₁ ◁ C₂×C₄ ◁ C₈×Q₈

N	H ≅ G/N	type
2. #17986129	C₂	semidirect
3. #17986126	C₂	semidirect
4. Q₈×C₇₁⋊C₈	C₂	semidirect
7. C₈×Q₈×C₇₁	C₂	semidirect
74. C₇₁	#2984	semidirect
79. C₈×Q₈	D₇₁	semidirect

H ≅ G/N

type

2. #17986129

C₂

semidirect

3. #17986126

C₂

semidirect

4. Q₈×C₇₁⋊C₈

C₂

semidirect

7. C₈×Q₈×C₇₁

C₂

semidirect

74. C₇₁

#2984

semidirect

79. C₈×Q₈

D₇₁

semidirect

Subgroups — #22597428

Conjugacy classes of subgroups

Complements