Group #22601191 — subgroups

Subgroups — Q₈×#11765850 #22601191

Order 9088.

404 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 404 conjugacy classes of subgroups — too many to draw. Only its 149 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	Q₈×D₂₈₄	C₂	1	1	yes	maximal, characteristic, complemented
2	#17986402	C₂	1	3	yes	maximal, complemented
3	#17986279	C₂	1	3	yes	maximal, complemented
4	#17986276	C₂	1	3	yes	maximal, complemented
5	Q₈×C₇₁⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄×#11765850	C₂	1	3	yes	maximal
7	D₄×Q₈×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×Q₈×D₇₁	—	2	1	no	—
9	C₂×#11765850	C₂²	1	1	yes	characteristic
10	#14556128	C₂²	1	3	yes	—
11	#14556108	—	2	3	no	—
12	C₄×D₂₈₄	C₂²	1	3	yes	—
13	#14556061	C₂²	1	3	yes	—
14	#14556061	C₂²	1	6	yes	—
15	C₄⋊(C₇₁⋊C₈)	C₂²	1	3	yes	—
16	#14556049	C₂²	1	3	yes	—
17	#14556048	C₂²	1	3	yes	—
18	C₄×C₇₁⋊C₈	C₂²	1	3	yes	—
19	C₂²×Q₈×C₇₁	—	2	1	no	—
20	C₄⋊Q₈×C₇₁	C₂²	1	3	yes	—
21	C₂²⋊Q₈×C₇₁	—	2	3	no	—
22	C₄×Q₈×C₇₁	C₂²	1	1	yes	characteristic
23	C₄×D₄×C₇₁	C₂²	1	3	yes	—
24	C₂×Dic₁₄₂	—	2	3	no	—
25	Q₈×D₇₁	—	4	4	no	—
26	C₂×D₂₈₄	C₂³	1	1	yes	characteristic
27	C₂×C₄×D₇₁	—	2	3	no	—
28	#11765850	Q₈	1	4	yes	direct product factor
29	C₂×C₇₁⋊C₈	C₂³	1	3	yes	—
30	C₂×C₇₁⋊C₈	C₂³	1	1	yes	characteristic
31	C₄⋊Dic₇₁	—	2	3	no	—
32	C₄⋊Dic₇₁	C₂³	1	3	yes	—
33	(C₂×D₇₁)⋊C₄	—	2	3	no	—
34	C₂×Q₈×C₇₁	—	2	3	no	—
35	C₂×Q₈×C₇₁	—	4	4	no	—
36	C₂×Q₈×C₇₁	D₄	1	1	yes	characteristic
37	C₂×D₄×C₇₁	C₂³	1	1	yes	characteristic
38	C₂²×C₄×C₇₁	—	2	3	no	—
39	C₄⋊C₄×C₇₁	—	2	3	no	—
40	C₄⋊C₄×C₇₁	D₄	1	3	yes	—
41	C₄⋊C₄×C₇₁	C₂³	1	3	yes	—
42	C₂²⋊C₄×C₇₁	—	2	3	no	—
43	C₄²×C₇₁	C₂³	1	3	yes	—
44	C₂³×C₇₁	—	2	1	no	—
45	C₂²×D₇₁	—	2	1	no	—
46	Q₈×C₇₁	—	4	6	no	—
47	Q₈×C₇₁	D₈	1	4	yes	complemented
48	D₄×C₇₁	—	2	1	no	—
49	D₄×C₇₁	C₂×Q₈	1	2	yes	complemented
50	C₂×C₄×C₇₁	—	2	3	no	—
51	C₂×C₄×C₇₁	—	4	6	no	—
52	C₂×C₄×C₇₁	C₂×D₄	1	3	yes	—
53	C₂×C₄×C₇₁	C₂×D₄	1	3	yes	—
54	C₂×C₄×C₇₁	C₂⁴	1	1	yes	characteristic
55	C₂×Dic₇₁	—	2	3	no	—
56	D₂₈₄	—	2	1	no	—
57	D₂₈₄	C₂×Q₈	1	2	yes	complemented
58	C₄×D₇₁	—	4	6	no	—
59	Dic₁₄₂	—	4	6	no	—
60	C₇₁⋊C₈	—	2	3	no	—
61	C₇₁⋊C₈	C₂×Q₈	1	2	yes	complemented
62	C₂²×C₇₁	—	2	2	no	—
63	C₂²×C₇₁	—	4	1	no	—
64	C₂²×C₇₁	—	4	1	no	—
65	C₂²×C₇₁	C₂²×D₄	1	1	yes	characteristic
66	C₂×D₇₁	—	2	2	no	—
67	C₂×D₇₁	—	4	1	no	—
68	C₂×D₇₁	—	4	1	no	—
69	C₄×C₇₁	—	2	3	no	—
70	C₄×C₇₁	—	4	3	no	—
71	C₄×C₇₁	C₂×D₈	1	6	yes	—
72	C₄×C₇₁	C₂²×Q₈	1	1	yes	characteristic
73	C₄×C₇₁	2₋¹⁺⁴	1	1	yes	characteristic
74	Dic₇₁	—	4	3	no	—
75	C₂×C₇₁	—	4	2	no	—
76	C₂×C₇₁	D₄×Q₈	1	1	yes	characteristic
77	C₂×C₇₁	C₂²×D₈	1	1	yes	characteristic
78	C₂×C₇₁	#578	1	1	yes	characteristic
79	D₇₁	—	4	2	no	—
80	Q₈×D₈	—	71	1	no	maximal, Sylow
81	C₇₁	Q₈×D₈	1	1	yes	characteristic, complemented, Sylow
82	D₄×Q₈	—	71	1	no	—
83	D₄×Q₈	D₇₁	1	1	yes	characteristic, complemented
84	#500	—	71	3	no	—
85	#474	—	71	3	no	—
86	#474	—	71	3	no	—
87	C₈×Q₈	—	71	1	no	—
88	C₄×D₈	—	71	3	no	—
89	C₂²×Q₈	—	2	1	no	—
90	C₂²×Q₈	—	142	1	no	—
91	C₂×D₈	—	71	1	no	—
92	C₄⋊Q₈	—	71	3	no	—
93	C₄⋊Q₈	C₂×D₇₁	1	3	yes	—
94	C₂²⋊Q₈	—	2	3	no	—
95	C₂²⋊Q₈	—	142	3	no	—
96	C₄×Q₈	C₂×D₇₁	1	1	yes	characteristic
97	C₄×D₄	—	71	3	no	—
98	C₄×D₄	C₂×D₇₁	1	3	yes	—
99	#107	—	71	3	no	—
100	#107	—	71	6	no	—
101	C₄⋊C₈	—	71	3	no	—
102	#102	—	71	3	no	—
103	#102	—	71	3	no	—
104	C₄×C₈	—	71	3	no	—
105	C₂×Q₈	—	2	3	no	—
106	C₂×Q₈	—	4	4	no	—
107	C₂×Q₈	—	284	4	no	—
108	C₂×Q₈	—	142	3	no	—
109	C₂×Q₈	C₂²⋊D₇₁	1	1	yes	characteristic
110	C₂×D₄	—	71	1	no	—
111	C₂×D₄	C₂²×D₇₁	1	1	yes	characteristic
112	C₂²×C₄	—	2	3	no	—
113	C₂²×C₄	—	142	3	no	—
114	D₈	—	71	4	no	—
115	C₂×C₈	—	71	1	no	—
116	C₂×C₈	—	71	3	no	—
117	C₄⋊C₄	—	2	3	no	—
118	C₄⋊C₄	—	71	3	no	—
119	C₄⋊C₄	—	142	3	no	—
120	C₄⋊C₄	C₂²⋊D₇₁	1	3	yes	—
121	C₄⋊C₄	C₂²×D₇₁	1	3	yes	—
122	C₂²⋊C₄	—	2	3	no	—
123	C₂²⋊C₄	—	142	3	no	—
124	C₄²	C₂²×D₇₁	1	3	yes	—
125	C₂³	—	2	1	no	—
126	C₂³	—	142	1	no	—
127	Q₈	—	4	6	no	—
128	Q₈	—	284	6	no	—
129	Q₈	#11765850	1	4	yes	direct product factor
130	D₄	—	2	1	no	—
131	D₄	—	71	2	no	—
132	D₄	—	142	1	no	—
133	D₄	Q₈×D₇₁	1	2	yes	complemented
134	C₂×C₄	—	2	3	no	—
135	C₂×C₄	—	4	6	no	—
136	C₂×C₄	—	284	6	no	—
137	C₂×C₄	—	142	3	no	—
138	C₂×C₄	C₂×C₂²⋊D₇₁	1	3	yes	—
139	C₂×C₄	C₂×C₂²⋊D₇₁	1	3	yes	—
140	C₂×C₄	C₂³×D₇₁	1	1	yes	characteristic
141	C₈	—	71	2	no	—
142	C₈	—	142	3	no	—
143	C₂²	—	2	2	no	—
144	C₂²	—	4	1	no	—
145	C₂²	—	4	1	no	—
146	C₂²	—	284	1	no	—
147	C₂²	—	284	1	no	—
148	C₂²	—	142	2	no	—
149	C₂²	C₂²×C₂²⋊D₇₁	1	1	yes	characteristic
150	C₄	—	2	3	no	—
151	C₄	—	4	3	no	—
152	C₄	—	284	3	no	—
153	C₄	C₂×#11765850	1	6	yes	—
154	C₄	C₂×Q₈×D₇₁	1	1	yes	characteristic
155	C₄	#14556176	1	1	yes	characteristic
156	C₂	—	4	2	no	—
157	C₂	—	284	2	no	—
158	C₂	Q₈×C₂²⋊D₇₁	1	1	yes	characteristic
159	C₂	C₂²×#11765850	1	1	yes	characteristic
160	C₂	#17986814	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

Q₈×D₂₈₄

C₂

yes

maximal, characteristic, complemented

#17986402

C₂

yes

maximal, complemented

#17986279

C₂

yes

maximal, complemented

#17986276

C₂

yes

maximal, complemented

Q₈×C₇₁⋊C₈

C₂

yes

maximal, characteristic, complemented

C₄×#11765850

C₂

yes

maximal

D₄×Q₈×C₇₁

C₂

yes

maximal, characteristic, complemented

C₂×Q₈×D₇₁

—

C₂×#11765850

C₂²

yes

characteristic

#14556128

C₂²

yes

—

#14556108

—

C₄×D₂₈₄

C₂²

yes

—

#14556061

C₂²

yes

—

#14556061

C₂²

yes

—

C₄⋊(C₇₁⋊C₈)

C₂²

yes

—

#14556049

C₂²

yes

—

#14556048

C₂²

yes

—

C₄×C₇₁⋊C₈

C₂²

yes

—

C₂²×Q₈×C₇₁

—

C₄⋊Q₈×C₇₁

C₂²

yes

—

C₂²⋊Q₈×C₇₁

—

C₄×Q₈×C₇₁

C₂²

yes

characteristic

C₄×D₄×C₇₁

C₂²

yes

—

C₂×Dic₁₄₂

—

Q₈×D₇₁

—

C₂×D₂₈₄

C₂³

yes

characteristic

C₂×C₄×D₇₁

—

#11765850

Q₈

yes

direct product factor

C₂×C₇₁⋊C₈

C₂³

yes

—

C₂×C₇₁⋊C₈

C₂³

yes

characteristic

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

C₂³

yes

—

(C₂×D₇₁)⋊C₄

—

C₂×Q₈×C₇₁

—

C₂×Q₈×C₇₁

—

C₂×Q₈×C₇₁

D₄

yes

characteristic

C₂×D₄×C₇₁

C₂³

yes

characteristic

C₂²×C₄×C₇₁

—

C₄⋊C₄×C₇₁

—

C₄⋊C₄×C₇₁

D₄

yes

—

C₄⋊C₄×C₇₁

C₂³

yes

—

C₂²⋊C₄×C₇₁

—

C₄²×C₇₁

C₂³

yes

—

C₂³×C₇₁

—

C₂²×D₇₁

—

Q₈×C₇₁

—

Q₈×C₇₁

D₈

yes

complemented

D₄×C₇₁

—

D₄×C₇₁

C₂×Q₈

yes

complemented

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

—

D₂₈₄

—

D₂₈₄

C₂×Q₈

yes

complemented

C₄×D₇₁

—

Dic₁₄₂

—

C₇₁⋊C₈

—

C₇₁⋊C₈

C₂×Q₈

yes

complemented

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

C₂²×D₄

yes

characteristic

C₂×D₇₁

—

C₂×D₇₁

—

C₂×D₇₁

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

C₂×D₈

yes

—

C₄×C₇₁

C₂²×Q₈

yes

characteristic

C₄×C₇₁

2₋¹⁺⁴

yes

characteristic

Dic₇₁

—

C₂×C₇₁

—

C₂×C₇₁

D₄×Q₈

yes

characteristic

C₂×C₇₁

C₂²×D₈

yes

characteristic

C₂×C₇₁

#578

yes

characteristic

D₇₁

—

Q₈×D₈

—

maximal, Sylow

C₇₁

Q₈×D₈

yes

characteristic, complemented, Sylow

D₄×Q₈

—

D₄×Q₈

D₇₁

yes

characteristic, complemented

#500

—

#474

—

#474

—

C₈×Q₈

—

C₄×D₈

—

C₂²×Q₈

—

C₂²×Q₈

—

142

—

C₂×D₈

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₇₁

yes

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

142

—

C₄×Q₈

C₂×D₇₁

yes

characteristic

C₄×D₄

—

C₄×D₄

C₂×D₇₁

yes

—

#107

—

100

#107

—

101

C₄⋊C₈

—

102

#102

—

103

#102

—

104

C₄×C₈

—

105

C₂×Q₈

—

106

C₂×Q₈

—

107

C₂×Q₈

—

284

—

108

C₂×Q₈

—

142

—

109

C₂×Q₈

C₂²⋊D₇₁

yes

characteristic

110

C₂×D₄

—

111

C₂×D₄

C₂²×D₇₁

yes

characteristic

112

C₂²×C₄

—

113

C₂²×C₄

—

142

—

114

D₈

—

115

C₂×C₈

—

116

C₂×C₈

—

117

C₄⋊C₄

—

118

C₄⋊C₄

—

119

C₄⋊C₄

—

142

—

120

C₄⋊C₄

C₂²⋊D₇₁

yes

—

121

C₄⋊C₄

C₂²×D₇₁

yes

—

122

C₂²⋊C₄

—

123

C₂²⋊C₄

—

142

—

124

C₄²

C₂²×D₇₁

yes

—

125

C₂³

—

126

C₂³

—

142

—

127

Q₈

—

128

Q₈

—

284

—

129

Q₈

#11765850

yes

direct product factor

130

D₄

—

131

D₄

—

132

D₄

—

142

—

133

D₄

Q₈×D₇₁

yes

complemented

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

284

—

137

C₂×C₄

—

142

—

138

C₂×C₄

C₂×C₂²⋊D₇₁

yes

—

139

C₂×C₄

C₂×C₂²⋊D₇₁

yes

—

140

C₂×C₄

C₂³×D₇₁

yes

characteristic

141

C₈

—

142

C₈

—

142

—

143

C₂²

—

144

C₂²

—

145

C₂²

—

146

C₂²

—

284

—

147

C₂²

—

284

—

148

C₂²

—

142

—

149

C₂²

C₂²×C₂²⋊D₇₁

yes

characteristic

150

C₄

—

151

C₄

—

152

C₄

—

284

—

153

C₄

C₂×#11765850

yes

—

154

C₄

C₂×Q₈×D₇₁

yes

characteristic

155

C₄

#14556176

yes

characteristic

156

C₂

—

157

C₂

—

284

—

158

C₂

Q₈×C₂²⋊D₇₁

yes

characteristic

159

C₂

C₂²×#11765850

yes

characteristic

160

C₂

#17986814

yes

characteristic

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₇₁
commutator	C₂×C₄×C₇₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₇₁
Fitting	D₄×Q₈×C₇₁	C₂
socle	C₂²×C₇₁	C₂²×D₄
hypercenter	D₄×Q₈	D₇₁
hypocenter	C₇₁	Q₈×D₈
LCS	G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ D₄×Q₈

subgroup

G/H

center

C₂²

C₂²×C₂²⋊D₇₁

commutator

C₂×C₄×C₇₁

C₂⁴

Frattini

C₂×C₄

C₂³×D₇₁

Fitting

D₄×Q₈×C₇₁

C₂

socle

C₂²×C₇₁

C₂²×D₄

hypercenter

D₄×Q₈

D₇₁

hypocenter

C₇₁

Q₈×D₈

LCS

G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂² ◁ C₄×Q₈ ◁ D₄×Q₈

N	H ≅ G/N	type
1. Q₈×D₂₈₄	C₂	semidirect
2. #17986402	C₂	semidirect
3. #17986279	C₂	semidirect
4. #17986276	C₂	semidirect
5. Q₈×C₇₁⋊C₈	C₂	semidirect
7. D₄×Q₈×C₇₁	C₂	semidirect
28. #11765850	Q₈	direct (N × H)
47. Q₈×C₇₁	D₈	semidirect
49. D₄×C₇₁	C₂×Q₈	semidirect
57. D₂₈₄	C₂×Q₈	semidirect
61. C₇₁⋊C₈	C₂×Q₈	semidirect
81. C₇₁	Q₈×D₈	semidirect
83. D₄×Q₈	D₇₁	semidirect
129. Q₈	#11765850	direct (N × H)
133. D₄	Q₈×D₇₁	semidirect

H ≅ G/N

type

1. Q₈×D₂₈₄

C₂

semidirect

2. #17986402

C₂

semidirect

3. #17986279

C₂

semidirect

4. #17986276

C₂

semidirect

5. Q₈×C₇₁⋊C₈

C₂

semidirect

7. D₄×Q₈×C₇₁

C₂

semidirect

28. #11765850

Q₈

direct (N × H)

47. Q₈×C₇₁

D₈

semidirect

49. D₄×C₇₁

C₂×Q₈

semidirect

57. D₂₈₄

C₂×Q₈

semidirect

61. C₇₁⋊C₈

C₂×Q₈

semidirect

81. C₇₁

Q₈×D₈

semidirect

83. D₄×Q₈

D₇₁

semidirect

129. Q₈

#11765850

direct (N × H)

133. D₄

Q₈×D₇₁

semidirect

Subgroups — Q₈×#11765850 #22601191

Conjugacy classes of subgroups

Complements

Subgroups — Q8×#11765850 #22601191

Conjugacy classes of subgroups

Complements

Subgroups — Q₈×#11765850 #22601191