Group #22601196 — subgroups

Subgroups — Q₈×#11765852 #22601196

Order 9088.

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

normal subgroups only

This group has 370 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
2	#17986395	C₂	1	3	yes	maximal, complemented
3	#17986286	C₂	1	3	yes	maximal
4	#17986284	C₂	1	3	yes	maximal, complemented
5	Q₈×C₇₁⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄×#11765852	C₂	1	3	yes	maximal
7	Q₈²×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×Q₈×D₇₁	—	2	1	no	—
9	C₂×#11765852	C₂²	1	1	yes	characteristic
10	#14556128	C₂²	1	3	yes	—
11	#14556108	—	2	3	no	—
12	C₄×D₂₈₄	C₂²	1	3	yes	—
13	#14556059	C₂²	1	3	yes	—
14	#14556059	C₂²	1	6	yes	—
15	C₄⋊(C₇₁⋊C₈)	C₂²	1	3	yes	—
16	#14556052	C₂²	1	3	yes	—
17	#14556048	C₂²	1	3	yes	—
18	C₄×C₇₁⋊C₈	C₂²	1	3	yes	—
19	C₄⋊Q₈×C₇₁	—	2	3	no	—
20	C₄⋊Q₈×C₇₁	C₂²	1	3	yes	—
21	C₄×Q₈×C₇₁	—	2	1	no	—
22	C₄×Q₈×C₇₁	C₂²	1	3	yes	—
23	C₄×Q₈×C₇₁	C₂²	1	1	yes	characteristic
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	#11765852	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	D₄	1	1	yes	characteristic
36	C₂×Q₈×C₇₁	C₂³	1	1	yes	characteristic
37	C₄⋊C₄×C₇₁	—	2	3	no	—
38	C₄⋊C₄×C₇₁	—	2	3	no	—
39	C₄⋊C₄×C₇₁	D₄	1	3	yes	—
40	C₄⋊C₄×C₇₁	C₂³	1	3	yes	—
41	C₄²×C₇₁	—	2	3	no	—
42	C₄²×C₇₁	C₂³	1	3	yes	—
43	C₂²×D₇₁	—	2	1	no	—
44	Q₈×C₇₁	—	2	1	no	—
45	Q₈×C₇₁	—	8	3	no	—
46	Q₈×C₇₁	SD₁₆	1	4	yes	complemented
47	Q₈×C₇₁	C₂×Q₈	1	2	yes	complemented
48	C₂×C₄×C₇₁	—	2	1	no	—
49	C₂×C₄×C₇₁	—	2	3	no	—
50	C₂×C₄×C₇₁	C₂×D₄	1	3	yes	—
51	C₂×C₄×C₇₁	C₂×D₄	1	3	yes	—
52	C₂×C₄×C₇₁	C₂⁴	1	1	yes	characteristic
53	C₂×Dic₇₁	—	2	3	no	—
54	D₂₈₄	—	2	1	no	—
55	D₂₈₄	C₂×Q₈	1	2	yes	—
56	C₄×D₇₁	—	4	6	no	—
57	Dic₁₄₂	—	4	6	no	—
58	C₇₁⋊C₈	—	2	3	no	—
59	C₇₁⋊C₈	C₂×Q₈	1	2	yes	complemented
60	C₂²×C₇₁	C₂²×D₄	1	1	yes	characteristic
61	C₂×D₇₁	—	2	2	no	—
62	C₂×D₇₁	—	4	1	no	—
63	C₂×D₇₁	—	4	1	no	—
64	C₄×C₇₁	—	2	2	no	—
65	C₄×C₇₁	—	2	3	no	—
66	C₄×C₇₁	—	4	3	no	—
67	C₄×C₇₁	C₂×SD₁₆	1	6	yes	—
68	C₄×C₇₁	C₂²×Q₈	1	1	yes	characteristic
69	C₄×C₇₁	2₋¹⁺⁴	1	1	yes	characteristic
70	Dic₇₁	—	4	3	no	—
71	C₂×C₇₁	D₄×Q₈	1	1	yes	characteristic
72	C₂×C₇₁	C₂²×SD₁₆	1	1	yes	characteristic
73	C₂×C₇₁	#577	1	1	yes	characteristic
74	D₇₁	—	4	2	no	—
75	Q₈×SD₁₆	—	71	1	no	maximal, Sylow
76	C₇₁	Q₈×SD₁₆	1	1	yes	characteristic, complemented, Sylow
77	Q₈²	D₇₁	1	1	yes	characteristic, complemented
78	D₄×Q₈	—	71	1	no	—
79	#498	—	71	3	no	—
80	#476	—	71	3	no	—
81	#475	—	71	3	no	—
82	C₈×Q₈	—	71	1	no	—
83	C₄×SD₁₆	—	71	3	no	—
84	C₂²×Q₈	—	142	1	no	—
85	C₂×SD₁₆	—	71	1	no	—
86	C₄⋊Q₈	—	2	3	no	—
87	C₄⋊Q₈	—	71	3	no	—
88	C₄⋊Q₈	C₂×D₇₁	1	3	yes	—
89	C₂²⋊Q₈	—	142	3	no	—
90	C₄×Q₈	—	2	1	no	—
91	C₄×Q₈	C₂×D₇₁	1	3	yes	—
92	C₄×Q₈	C₂×D₇₁	1	1	yes	characteristic
93	C₄×D₄	—	71	3	no	—
94	#106	—	71	3	no	—
95	#106	—	71	6	no	—
96	C₄⋊C₈	—	71	3	no	—
97	#103	—	71	3	no	—
98	#102	—	71	3	no	—
99	C₄×C₈	—	71	3	no	—
100	C₂×Q₈	—	2	3	no	—
101	C₂×Q₈	—	284	4	no	—
102	C₂×Q₈	—	142	3	no	—
103	C₂×Q₈	C₂²⋊D₇₁	1	1	yes	characteristic
104	C₂×Q₈	C₂²×D₇₁	1	1	yes	characteristic
105	C₂×D₄	—	71	1	no	—
106	C₂²×C₄	—	142	3	no	—
107	SD₁₆	—	71	4	no	—
108	C₂×C₈	—	71	1	no	—
109	C₂×C₈	—	71	3	no	—
110	C₄⋊C₄	—	2	3	no	—
111	C₄⋊C₄	—	2	3	no	—
112	C₄⋊C₄	—	71	3	no	—
113	C₄⋊C₄	—	142	3	no	—
114	C₄⋊C₄	C₂²⋊D₇₁	1	3	yes	—
115	C₄⋊C₄	C₂²×D₇₁	1	3	yes	—
116	C₂²⋊C₄	—	142	3	no	—
117	C₄²	—	2	3	no	—
118	C₄²	C₂²×D₇₁	1	3	yes	—
119	C₂³	—	142	1	no	—
120	Q₈	—	2	1	no	—
121	Q₈	—	8	3	no	—
122	Q₈	—	284	6	no	—
123	Q₈	#11765852	1	4	yes	direct product factor
124	Q₈	Q₈×D₇₁	1	2	yes	complemented
125	D₄	—	71	2	no	—
126	D₄	—	142	1	no	—
127	C₂×C₄	—	2	1	no	—
128	C₂×C₄	—	2	3	no	—
129	C₂×C₄	—	284	6	no	—
130	C₂×C₄	—	142	3	no	—
131	C₂×C₄	C₂×C₂²⋊D₇₁	1	3	yes	—
132	C₂×C₄	C₂×C₂²⋊D₇₁	1	3	yes	—
133	C₂×C₄	C₂³×D₇₁	1	1	yes	characteristic
134	C₈	—	71	2	no	—
135	C₈	—	142	3	no	—
136	C₂²	—	284	1	no	—
137	C₂²	—	284	1	no	—
138	C₂²	—	142	2	no	—
139	C₂²	C₂²×C₂²⋊D₇₁	1	1	yes	characteristic
140	C₄	—	2	2	no	—
141	C₄	—	2	3	no	—
142	C₄	—	4	3	no	—
143	C₄	—	284	3	no	—
144	C₄	C₂×#11765852	1	6	yes	—
145	C₄	C₂×Q₈×D₇₁	1	1	yes	characteristic
146	C₄	#14556176	1	1	yes	characteristic
147	C₂	—	284	2	no	—
148	C₂	Q₈×C₂²⋊D₇₁	1	1	yes	characteristic
149	C₂	C₂²×#11765852	1	1	yes	characteristic
150	C₂	#17986810	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

Q₈×D₂₈₄

C₂

yes

maximal, characteristic

#17986395

C₂

yes

maximal, complemented

#17986286

C₂

yes

maximal

#17986284

C₂

yes

maximal, complemented

Q₈×C₇₁⋊C₈

C₂

yes

maximal, characteristic, complemented

C₄×#11765852

C₂

yes

maximal

Q₈²×C₇₁

C₂

yes

maximal, characteristic, complemented

C₂×Q₈×D₇₁

—

C₂×#11765852

C₂²

yes

characteristic

#14556128

C₂²

yes

—

#14556108

—

C₄×D₂₈₄

C₂²

yes

—

#14556059

C₂²

yes

—

#14556059

C₂²

yes

—

C₄⋊(C₇₁⋊C₈)

C₂²

yes

—

#14556052

C₂²

yes

—

#14556048

C₂²

yes

—

C₄×C₇₁⋊C₈

C₂²

yes

—

C₄⋊Q₈×C₇₁

—

C₄⋊Q₈×C₇₁

C₂²

yes

—

C₄×Q₈×C₇₁

—

C₄×Q₈×C₇₁

C₂²

yes

—

C₄×Q₈×C₇₁

C₂²

yes

characteristic

C₂×Dic₁₄₂

—

Q₈×D₇₁

—

C₂×D₂₈₄

C₂³

yes

characteristic

C₂×C₄×D₇₁

—

#11765852

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

D₄

yes

characteristic

C₂×Q₈×C₇₁

C₂³

yes

characteristic

C₄⋊C₄×C₇₁

—

C₄⋊C₄×C₇₁

—

C₄⋊C₄×C₇₁

D₄

yes

—

C₄⋊C₄×C₇₁

C₂³

yes

—

C₄²×C₇₁

—

C₄²×C₇₁

C₂³

yes

—

C₂²×D₇₁

—

Q₈×C₇₁

—

Q₈×C₇₁

—

Q₈×C₇₁

SD₁₆

yes

complemented

Q₈×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

—

C₄×D₇₁

—

Dic₁₄₂

—

C₇₁⋊C₈

—

C₇₁⋊C₈

C₂×Q₈

yes

complemented

C₂²×C₇₁

C₂²×D₄

yes

characteristic

C₂×D₇₁

—

C₂×D₇₁

—

C₂×D₇₁

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

C₂×SD₁₆

yes

—

C₄×C₇₁

C₂²×Q₈

yes

characteristic

C₄×C₇₁

2₋¹⁺⁴

yes

characteristic

Dic₇₁

—

C₂×C₇₁

D₄×Q₈

yes

characteristic

C₂×C₇₁

C₂²×SD₁₆

yes

characteristic

C₂×C₇₁

#577

yes

characteristic

D₇₁

—

Q₈×SD₁₆

—

maximal, Sylow

C₇₁

Q₈×SD₁₆

yes

characteristic, complemented, Sylow

Q₈²

D₇₁

yes

characteristic, complemented

D₄×Q₈

—

#498

—

#476

—

#475

—

C₈×Q₈

—

C₄×SD₁₆

—

C₂²×Q₈

—

142

—

C₂×SD₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₇₁

yes

—

C₂²⋊Q₈

—

142

—

C₄×Q₈

—

C₄×Q₈

C₂×D₇₁

yes

—

C₄×Q₈

C₂×D₇₁

yes

characteristic

C₄×D₄

—

#106

—

#106

—

C₄⋊C₈

—

#103

—

#102

—

C₄×C₈

—

100

C₂×Q₈

—

101

C₂×Q₈

—

284

—

102

C₂×Q₈

—

142

—

103

C₂×Q₈

C₂²⋊D₇₁

yes

characteristic

104

C₂×Q₈

C₂²×D₇₁

yes

characteristic

105

C₂×D₄

—

106

C₂²×C₄

—

142

—

107

SD₁₆

—

108

C₂×C₈

—

109

C₂×C₈

—

110

C₄⋊C₄

—

111

C₄⋊C₄

—

112

C₄⋊C₄

—

113

C₄⋊C₄

—

142

—

114

C₄⋊C₄

C₂²⋊D₇₁

yes

—

115

C₄⋊C₄

C₂²×D₇₁

yes

—

116

C₂²⋊C₄

—

142

—

117

C₄²

—

118

C₄²

C₂²×D₇₁

yes

—

119

C₂³

—

142

—

120

Q₈

—

121

Q₈

—

122

Q₈

—

284

—

123

Q₈

#11765852

yes

direct product factor

124

Q₈

Q₈×D₇₁

yes

complemented

125

D₄

—

126

D₄

—

142

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

284

—

130

C₂×C₄

—

142

—

131

C₂×C₄

C₂×C₂²⋊D₇₁

yes

—

132

C₂×C₄

C₂×C₂²⋊D₇₁

yes

—

133

C₂×C₄

C₂³×D₇₁

yes

characteristic

134

C₈

—

135

C₈

—

142

—

136

C₂²

—

284

—

137

C₂²

—

284

—

138

C₂²

—

142

—

139

C₂²

C₂²×C₂²⋊D₇₁

yes

characteristic

140

C₄

—

141

C₄

—

142

C₄

—

143

C₄

—

284

—

144

C₄

C₂×#11765852

yes

—

145

C₄

C₂×Q₈×D₇₁

yes

characteristic

146

C₄

#14556176

yes

characteristic

147

C₂

—

284

—

148

C₂

Q₈×C₂²⋊D₇₁

yes

characteristic

149

C₂

C₂²×#11765852

yes

characteristic

150

C₂

#17986810

yes

characteristic

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₇₁
commutator	C₂×C₄×C₇₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₇₁
Fitting	Q₈²×C₇₁	C₂
socle	C₂²×C₇₁	C₂²×D₄
hypercenter	Q₈²	D₇₁
hypocenter	C₇₁	Q₈×SD₁₆
LCS	G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ Q₈²

subgroup

G/H

center

C₂²

C₂²×C₂²⋊D₇₁

commutator

C₂×C₄×C₇₁

C₂⁴

Frattini

C₂×C₄

C₂³×D₇₁

Fitting

Q₈²×C₇₁

C₂

socle

C₂²×C₇₁

C₂²×D₄

hypercenter

Q₈²

D₇₁

hypocenter

C₇₁

Q₈×SD₁₆

LCS

G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂² ◁ C₄×Q₈ ◁ Q₈²

N	H ≅ G/N	type
2. #17986395	C₂	semidirect
4. #17986284	C₂	semidirect
5. Q₈×C₇₁⋊C₈	C₂	semidirect
7. Q₈²×C₇₁	C₂	semidirect
28. #11765852	Q₈	direct (N × H)
46. Q₈×C₇₁	SD₁₆	semidirect
47. Q₈×C₇₁	C₂×Q₈	semidirect
59. C₇₁⋊C₈	C₂×Q₈	semidirect
76. C₇₁	Q₈×SD₁₆	semidirect
77. Q₈²	D₇₁	semidirect
123. Q₈	#11765852	direct (N × H)
124. Q₈	Q₈×D₇₁	semidirect

H ≅ G/N

type

2. #17986395

C₂

semidirect

4. #17986284

C₂

semidirect

5. Q₈×C₇₁⋊C₈

C₂

semidirect

7. Q₈²×C₇₁

C₂

semidirect

28. #11765852

Q₈

direct (N × H)

46. Q₈×C₇₁

SD₁₆

semidirect

47. Q₈×C₇₁

C₂×Q₈

semidirect

59. C₇₁⋊C₈

C₂×Q₈

semidirect

76. C₇₁

Q₈×SD₁₆

semidirect

77. Q₈²

D₇₁

semidirect

123. Q₈

#11765852

direct (N × H)

124. Q₈

Q₈×D₇₁

semidirect

Subgroups — Q₈×#11765852 #22601196

Conjugacy classes of subgroups

Complements

Subgroups — Q8×#11765852 #22601196

Conjugacy classes of subgroups

Complements

Subgroups — Q₈×#11765852 #22601196