Group #22597294 — subgroups

Subgroups — Q₈×#11765849 #22597294

Order 9088.

388 conjugacy classes of subgroups; only the 241 normal subgroups are shown.

normal subgroups only

This group has 388 conjugacy classes of subgroups — too many to draw. Only its 241 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	#17986130	C₂	1	6	yes	maximal, complemented
2	Q₈×C₇₁⋊C₈	C₂	1	2	yes	maximal, complemented
3	#17986023	C₂	1	3	yes	maximal
4	C₄×#11765849	C₂	1	3	yes	maximal
5	C₂×C₄×Q₈×C₇₁	C₂	1	1	yes	maximal, characteristic
6	C₂×#11765849	C₂²	1	3	yes	—
7	C₂×#11765849	C₂²	1	1	yes	characteristic
8	C₄⋊(C₇₁⋊C₈)	C₂²	1	6	yes	—
9	C₄⋊(C₇₁⋊C₈)	C₂²	1	6	yes	—
10	#14556039	C₂²	1	6	yes	—
11	C₄×C₇₁⋊C₈	C₂²	1	6	yes	—
12	C₂²×Q₈×C₇₁	C₄	1	1	yes	characteristic
13	C₄×Q₈×C₇₁	C₄	1	3	yes	—
14	C₄×Q₈×C₇₁	C₄	1	1	yes	characteristic
15	C₄×Q₈×C₇₁	C₂²	1	1	yes	characteristic
16	C₄×Q₈×C₇₁	C₂²	1	3	yes	—
17	C₂×C₄⋊C₄×C₇₁	C₄	1	3	yes	—
18	C₂×C₄²×C₇₁	C₂²	1	3	yes	—
19	#11765849	—	2	6	no	—
20	#11765849	Q₈	1	4	yes	direct product factor
21	C₂×C₇₁⋊C₈	C₂³	1	6	yes	—
22	C₂×C₇₁⋊C₈	C₂³	1	2	yes	—
23	C₂×Q₈×C₇₁	—	2	4	no	—
24	C₂×Q₈×C₇₁	C₂×C₄	1	3	yes	—
25	C₂×Q₈×C₇₁	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₄×C₇₁	C₂×C₄	1	3	yes	—
27	C₂²×C₄×C₇₁	C₂×C₄	1	3	yes	—
28	C₂²×C₄×C₇₁	C₂³	1	1	yes	characteristic
29	C₄⋊C₄×C₇₁	C₂×C₄	1	6	yes	—
30	C₄⋊C₄×C₇₁	C₂×C₄	1	3	yes	—
31	C₄⋊C₄×C₇₁	C₂×C₄	1	3	yes	—
32	C₄²×C₇₁	C₂×C₄	1	3	yes	—
33	C₄²×C₇₁	C₂×C₄	1	3	yes	—
34	C₄²×C₇₁	C₂³	1	3	yes	—
35	C₄²×C₇₁	C₂³	1	3	yes	—
36	C₂³×C₇₁	C₂²×C₄	1	1	yes	characteristic
37	Q₈×C₇₁	—	2	6	no	—
38	Q₈×C₇₁	M₁₆	1	4	yes	complemented
39	C₂×C₄×C₇₁	—	2	6	no	—
40	C₂×C₄×C₇₁	—	4	3	no	—
41	C₂×C₄×C₇₁	C₂²×C₄	1	3	yes	—
42	C₂×C₄×C₇₁	C₂²×C₄	1	3	yes	—
43	C₂×C₄×C₇₁	C₂²×C₄	1	3	yes	—
44	C₂×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₄×C₇₁	C₂²×C₄	1	3	yes	—
46	C₂×C₄×C₇₁	C₂×Q₈	1	2	yes	—
47	C₂×C₄×C₇₁	#41	1	2	yes	—
48	C₂×C₄×C₇₁	C₂⁴	1	1	yes	characteristic
49	C₇₁⋊C₈	—	2	6	no	—
50	C₇₁⋊C₈	C₂×Q₈	1	4	yes	complemented
51	C₂²×C₇₁	—	2	1	no	—
52	C₂²×C₇₁	—	2	1	no	—
53	C₂²×C₇₁	C₄×Q₈	1	2	yes	—
54	C₂²×C₇₁	C₂³×C₄	1	1	yes	characteristic
55	C₄×C₇₁	—	2	3	no	—
56	C₄×C₇₁	—	2	3	no	—
57	C₄×C₇₁	—	2	3	no	—
58	C₄×C₇₁	C₄×Q₈	1	2	yes	—
59	C₄×C₇₁	C₂×M₁₆	1	6	yes	—
60	C₄×C₇₁	C₂²×Q₈	1	1	yes	characteristic
61	C₄×C₇₁	C₂×#41	1	1	yes	characteristic
62	C₂×C₇₁	—	2	2	no	—
63	C₂×C₇₁	C₂×C₄×Q₈	1	1	yes	characteristic
64	C₂×C₇₁	C₂²×M₁₆	1	1	yes	characteristic
65	C₂×C₇₁	#568	1	1	yes	characteristic
66	Q₈×M₁₆	—	71	1	no	maximal, Sylow
67	C₇₁	Q₈×M₁₆	1	1	yes	characteristic, complemented, Sylow
68	C₂×C₄×Q₈	D₇₁	1	1	yes	characteristic
69	#446	—	71	6	no	—
70	C₈×Q₈	—	71	2	no	—
71	#423	—	71	3	no	—
72	C₄×M₁₆	—	71	3	no	—
73	C₂²×Q₈	Dic₇₁	1	1	yes	characteristic
74	C₂×M₁₆	—	71	1	no	—
75	C₂×M₁₆	—	71	3	no	—
76	C₄×Q₈	Dic₇₁	1	3	yes	—
77	C₄×Q₈	Dic₇₁	1	1	yes	characteristic
78	C₄×Q₈	C₂×D₇₁	1	1	yes	characteristic
79	C₄×Q₈	C₂×D₇₁	1	3	yes	—
80	C₂×C₄⋊C₄	Dic₇₁	1	3	yes	—
81	C₂×C₄²	C₂×D₇₁	1	3	yes	—
82	C₄⋊C₈	—	71	6	no	—
83	C₄⋊C₈	—	71	6	no	—
84	#97	—	71	6	no	—
85	C₄×C₈	—	71	6	no	—
86	C₂×Q₈	—	2	4	no	—
87	C₂×Q₈	C₂×Dic₇₁	1	3	yes	—
88	C₂×Q₈	C₂×Dic₇₁	1	1	yes	characteristic
89	C₂²×C₄	C₂×Dic₇₁	1	3	yes	—
90	C₂²×C₄	C₂×Dic₇₁	1	3	yes	—
91	C₂²×C₄	C₂²×D₇₁	1	1	yes	characteristic
92	M₁₆	—	71	4	no	—
93	M₁₆	—	142	6	no	—
94	C₂×C₈	—	71	2	no	—
95	C₂×C₈	—	71	6	no	—
96	C₄⋊C₄	C₂×Dic₇₁	1	6	yes	—
97	C₄⋊C₄	C₂×Dic₇₁	1	3	yes	—
98	C₄⋊C₄	C₂×Dic₇₁	1	3	yes	—
99	C₄²	C₂×Dic₇₁	1	3	yes	—
100	C₄²	C₂×Dic₇₁	1	3	yes	—
101	C₄²	C₂²×D₇₁	1	3	yes	—
102	C₄²	C₂²×D₇₁	1	3	yes	—
103	C₂³	C₂²×Dic₇₁	1	1	yes	characteristic
104	Q₈	—	2	6	no	—
105	Q₈	#11765849	1	4	yes	direct product factor
106	C₂×C₄	—	2	6	no	—
107	C₂×C₄	—	4	3	no	—
108	C₂×C₄	C₂²×Dic₇₁	1	3	yes	—
109	C₂×C₄	C₂²×Dic₇₁	1	3	yes	—
110	C₂×C₄	C₂²×Dic₇₁	1	3	yes	—
111	C₂×C₄	C₂²×Dic₇₁	1	1	yes	characteristic
112	C₂×C₄	C₂²×Dic₇₁	1	3	yes	—
113	C₂×C₄	Q₈×D₇₁	1	2	yes	—
114	C₂×C₄	#11765866	1	2	yes	—
115	C₂×C₄	C₂³×D₇₁	1	1	yes	characteristic
116	C₈	—	71	4	no	—
117	C₈	—	142	6	no	—
118	C₂²	—	2	1	no	—
119	C₂²	—	2	1	no	—
120	C₂²	Q₈×Dic₇₁	1	2	yes	—
121	C₂²	C₂³×Dic₇₁	1	1	yes	characteristic
122	C₄	—	2	3	no	—
123	C₄	—	2	3	no	—
124	C₄	—	2	3	no	—
125	C₄	Q₈×Dic₇₁	1	2	yes	—
126	C₄	C₂×#11765849	1	6	yes	—
127	C₄	C₂×Q₈×D₇₁	1	1	yes	characteristic
128	C₄	C₂×#11765866	1	1	yes	characteristic
129	C₂	—	2	2	no	—
130	C₂	C₂×Q₈×Dic₇₁	1	1	yes	characteristic
131	C₂	C₂²×#11765849	1	1	yes	characteristic
132	C₂	#17986766	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17986130

C₂

yes

maximal, complemented

Q₈×C₇₁⋊C₈

C₂

yes

maximal, complemented

#17986023

C₂

yes

maximal

C₄×#11765849

C₂

yes

maximal

C₂×C₄×Q₈×C₇₁

C₂

yes

maximal, characteristic

C₂×#11765849

C₂²

yes

—

C₂×#11765849

C₂²

yes

characteristic

C₄⋊(C₇₁⋊C₈)

C₂²

yes

—

C₄⋊(C₇₁⋊C₈)

C₂²

yes

—

#14556039

C₂²

yes

—

C₄×C₇₁⋊C₈

C₂²

yes

—

C₂²×Q₈×C₇₁

C₄

yes

characteristic

C₄×Q₈×C₇₁

C₄

yes

—

C₄×Q₈×C₇₁

C₄

yes

characteristic

C₄×Q₈×C₇₁

C₂²

yes

characteristic

C₄×Q₈×C₇₁

C₂²

yes

—

C₂×C₄⋊C₄×C₇₁

C₄

yes

—

C₂×C₄²×C₇₁

C₂²

yes

—

#11765849

—

#11765849

Q₈

yes

direct product factor

C₂×C₇₁⋊C₈

C₂³

yes

—

C₂×C₇₁⋊C₈

C₂³

yes

—

C₂×Q₈×C₇₁

—

C₂×Q₈×C₇₁

C₂×C₄

yes

—

C₂×Q₈×C₇₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₇₁

C₂×C₄

yes

—

C₂²×C₄×C₇₁

C₂×C₄

yes

—

C₂²×C₄×C₇₁

C₂³

yes

characteristic

C₄⋊C₄×C₇₁

C₂×C₄

yes

—

C₄⋊C₄×C₇₁

C₂×C₄

yes

—

C₄⋊C₄×C₇₁

C₂×C₄

yes

—

C₄²×C₇₁

C₂×C₄

yes

—

C₄²×C₇₁

C₂×C₄

yes

—

C₄²×C₇₁

C₂³

yes

—

C₄²×C₇₁

C₂³

yes

—

C₂³×C₇₁

C₂²×C₄

yes

characteristic

Q₈×C₇₁

—

Q₈×C₇₁

M₁₆

yes

complemented

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

C₂²×C₄

yes

—

C₂×C₄×C₇₁

C₂²×C₄

yes

—

C₂×C₄×C₇₁

C₂²×C₄

yes

—

C₂×C₄×C₇₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₇₁

C₂²×C₄

yes

—

C₂×C₄×C₇₁

C₂×Q₈

yes

—

C₂×C₄×C₇₁

#41

yes

—

C₂×C₄×C₇₁

C₂⁴

yes

characteristic

C₇₁⋊C₈

—

C₇₁⋊C₈

C₂×Q₈

yes

complemented

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

C₄×Q₈

yes

—

C₂²×C₇₁

C₂³×C₄

yes

characteristic

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

C₄×Q₈

yes

—

C₄×C₇₁

C₂×M₁₆

yes

—

C₄×C₇₁

C₂²×Q₈

yes

characteristic

C₄×C₇₁

C₂×#41

yes

characteristic

C₂×C₇₁

—

C₂×C₇₁

C₂×C₄×Q₈

yes

characteristic

C₂×C₇₁

C₂²×M₁₆

yes

characteristic

C₂×C₇₁

#568

yes

characteristic

Q₈×M₁₆

—

maximal, Sylow

C₇₁

Q₈×M₁₆

yes

characteristic, complemented, Sylow

C₂×C₄×Q₈

D₇₁

yes

characteristic

#446

—

C₈×Q₈

—

#423

—

C₄×M₁₆

—

C₂²×Q₈

Dic₇₁

yes

characteristic

C₂×M₁₆

—

C₂×M₁₆

—

C₄×Q₈

Dic₇₁

yes

—

C₄×Q₈

Dic₇₁

yes

characteristic

C₄×Q₈

C₂×D₇₁

yes

characteristic

C₄×Q₈

C₂×D₇₁

yes

—

C₂×C₄⋊C₄

Dic₇₁

yes

—

C₂×C₄²

C₂×D₇₁

yes

—

C₄⋊C₈

—

C₄⋊C₈

—

#97

—

C₄×C₈

—

C₂×Q₈

—

C₂×Q₈

C₂×Dic₇₁

yes

—

C₂×Q₈

C₂×Dic₇₁

yes

characteristic

C₂²×C₄

C₂×Dic₇₁

yes

—

C₂²×C₄

C₂×Dic₇₁

yes

—

C₂²×C₄

C₂²×D₇₁

yes

characteristic

M₁₆

—

M₁₆

—

142

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

C₂×Dic₇₁

yes

—

C₄⋊C₄

C₂×Dic₇₁

yes

—

C₄⋊C₄

C₂×Dic₇₁

yes

—

C₄²

C₂×Dic₇₁

yes

—

100

C₄²

C₂×Dic₇₁

yes

—

101

C₄²

C₂²×D₇₁

yes

—

102

C₄²

C₂²×D₇₁

yes

—

103

C₂³

C₂²×Dic₇₁

yes

characteristic

104

Q₈

—

105

Q₈

#11765849

yes

direct product factor

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

C₂²×Dic₇₁

yes

—

109

C₂×C₄

C₂²×Dic₇₁

yes

—

110

C₂×C₄

C₂²×Dic₇₁

yes

—

111

C₂×C₄

C₂²×Dic₇₁

yes

characteristic

112

C₂×C₄

C₂²×Dic₇₁

yes

—

113

C₂×C₄

Q₈×D₇₁

yes

—

114

C₂×C₄

#11765866

yes

—

115

C₂×C₄

C₂³×D₇₁

yes

characteristic

116

C₈

—

117

C₈

—

142

—

118

C₂²

—

119

C₂²

—

120

C₂²

Q₈×Dic₇₁

yes

—

121

C₂²

C₂³×Dic₇₁

yes

characteristic

122

C₄

—

123

C₄

—

124

C₄

—

125

C₄

Q₈×Dic₇₁

yes

—

126

C₄

C₂×#11765849

yes

—

127

C₄

C₂×Q₈×D₇₁

yes

characteristic

128

C₄

C₂×#11765866

yes

characteristic

129

C₂

—

130

C₂

C₂×Q₈×Dic₇₁

yes

characteristic

131

C₂

C₂²×#11765849

yes

characteristic

132

C₂

#17986766

yes

characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₇₁
commutator, socle	C₂²×C₇₁	C₂³×C₄
Fitting	C₂×C₄×Q₈×C₇₁	C₂
hypercenter	C₂×C₄×Q₈	D₇₁
hypocenter	C₇₁	Q₈×M₁₆
LCS	G ▷ C₂²×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×Q₈

subgroup

G/H

center, Frattini

C₂×C₄

C₂³×D₇₁

commutator, socle

C₂²×C₇₁

C₂³×C₄

Fitting

C₂×C₄×Q₈×C₇₁

C₂

hypercenter

C₂×C₄×Q₈

D₇₁

hypocenter

C₇₁

Q₈×M₁₆

LCS

G ▷ C₂²×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄×Q₈

N	H ≅ G/N	type
1. #17986130	C₂	semidirect
2. Q₈×C₇₁⋊C₈	C₂	semidirect
20. #11765849	Q₈	direct (N × H)
38. Q₈×C₇₁	M₁₆	semidirect
50. C₇₁⋊C₈	C₂×Q₈	semidirect
67. C₇₁	Q₈×M₁₆	semidirect
105. Q₈	#11765849	direct (N × H)

H ≅ G/N

type

1. #17986130

C₂

semidirect

2. Q₈×C₇₁⋊C₈

C₂

semidirect

20. #11765849

Q₈

direct (N × H)

38. Q₈×C₇₁

M₁₆

semidirect

50. C₇₁⋊C₈

C₂×Q₈

semidirect

67. C₇₁

Q₈×M₁₆

semidirect

105. Q₈

#11765849

direct (N × H)

Subgroups — Q₈×#11765849 #22597294

Conjugacy classes of subgroups

Complements

Subgroups — Q8×#11765849 #22597294

Conjugacy classes of subgroups

Complements

Subgroups — Q₈×#11765849 #22597294