Group #22601201 — subgroups

Subgroups — Q₈×#11765854 #22601201

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×Dic₁₄₂	C₂	1	1	yes	maximal, characteristic, complemented
2	Q₈×D₂₈₄	C₂	1	1	yes	maximal, characteristic
3	#17986397	C₂	1	3	yes	maximal, complemented
4	#17986290	C₂	1	3	yes	maximal
5	#17986285	C₂	1	3	yes	maximal, complemented
6	C₄×#11765854	C₂	1	3	yes	maximal
7	C₈×Q₈×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×Q₈×D₇₁	—	2	1	no	—
9	C₂×#11765854	C₂²	1	1	yes	characteristic
10	#14556128	C₂²	1	3	yes	—
11	#14556128	C₂²	1	3	yes	—
12	#14556127	—	2	3	no	—
13	#14556108	—	2	3	no	—
14	Q₈×Dic₇₁	—	2	1	no	—
15	C₄×Dic₁₄₂	C₂²	1	3	yes	—
16	C₄×D₂₈₄	C₂²	1	3	yes	—
17	#14556060	C₂²	1	3	yes	—
18	#14556060	C₂²	1	6	yes	—
19	#14556053	C₂²	1	3	yes	—
20	#14556050	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	—	2	3	no	—
25	C₂×Dic₁₄₂	—	2	3	no	—
26	C₂×Dic₁₄₂	C₂³	1	1	yes	characteristic
27	Q₈×D₇₁	—	4	4	no	—
28	C₂×D₂₈₄	C₂³	1	1	yes	characteristic
29	C₂×C₄×D₇₁	—	2	3	no	—
30	#11765854	Q₈	1	4	yes	direct product factor
31	C₄⋊Dic₇₁	—	2	3	no	—
32	C₄⋊Dic₇₁	—	2	3	no	—
33	C₄⋊Dic₇₁	C₂³	1	3	yes	—
34	C₄⋊Dic₇₁	C₂³	1	3	yes	—
35	Dic₇₁⋊C₄	—	2	3	no	—
36	(C₂×D₇₁)⋊C₄	—	2	3	no	—
37	C₄×Dic₇₁	—	2	3	no	—
38	C₂×Q₈×C₇₁	D₄	1	1	yes	characteristic
39	C₂×C₈×C₇₁	C₂³	1	3	yes	—
40	C₂×C₈×C₇₁	C₂³	1	1	yes	characteristic
41	C₄⋊C₄×C₇₁	D₄	1	3	yes	—
42	C₄²×C₇₁	C₂³	1	3	yes	—
43	C₂²×D₇₁	—	2	1	no	—
44	Q₈×C₇₁	SD₁₆	1	4	yes	complemented
45	C₂×C₄×C₇₁	C₂×D₄	1	3	yes	—
46	C₂×C₄×C₇₁	C₂×D₄	1	3	yes	—
47	C₂×C₄×C₇₁	C₂⁴	1	1	yes	characteristic
48	C₂×Dic₇₁	—	2	1	no	—
49	C₂×Dic₇₁	—	2	3	no	—
50	C₂×Dic₇₁	—	2	3	no	—
51	D₂₈₄	—	2	1	no	—
52	D₂₈₄	C₂×Q₈	1	2	yes	—
53	C₄×D₇₁	—	4	6	no	—
54	Dic₁₄₂	—	2	1	no	—
55	Dic₁₄₂	—	4	6	no	—
56	Dic₁₄₂	—	8	3	no	—
57	Dic₁₄₂	C₂×Q₈	1	2	yes	complemented
58	C₈×C₇₁	—	2	3	no	—
59	C₈×C₇₁	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	3	no	—
65	C₄×C₇₁	C₂×SD₁₆	1	6	yes	—
66	C₄×C₇₁	C₂²×Q₈	1	1	yes	characteristic
67	C₄×C₇₁	2₋¹⁺⁴	1	1	yes	characteristic
68	Dic₇₁	—	2	2	no	—
69	Dic₇₁	—	4	3	no	—
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	—	71	1	no	—
78	D₄×Q₈	—	71	1	no	—
79	#498	—	71	3	no	—
80	#476	—	71	3	no	—
81	#475	—	71	3	no	—
82	C₈×Q₈	D₇₁	1	1	yes	characteristic, complemented
83	C₄×SD₁₆	—	71	3	no	—
84	C₂²×Q₈	—	142	1	no	—
85	C₂×SD₁₆	—	71	1	no	—
86	C₄⋊Q₈	—	71	3	no	—
87	C₄⋊Q₈	—	71	3	no	—
88	C₄⋊Q₈	—	142	3	no	—
89	C₂²⋊Q₈	—	142	3	no	—
90	C₄×Q₈	—	71	3	no	—
91	C₄×Q₈	—	142	1	no	—
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	C₂×D₇₁	1	3	yes	—
97	#103	—	71	3	no	—
98	#102	—	71	3	no	—
99	C₄×C₈	C₂×D₇₁	1	3	yes	—
100	C₂×Q₈	—	284	4	no	—
101	C₂×Q₈	—	71	1	no	—
102	C₂×Q₈	—	142	3	no	—
103	C₂×Q₈	—	142	3	no	—
104	C₂×Q₈	D₂₈₄	1	1	yes	characteristic
105	C₂×D₄	—	71	1	no	—
106	C₂²×C₄	—	142	3	no	—
107	SD₁₆	—	71	4	no	—
108	C₂×C₈	C₂²×D₇₁	1	3	yes	—
109	C₂×C₈	C₂²×D₇₁	1	1	yes	characteristic
110	C₄⋊C₄	—	71	3	no	—
111	C₄⋊C₄	—	71	3	no	—
112	C₄⋊C₄	—	142	3	no	—
113	C₄⋊C₄	—	142	3	no	—
114	C₄⋊C₄	—	142	3	no	—
115	C₄⋊C₄	D₂₈₄	1	3	yes	—
116	C₂²⋊C₄	—	142	3	no	—
117	C₄²	—	142	3	no	—
118	C₄²	C₂²×D₇₁	1	3	yes	—
119	C₂³	—	142	1	no	—
120	Q₈	—	284	6	no	—
121	Q₈	—	568	3	no	—
122	Q₈	—	71	2	no	—
123	Q₈	—	142	1	no	—
124	Q₈	#11765854	1	4	yes	direct product factor
125	D₄	—	71	2	no	—
126	D₄	—	142	1	no	—
127	C₂×C₄	—	284	6	no	—
128	C₂×C₄	—	142	3	no	—
129	C₂×C₄	—	142	3	no	—
130	C₂×C₄	—	142	1	no	—
131	C₂×C₄	C₂×D₂₈₄	1	3	yes	—
132	C₂×C₄	C₂×D₂₈₄	1	3	yes	—
133	C₂×C₄	C₂³×D₇₁	1	1	yes	characteristic
134	C₈	—	2	3	no	—
135	C₈	Q₈×D₇₁	1	2	yes	complemented
136	C₂²	—	284	1	no	—
137	C₂²	—	284	1	no	—
138	C₂²	—	142	2	no	—
139	C₂²	C₂²×D₂₈₄	1	1	yes	characteristic
140	C₄	—	2	3	no	—
141	C₄	—	284	3	no	—
142	C₄	—	284	3	no	—
143	C₄	—	142	2	no	—
144	C₄	C₂×#11765854	1	6	yes	—
145	C₄	C₂×Q₈×D₇₁	1	1	yes	characteristic
146	C₄	#14556175	1	1	yes	characteristic
147	C₂	—	284	2	no	—
148	C₂	Q₈×D₂₈₄	1	1	yes	characteristic
149	C₂	C₂²×#11765854	1	1	yes	characteristic
150	C₂	#17986811	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

Q₈×Dic₁₄₂

C₂

yes

maximal, characteristic, complemented

Q₈×D₂₈₄

C₂

yes

maximal, characteristic

#17986397

C₂

yes

maximal, complemented

#17986290

C₂

yes

maximal

#17986285

C₂

yes

maximal, complemented

C₄×#11765854

C₂

yes

maximal

C₈×Q₈×C₇₁

C₂

yes

maximal, characteristic, complemented

C₂×Q₈×D₇₁

—

C₂×#11765854

C₂²

yes

characteristic

#14556128

C₂²

yes

—

#14556128

C₂²

yes

—

#14556127

—

#14556108

—

Q₈×Dic₇₁

—

C₄×Dic₁₄₂

C₂²

yes

—

C₄×D₂₈₄

C₂²

yes

—

#14556060

C₂²

yes

—

#14556060

C₂²

yes

—

#14556053

C₂²

yes

—

#14556050

C₂²

yes

—

C₄×Q₈×C₇₁

C₂²

yes

characteristic

C₄⋊C₈×C₇₁

C₂²

yes

—

C₄×C₈×C₇₁

C₂²

yes

—

C₂×Dic₁₄₂

—

C₂×Dic₁₄₂

—

C₂×Dic₁₄₂

C₂³

yes

characteristic

Q₈×D₇₁

—

C₂×D₂₈₄

C₂³

yes

characteristic

C₂×C₄×D₇₁

—

#11765854

Q₈

yes

direct product factor

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

C₂³

yes

—

C₄⋊Dic₇₁

C₂³

yes

—

Dic₇₁⋊C₄

—

(C₂×D₇₁)⋊C₄

—

C₄×Dic₇₁

—

C₂×Q₈×C₇₁

D₄

yes

characteristic

C₂×C₈×C₇₁

C₂³

yes

—

C₂×C₈×C₇₁

C₂³

yes

characteristic

C₄⋊C₄×C₇₁

D₄

yes

—

C₄²×C₇₁

C₂³

yes

—

C₂²×D₇₁

—

Q₈×C₇₁

SD₁₆

yes

complemented

C₂×C₄×C₇₁

C₂×D₄

yes

—

C₂×C₄×C₇₁

C₂×D₄

yes

—

C₂×C₄×C₇₁

C₂⁴

yes

characteristic

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₂×Dic₇₁

—

D₂₈₄

—

D₂₈₄

C₂×Q₈

yes

—

C₄×D₇₁

—

Dic₁₄₂

—

Dic₁₄₂

—

Dic₁₄₂

—

Dic₁₄₂

C₂×Q₈

yes

complemented

C₈×C₇₁

—

C₈×C₇₁

C₂×Q₈

yes

complemented

C₂²×C₇₁

C₂²×D₄

yes

characteristic

C₂×D₇₁

—

C₂×D₇₁

—

C₂×D₇₁

—

C₄×C₇₁

—

C₄×C₇₁

C₂×SD₁₆

yes

—

C₄×C₇₁

C₂²×Q₈

yes

characteristic

C₄×C₇₁

2₋¹⁺⁴

yes

characteristic

Dic₇₁

—

Dic₇₁

—

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

—

D₄×Q₈

—

#498

—

#476

—

#475

—

C₈×Q₈

D₇₁

yes

characteristic, complemented

C₄×SD₁₆

—

C₂²×Q₈

—

142

—

C₂×SD₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

142

—

C₂²⋊Q₈

—

142

—

C₄×Q₈

—

C₄×Q₈

—

142

—

C₄×Q₈

C₂×D₇₁

yes

characteristic

C₄×D₄

—

#106

—

#106

—

C₄⋊C₈

C₂×D₇₁

yes

—

#103

—

#102

—

C₄×C₈

C₂×D₇₁

yes

—

100

C₂×Q₈

—

284

—

101

C₂×Q₈

—

102

C₂×Q₈

—

142

—

103

C₂×Q₈

—

142

—

104

C₂×Q₈

D₂₈₄

yes

characteristic

105

C₂×D₄

—

106

C₂²×C₄

—

142

—

107

SD₁₆

—

108

C₂×C₈

C₂²×D₇₁

yes

—

109

C₂×C₈

C₂²×D₇₁

yes

characteristic

110

C₄⋊C₄

—

111

C₄⋊C₄

—

112

C₄⋊C₄

—

142

—

113

C₄⋊C₄

—

142

—

114

C₄⋊C₄

—

142

—

115

C₄⋊C₄

D₂₈₄

yes

—

116

C₂²⋊C₄

—

142

—

117

C₄²

—

142

—

118

C₄²

C₂²×D₇₁

yes

—

119

C₂³

—

142

—

120

Q₈

—

284

—

121

Q₈

—

568

—

122

Q₈

—

123

Q₈

—

142

—

124

Q₈

#11765854

yes

direct product factor

125

D₄

—

126

D₄

—

142

—

127

C₂×C₄

—

284

—

128

C₂×C₄

—

142

—

129

C₂×C₄

—

142

—

130

C₂×C₄

—

142

—

131

C₂×C₄

C₂×D₂₈₄

yes

—

132

C₂×C₄

C₂×D₂₈₄

yes

—

133

C₂×C₄

C₂³×D₇₁

yes

characteristic

134

C₈

—

135

C₈

Q₈×D₇₁

yes

complemented

136

C₂²

—

284

—

137

C₂²

—

284

—

138

C₂²

—

142

—

139

C₂²

C₂²×D₂₈₄

yes

characteristic

140

C₄

—

141

C₄

—

284

—

142

C₄

—

284

—

143

C₄

—

142

—

144

C₄

C₂×#11765854

yes

—

145

C₄

C₂×Q₈×D₇₁

yes

characteristic

146

C₄

#14556175

yes

characteristic

147

C₂

—

284

—

148

C₂

Q₈×D₂₈₄

yes

characteristic

149

C₂

C₂²×#11765854

yes

characteristic

150

C₂

#17986811

yes

characteristic

	subgroup	G/H
center	C₂²	C₂²×D₂₈₄
commutator	C₂×C₄×C₇₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₇₁
Fitting	C₈×Q₈×C₇₁	C₂
socle	C₂²×C₇₁	C₂²×D₄
hypercenter	C₈×Q₈	D₇₁
hypocenter	C₇₁	Q₈×SD₁₆
LCS	G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ C₈×Q₈

subgroup

G/H

center

C₂²

C₂²×D₂₈₄

commutator

C₂×C₄×C₇₁

C₂⁴

Frattini

C₂×C₄

C₂³×D₇₁

Fitting

C₈×Q₈×C₇₁

C₂

socle

C₂²×C₇₁

C₂²×D₄

hypercenter

C₈×Q₈

D₇₁

hypocenter

C₇₁

Q₈×SD₁₆

LCS

G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂² ◁ C₄×Q₈ ◁ C₈×Q₈

N	H ≅ G/N	type
1. Q₈×Dic₁₄₂	C₂	semidirect
3. #17986397	C₂	semidirect
5. #17986285	C₂	semidirect
7. C₈×Q₈×C₇₁	C₂	semidirect
30. #11765854	Q₈	direct (N × H)
44. Q₈×C₇₁	SD₁₆	semidirect
57. Dic₁₄₂	C₂×Q₈	semidirect
59. C₈×C₇₁	C₂×Q₈	semidirect
76. C₇₁	Q₈×SD₁₆	semidirect
82. C₈×Q₈	D₇₁	semidirect
124. Q₈	#11765854	direct (N × H)
135. C₈	Q₈×D₇₁	semidirect

H ≅ G/N

type

1. Q₈×Dic₁₄₂

C₂

semidirect

3. #17986397

C₂

semidirect

5. #17986285

C₂

semidirect

7. C₈×Q₈×C₇₁

C₂

semidirect

30. #11765854

Q₈

direct (N × H)

44. Q₈×C₇₁

SD₁₆

semidirect

57. Dic₁₄₂

C₂×Q₈

semidirect

59. C₈×C₇₁

C₂×Q₈

semidirect

76. C₇₁

Q₈×SD₁₆

semidirect

82. C₈×Q₈

D₇₁

semidirect

124. Q₈

#11765854

direct (N × H)

135. C₈

Q₈×D₇₁

semidirect

Subgroups — Q₈×#11765854 #22601201

Conjugacy classes of subgroups

Complements

Subgroups — Q8×#11765854 #22601201

Conjugacy classes of subgroups

Complements

Subgroups — Q₈×#11765854 #22601201