Group #22553939 — subgroups

Subgroups — #22553939

Order 8384.

206 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 206 conjugacy classes of subgroups — too many to draw. Only its 103 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	#16862245	C₂	1	1	yes	maximal, characteristic, complemented
2	#16862244	C₂	1	2	yes	maximal, complemented
3	#16862238	C₂	1	1	yes	maximal, characteristic, complemented
4	#16862237	C₂	1	1	yes	maximal, characteristic, complemented
5	#16862236	C₂	1	2	yes	maximal, complemented
6	#16862224	C₂	1	2	yes	maximal
7	#16862222	C₂	1	2	yes	maximal
8	#16862204	C₂	1	1	yes	maximal, characteristic
9	#16862203	C₂	1	1	yes	maximal, characteristic
10	C₄⋊C₄×D₁₃₁	C₂	1	1	yes	maximal, characteristic
11	C₄⋊Q₈×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×Dic₂₆₂	C₂²	1	2	yes	—
13	C₂×C₄×D₁₃₁	C₂²	1	2	yes	—
14	C₂×C₄×D₁₃₁	C₂²	1	1	yes	characteristic
15	C₄⋊Dic₁₃₁	C₂²	1	2	yes	—
16	C₄⋊Dic₁₃₁	C₂²	1	1	yes	characteristic
17	C₄⋊Dic₁₃₁	C₂²	1	1	yes	characteristic
18	Dic₁₃₁⋊C₄	C₂²	1	2	yes	—
19	Dic₁₃₁⋊C₄	C₂²	1	2	yes	—
20	Dic₁₃₁⋊C₄	C₂²	1	2	yes	—
21	Dic₁₃₁⋊C₄	C₂²	1	2	yes	—
22	Dic₁₃₁⋊C₄	C₂²	1	1	yes	characteristic
23	Dic₁₃₁⋊C₄	C₂²	1	2	yes	—
24	Dic₁₃₁⋊C₄	C₂²	1	1	yes	characteristic
25	(C₂×D₁₃₁)⋊C₄	C₂²	1	1	yes	characteristic
26	(C₂×D₁₃₁)⋊C₄	C₂²	1	2	yes	—
27	(C₂×D₁₃₁)⋊C₄	C₂²	1	1	yes	characteristic
28	C₄×Dic₁₃₁	C₂²	1	2	yes	—
29	C₄×Dic₁₃₁	C₂²	1	1	yes	characteristic
30	C₂×Q₈×C₁₃₁	C₂²	1	2	yes	—
31	C₄⋊C₄×C₁₃₁	C₂²	1	2	yes	—
32	C₄⋊C₄×C₁₃₁	C₂²	1	1	yes	characteristic
33	C₄⋊C₄×C₁₃₁	C₂²	1	1	yes	characteristic
34	C₄²×C₁₃₁	C₂²	1	1	yes	characteristic
35	C₂²×D₁₃₁	C₂³	1	1	yes	characteristic
36	Dic₂₆₂	—	4	2	no	—
37	C₂×Dic₁₃₁	C₂³	1	2	yes	—
38	C₂×Dic₁₃₁	C₂³	1	1	yes	characteristic
39	C₂×Dic₁₃₁	C₂³	1	1	yes	characteristic
40	C₂×Dic₁₃₁	C₂³	1	2	yes	—
41	C₂×Dic₁₃₁	C₂³	1	1	yes	characteristic
42	C₄×D₁₃₁	—	4	2	no	—
43	C₄×D₁₃₁	Q₈	1	4	yes	complemented
44	Q₈×C₁₃₁	—	4	2	no	—
45	C₂×C₄×C₁₃₁	C₂³	1	1	yes	characteristic
46	C₂×C₄×C₁₃₁	C₂³	1	1	yes	characteristic
47	C₂×C₄×C₁₃₁	C₂³	1	2	yes	—
48	C₂×C₄×C₁₃₁	C₂³	1	2	yes	—
49	C₂×C₄×C₁₃₁	C₂³	1	1	yes	characteristic
50	C₂²×C₁₃₁	C₂⁴	1	1	yes	characteristic
51	C₂×D₁₃₁	—	2	1	no	—
52	C₂×D₁₃₁	—	2	1	no	—
53	C₂×D₁₃₁	C₂×Q₈	1	2	yes	—
54	C₄×C₁₃₁	—	2	1	no	—
55	C₄×C₁₃₁	—	2	2	no	—
56	C₄×C₁₃₁	—	2	2	no	—
57	C₄×C₁₃₁	—	2	1	no	—
58	C₄×C₁₃₁	C₂×Q₈	1	2	yes	—
59	Dic₁₃₁	—	2	2	no	—
60	Dic₁₃₁	—	2	1	no	—
61	Dic₁₃₁	—	2	2	no	—
62	Dic₁₃₁	—	2	1	no	—
63	Dic₁₃₁	C₂×Q₈	1	2	yes	—
64	C₂×C₁₃₁	C₂²×Q₈	1	1	yes	characteristic
65	C₂×C₁₃₁	H(5,𝔽₂)	1	1	yes	characteristic
66	C₂×C₁₃₁	2₋¹⁺⁴	1	1	yes	characteristic
67	D₁₃₁	—	2	2	no	—
68	C₁₃₁	#544	1	1	yes	characteristic, complemented, Sylow
69	#544	—	131	1	no	maximal, Sylow
70	C₄⋊Q₈	—	131	2	no	—
71	C₄⋊Q₈	—	131	1	no	—
72	C₄⋊Q₈	D₁₃₁	1	1	yes	characteristic, complemented
73	#125	—	131	2	no	—
74	#125	—	131	1	no	—
75	#125	—	131	1	no	—
76	C₂²⋊Q₈	—	131	2	no	—
77	C₂²⋊Q₈	—	131	2	no	—
78	#117	—	131	1	no	—
79	#117	—	131	1	no	—
80	C₂×C₄⋊C₄	—	131	1	no	—
81	C₂×Q₈	—	131	2	no	—
82	C₂×Q₈	C₂×D₁₃₁	1	2	yes	—
83	C₂²×C₄	—	131	2	no	—
84	C₂²×C₄	—	131	1	no	—
85	C₄⋊C₄	—	131	2	no	—
86	C₄⋊C₄	—	131	1	no	—
87	C₄⋊C₄	—	131	1	no	—
88	C₄⋊C₄	—	131	2	no	—
89	C₄⋊C₄	—	131	2	no	—
90	C₄⋊C₄	—	131	2	no	—
91	C₄⋊C₄	—	131	2	no	—
92	C₄⋊C₄	—	131	1	no	—
93	C₄⋊C₄	—	131	1	no	—
94	C₄⋊C₄	—	131	2	no	—
95	C₄⋊C₄	C₂×D₁₃₁	1	1	yes	characteristic
96	C₄⋊C₄	C₂×D₁₃₁	1	1	yes	characteristic
97	C₄⋊C₄	C₂×D₁₃₁	1	2	yes	—
98	C₂²⋊C₄	—	131	2	no	—
99	C₂²⋊C₄	—	131	1	no	—
100	C₂²⋊C₄	—	131	1	no	—
101	C₄²	—	131	1	no	—
102	C₄²	—	131	2	no	—
103	C₄²	C₂×D₁₃₁	1	1	yes	characteristic
104	C₂³	—	131	1	no	—
105	Q₈	—	4	2	no	—
106	Q₈	—	524	2	no	—
107	C₂×C₄	—	524	2	no	—
108	C₂×C₄	—	131	1	no	—
109	C₂×C₄	—	131	2	no	—
110	C₂×C₄	—	131	1	no	—
111	C₂×C₄	—	131	2	no	—
112	C₂×C₄	—	131	1	no	—
113	C₂×C₄	—	131	4	no	—
114	C₂×C₄	C₂²×D₁₃₁	1	2	yes	—
115	C₂×C₄	C₂²×D₁₃₁	1	1	yes	characteristic
116	C₂×C₄	C₂²×D₁₃₁	1	1	yes	characteristic
117	C₂×C₄	C₂²×D₁₃₁	1	2	yes	—
118	C₂×C₄	C₂²×D₁₃₁	1	1	yes	characteristic
119	C₂²	—	262	1	no	—
120	C₂²	—	262	1	no	—
121	C₂²	—	131	2	no	—
122	C₂²	C₂³×D₁₃₁	1	1	yes	characteristic
123	C₄	—	2	2	no	—
124	C₄	—	2	1	no	—
125	C₄	—	2	2	no	—
126	C₄	—	2	1	no	—
127	C₄	—	262	2	no	—
128	C₄	—	262	1	no	—
129	C₄	—	262	1	no	—
130	C₄	—	262	2	no	—
131	C₄	—	131	2	no	—
132	C₄	Q₈×D₁₃₁	1	2	yes	—
133	C₂	—	262	2	no	—
134	C₂	C₂×Q₈×D₁₃₁	1	1	yes	characteristic
135	C₂	#16862291	1	1	yes	characteristic
136	C₂	#16862294	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16862245

C₂

yes

maximal, characteristic, complemented

#16862244

C₂

yes

maximal, complemented

#16862238

C₂

yes

maximal, characteristic, complemented

#16862237

C₂

yes

maximal, characteristic, complemented

#16862236

C₂

yes

maximal, complemented

#16862224

C₂

yes

maximal

#16862222

C₂

yes

maximal

#16862204

C₂

yes

maximal, characteristic

#16862203

C₂

yes

maximal, characteristic

C₄⋊C₄×D₁₃₁

C₂

yes

maximal, characteristic

C₄⋊Q₈×C₁₃₁

C₂

yes

maximal, characteristic, complemented

C₂×Dic₂₆₂

C₂²

yes

—

C₂×C₄×D₁₃₁

C₂²

yes

—

C₂×C₄×D₁₃₁

C₂²

yes

characteristic

C₄⋊Dic₁₃₁

C₂²

yes

—

C₄⋊Dic₁₃₁

C₂²

yes

characteristic

C₄⋊Dic₁₃₁

C₂²

yes

characteristic

Dic₁₃₁⋊C₄

C₂²

yes

—

Dic₁₃₁⋊C₄

C₂²

yes

—

Dic₁₃₁⋊C₄

C₂²

yes

—

Dic₁₃₁⋊C₄

C₂²

yes

—

Dic₁₃₁⋊C₄

C₂²

yes

characteristic

Dic₁₃₁⋊C₄

C₂²

yes

—

Dic₁₃₁⋊C₄

C₂²

yes

characteristic

(C₂×D₁₃₁)⋊C₄

C₂²

yes

characteristic

(C₂×D₁₃₁)⋊C₄

C₂²

yes

—

(C₂×D₁₃₁)⋊C₄

C₂²

yes

characteristic

C₄×Dic₁₃₁

C₂²

yes

—

C₄×Dic₁₃₁

C₂²

yes

characteristic

C₂×Q₈×C₁₃₁

C₂²

yes

—

C₄⋊C₄×C₁₃₁

C₂²

yes

—

C₄⋊C₄×C₁₃₁

C₂²

yes

characteristic

C₄⋊C₄×C₁₃₁

C₂²

yes

characteristic

C₄²×C₁₃₁

C₂²

yes

characteristic

C₂²×D₁₃₁

C₂³

yes

characteristic

Dic₂₆₂

—

C₂×Dic₁₃₁

C₂³

yes

—

C₂×Dic₁₃₁

C₂³

yes

characteristic

C₂×Dic₁₃₁

C₂³

yes

characteristic

C₂×Dic₁₃₁

C₂³

yes

—

C₂×Dic₁₃₁

C₂³

yes

characteristic

C₄×D₁₃₁

—

C₄×D₁₃₁

Q₈

yes

complemented

Q₈×C₁₃₁

—

C₂×C₄×C₁₃₁

C₂³

yes

characteristic

C₂×C₄×C₁₃₁

C₂³

yes

characteristic

C₂×C₄×C₁₃₁

C₂³

yes

—

C₂×C₄×C₁₃₁

C₂³

yes

—

C₂×C₄×C₁₃₁

C₂³

yes

characteristic

C₂²×C₁₃₁

C₂⁴

yes

characteristic

C₂×D₁₃₁

—

C₂×D₁₃₁

—

C₂×D₁₃₁

C₂×Q₈

yes

—

C₄×C₁₃₁

—

C₄×C₁₃₁

—

C₄×C₁₃₁

—

C₄×C₁₃₁

—

C₄×C₁₃₁

C₂×Q₈

yes

—

Dic₁₃₁

—

Dic₁₃₁

—

Dic₁₃₁

—

Dic₁₃₁

—

Dic₁₃₁

C₂×Q₈

yes

—

C₂×C₁₃₁

C₂²×Q₈

yes

characteristic

C₂×C₁₃₁

H(5,𝔽₂)

yes

characteristic

C₂×C₁₃₁

2₋¹⁺⁴

yes

characteristic

D₁₃₁

—

C₁₃₁

#544

yes

characteristic, complemented, Sylow

#544

—

131

maximal, Sylow

C₄⋊Q₈

—

131

—

C₄⋊Q₈

—

131

—

C₄⋊Q₈

D₁₃₁

yes

characteristic, complemented

#125

—

131

—

#125

—

131

—

#125

—

131

—

C₂²⋊Q₈

—

131

—

C₂²⋊Q₈

—

131

—

#117

—

131

—

#117

—

131

—

C₂×C₄⋊C₄

—

131

—

C₂×Q₈

—

131

—

C₂×Q₈

C₂×D₁₃₁

yes

—

C₂²×C₄

—

131

—

C₂²×C₄

—

131

—

C₄⋊C₄

—

131

—

C₄⋊C₄

—

131

—

C₄⋊C₄

—

131

—

C₄⋊C₄

—

131

—

C₄⋊C₄

—

131

—

C₄⋊C₄

—

131

—

C₄⋊C₄

—

131

—

C₄⋊C₄

—

131

—

C₄⋊C₄

—

131

—

C₄⋊C₄

—

131

—

C₄⋊C₄

C₂×D₁₃₁

yes

characteristic

C₄⋊C₄

C₂×D₁₃₁

yes

characteristic

C₄⋊C₄

C₂×D₁₃₁

yes

—

C₂²⋊C₄

—

131

—

C₂²⋊C₄

—

131

—

100

C₂²⋊C₄

—

131

—

101

C₄²

—

131

—

102

C₄²

—

131

—

103

C₄²

C₂×D₁₃₁

yes

characteristic

104

C₂³

—

131

—

105

Q₈

—

106

Q₈

—

524

—

107

C₂×C₄

—

524

—

108

C₂×C₄

—

131

—

109

C₂×C₄

—

131

—

110

C₂×C₄

—

131

—

111

C₂×C₄

—

131

—

112

C₂×C₄

—

131

—

113

C₂×C₄

—

131

—

114

C₂×C₄

C₂²×D₁₃₁

yes

—

115

C₂×C₄

C₂²×D₁₃₁

yes

characteristic

116

C₂×C₄

C₂²×D₁₃₁

yes

characteristic

117

C₂×C₄

C₂²×D₁₃₁

yes

—

118

C₂×C₄

C₂²×D₁₃₁

yes

characteristic

119

C₂²

—

262

—

120

C₂²

—

262

—

121

C₂²

—

131

—

122

C₂²

C₂³×D₁₃₁

yes

characteristic

123

C₄

—

124

C₄

—

125

C₄

—

126

C₄

—

127

C₄

—

262

—

128

C₄

—

262

—

129

C₄

—

262

—

130

C₄

—

262

—

131

C₄

—

131

—

132

C₄

Q₈×D₁₃₁

yes

—

133

C₂

—

262

—

134

C₂

C₂×Q₈×D₁₃₁

yes

characteristic

135

C₂

#16862291

yes

characteristic

136

C₂

#16862294

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₁₃₁
commutator, socle	C₂²×C₁₃₁	C₂⁴
Fitting	C₄⋊Q₈×C₁₃₁	C₂
hypercenter	C₄⋊Q₈	D₁₃₁
hypocenter	C₁₃₁	#544
LCS	G ▷ C₂²×C₁₃₁ ▷ C₁₃₁
UCS	C₁ ◁ C₂² ◁ C₄⋊Q₈

subgroup

G/H

center, Frattini

C₂²

C₂³×D₁₃₁

commutator, socle

C₂²×C₁₃₁

C₂⁴

Fitting

C₄⋊Q₈×C₁₃₁

C₂

hypercenter

C₄⋊Q₈

D₁₃₁

C₁₃₁

G ▷ C₂²×C₁₃₁ ▷ C₁₃₁

C₁ ◁ C₂² ◁ C₄⋊Q₈

N	H ≅ G/N	type
1. #16862245	C₂	semidirect
2. #16862244	C₂	semidirect
3. #16862238	C₂	semidirect
4. #16862237	C₂	semidirect
5. #16862236	C₂	semidirect
11. C₄⋊Q₈×C₁₃₁	C₂	semidirect
43. C₄×D₁₃₁	Q₈	semidirect
68. C₁₃₁	#544	semidirect
72. C₄⋊Q₈	D₁₃₁	semidirect

H ≅ G/N

type

1. #16862245

C₂

semidirect

2. #16862244

C₂

semidirect

3. #16862238

C₂

semidirect

4. #16862237

C₂

semidirect

5. #16862236

C₂

semidirect

11. C₄⋊Q₈×C₁₃₁

C₂

semidirect

43. C₄×D₁₃₁

Q₈

semidirect

68. C₁₃₁

#544

semidirect

72. C₄⋊Q₈

D₁₃₁

semidirect

Subgroups — #22553939

Conjugacy classes of subgroups

Complements