Group #27133273 — subgroups

Subgroups — #27133273

Order 12224.

192 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 192 conjugacy classes of subgroups — too many to draw. Only its 95 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	#20259320	C₂	1	1	yes	maximal, characteristic, complemented
2	#20259315	C₂	1	2	yes	maximal
3	#20259315	C₂	1	2	yes	maximal
4	#20259313	C₂	1	1	yes	maximal, characteristic, complemented
5	#20259313	C₂	1	2	yes	maximal, complemented
6	#20259312	C₂	1	2	yes	maximal, complemented
7	#20259301	C₂	1	1	yes	maximal, characteristic
8	Q₈×Dic₁₉₁	C₂	1	1	yes	maximal, characteristic, complemented
9	#20259287	C₂	1	1	yes	maximal, characteristic, complemented
10	#20259281	C₂	1	1	yes	maximal, characteristic
11	C₂²⋊Q₈×C₁₉₁	C₂	1	1	yes	maximal, characteristic
12	C₂×Dic₃₈₂	C₂²	1	1	yes	characteristic
13	C₄⋊Dic₁₉₁	C₂²	1	1	yes	characteristic
14	C₄⋊Dic₁₉₁	C₂²	1	1	yes	characteristic
15	C₄⋊Dic₁₉₁	C₂²	1	1	yes	characteristic
16	C₄⋊Dic₁₉₁	C₂²	1	2	yes	—
17	C₄⋊Dic₁₉₁	C₂²	1	2	yes	—
18	Dic₁₉₁⋊C₄	C₂²	1	2	yes	—
19	Dic₁₉₁⋊C₄	C₂²	1	2	yes	—
20	Dic₁₉₁⋊C₄	C₂²	1	2	yes	—
21	Dic₁₉₁⋊C₄	C₂²	1	2	yes	—
22	Dic₁₉₁⋊C₄	C₂²	1	2	yes	—
23	C₂²⋊Dic₁₉₁	C₂²	1	1	yes	characteristic
24	C₂²⋊Dic₁₉₁	C₂²	1	1	yes	characteristic
25	C₂²⋊Dic₁₉₁	C₂²	1	2	yes	—
26	C₄×Dic₁₉₁	C₂²	1	1	yes	characteristic
27	C₄×Dic₁₉₁	C₂²	1	1	yes	characteristic
28	C₄×Dic₁₉₁	C₂²	1	2	yes	—
29	C₄×Dic₁₉₁	C₂²	1	2	yes	—
30	C₂×Q₈×C₁₉₁	C₂²	1	1	yes	characteristic
31	C₂²×C₄×C₁₉₁	C₂²	1	1	yes	characteristic
32	C₄⋊C₄×C₁₉₁	C₂²	1	1	yes	characteristic
33	C₄⋊C₄×C₁₉₁	C₂²	1	2	yes	—
34	C₂²⋊C₄×C₁₉₁	C₂²	1	2	yes	—
35	Dic₃₈₂	—	2	2	no	—
36	C₂×Dic₁₉₁	C₂³	1	1	yes	characteristic
37	C₂×Dic₁₉₁	C₂³	1	1	yes	characteristic
38	C₂×Dic₁₉₁	C₂³	1	1	yes	characteristic
39	C₂×Dic₁₉₁	C₂³	1	2	yes	—
40	C₂×Dic₁₉₁	C₂³	1	1	yes	characteristic
41	C₂×Dic₁₉₁	C₂³	1	2	yes	—
42	C₂³×C₁₉₁	C₂³	1	1	yes	characteristic
43	Q₈×C₁₉₁	—	2	2	no	—
44	C₂×C₄×C₁₉₁	—	2	2	no	—
45	C₂×C₄×C₁₉₁	C₂³	1	2	yes	—
46	C₂×C₄×C₁₉₁	C₂³	1	2	yes	—
47	C₂×C₄×C₁₉₁	C₂³	1	1	yes	characteristic
48	C₂×C₄×C₁₉₁	C₂³	1	1	yes	characteristic
49	C₂²×C₁₉₁	—	2	1	no	—
50	C₂²×C₁₉₁	—	2	1	no	—
51	C₂²×C₁₉₁	—	2	1	no	—
52	C₂²×C₁₉₁	C₂⁴	1	1	yes	characteristic
53	C₄×C₁₉₁	—	2	1	no	—
54	C₄×C₁₉₁	—	2	2	no	—
55	C₄×C₁₉₁	—	2	2	no	—
56	C₄×C₁₉₁	#41	1	2	yes	—
57	Dic₁₉₁	—	2	2	no	—
58	Dic₁₉₁	—	2	1	no	—
59	Dic₁₉₁	—	2	1	no	—
60	Dic₁₉₁	—	2	1	no	—
61	Dic₁₉₁	—	2	1	no	—
62	Dic₁₉₁	—	2	2	no	—
63	C₂×C₁₉₁	—	4	1	no	—
64	C₂×C₁₉₁	C₂×#41	1	1	yes	characteristic
65	C₂×C₁₉₁	2₋¹⁺⁴	1	1	yes	characteristic
66	C₂×C₁₉₁	2₋¹⁺⁴	1	1	yes	characteristic
67	C₁₉₁	#541	1	1	yes	characteristic, complemented, Sylow
68	#541	—	191	1	no	maximal, Sylow
69	C₄⋊Q₈	—	191	1	no	—
70	#126	—	191	2	no	—
71	#126	—	191	2	no	—
72	#125	—	191	2	no	—
73	#125	—	191	1	no	—
74	#125	—	191	2	no	—
75	C₂²⋊Q₈	—	191	1	no	—
76	C₂²⋊Q₈	D₁₉₁	1	1	yes	characteristic
77	C₄×Q₈	—	191	1	no	—
78	C₄×Q₈	—	191	1	no	—
79	#117	—	191	1	no	—
80	C₂×Q₈	—	191	1	no	—
81	C₂×Q₈	C₂×D₁₉₁	1	1	yes	characteristic
82	C₂²×C₄	C₂×D₁₉₁	1	1	yes	characteristic
83	C₄⋊C₄	—	191	2	no	—
84	C₄⋊C₄	—	191	2	no	—
85	C₄⋊C₄	—	191	2	no	—
86	C₄⋊C₄	—	191	2	no	—
87	C₄⋊C₄	—	191	2	no	—
88	C₄⋊C₄	—	191	1	no	—
89	C₄⋊C₄	—	191	1	no	—
90	C₄⋊C₄	—	191	1	no	—
91	C₄⋊C₄	—	191	2	no	—
92	C₄⋊C₄	—	191	2	no	—
93	C₄⋊C₄	C₂×D₁₉₁	1	1	yes	characteristic
94	C₄⋊C₄	C₂×D₁₉₁	1	2	yes	—
95	C₂²⋊C₄	—	191	1	no	—
96	C₂²⋊C₄	—	191	1	no	—
97	C₂²⋊C₄	—	191	2	no	—
98	C₂²⋊C₄	C₂×D₁₉₁	1	2	yes	—
99	C₄²	—	191	1	no	—
100	C₄²	—	191	1	no	—
101	C₄²	—	191	2	no	—
102	C₄²	—	191	2	no	—
103	C₂³	C₂²×D₁₉₁	1	1	yes	characteristic
104	Q₈	—	2	2	no	—
105	Q₈	—	382	2	no	—
106	C₂×C₄	—	2	2	no	—
107	C₂×C₄	—	191	1	no	—
108	C₂×C₄	—	191	1	no	—
109	C₂×C₄	—	191	1	no	—
110	C₂×C₄	—	191	1	no	—
111	C₂×C₄	—	191	2	no	—
112	C₂×C₄	—	191	2	no	—
113	C₂×C₄	C₂²×D₁₉₁	1	2	yes	—
114	C₂×C₄	C₂²×D₁₉₁	1	2	yes	—
115	C₂×C₄	C₂²×D₁₉₁	1	1	yes	characteristic
116	C₂×C₄	C₂²×D₁₉₁	1	1	yes	characteristic
117	C₂²	—	2	1	no	—
118	C₂²	—	2	1	no	—
119	C₂²	—	2	1	no	—
120	C₂²	C₂³×D₁₉₁	1	1	yes	characteristic
121	C₄	—	2	1	no	—
122	C₄	—	2	2	no	—
123	C₄	—	2	2	no	—
124	C₄	—	382	2	no	—
125	C₄	—	382	1	no	—
126	C₄	—	382	1	no	—
127	C₄	—	382	1	no	—
128	C₄	—	382	1	no	—
129	C₄	—	382	2	no	—
130	C₄	#15690286	1	2	yes	—
131	C₂	—	4	1	no	—
132	C₂	C₂×#15690286	1	1	yes	characteristic
133	C₂	#20259369	1	1	yes	characteristic
134	C₂	#20259370	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20259320

C₂

yes

maximal, characteristic, complemented

#20259315

C₂

yes

maximal

#20259315

C₂

yes

maximal

#20259313

C₂

yes

maximal, characteristic, complemented

#20259313

C₂

yes

maximal, complemented

#20259312

C₂

yes

maximal, complemented

#20259301

C₂

yes

maximal, characteristic

Q₈×Dic₁₉₁

C₂

yes

maximal, characteristic, complemented

#20259287

C₂

yes

maximal, characteristic, complemented

#20259281

C₂

yes

maximal, characteristic

C₂²⋊Q₈×C₁₉₁

C₂

yes

maximal, characteristic

C₂×Dic₃₈₂

C₂²

yes

characteristic

C₄⋊Dic₁₉₁

C₂²

yes

characteristic

C₄⋊Dic₁₉₁

C₂²

yes

characteristic

C₄⋊Dic₁₉₁

C₂²

yes

characteristic

C₄⋊Dic₁₉₁

C₂²

yes

—

C₄⋊Dic₁₉₁

C₂²

yes

—

Dic₁₉₁⋊C₄

C₂²

yes

—

Dic₁₉₁⋊C₄

C₂²

yes

—

Dic₁₉₁⋊C₄

C₂²

yes

—

Dic₁₉₁⋊C₄

C₂²

yes

—

Dic₁₉₁⋊C₄

C₂²

yes

—

C₂²⋊Dic₁₉₁

C₂²

yes

characteristic

C₂²⋊Dic₁₉₁

C₂²

yes

characteristic

C₂²⋊Dic₁₉₁

C₂²

yes

—

C₄×Dic₁₉₁

C₂²

yes

characteristic

C₄×Dic₁₉₁

C₂²

yes

characteristic

C₄×Dic₁₉₁

C₂²

yes

—

C₄×Dic₁₉₁

C₂²

yes

—

C₂×Q₈×C₁₉₁

C₂²

yes

characteristic

C₂²×C₄×C₁₉₁

C₂²

yes

characteristic

C₄⋊C₄×C₁₉₁

C₂²

yes

characteristic

C₄⋊C₄×C₁₉₁

C₂²

yes

—

C₂²⋊C₄×C₁₉₁

C₂²

yes

—

Dic₃₈₂

—

C₂×Dic₁₉₁

C₂³

yes

characteristic

C₂×Dic₁₉₁

C₂³

yes

characteristic

C₂×Dic₁₉₁

C₂³

yes

characteristic

C₂×Dic₁₉₁

C₂³

yes

—

C₂×Dic₁₉₁

C₂³

yes

characteristic

C₂×Dic₁₉₁

C₂³

yes

—

C₂³×C₁₉₁

C₂³

yes

characteristic

Q₈×C₁₉₁

—

C₂×C₄×C₁₉₁

—

C₂×C₄×C₁₉₁

C₂³

yes

—

C₂×C₄×C₁₉₁

C₂³

yes

—

C₂×C₄×C₁₉₁

C₂³

yes

characteristic

C₂×C₄×C₁₉₁

C₂³

yes

characteristic

C₂²×C₁₉₁

—

C₂²×C₁₉₁

—

C₂²×C₁₉₁

—

C₂²×C₁₉₁

C₂⁴

yes

characteristic

C₄×C₁₉₁

—

C₄×C₁₉₁

—

C₄×C₁₉₁

—

C₄×C₁₉₁

#41

yes

—

Dic₁₉₁

—

Dic₁₉₁

—

Dic₁₉₁

—

Dic₁₉₁

—

Dic₁₉₁

—

Dic₁₉₁

—

C₂×C₁₉₁

—

C₂×C₁₉₁

C₂×#41

yes

characteristic

C₂×C₁₉₁

2₋¹⁺⁴

yes

characteristic

C₂×C₁₉₁

2₋¹⁺⁴

yes

characteristic

C₁₉₁

#541

yes

characteristic, complemented, Sylow

#541

—

191

maximal, Sylow

C₄⋊Q₈

—

191

—

#126

—

191

—

#126

—

191

—

#125

—

191

—

#125

—

191

—

#125

—

191

—

C₂²⋊Q₈

—

191

—

C₂²⋊Q₈

D₁₉₁

yes

characteristic

C₄×Q₈

—

191

—

C₄×Q₈

—

191

—

#117

—

191

—

C₂×Q₈

—

191

—

C₂×Q₈

C₂×D₁₉₁

yes

characteristic

C₂²×C₄

C₂×D₁₉₁

yes

characteristic

C₄⋊C₄

—

191

—

C₄⋊C₄

—

191

—

C₄⋊C₄

—

191

—

C₄⋊C₄

—

191

—

C₄⋊C₄

—

191

—

C₄⋊C₄

—

191

—

C₄⋊C₄

—

191

—

C₄⋊C₄

—

191

—

C₄⋊C₄

—

191

—

C₄⋊C₄

—

191

—

C₄⋊C₄

C₂×D₁₉₁

yes

characteristic

C₄⋊C₄

C₂×D₁₉₁

yes

—

C₂²⋊C₄

—

191

—

C₂²⋊C₄

—

191

—

C₂²⋊C₄

—

191

—

C₂²⋊C₄

C₂×D₁₉₁

yes

—

C₄²

—

191

—

100

C₄²

—

191

—

101

C₄²

—

191

—

102

C₄²

—

191

—

103

C₂³

C₂²×D₁₉₁

yes

characteristic

104

Q₈

—

105

Q₈

—

382

—

106

C₂×C₄

—

107

C₂×C₄

—

191

—

108

C₂×C₄

—

191

—

109

C₂×C₄

—

191

—

110

C₂×C₄

—

191

—

111

C₂×C₄

—

191

—

112

C₂×C₄

—

191

—

113

C₂×C₄

C₂²×D₁₉₁

yes

—

114

C₂×C₄

C₂²×D₁₉₁

yes

—

115

C₂×C₄

C₂²×D₁₉₁

yes

characteristic

116

C₂×C₄

C₂²×D₁₉₁

yes

characteristic

117

C₂²

—

118

C₂²

—

119

C₂²

—

120

C₂²

C₂³×D₁₉₁

yes

characteristic

121

C₄

—

122

C₄

—

123

C₄

—

124

C₄

—

382

—

125

C₄

—

382

—

126

C₄

—

382

—

127

C₄

—

382

—

128

C₄

—

382

—

129

C₄

—

382

—

130

C₄

#15690286

yes

—

131

C₂

—

132

C₂

C₂×#15690286

yes

characteristic

133

C₂

#20259369

yes

characteristic

134

C₂

#20259370

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₁₉₁
commutator, socle, cosocle	C₂²×C₁₉₁	C₂⁴
Fitting	C₂²⋊Q₈×C₁₉₁	C₂
prime radical	C₂³×C₁₉₁	C₂³
hypercenter	C₂²⋊Q₈	D₁₉₁
hypocenter	C₁₉₁	#541
LCS	G ▷ C₂²×C₁₉₁ ▷ C₁₉₁
UCS	C₁ ◁ C₂² ◁ C₂²⋊Q₈

subgroup

G/H

center, Frattini

C₂²

C₂³×D₁₉₁

commutator, socle, cosocle

C₂²×C₁₉₁

C₂⁴

Fitting

C₂²⋊Q₈×C₁₉₁

C₂

prime radical

C₂³×C₁₉₁

C₂³

hypercenter

C₂²⋊Q₈

D₁₉₁

C₁₉₁

G ▷ C₂²×C₁₉₁ ▷ C₁₉₁

C₁ ◁ C₂² ◁ C₂²⋊Q₈

N	H ≅ G/N	type
1. #20259320	C₂	semidirect
4. #20259313	C₂	semidirect
5. #20259313	C₂	semidirect
6. #20259312	C₂	semidirect
8. Q₈×Dic₁₉₁	C₂	semidirect
9. #20259287	C₂	semidirect
67. C₁₉₁	#541	semidirect

H ≅ G/N

type

1. #20259320

C₂

semidirect

4. #20259313

C₂

semidirect

5. #20259313

C₂

semidirect

6. #20259312

C₂

semidirect

8. Q₈×Dic₁₉₁

C₂

semidirect

9. #20259287

C₂

semidirect

67. C₁₉₁

#541

semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#20259320	2
#20259315	2
#20259313	2
#20259312	2
#20259301	2
Q₈×Dic₁₉₁	2
#20259287	2
#20259281	2
C₂²⋊Q₈×C₁₉₁	2
#541	191

subgroup

index

Q₈×Dic₁₉₁

C₂²⋊Q₈×C₁₉₁

191

Subgroups — #27133273

Conjugacy classes of subgroups

Complements

Maximal subgroups