Group #37573668 — subgroups

Subgroups — #37573668

Order 24448.

258 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 258 conjugacy classes of subgroups — too many to draw. Only its 87 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	C₂×#20259245	C₂	1	2	yes	maximal
2	#27132569	C₂	1	1	yes	maximal, characteristic
3	#27132425	C₂	1	2	yes	maximal
4	#27132418	C₂	1	1	yes	maximal, characteristic
5	C₂×#125×C₁₉₁	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₃₈₂	C₂²	1	1	yes	characteristic
7	C₂²×C₁₉₁⋊C₈	C₂²	1	2	yes	—
8	C₂×C₄⋊Dic₁₉₁	C₂²	1	1	yes	characteristic
9	#20259245	—	2	4	no	—
10	C₁₉₁⋊#95	—	2	2	no	—
11	#125×C₁₉₁	—	2	4	no	—
12	C₂×C₄⋊C₄×C₁₉₁	—	2	2	no	—
13	C₂×C₄⋊C₄×C₁₉₁	C₂²	1	2	yes	—
14	C₂×C₄²×C₁₉₁	C₂²	1	1	yes	characteristic
15	C₂×Dic₃₈₂	—	2	2	no	—
16	C₂×Dic₃₈₂	—	2	1	no	—
17	C₂×Dic₃₈₂	—	2	2	no	—
18	C₂×Dic₃₈₂	Q₈	1	2	yes	—
19	C₂²×Dic₁₉₁	—	2	1	no	—
20	C₂²×Dic₁₉₁	—	2	1	no	—
21	C₂×C₁₉₁⋊C₈	—	2	2	no	—
22	C₂×C₁₉₁⋊C₈	—	2	2	no	—
23	C₂×C₁₉₁⋊C₈	—	2	2	no	—
24	C₄⋊Dic₁₉₁	—	2	2	no	—
25	C₂²×C₄×C₁₉₁	—	2	2	no	—
26	C₂²×C₄×C₁₉₁	D₄	1	2	yes	—
27	C₂²×C₄×C₁₉₁	C₂³	1	1	yes	characteristic
28	C₄⋊C₄×C₁₉₁	—	2	8	no	—
29	C₄⋊C₄×C₁₉₁	—	2	2	no	—
30	C₄⋊C₄×C₁₉₁	D₄	1	4	yes	—
31	C₄²×C₁₉₁	—	2	2	no	—
32	Dic₃₈₂	—	2	4	no	—
33	Dic₃₈₂	—	4	2	no	—
34	C₂×Dic₁₉₁	—	2	2	no	—
35	C₂×Dic₁₉₁	—	4	1	no	—
36	C₂×Dic₁₉₁	—	4	1	no	—
37	C₂×Dic₁₉₁	—	4	1	no	—
38	C₂×Dic₁₉₁	—	4	2	no	—
39	C₁₉₁⋊C₈	—	4	2	no	—
40	C₂³×C₁₉₁	C₂×D₄	1	1	yes	characteristic
41	C₂×C₄×C₁₉₁	—	2	2	no	—
42	C₂×C₄×C₁₉₁	—	2	4	no	—
43	C₂×C₄×C₁₉₁	—	2	2	no	—
44	C₂×C₄×C₁₉₁	—	2	2	no	—
45	C₂×C₄×C₁₉₁	—	4	2	no	—
46	C₂×C₄×C₁₉₁	—	4	2	no	—
47	C₂×C₄×C₁₉₁	C₂×D₄	1	2	yes	—
48	C₂×C₄×C₁₉₁	C₂×Q₈	1	1	yes	characteristic
49	C₂×C₄×C₁₉₁	#41	1	2	yes	—
50	C₂×C₄×C₁₉₁	#41	1	1	yes	characteristic
51	C₂²×C₁₉₁	#120	1	1	yes	characteristic
52	C₂²×C₁₉₁	C₂²⋊Q₈	1	1	yes	characteristic
53	C₂²×C₁₉₁	#124	1	1	yes	characteristic
54	C₂²×C₁₉₁	#135	1	2	yes	—
55	C₂²×C₁₉₁	#137	1	2	yes	—
56	C₄×C₁₉₁	—	4	2	no	—
57	C₄×C₁₉₁	—	4	4	no	—
58	C₄×C₁₉₁	C₂²⋊Q₈	1	2	yes	—
59	C₄×C₁₉₁	#124	1	2	yes	—
60	Dic₁₉₁	—	4	2	no	—
61	Dic₁₉₁	—	8	1	no	—
62	C₂×C₁₉₁	#393	1	1	yes	characteristic
63	C₂×C₁₉₁	#452	1	2	yes	—
64	C₂×C₁₉₁	#479	1	2	yes	—
65	C₂×C₁₉₁	#488	1	1	yes	characteristic
66	C₂×C₁₉₁	#491	1	1	yes	characteristic
67	C₁₉₁	#2026	1	1	yes	characteristic, complemented, Sylow
68	#2026	—	191	1	no	maximal, Sylow
69	C₂×#125	D₁₉₁	1	1	yes	characteristic
70	C₂×#103	—	191	2	no	—
71	#391	—	191	1	no	—
72	#340	—	191	2	no	—
73	#336	—	191	1	no	—
74	C₂²×Q₈	—	191	1	no	—
75	C₂²×C₈	—	191	2	no	—
76	#125	—	2	4	no	—
77	C₂×C₄⋊C₄	—	2	2	no	—
78	C₂×C₄⋊C₄	—	191	1	no	—
79	C₂×C₄⋊C₄	C₂×D₁₉₁	1	2	yes	—
80	C₂×C₄²	C₂×D₁₉₁	1	1	yes	characteristic
81	#103	—	382	4	no	—
82	#95	—	382	2	no	—
83	C₂×Q₈	—	382	1	no	—
84	C₂×Q₈	—	382	2	no	—
85	C₂×Q₈	—	382	2	no	—
86	C₂×Q₈	—	191	2	no	—
87	C₂²×C₄	—	2	2	no	—
88	C₂²×C₄	—	382	1	no	—
89	C₂²×C₄	—	382	1	no	—
90	C₂²×C₄	C₂²⋊D₁₉₁	1	2	yes	—
91	C₂²×C₄	C₂²×D₁₉₁	1	1	yes	characteristic
92	C₂×C₈	—	382	2	no	—
93	C₂×C₈	—	382	2	no	—
94	C₂×C₈	—	382	2	no	—
95	C₄⋊C₄	—	2	8	no	—
96	C₄⋊C₄	—	2	2	no	—
97	C₄⋊C₄	—	382	2	no	—
98	C₄⋊C₄	C₂²⋊D₁₉₁	1	4	yes	—
99	C₄²	—	2	2	no	—
100	C₂³	C₂×C₂²⋊D₁₉₁	1	1	yes	characteristic
101	Q₈	—	382	4	no	—
102	Q₈	—	764	2	no	—
103	C₂×C₄	—	2	2	no	—
104	C₂×C₄	—	2	2	no	—
105	C₂×C₄	—	2	2	no	—
106	C₂×C₄	—	2	4	no	—
107	C₂×C₄	—	4	2	no	—
108	C₂×C₄	—	4	2	no	—
109	C₂×C₄	—	382	2	no	—
110	C₂×C₄	—	764	2	no	—
111	C₂×C₄	—	764	1	no	—
112	C₂×C₄	—	764	1	no	—
113	C₂×C₄	—	764	1	no	—
114	C₂×C₄	C₂×C₂²⋊D₁₉₁	1	2	yes	—
115	C₂×C₄	Q₈×D₁₉₁	1	1	yes	characteristic
116	C₂×C₄	#15690288	1	2	yes	—
117	C₂×C₄	#15690288	1	1	yes	characteristic
118	C₈	—	764	2	no	—
119	C₂²	#20259291	1	1	yes	characteristic
120	C₂²	#20259298	1	1	yes	characteristic
121	C₂²	#20259311	1	1	yes	characteristic
122	C₂²	#20259342	1	2	yes	—
123	C₂²	#20259352	1	2	yes	—
124	C₄	—	4	2	no	—
125	C₄	—	4	4	no	—
126	C₄	—	1528	1	no	—
127	C₄	—	764	2	no	—
128	C₄	#20259298	1	2	yes	—
129	C₄	#20259311	1	2	yes	—
130	C₂	#27132574	1	1	yes	characteristic
131	C₂	#27132825	1	2	yes	—
132	C₂	#27132967	1	2	yes	—
133	C₂	#27133020	1	1	yes	characteristic
134	C₂	#27133033	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#20259245

C₂

yes

maximal

#27132569

C₂

yes

maximal, characteristic

#27132425

C₂

yes

maximal

#27132418

C₂

yes

maximal, characteristic

C₂×#125×C₁₉₁

C₂

yes

maximal, characteristic

C₂²×Dic₃₈₂

C₂²

yes

characteristic

C₂²×C₁₉₁⋊C₈

C₂²

yes

—

C₂×C₄⋊Dic₁₉₁

C₂²

yes

characteristic

#20259245

—

C₁₉₁⋊#95

—

#125×C₁₉₁

—

C₂×C₄⋊C₄×C₁₉₁

—

C₂×C₄⋊C₄×C₁₉₁

C₂²

yes

—

C₂×C₄²×C₁₉₁

C₂²

yes

characteristic

C₂×Dic₃₈₂

—

C₂×Dic₃₈₂

—

C₂×Dic₃₈₂

—

C₂×Dic₃₈₂

Q₈

yes

—

C₂²×Dic₁₉₁

—

C₂²×Dic₁₉₁

—

C₂×C₁₉₁⋊C₈

—

C₂×C₁₉₁⋊C₈

—

C₂×C₁₉₁⋊C₈

—

C₄⋊Dic₁₉₁

—

C₂²×C₄×C₁₉₁

—

C₂²×C₄×C₁₉₁

D₄

yes

—

C₂²×C₄×C₁₉₁

C₂³

yes

characteristic

C₄⋊C₄×C₁₉₁

—

C₄⋊C₄×C₁₉₁

—

C₄⋊C₄×C₁₉₁

D₄

yes

—

C₄²×C₁₉₁

—

Dic₃₈₂

—

Dic₃₈₂

—

C₂×Dic₁₉₁

—

C₂×Dic₁₉₁

—

C₂×Dic₁₉₁

—

C₂×Dic₁₉₁

—

C₂×Dic₁₉₁

—

C₁₉₁⋊C₈

—

C₂³×C₁₉₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₉₁

—

C₂×C₄×C₁₉₁

—

C₂×C₄×C₁₉₁

—

C₂×C₄×C₁₉₁

—

C₂×C₄×C₁₉₁

—

C₂×C₄×C₁₉₁

—

C₂×C₄×C₁₉₁

C₂×D₄

yes

—

C₂×C₄×C₁₉₁

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₉₁

#41

yes

—

C₂×C₄×C₁₉₁

yes

C₂²×C₁₉₁

yes

C₂²×C₁₉₁

C₂²⋊Q₈

yes

C₂²×C₁₉₁

yes

C₂²×C₁₉₁

yes

—

C₂²×C₁₉₁

#137

yes

—

C₄×C₁₉₁

—

C₄×C₁₉₁

—

C₄×C₁₉₁

C₂²⋊Q₈

yes

—

C₄×C₁₉₁

#124

yes

—

Dic₁₉₁

—

Dic₁₉₁

—

C₂×C₁₉₁

#393

yes

characteristic

C₂×C₁₉₁

#452

yes

—

C₂×C₁₉₁

#479

yes

—

C₂×C₁₉₁

yes

C₂×C₁₉₁

yes

C₁₉₁

yes

characteristic, complemented, Sylow

#2026

—

191

maximal, Sylow

C₂×#125

D₁₉₁

yes

characteristic

C₂×#103

—

191

—

#391

—

191

—

#340

—

191

—

#336

—

191

—

C₂²×Q₈

—

191

—

C₂²×C₈

—

191

—

#125

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

191

—

C₂×C₄⋊C₄

C₂×D₁₉₁

yes

—

C₂×C₄²

C₂×D₁₉₁

yes

characteristic

#103

—

382

—

#95

—

382

—

C₂×Q₈

—

382

—

C₂×Q₈

—

382

—

C₂×Q₈

—

382

—

C₂×Q₈

—

191

—

C₂²×C₄

—

C₂²×C₄

—

382

—

C₂²×C₄

—

382

—

C₂²×C₄

C₂²⋊D₁₉₁

yes

—

C₂²×C₄

C₂²×D₁₉₁

yes

characteristic

C₂×C₈

—

382

—

C₂×C₈

—

382

—

C₂×C₈

—

382

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

382

—

C₄⋊C₄

C₂²⋊D₁₉₁

yes

—

C₄²

—

100

C₂³

C₂×C₂²⋊D₁₉₁

yes

characteristic

101

Q₈

—

382

—

102

Q₈

—

764

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

382

—

110

C₂×C₄

—

764

—

111

C₂×C₄

—

764

—

112

C₂×C₄

—

764

—

113

C₂×C₄

—

764

—

114

C₂×C₄

C₂×C₂²⋊D₁₉₁

yes

—

115

C₂×C₄

Q₈×D₁₉₁

yes

characteristic

116

C₂×C₄

#15690288

yes

—

117

C₂×C₄

#15690288

yes

characteristic

118

C₈

—

764

—

119

C₂²

#20259291

yes

characteristic

120

C₂²

#20259298

yes

characteristic

121

C₂²

#20259311

yes

characteristic

122

C₂²

#20259342

yes

—

123

C₂²

#20259352

yes

—

124

C₄

—

125

C₄

—

126

C₄

—

1528

—

127

C₄

—

764

—

128

C₄

#20259298

yes

—

129

C₄

#20259311

yes

—

130

C₂

#27132574

yes

characteristic

131

C₂

#27132825

yes

—

132

C₂

#27132967

yes

—

133

C₂

#27133020

yes

characteristic

134

C₂

#27133033

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₁₉₁
commutator, cosocle	C₂²×C₄×C₁₉₁	C₂³
Frattini	C₂²×C₄	C₂²×D₁₉₁
Fitting	C₂×#125×C₁₉₁	C₂
socle, prime radical	C₂³×C₁₉₁	C₂×D₄
hypercenter	C₂×#125	D₁₉₁
hypocenter	C₁₉₁	#2026
LCS	G ▷ C₂²×C₄×C₁₉₁ ▷ C₂×C₁₉₁ ▷ C₁₉₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#125

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₁₉₁

commutator, cosocle

C₂²×C₄×C₁₉₁

C₂³

Frattini

C₂²×C₄

C₂²×D₁₉₁

Fitting

C₂×#125×C₁₉₁

C₂

socle, prime radical

C₂³×C₁₉₁

C₂×D₄

D₁₉₁

C₁₉₁

G ▷ C₂²×C₄×C₁₉₁ ▷ C₂×C₁₉₁ ▷ C₁₉₁

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#125

N	H ≅ G/N	type
67. C₁₉₁	#2026	semidirect

H ≅ G/N

type

67. C₁₉₁

#2026

semidirect

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

subgroup	index
C₂×#20259245	2
#27132569	2
#27132425	2
#27132418	2
C₂×#125×C₁₉₁	2
#2026	191

subgroup

index

C₂×#125×C₁₉₁

191

Subgroups — #37573668

Conjugacy classes of subgroups

Complements

Maximal subgroups