Group #37572686 — subgroups

Subgroups — #37572686

Order 24448.

256 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 117 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×#20259255	C₂	1	2	yes	maximal
2	C₂×#20259253	C₂	1	2	yes	maximal
3	C₂×#20259233	C₂	1	1	yes	maximal, characteristic
4	#27132529	C₂	1	1	yes	maximal, characteristic
5	#384×C₁₉₁	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₁₉₁⋊C₈	C₂²	1	2	yes	—
7	C₂×C₄⋊Dic₁₉₁	C₂²	1	1	yes	characteristic
8	C₂×C₄⋊Dic₁₉₁	C₂²	1	1	yes	characteristic
9	#20259255	—	2	4	no	—
10	#20259253	—	2	4	no	—
11	#20259233	C₄	1	4	yes	complemented
12	C₁₉₁⋊#95	—	2	2	no	—
13	C₂×C₄⋊C₄×C₁₉₁	—	2	2	no	—
14	C₂×C₄⋊C₄×C₁₉₁	C₂²	1	2	yes	—
15	C₂×C₄²×C₁₉₁	C₂²	1	1	yes	characteristic
16	C₂²×Dic₁₉₁	—	2	1	no	—
17	C₂²×Dic₁₉₁	—	2	1	no	—
18	C₂×C₁₉₁⋊C₈	C₂×C₄	1	4	yes	—
19	C₂×C₁₉₁⋊C₈	D₄	1	4	yes	—
20	C₂×C₁₉₁⋊C₈	Q₈	1	4	yes	—
21	C₄⋊Dic₁₉₁	—	2	2	no	—
22	C₄⋊Dic₁₉₁	—	2	2	no	—
23	C₂²×C₄×C₁₉₁	—	2	2	no	—
24	C₂²×C₄×C₁₉₁	D₄	1	2	yes	—
25	C₂²×C₄×C₁₉₁	C₂³	1	1	yes	characteristic
26	C₄⋊C₄×C₁₉₁	—	2	2	no	—
27	C₄⋊C₄×C₁₉₁	—	2	2	no	—
28	C₄⋊C₄×C₁₉₁	—	4	4	no	—
29	C₄²×C₁₉₁	C₂×C₄	1	2	yes	—
30	C₄²×C₁₉₁	Q₈	1	2	yes	—
31	C₂×Dic₁₉₁	—	4	1	no	—
32	C₂×Dic₁₉₁	—	4	1	no	—
33	C₂×Dic₁₉₁	—	4	2	no	—
34	C₂×Dic₁₉₁	—	4	2	no	—
35	C₁₉₁⋊C₈	C₄⋊C₄	1	8	yes	complemented
36	C₂³×C₁₉₁	C₂×D₄	1	1	yes	characteristic
37	C₂×C₄×C₁₉₁	—	2	2	no	—
38	C₂×C₄×C₁₉₁	—	2	2	no	—
39	C₂×C₄×C₁₉₁	—	4	2	no	—
40	C₂×C₄×C₁₉₁	—	4	4	no	—
41	C₂×C₄×C₁₉₁	C₄⋊C₄	1	4	yes	—
42	C₂×C₄×C₁₉₁	C₂²×C₄	1	1	yes	characteristic
43	C₂×C₄×C₁₉₁	C₂×D₄	1	2	yes	—
44	C₂×C₄×C₁₉₁	C₂×Q₈	1	2	yes	—
45	C₂×C₄×C₁₉₁	C₂×Q₈	1	1	yes	characteristic
46	C₂²×C₁₉₁	C₂×C₄⋊C₄	1	1	yes	characteristic
47	C₂²×C₁₉₁	#127	1	1	yes	characteristic
48	C₂²×C₁₉₁	C₄⋊Q₈	1	1	yes	characteristic
49	C₂²×C₁₉₁	AGL(1,ℤ/8)	1	2	yes	—
50	C₂²×C₁₉₁	#137	1	2	yes	—
51	C₄×C₁₉₁	—	2	4	no	—
52	C₄×C₁₉₁	—	8	2	no	—
53	C₄×C₁₉₁	C₂×C₄⋊C₄	1	2	yes	—
54	C₄×C₁₉₁	C₄⋊Q₈	1	2	yes	—
55	Dic₁₉₁	—	8	1	no	—
56	Dic₁₉₁	—	8	1	no	—
57	C₂×C₁₉₁	#384	1	1	yes	characteristic
58	C₂×C₁₉₁	#428	1	2	yes	—
59	C₂×C₁₉₁	#496	1	1	yes	characteristic
60	C₂×C₁₉₁	#497	1	1	yes	characteristic
61	C₂×C₁₉₁	#501	1	2	yes	—
62	C₁₉₁	#1847	1	1	yes	characteristic, complemented, Sylow
63	#1847	—	191	1	no	maximal, Sylow
64	C₂×#107	—	191	2	no	—
65	C₂×#106	—	191	2	no	—
66	C₂×#97	—	191	1	no	—
67	#384	D₁₉₁	1	1	yes	characteristic
68	#382	—	191	1	no	—
69	C₂²×C₈	—	191	2	no	—
70	C₂×C₄⋊C₄	—	2	2	no	—
71	C₂×C₄⋊C₄	—	191	1	no	—
72	C₂×C₄⋊C₄	—	191	1	no	—
73	C₂×C₄⋊C₄	C₂×D₁₉₁	1	2	yes	—
74	C₂×C₄²	C₂×D₁₉₁	1	1	yes	characteristic
75	#107	—	382	4	no	—
76	#106	—	382	4	no	—
77	#97	—	191	4	no	—
78	#95	—	382	2	no	—
79	C₂²×C₄	—	2	2	no	—
80	C₂²×C₄	—	382	1	no	—
81	C₂²×C₄	—	382	1	no	—
82	C₂²×C₄	C₂²⋊D₁₉₁	1	2	yes	—
83	C₂²×C₄	C₂²×D₁₉₁	1	1	yes	characteristic
84	C₂×C₈	—	191	4	no	—
85	C₂×C₈	—	191	4	no	—
86	C₂×C₈	—	191	4	no	—
87	C₄⋊C₄	—	2	2	no	—
88	C₄⋊C₄	—	2	2	no	—
89	C₄⋊C₄	—	4	4	no	—
90	C₄⋊C₄	—	382	2	no	—
91	C₄⋊C₄	—	382	2	no	—
92	C₄²	C₄×D₁₉₁	1	2	yes	—
93	C₄²	Dic₃₈₂	1	2	yes	—
94	C₂³	C₂×C₂²⋊D₁₉₁	1	1	yes	characteristic
95	C₂×C₄	—	2	2	no	—
96	C₂×C₄	—	2	2	no	—
97	C₂×C₄	—	4	4	no	—
98	C₂×C₄	—	4	2	no	—
99	C₂×C₄	—	764	2	no	—
100	C₂×C₄	—	764	2	no	—
101	C₂×C₄	—	764	1	no	—
102	C₂×C₄	—	764	1	no	—
103	C₂×C₄	Dic₁₉₁⋊C₄	1	4	yes	—
104	C₂×C₄	C₂×C₄×D₁₉₁	1	1	yes	characteristic
105	C₂×C₄	D₄×D₁₉₁	1	2	yes	—
106	C₂×C₄	Q₈×D₁₉₁	1	2	yes	—
107	C₂×C₄	C₂×Dic₃₈₂	1	1	yes	characteristic
108	C₈	—	191	8	no	—
109	C₂²	C₂×Dic₁₉₁⋊C₄	1	1	yes	characteristic
110	C₂²	#20259318	1	1	yes	characteristic
111	C₂²	#20259320	1	1	yes	characteristic
112	C₂²	#20259348	1	2	yes	—
113	C₂²	#20259353	1	2	yes	—
114	C₄	—	2	4	no	—
115	C₄	—	8	2	no	—
116	C₄	—	1528	1	no	—
117	C₄	—	1528	1	no	—
118	C₄	C₄⋊C₄×D₁₉₁	1	2	yes	—
119	C₄	#20259321	1	2	yes	—
120	C₂	#27132532	1	1	yes	characteristic
121	C₂	#27132703	1	2	yes	—
122	C₂	#27133050	1	1	yes	characteristic
123	C₂	#27133055	1	1	yes	characteristic
124	C₂	#27133069	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#20259255

C₂

yes

maximal

C₂×#20259253

C₂

yes

maximal

C₂×#20259233

C₂

yes

maximal, characteristic

#27132529

C₂

yes

maximal, characteristic

#384×C₁₉₁

C₂

yes

maximal, characteristic

C₂²×C₁₉₁⋊C₈

C₂²

yes

—

C₂×C₄⋊Dic₁₉₁

C₂²

yes

characteristic

C₂×C₄⋊Dic₁₉₁

C₂²

yes

—

—

C₄

yes

complemented

C₁₉₁⋊#95

—

C₂×C₄⋊C₄×C₁₉₁

—

C₂×C₄⋊C₄×C₁₉₁

C₂²

yes

—

C₂×C₄²×C₁₉₁

C₂²

yes

characteristic

C₂²×Dic₁₉₁

—

C₂²×Dic₁₉₁

—

C₂×C₁₉₁⋊C₈

C₂×C₄

yes

—

C₂×C₁₉₁⋊C₈

D₄

yes

—

C₂×C₁₉₁⋊C₈

Q₈

yes

—

C₄⋊Dic₁₉₁

—

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

—

C₄²×C₁₉₁

C₂×C₄

yes

—

C₄²×C₁₉₁

Q₈

yes

—

C₂×Dic₁₉₁

—

C₂×Dic₁₉₁

—

C₂×Dic₁₉₁

—

C₂×Dic₁₉₁

—

C₁₉₁⋊C₈

C₄⋊C₄

yes

complemented

C₂³×C₁₉₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₉₁

—

C₂×C₄×C₁₉₁

—

C₂×C₄×C₁₉₁

—

C₂×C₄×C₁₉₁

—

C₂×C₄×C₁₉₁

C₄⋊C₄

yes

—

C₂×C₄×C₁₉₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₉₁

C₂×D₄

yes

—

C₂×C₄×C₁₉₁

C₂×Q₈

yes

—

C₂×C₄×C₁₉₁

C₂×Q₈

yes

characteristic

C₂²×C₁₉₁

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₁₉₁

#127

yes

characteristic

C₂²×C₁₉₁

C₄⋊Q₈

yes

characteristic

C₂²×C₁₉₁

AGL(1,ℤ/8)

yes

—

C₂²×C₁₉₁

#137

yes

—

C₄×C₁₉₁

—

C₄×C₁₉₁

—

C₄×C₁₉₁

C₂×C₄⋊C₄

yes

—

C₄×C₁₉₁

C₄⋊Q₈

yes

—

Dic₁₉₁

—

Dic₁₉₁

—

C₂×C₁₉₁

#384

yes

characteristic

C₂×C₁₉₁

#428

yes

—

C₂×C₁₉₁

yes

C₂×C₁₉₁

yes

C₂×C₁₉₁

yes

—

C₁₉₁

#1847

yes

characteristic, complemented, Sylow

#1847

—

191

maximal, Sylow

C₂×#107

—

191

—

C₂×#106

—

191

—

C₂×#97

—

191

—

#384

D₁₉₁

yes

characteristic

#382

—

191

—

C₂²×C₈

—

191

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

191

—

C₂×C₄⋊C₄

—

191

—

C₂×C₄⋊C₄

C₂×D₁₉₁

yes

—

C₂×C₄²

C₂×D₁₉₁

yes

characteristic

#107

—

382

—

#106

—

382

—

#97

—

191

—

#95

—

382

—

C₂²×C₄

—

C₂²×C₄

—

382

—

C₂²×C₄

—

382

—

C₂²×C₄

C₂²⋊D₁₉₁

yes

—

C₂²×C₄

C₂²×D₁₉₁

yes

characteristic

C₂×C₈

—

191

—

C₂×C₈

—

191

—

C₂×C₈

—

191

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

382

—

C₄⋊C₄

—

382

—

C₄²

C₄×D₁₉₁

yes

—

C₄²

Dic₃₈₂

yes

—

C₂³

C₂×C₂²⋊D₁₉₁

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

764

—

100

C₂×C₄

—

764

—

101

C₂×C₄

—

764

—

102

C₂×C₄

—

764

—

103

C₂×C₄

Dic₁₉₁⋊C₄

yes

—

104

C₂×C₄

C₂×C₄×D₁₉₁

yes

characteristic

105

C₂×C₄

D₄×D₁₉₁

yes

—

106

C₂×C₄

Q₈×D₁₉₁

yes

—

107

C₂×C₄

C₂×Dic₃₈₂

yes

characteristic

108

C₈

—

191

—

109

C₂²

C₂×Dic₁₉₁⋊C₄

yes

characteristic

110

C₂²

#20259318

yes

characteristic

111

C₂²

#20259320

yes

characteristic

112

C₂²

#20259348

yes

—

113

C₂²

#20259353

yes

—

114

C₄

—

115

C₄

—

116

C₄

—

1528

—

117

C₄

—

1528

—

118

C₄

C₄⋊C₄×D₁₉₁

yes

—

119

C₄

#20259321

yes

—

120

C₂

#27132532

yes

characteristic

121

C₂

#27132703

yes

—

122

C₂

#27133050

yes

characteristic

123

C₂

#27133055

yes

characteristic

124

C₂

#27133069

yes

—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₁₉₁
commutator	C₂×C₄×C₁₉₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₉₁
Fitting	#384×C₁₉₁	C₂
socle, prime radical	C₂³×C₁₉₁	C₂×D₄
cosocle	C₂²×C₄×C₁₉₁	C₂³
hypercenter	#384	D₁₉₁
hypocenter	C₁₉₁	#1847
LCS	G ▷ C₂×C₄×C₁₉₁ ▷ C₂×C₁₉₁ ▷ C₁₉₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #384

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₁₉₁

commutator

C₂×C₄×C₁₉₁

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₉₁

Fitting

#384×C₁₉₁

C₂

socle, prime radical

C₂³×C₁₉₁

C₂×D₄

cosocle

C₂²×C₄×C₁₉₁

C₂³

D₁₉₁

C₁₉₁

G ▷ C₂×C₄×C₁₉₁ ▷ C₂×C₁₉₁ ▷ C₁₉₁

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #384

N	H ≅ G/N	type
11. #20259233	C₄	semidirect
35. C₁₉₁⋊C₈	C₄⋊C₄	semidirect
62. C₁₉₁	#1847	semidirect

H ≅ G/N

type

11. #20259233

C₄

semidirect

35. C₁₉₁⋊C₈

C₄⋊C₄

semidirect

62. C₁₉₁

#1847

semidirect

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

subgroup	index
C₂×#20259255	2
C₂×#20259253	2
C₂×#20259233	2
#27132529	2
#384×C₁₉₁	2
#1847	191

subgroup

index

#384×C₁₉₁

191

Subgroups — #37572686

Conjugacy classes of subgroups

Complements

Maximal subgroups