Group #37573497 — subgroups

Subgroups — #37573497

Order 24448.

284 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 284 conjugacy classes of subgroups — too many to draw. Only its 101 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	#27133220	C₂	1	1	yes	maximal, characteristic
2	#27132687	C₂	1	1	yes	maximal, characteristic
3	#27132423	C₂	1	4	yes	maximal
4	C₄³×C₁₉₁	C₂	1	1	yes	maximal, characteristic
5	C₂×#15690271	C₂²	1	2	yes	—
6	#20259322	C₄	1	2	yes	complemented
7	#20259314	C₄	1	2	yes	complemented
8	#20259301	—	2	2	no	—
9	C₄×Dic₃₈₂	—	2	2	no	—
10	#20259281	C₂²	1	2	yes	—
11	C₄⋊(C₁₉₁⋊C₈)	—	2	2	no	—
12	C₂×C₄²×C₁₉₁	—	2	2	no	—
13	C₂×C₄²×C₁₉₁	C₂²	1	2	yes	—
14	C₂×C₄²×C₁₉₁	C₂²	1	1	yes	characteristic
15	C₂×Dic₃₈₂	—	2	2	no	—
16	#15690271	—	2	4	no	—
17	C₂×C₁₉₁⋊C₈	—	2	2	no	—
18	C₄⋊Dic₁₉₁	—	2	2	no	—
19	C₄⋊Dic₁₉₁	C₂×C₄	1	4	yes	—
20	Dic₁₉₁⋊C₄	—	2	4	no	—
21	C₂²⋊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	C₄²×C₁₉₁	—	2	8	no	—
27	C₄²×C₁₉₁	—	2	2	no	—
28	C₄²×C₁₉₁	C₂×C₄	1	1	yes	characteristic
29	C₄²×C₁₉₁	C₂×C₄	1	1	yes	characteristic
30	C₄²×C₁₉₁	D₄	1	2	yes	—
31	C₄²×C₁₉₁	Q₈	1	4	yes	complemented
32	Dic₃₈₂	—	4	4	no	—
33	C₂×Dic₁₉₁	—	2	4	no	—
34	C₁₉₁⋊C₈	—	4	2	no	—
35	C₂³×C₁₉₁	C₂×D₄	1	1	yes	characteristic
36	C₂×C₄×C₁₉₁	—	2	8	no	—
37	C₂×C₄×C₁₉₁	—	2	4	no	—
38	C₂×C₄×C₁₉₁	—	2	4	no	—
39	C₂×C₄×C₁₉₁	C₂²⋊C₄	1	2	yes	—
40	C₂×C₄×C₁₉₁	C₂²⋊C₄	1	2	yes	—
41	C₂×C₄×C₁₉₁	C₂²×C₄	1	1	yes	characteristic
42	C₂×C₄×C₁₉₁	C₂×D₄	1	1	yes	characteristic
43	C₂×C₄×C₁₉₁	C₂×Q₈	1	2	yes	—
44	C₂×C₄×C₁₉₁	#41	1	2	yes	—
45	C₂²×C₁₉₁	—	2	2	no	—
46	C₂²×C₁₉₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
47	C₂²×C₁₉₁	C₂²⋊Q₈	1	2	yes	—
48	C₄×C₁₉₁	—	2	8	no	—
49	C₄×C₁₉₁	—	2	2	no	—
50	C₄×C₁₉₁	#104	1	4	yes	—
51	C₄×C₁₉₁	C₄×Q₈	1	2	yes	—
52	C₄×C₁₉₁	#124	1	1	yes	characteristic
53	C₄×C₁₉₁	C₄⋊Q₈	1	1	yes	characteristic
54	Dic₁₉₁	—	4	4	no	—
55	C₂×C₁₉₁	—	2	2	no	—
56	C₂×C₁₉₁	#391	1	1	yes	characteristic
57	C₂×C₁₉₁	C₂×#104	1	2	yes	—
58	C₁₉₁	#1994	1	1	yes	characteristic, complemented, Sylow
59	#1994	—	191	1	no	maximal, Sylow
60	#533	—	191	1	no	—
61	#423	—	191	1	no	—
62	C₄³	D₁₉₁	1	1	yes	characteristic
63	#339	—	191	4	no	—
64	C₂×M₁₆	—	191	2	no	—
65	C₄⋊Q₈	—	191	2	no	—
66	#125	—	191	2	no	—
67	C₂²⋊Q₈	—	382	2	no	—
68	C₄×Q₈	—	382	2	no	—
69	#117	—	191	2	no	—
70	C₂×C₄²	—	2	2	no	—
71	C₂×C₄²	C₂×D₁₉₁	1	2	yes	—
72	C₂×C₄²	C₂×D₁₉₁	1	1	yes	characteristic
73	C₄⋊C₈	—	382	2	no	—
74	C₂×Q₈	—	382	2	no	—
75	C₂²×C₄	—	2	2	no	—
76	C₂²×C₄	C₂²⋊D₁₉₁	1	2	yes	—
77	C₂²×C₄	C₂²×D₁₉₁	1	1	yes	characteristic
78	M₁₆	—	382	4	no	—
79	C₂×C₈	—	382	2	no	—
80	C₄⋊C₄	—	382	4	no	—
81	C₄⋊C₄	—	382	2	no	—
82	C₄⋊C₄	—	191	4	no	—
83	C₂²⋊C₄	—	382	2	no	—
84	C₄²	—	2	8	no	—
85	C₄²	—	2	2	no	—
86	C₄²	—	382	2	no	—
87	C₄²	C₄×D₁₉₁	1	1	yes	characteristic
88	C₄²	C₄×D₁₉₁	1	1	yes	characteristic
89	C₄²	D₇₆₄	1	2	yes	—
90	C₄²	Dic₃₈₂	1	4	yes	complemented
91	C₂³	C₂×C₂²⋊D₁₉₁	1	1	yes	characteristic
92	Q₈	—	764	4	no	—
93	C₂×C₄	—	2	8	no	—
94	C₂×C₄	—	2	4	no	—
95	C₂×C₄	—	2	4	no	—
96	C₂×C₄	—	382	4	no	—
97	C₂×C₄	(C₂×D₁₉₁)⋊C₄	1	2	yes	—
98	C₂×C₄	(C₂×D₁₉₁)⋊C₄	1	2	yes	—
99	C₂×C₄	C₂×C₄×D₁₉₁	1	1	yes	characteristic
100	C₂×C₄	C₂×D₇₆₄	1	1	yes	characteristic
101	C₂×C₄	C₂×Dic₃₈₂	1	2	yes	—
102	C₂×C₄	#15690287	1	2	yes	—
103	C₈	—	764	2	no	—
104	C₂²	—	2	2	no	—
105	C₂²	C₂×(C₂×D₁₉₁)⋊C₄	1	1	yes	characteristic
106	C₂²	#20259301	1	2	yes	—
107	C₄	—	2	8	no	—
108	C₄	—	2	2	no	—
109	C₄	—	764	4	no	—
110	C₄	#20259250	1	4	yes	—
111	C₄	C₄×Dic₃₈₂	1	2	yes	—
112	C₄	#20259310	1	1	yes	characteristic
113	C₄	#20259322	1	1	yes	characteristic
114	C₂	—	2	2	no	—
115	C₂	#27132569	1	1	yes	characteristic
116	C₂	C₂×#20259250	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#27133220

C₂

yes

maximal, characteristic

#27132687

C₂

yes

maximal, characteristic

#27132423

C₂

yes

maximal

C₄³×C₁₉₁

C₂

yes

maximal, characteristic

C₂×#15690271

C₂²

yes

—

#20259322

C₄

yes

complemented

#20259314

C₄

yes

complemented

#20259301

—

C₄×Dic₃₈₂

—

#20259281

C₂²

yes

—

C₄⋊(C₁₉₁⋊C₈)

—

C₂×C₄²×C₁₉₁

—

C₂×C₄²×C₁₉₁

C₂²

yes

—

C₂×C₄²×C₁₉₁

C₂²

yes

characteristic

C₂×Dic₃₈₂

—

#15690271

—

C₂×C₁₉₁⋊C₈

—

C₄⋊Dic₁₉₁

—

C₄⋊Dic₁₉₁

C₂×C₄

yes

—

Dic₁₉₁⋊C₄

—

C₂²⋊Dic₁₉₁

—

C₄×Dic₁₉₁

—

C₂²×C₄×C₁₉₁

—

C₂²×C₄×C₁₉₁

D₄

yes

—

C₂²×C₄×C₁₉₁

C₂³

yes

characteristic

C₄²×C₁₉₁

—

C₄²×C₁₉₁

—

C₄²×C₁₉₁

C₂×C₄

yes

characteristic

C₄²×C₁₉₁

C₂×C₄

yes

characteristic

C₄²×C₁₉₁

D₄

yes

—

C₄²×C₁₉₁

Q₈

yes

complemented

Dic₃₈₂

—

C₂×Dic₁₉₁

—

C₁₉₁⋊C₈

—

C₂³×C₁₉₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₉₁

—

C₂×C₄×C₁₉₁

—

C₂×C₄×C₁₉₁

—

C₂×C₄×C₁₉₁

C₂²⋊C₄

yes

—

C₂×C₄×C₁₉₁

C₂²⋊C₄

yes

—

C₂×C₄×C₁₉₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₉₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₉₁

C₂×Q₈

yes

—

C₂×C₄×C₁₉₁

#41

yes

—

C₂²×C₁₉₁

—

C₂²×C₁₉₁

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₁₉₁

C₂²⋊Q₈

yes

—

C₄×C₁₉₁

—

C₄×C₁₉₁

—

C₄×C₁₉₁

#104

yes

—

C₄×C₁₉₁

C₄×Q₈

yes

—

C₄×C₁₉₁

#124

yes

characteristic

C₄×C₁₉₁

C₄⋊Q₈

yes

characteristic

Dic₁₉₁

—

C₂×C₁₉₁

—

C₂×C₁₉₁

#391

yes

characteristic

C₂×C₁₉₁

C₂×#104

yes

—

C₁₉₁

#1994

yes

characteristic, complemented, Sylow

#1994

—

191

maximal, Sylow

#533

—

191

—

#423

—

191

—

C₄³

D₁₉₁

yes

characteristic

#339

—

191

—

C₂×M₁₆

—

191

—

C₄⋊Q₈

—

191

—

#125

—

191

—

C₂²⋊Q₈

—

382

—

C₄×Q₈

—

382

—

#117

—

191

—

C₂×C₄²

—

C₂×C₄²

C₂×D₁₉₁

yes

—

C₂×C₄²

C₂×D₁₉₁

yes

characteristic

C₄⋊C₈

—

382

—

C₂×Q₈

—

382

—

C₂²×C₄

—

C₂²×C₄

C₂²⋊D₁₉₁

yes

—

C₂²×C₄

C₂²×D₁₉₁

yes

characteristic

M₁₆

—

382

—

C₂×C₈

—

382

—

C₄⋊C₄

—

382

—

C₄⋊C₄

—

382

—

C₄⋊C₄

—

191

—

C₂²⋊C₄

—

382

—

C₄²

—

C₄²

—

C₄²

—

382

—

C₄²

C₄×D₁₉₁

yes

characteristic

C₄²

C₄×D₁₉₁

yes

characteristic

C₄²

D₇₆₄

yes

—

C₄²

Dic₃₈₂

yes

complemented

C₂³

C₂×C₂²⋊D₁₉₁

yes

characteristic

Q₈

—

764

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

382

—

C₂×C₄

(C₂×D₁₉₁)⋊C₄

yes

—

C₂×C₄

(C₂×D₁₉₁)⋊C₄

yes

—

C₂×C₄

C₂×C₄×D₁₉₁

yes

characteristic

100

C₂×C₄

C₂×D₇₆₄

yes

characteristic

101

C₂×C₄

C₂×Dic₃₈₂

yes

—

102

C₂×C₄

#15690287

yes

—

103

C₈

—

764

—

104

C₂²

—

105

C₂²

C₂×(C₂×D₁₉₁)⋊C₄

yes

characteristic

106

C₂²

#20259301

yes

—

107

C₄

—

108

C₄

—

109

C₄

—

764

—

110

C₄

#20259250

yes

—

111

C₄

C₄×Dic₃₈₂

yes

—

112

C₄

#20259310

yes

characteristic

113

C₄

#20259322

yes

characteristic

114

C₂

—

115

C₂

#27132569

yes

characteristic

116

C₂

C₂×#20259250

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×D₇₆₄
commutator	C₂×C₄×C₁₉₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₉₁
Fitting	C₄³×C₁₉₁	C₂
socle	C₂²×C₁₉₁	C₂×C₂²⋊C₄
prime radical	C₂³×C₁₉₁	C₂×D₄
cosocle	C₂²×C₄×C₁₉₁	C₂³
hypercenter	C₄³	D₁₉₁
hypocenter	C₁₉₁	#1994
LCS	G ▷ C₂×C₄×C₁₉₁ ▷ C₂×C₁₉₁ ▷ C₁₉₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄³

subgroup

G/H

center

C₂×C₄

C₂×D₇₆₄

commutator

C₂×C₄×C₁₉₁

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₉₁

Fitting

C₄³×C₁₉₁

C₂

socle

C₂²×C₁₉₁

C₂×C₂²⋊C₄

prime radical

C₂³×C₁₉₁

C₂×D₄

cosocle

C₂²×C₄×C₁₉₁

C₂³

hypercenter

C₄³

D₁₉₁

hypocenter

C₁₉₁

#1994

LCS

G ▷ C₂×C₄×C₁₉₁ ▷ C₂×C₁₉₁ ▷ C₁₉₁

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄³

N	H ≅ G/N	type
6. #20259322	C₄	semidirect
7. #20259314	C₄	semidirect
31. C₄²×C₁₉₁	Q₈	semidirect
58. C₁₉₁	#1994	semidirect
90. C₄²	Dic₃₈₂	semidirect

H ≅ G/N

type

6. #20259322

C₄

semidirect

7. #20259314

C₄

semidirect

31. C₄²×C₁₉₁

Q₈

semidirect

58. C₁₉₁

#1994

semidirect

90. C₄²

Dic₃₈₂

semidirect

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

subgroup	index
#27133220	2
#27132687	2
#27132423	2
C₄³×C₁₉₁	2
#1994	191

subgroup

index

C₄³×C₁₉₁

191

Subgroups — #37573497

Conjugacy classes of subgroups

Complements

Maximal subgroups