Group #37570403 — subgroups

Subgroups — #37570403

Order 24448.

188 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 188 conjugacy classes of subgroups — too many to draw. Only its 105 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×C₄⋊(C₁₉₁⋊C₈)	C₂	1	2	yes	maximal
2	#384×C₁₉₁	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₁₉₁⋊C₈	—	2	2	no	—
4	C₄⋊(C₁₉₁⋊C₈)	—	2	2	no	—
5	C₄⋊(C₁₉₁⋊C₈)	C₄	1	4	yes	complemented
6	C₂×C₄⋊C₄×C₁₉₁	—	2	2	no	—
7	C₂×C₄⋊C₄×C₁₉₁	C₄	1	2	yes	—
8	C₂×C₄²×C₁₉₁	C₂²	1	1	yes	characteristic
9	C₂×C₁₉₁⋊C₈	—	2	4	no	—
10	C₂×C₁₉₁⋊C₈	—	4	2	no	—
11	C₂×C₁₉₁⋊C₈	—	4	2	no	—
12	C₂²×C₄×C₁₉₁	—	2	2	no	—
13	C₂²×C₄×C₁₉₁	C₂×C₄	1	1	yes	characteristic
14	C₂²×C₄×C₁₉₁	D₄	1	2	yes	—
15	C₄⋊C₄×C₁₉₁	—	4	2	no	—
16	C₄⋊C₄×C₁₉₁	—	4	2	no	—
17	C₄⋊C₄×C₁₉₁	—	4	2	no	—
18	C₄²×C₁₉₁	C₂×C₄	1	2	yes	—
19	C₄²×C₁₉₁	D₄	1	1	yes	characteristic
20	C₄²×C₁₉₁	Q₈	1	1	yes	characteristic
21	C₁₉₁⋊C₈	—	4	4	no	—
22	C₂³×C₁₉₁	C₂²⋊C₄	1	1	yes	characteristic
23	C₂×C₄×C₁₉₁	—	2	1	no	—
24	C₂×C₄×C₁₉₁	—	2	1	no	—
25	C₂×C₄×C₁₉₁	—	4	2	no	—
26	C₂×C₄×C₁₉₁	—	4	4	no	—
27	C₂×C₄×C₁₉₁	C₄²	1	1	yes	characteristic
28	C₂×C₄×C₁₉₁	C₂²⋊C₄	1	2	yes	—
29	C₂×C₄×C₁₉₁	C₄⋊C₄	1	2	yes	—
30	C₂×C₄×C₁₉₁	C₄⋊C₄	1	1	yes	characteristic
31	C₂×C₄×C₁₉₁	D₈	1	2	yes	—
32	C₂×C₄×C₁₉₁	SD₁₆	1	2	yes	—
33	C₂×C₄×C₁₉₁	SD₁₆	1	2	yes	—
34	C₂×C₄×C₁₉₁	Q₁₆	1	2	yes	—
35	C₂²×C₁₉₁	#95	1	1	yes	characteristic
36	C₂²×C₁₉₁	#100	1	1	yes	characteristic
37	C₂²×C₁₉₁	#101	1	1	yes	characteristic
38	C₂²×C₁₉₁	#102	1	2	yes	—
39	C₂²×C₁₉₁	#103	1	2	yes	—
40	C₄×C₁₉₁	—	2	1	no	—
41	C₄×C₁₉₁	—	2	1	no	—
42	C₄×C₁₉₁	—	8	2	no	—
43	C₄×C₁₉₁	#102	1	2	yes	—
44	C₄×C₁₉₁	#103	1	2	yes	—
45	C₄×C₁₉₁	#106	1	2	yes	—
46	C₄×C₁₉₁	#107	1	2	yes	—
47	C₂×C₁₉₁	#331	1	1	yes	characteristic
48	C₂×C₁₉₁	#332	1	2	yes	—
49	C₂×C₁₉₁	#333	1	1	yes	characteristic
50	C₂×C₁₉₁	#340	1	2	yes	—
51	C₂×C₁₉₁	#343	1	1	yes	characteristic
52	C₁₉₁	#1296	1	1	yes	characteristic, complemented, Sylow
53	#1296	—	191	1	no	maximal, Sylow
54	C₂×C₄⋊C₈	—	191	2	no	—
55	#384	D₁₉₁	1	1	yes	characteristic
56	C₂²×C₈	—	382	2	no	—
57	C₂×C₄⋊C₄	—	2	2	no	—
58	C₂×C₄⋊C₄	Dic₁₉₁	1	2	yes	—
59	C₂×C₄²	C₂×D₁₉₁	1	1	yes	characteristic
60	C₄⋊C₈	—	382	2	no	—
61	C₄⋊C₈	—	191	4	no	—
62	C₂²×C₄	—	2	2	no	—
63	C₂²×C₄	C₂×Dic₁₉₁	1	1	yes	characteristic
64	C₂²×C₄	C₂²⋊D₁₉₁	1	2	yes	—
65	C₂×C₈	—	382	4	no	—
66	C₂×C₈	—	764	2	no	—
67	C₂×C₈	—	764	2	no	—
68	C₄⋊C₄	—	4	2	no	—
69	C₄⋊C₄	—	4	2	no	—
70	C₄⋊C₄	—	4	2	no	—
71	C₄²	C₄×D₁₉₁	1	2	yes	—
72	C₄²	D₇₆₄	1	1	yes	characteristic
73	C₄²	Dic₃₈₂	1	1	yes	characteristic
74	C₂³	C₂²⋊Dic₁₉₁	1	1	yes	characteristic
75	C₂×C₄	—	2	1	no	—
76	C₂×C₄	—	2	1	no	—
77	C₂×C₄	—	4	2	no	—
78	C₂×C₄	—	4	4	no	—
79	C₂×C₄	C₄×Dic₁₉₁	1	1	yes	characteristic
80	C₂×C₄	(C₂×D₁₉₁)⋊C₄	1	2	yes	—
81	C₂×C₄	Dic₁₉₁⋊C₄	1	2	yes	—
82	C₂×C₄	C₄⋊Dic₁₉₁	1	1	yes	characteristic
83	C₂×C₄	#15690272	1	2	yes	—
84	C₂×C₄	#15690274	1	2	yes	—
85	C₂×C₄	#15690275	1	2	yes	—
86	C₂×C₄	#15690277	1	2	yes	—
87	C₈	—	764	4	no	—
88	C₂²	C₁₉₁⋊#95	1	1	yes	characteristic
89	C₂²	#20259239	1	1	yes	characteristic
90	C₂²	#20259241	1	1	yes	characteristic
91	C₂²	#20259243	1	2	yes	—
92	C₂²	#20259246	1	2	yes	—
93	C₄	—	2	1	no	—
94	C₄	—	2	1	no	—
95	C₄	—	8	2	no	—
96	C₄	#20259242	1	2	yes	—
97	C₄	#20259245	1	2	yes	—
98	C₄	#20259253	1	2	yes	—
99	C₄	#20259255	1	2	yes	—
100	C₂	#27132407	1	1	yes	characteristic
101	C₂	#27132409	1	2	yes	—
102	C₂	#27132412	1	1	yes	characteristic
103	C₂	#27132425	1	2	yes	—
104	C₂	#27132432	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊(C₁₉₁⋊C₈)

C₂

yes

maximal

#384×C₁₉₁

C₂

yes

maximal, characteristic

C₂²×C₁₉₁⋊C₈

—

C₄⋊(C₁₉₁⋊C₈)

—

C₄⋊(C₁₉₁⋊C₈)

C₄

yes

complemented

C₂×C₄⋊C₄×C₁₉₁

—

C₂×C₄⋊C₄×C₁₉₁

C₄

yes

—

C₂×C₄²×C₁₉₁

C₂²

yes

characteristic

C₂×C₁₉₁⋊C₈

—

C₂×C₁₉₁⋊C₈

—

C₂×C₁₉₁⋊C₈

—

C₂²×C₄×C₁₉₁

—

C₂²×C₄×C₁₉₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₉₁

D₄

yes

—

C₄⋊C₄×C₁₉₁

—

C₄⋊C₄×C₁₉₁

—

C₄⋊C₄×C₁₉₁

—

C₄²×C₁₉₁

C₂×C₄

yes

—

C₄²×C₁₉₁

D₄

yes

characteristic

C₄²×C₁₉₁

Q₈

yes

characteristic

C₁₉₁⋊C₈

—

C₂³×C₁₉₁

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₉₁

—

C₂×C₄×C₁₉₁

—

C₂×C₄×C₁₉₁

—

C₂×C₄×C₁₉₁

—

C₂×C₄×C₁₉₁

C₄²

yes

characteristic

C₂×C₄×C₁₉₁

C₂²⋊C₄

yes

—

C₂×C₄×C₁₉₁

C₄⋊C₄

yes

—

C₂×C₄×C₁₉₁

C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₉₁

D₈

yes

—

C₂×C₄×C₁₉₁

SD₁₆

yes

—

C₂×C₄×C₁₉₁

SD₁₆

yes

—

C₂×C₄×C₁₉₁

Q₁₆

yes

—

C₂²×C₁₉₁

yes

C₂²×C₁₉₁

yes

C₂²×C₁₉₁

yes

C₂²×C₁₉₁

yes

—

C₂²×C₁₉₁

#103

yes

—

C₄×C₁₉₁

—

C₄×C₁₉₁

—

C₄×C₁₉₁

—

C₄×C₁₉₁

#102

yes

—

C₄×C₁₉₁

#103

yes

—

C₄×C₁₉₁

#106

yes

—

C₄×C₁₉₁

#107

yes

—

C₂×C₁₉₁

#331

yes

characteristic

C₂×C₁₉₁

#332

yes

—

C₂×C₁₉₁

#333

yes

characteristic

C₂×C₁₉₁

#340

yes

—

C₂×C₁₉₁

#343

yes

characteristic

C₁₉₁

#1296

yes

characteristic, complemented, Sylow

#1296

—

191

maximal, Sylow

C₂×C₄⋊C₈

—

191

—

#384

D₁₉₁

yes

characteristic

C₂²×C₈

—

382

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

Dic₁₉₁

yes

—

C₂×C₄²

C₂×D₁₉₁

yes

characteristic

C₄⋊C₈

—

382

—

C₄⋊C₈

—

191

—

C₂²×C₄

—

C₂²×C₄

C₂×Dic₁₉₁

yes

characteristic

C₂²×C₄

C₂²⋊D₁₉₁

yes

—

C₂×C₈

—

382

—

C₂×C₈

—

764

—

C₂×C₈

—

764

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

C₄×D₁₉₁

yes

—

C₄²

D₇₆₄

yes

characteristic

C₄²

Dic₃₈₂

yes

characteristic

C₂³

C₂²⋊Dic₁₉₁

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₄×Dic₁₉₁

yes

characteristic

C₂×C₄

(C₂×D₁₉₁)⋊C₄

yes

—

C₂×C₄

Dic₁₉₁⋊C₄

yes

—

C₂×C₄

C₄⋊Dic₁₉₁

yes

characteristic

C₂×C₄

#15690272

yes

—

C₂×C₄

#15690274

yes

—

C₂×C₄

#15690275

yes

—

C₂×C₄

#15690277

yes

—

C₈

—

764

—

C₂²

C₁₉₁⋊#95

yes

C₂²

yes

C₂²

yes

C₂²

yes

—

C₂²

#20259246

yes

—

C₄

—

C₄

—

C₄

—

C₄

#20259242

yes

—

C₄

#20259245

yes

—

C₄

#20259253

yes

—

C₄

#20259255

yes

—

100

C₂

#27132407

yes

characteristic

101

C₂

#27132409

yes

—

102

C₂

#27132412

yes

characteristic

103

C₂

#27132425

yes

—

104

C₂

#27132432

yes

characteristic

	subgroup	G/H
center	C₂³	C₂²⋊Dic₁₉₁
commutator	C₂×C₄×C₁₉₁	C₄²
Frattini	C₂×C₄²	C₂×D₁₉₁
Fitting	#384×C₁₉₁	C₂
socle, prime radical	C₂³×C₁₉₁	C₂²⋊C₄
cosocle	C₂×C₄²×C₁₉₁	C₂²
hypercenter	#384	D₁₉₁
hypocenter	C₁₉₁	#1296
LCS	G ▷ C₂×C₄×C₁₉₁ ▷ C₂²×C₁₉₁ ▷ C₁₉₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #384

subgroup

G/H

center

C₂³

C₂²⋊Dic₁₉₁

commutator

C₂×C₄×C₁₉₁

C₄²

Frattini

C₂×C₄²

C₂×D₁₉₁

Fitting

#384×C₁₉₁

C₂

socle, prime radical

C₂³×C₁₉₁

C₂²⋊C₄

cosocle

C₂×C₄²×C₁₉₁

C₂²

D₁₉₁

C₁₉₁

G ▷ C₂×C₄×C₁₉₁ ▷ C₂²×C₁₉₁ ▷ C₁₉₁

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #384

N	H ≅ G/N	type
5. C₄⋊(C₁₉₁⋊C₈)	C₄	semidirect
52. C₁₉₁	#1296	semidirect

H ≅ G/N

type

5. C₄⋊(C₁₉₁⋊C₈)

C₄

semidirect

52. C₁₉₁

#1296

semidirect

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

subgroup	index
C₂×C₄⋊(C₁₉₁⋊C₈)	2
#384×C₁₉₁	2
#1296	191

subgroup

index

C₂×C₄⋊(C₁₉₁⋊C₈)

#384×C₁₉₁

#1296

191

Subgroups — #37570403

Conjugacy classes of subgroups

Complements

Maximal subgroups