Group #37574327 — subgroups

Subgroups — #37574327

Order 24448.

138 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 138 conjugacy classes of subgroups — too many to draw. Only its 85 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	C₄⋊(C₁₉₁⋊C₁₆)	C₂	1	2	yes	maximal, complemented
2	C₂²⋊(C₁₉₁⋊C₁₆)	C₂	1	2	yes	maximal
3	#27132439	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄×C₁₉₁⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
5	#405×C₁₉₁	C₂	1	1	yes	maximal, characteristic
6	C₂×C₁₉₁⋊C₁₆	C₂²	1	2	yes	—
7	C₂×C₁₉₁⋊C₁₆	C₂²	1	2	yes	—
8	C₂×M₁₆×C₁₉₁	C₄	1	1	yes	characteristic
9	C₂²×C₈×C₁₉₁	C₂²	1	1	yes	characteristic
10	#117×C₁₉₁	C₄	1	1	yes	characteristic
11	#97×C₁₉₁	C₄	1	1	yes	characteristic
12	#97×C₁₉₁	C₄	1	1	yes	characteristic
13	C₄×C₈×C₁₉₁	C₂²	1	1	yes	characteristic
14	C₄×C₈×C₁₉₁	C₂²	1	1	yes	characteristic
15	C₁₉₁⋊C₁₆	—	2	2	no	—
16	C₁₉₁⋊C₁₆	—	2	2	no	—
17	C₂²×C₄×C₁₉₁	C₂×C₄	1	1	yes	characteristic
18	M₁₆×C₁₉₁	—	2	2	no	—
19	C₂×C₈×C₁₉₁	—	2	2	no	—
20	C₂×C₈×C₁₉₁	C₂×C₄	1	1	yes	characteristic
21	C₂×C₈×C₁₉₁	C₂×C₄	1	1	yes	characteristic
22	C₂×C₈×C₁₉₁	C₂×C₄	1	1	yes	characteristic
23	C₂×C₈×C₁₉₁	C₂³	1	1	yes	characteristic
24	C₄⋊C₄×C₁₉₁	C₈	1	2	yes	—
25	C₂²⋊C₄×C₁₉₁	C₈	1	2	yes	—
26	C₄²×C₁₉₁	C₂×C₄	1	1	yes	characteristic
27	C₄²×C₁₉₁	C₂×C₄	1	1	yes	characteristic
28	C₂³×C₁₉₁	C₂×C₈	1	1	yes	characteristic
29	C₂×C₄×C₁₉₁	—	2	1	no	—
30	C₂×C₄×C₁₉₁	—	2	1	no	—
31	C₂×C₄×C₁₉₁	C₂×C₈	1	2	yes	—
32	C₂×C₄×C₁₉₁	C₂×C₈	1	2	yes	—
33	C₂×C₄×C₁₉₁	C₂×C₈	1	1	yes	characteristic
34	C₂×C₄×C₁₉₁	C₂²×C₄	1	1	yes	characteristic
35	C₈×C₁₉₁	—	2	1	no	—
36	C₈×C₁₉₁	—	2	1	no	—
37	C₈×C₁₉₁	M₁₆	1	2	yes	—
38	C₈×C₁₉₁	#41	1	2	yes	—
39	C₂²×C₁₉₁	—	2	1	no	—
40	C₂²×C₁₉₁	—	2	2	no	—
41	C₂²×C₁₉₁	C₂²×C₈	1	1	yes	characteristic
42	C₄×C₁₉₁	—	2	1	no	—
43	C₄×C₁₉₁	—	2	2	no	—
44	C₄×C₁₉₁	—	2	2	no	—
45	C₄×C₁₉₁	#117	1	1	yes	characteristic
46	C₄×C₁₉₁	C₂×M₁₆	1	1	yes	characteristic
47	C₂×C₁₉₁	—	4	1	no	—
48	C₂×C₁₉₁	#431	1	1	yes	characteristic
49	C₂×C₁₉₁	#504	1	2	yes	—
50	C₁₉₁	#2164	1	1	yes	characteristic, complemented, Sylow
51	#2164	—	191	1	no	maximal, Sylow
52	#405	D₁₉₁	1	1	yes	characteristic
53	C₄⋊C₁₆	—	191	2	no	—
54	C₂²⋊C₁₆	—	191	2	no	—
55	#346	—	191	1	no	—
56	C₄×C₁₆	—	191	1	no	—
57	C₂×M₁₆	Dic₁₉₁	1	1	yes	characteristic
58	C₂²×C₈	C₂×D₁₉₁	1	1	yes	characteristic
59	#117	Dic₁₉₁	1	1	yes	characteristic
60	C₂×C₁₆	—	191	2	no	—
61	C₂×C₁₆	—	191	2	no	—
62	#97	Dic₁₉₁	1	1	yes	characteristic
63	#97	Dic₁₉₁	1	1	yes	characteristic
64	C₄×C₈	C₂×D₁₉₁	1	1	yes	characteristic
65	C₄×C₈	C₂×D₁₉₁	1	1	yes	characteristic
66	C₂²×C₄	C₂×Dic₁₉₁	1	1	yes	characteristic
67	M₁₆	—	2	2	no	—
68	C₂×C₈	—	2	2	no	—
69	C₂×C₈	C₂×Dic₁₉₁	1	1	yes	characteristic
70	C₂×C₈	C₂×Dic₁₉₁	1	1	yes	characteristic
71	C₂×C₈	C₂×Dic₁₉₁	1	1	yes	characteristic
72	C₂×C₈	C₂²×D₁₉₁	1	1	yes	characteristic
73	C₄⋊C₄	C₁₉₁⋊C₈	1	2	yes	—
74	C₂²⋊C₄	C₁₉₁⋊C₈	1	2	yes	—
75	C₄²	C₂×Dic₁₉₁	1	1	yes	characteristic
76	C₄²	C₂×Dic₁₉₁	1	1	yes	characteristic
77	C₁₆	—	382	2	no	—
78	C₁₆	—	382	2	no	—
79	C₂³	C₂×C₁₉₁⋊C₈	1	1	yes	characteristic
80	C₂×C₄	—	2	1	no	—
81	C₂×C₄	—	2	1	no	—
82	C₂×C₄	C₂×C₁₉₁⋊C₈	1	2	yes	—
83	C₂×C₄	C₂×C₁₉₁⋊C₈	1	2	yes	—
84	C₂×C₄	C₂×C₁₉₁⋊C₈	1	1	yes	characteristic
85	C₂×C₄	C₂²×Dic₁₉₁	1	1	yes	characteristic
86	C₈	—	2	1	no	—
87	C₈	—	2	1	no	—
88	C₈	#15690271	1	2	yes	—
89	C₈	#15690287	1	2	yes	—
90	C₂²	—	2	1	no	—
91	C₂²	—	2	2	no	—
92	C₂²	C₂²×C₁₉₁⋊C₈	1	1	yes	characteristic
93	C₄	—	2	1	no	—
94	C₄	—	2	2	no	—
95	C₄	—	2	2	no	—
96	C₄	#20259281	1	1	yes	characteristic
97	C₄	C₂×#15690271	1	1	yes	characteristic
98	C₂	—	4	1	no	—
99	C₂	#27132716	1	1	yes	characteristic
100	C₂	#27133080	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₄⋊(C₁₉₁⋊C₁₆)

C₂

yes

maximal, complemented

C₂²⋊(C₁₉₁⋊C₁₆)

C₂

yes

maximal

#27132439

C₂

yes

maximal, characteristic, complemented

C₄×C₁₉₁⋊C₁₆

C₂

yes

maximal, characteristic, complemented

#405×C₁₉₁

C₂

yes

maximal, characteristic

C₂×C₁₉₁⋊C₁₆

C₂²

yes

—

C₂×C₁₉₁⋊C₁₆

C₂²

yes

—

C₂×M₁₆×C₁₉₁

C₄

yes

characteristic

C₂²×C₈×C₁₉₁

C₂²

yes

characteristic

#117×C₁₉₁

C₄

yes

characteristic

#97×C₁₉₁

C₄

yes

characteristic

#97×C₁₉₁

C₄

yes

characteristic

C₄×C₈×C₁₉₁

C₂²

yes

characteristic

C₄×C₈×C₁₉₁

C₂²

yes

characteristic

C₁₉₁⋊C₁₆

—

C₁₉₁⋊C₁₆

—

C₂²×C₄×C₁₉₁

C₂×C₄

yes

characteristic

M₁₆×C₁₉₁

—

C₂×C₈×C₁₉₁

—

C₂×C₈×C₁₉₁

C₂×C₄

yes

characteristic

C₂×C₈×C₁₉₁

C₂×C₄

yes

characteristic

C₂×C₈×C₁₉₁

C₂×C₄

yes

characteristic

C₂×C₈×C₁₉₁

C₂³

yes

characteristic

C₄⋊C₄×C₁₉₁

C₈

yes

—

C₂²⋊C₄×C₁₉₁

C₈

yes

—

C₄²×C₁₉₁

C₂×C₄

yes

characteristic

C₄²×C₁₉₁

C₂×C₄

yes

characteristic

C₂³×C₁₉₁

C₂×C₈

yes

characteristic

C₂×C₄×C₁₉₁

—

C₂×C₄×C₁₉₁

—

C₂×C₄×C₁₉₁

C₂×C₈

yes

—

C₂×C₄×C₁₉₁

C₂×C₈

yes

—

C₂×C₄×C₁₉₁

C₂×C₈

yes

characteristic

C₂×C₄×C₁₉₁

C₂²×C₄

yes

characteristic

C₈×C₁₉₁

—

C₈×C₁₉₁

—

C₈×C₁₉₁

M₁₆

yes

—

C₈×C₁₉₁

#41

yes

—

C₂²×C₁₉₁

—

C₂²×C₁₉₁

—

C₂²×C₁₉₁

C₂²×C₈

yes

characteristic

C₄×C₁₉₁

—

C₄×C₁₉₁

—

C₄×C₁₉₁

—

C₄×C₁₉₁

#117

yes

characteristic

C₄×C₁₉₁

C₂×M₁₆

yes

characteristic

C₂×C₁₉₁

—

C₂×C₁₉₁

#431

yes

characteristic

C₂×C₁₉₁

#504

yes

—

C₁₉₁

#2164

yes

characteristic, complemented, Sylow

#2164

—

191

maximal, Sylow

#405

D₁₉₁

yes

characteristic

C₄⋊C₁₆

—

191

—

C₂²⋊C₁₆

—

191

—

#346

—

191

—

C₄×C₁₆

—

191

—

C₂×M₁₆

Dic₁₉₁

yes

characteristic

C₂²×C₈

C₂×D₁₉₁

yes

characteristic

#117

Dic₁₉₁

yes

characteristic

C₂×C₁₆

—

191

—

C₂×C₁₆

—

191

—

#97

Dic₁₉₁

yes

characteristic

#97

Dic₁₉₁

yes

characteristic

C₄×C₈

C₂×D₁₉₁

yes

characteristic

C₄×C₈

C₂×D₁₉₁

yes

characteristic

C₂²×C₄

C₂×Dic₁₉₁

yes

characteristic

M₁₆

—

C₂×C₈

—

C₂×C₈

C₂×Dic₁₉₁

yes

characteristic

C₂×C₈

C₂×Dic₁₉₁

yes

characteristic

C₂×C₈

C₂×Dic₁₉₁

yes

characteristic

C₂×C₈

C₂²×D₁₉₁

yes

characteristic

C₄⋊C₄

C₁₉₁⋊C₈

yes

—

C₂²⋊C₄

C₁₉₁⋊C₈

yes

—

C₄²

C₂×Dic₁₉₁

yes

characteristic

C₄²

C₂×Dic₁₉₁

yes

characteristic

C₁₆

—

382

—

C₁₆

—

382

—

C₂³

C₂×C₁₉₁⋊C₈

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂×C₁₉₁⋊C₈

yes

—

C₂×C₄

C₂×C₁₉₁⋊C₈

yes

—

C₂×C₄

C₂×C₁₉₁⋊C₈

yes

characteristic

C₂×C₄

C₂²×Dic₁₉₁

yes

characteristic

C₈

—

C₈

—

C₈

#15690271

yes

—

C₈

#15690287

yes

—

C₂²

—

C₂²

—

C₂²

C₂²×C₁₉₁⋊C₈

yes

characteristic

C₄

—

C₄

—

C₄

—

C₄

yes

C₄

yes

C₂

—

C₂

#27132716

yes

characteristic

100

C₂

#27133080

yes

—

	subgroup	G/H
center, Frattini	C₂×C₈	C₂²×D₁₉₁
commutator, socle	C₂²×C₁₉₁	C₂²×C₈
Fitting	#405×C₁₉₁	C₂
prime radical	C₂³×C₁₉₁	C₂×C₈
cosocle	C₂×C₈×C₁₉₁	C₂³
hypercenter	#405	D₁₉₁
hypocenter	C₁₉₁	#2164
LCS	G ▷ C₂²×C₁₉₁ ▷ C₁₉₁
UCS	C₁ ◁ C₂×C₈ ◁ #405

subgroup

G/H

center, Frattini

C₂×C₈

C₂²×D₁₉₁

commutator, socle

C₂²×C₁₉₁

C₂²×C₈

Fitting

#405×C₁₉₁

C₂

prime radical

C₂³×C₁₉₁

C₂×C₈

cosocle

C₂×C₈×C₁₉₁

C₂³

D₁₉₁

C₁₉₁

G ▷ C₂²×C₁₉₁ ▷ C₁₉₁

C₁ ◁ C₂×C₈ ◁ #405

N	H ≅ G/N	type
1. C₄⋊(C₁₉₁⋊C₁₆)	C₂	semidirect
3. #27132439	C₂	semidirect
4. C₄×C₁₉₁⋊C₁₆	C₂	semidirect
50. C₁₉₁	#2164	semidirect

H ≅ G/N

type

1. C₄⋊(C₁₉₁⋊C₁₆)

C₂

semidirect

3. #27132439

C₂

semidirect

4. C₄×C₁₉₁⋊C₁₆

C₂

semidirect

50. C₁₉₁

#2164

semidirect

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

subgroup	index
C₄⋊(C₁₉₁⋊C₁₆)	2
C₂²⋊(C₁₉₁⋊C₁₆)	2
#27132439	2
C₄×C₁₉₁⋊C₁₆	2
#405×C₁₉₁	2
#2164	191

subgroup

index

C₄⋊(C₁₉₁⋊C₁₆)

C₂²⋊(C₁₉₁⋊C₁₆)

#27132439

C₄×C₁₉₁⋊C₁₆

#405×C₁₉₁

#2164

191

Subgroups — #37574327

Conjugacy classes of subgroups

Complements

Maximal subgroups