Group #37572655 — subgroups

Subgroups — #37572655

Order 24448.

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

normal subgroups only

This group has 224 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	C₂×#20259233	C₂	1	1	yes	maximal, characteristic
2	#27132529	C₂	1	1	yes	maximal, characteristic
3	#27132425	C₂	1	4	yes	maximal
4	#384×C₁₉₁	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₁₉₁⋊C₈	C₂²	1	2	yes	—
6	C₂×C₄⋊Dic₁₉₁	C₂²	1	2	yes	—
7	#20259233	C₄	1	4	yes	complemented
8	C₁₉₁⋊#95	—	2	2	no	—
9	C₂×C₄⋊C₄×C₁₉₁	—	2	2	no	—
10	C₂×C₄⋊C₄×C₁₉₁	C₂²	1	2	yes	—
11	C₂×C₄²×C₁₉₁	C₂²	1	1	yes	characteristic
12	C₂²×Dic₁₉₁	—	2	2	no	—
13	C₂×C₁₉₁⋊C₈	—	2	2	no	—
14	C₂×C₁₉₁⋊C₈	—	2	2	no	—
15	C₂×C₁₉₁⋊C₈	C₂×C₄	1	4	yes	—
16	C₄⋊Dic₁₉₁	—	2	4	no	—
17	C₂²×C₄×C₁₉₁	—	2	2	no	—
18	C₂²×C₄×C₁₉₁	D₄	1	2	yes	—
19	C₂²×C₄×C₁₉₁	C₂³	1	1	yes	characteristic
20	C₄⋊C₄×C₁₉₁	—	2	4	no	—
21	C₄⋊C₄×C₁₉₁	—	4	4	no	—
22	C₄²×C₁₉₁	C₂×C₄	1	2	yes	—
23	C₄²×C₁₉₁	Q₈	1	2	yes	—
24	C₂×Dic₁₉₁	—	4	2	no	—
25	C₂×Dic₁₉₁	—	4	2	no	—
26	C₂×Dic₁₉₁	—	4	2	no	—
27	C₁₉₁⋊C₈	—	2	4	no	—
28	C₂³×C₁₉₁	C₂×D₄	1	1	yes	characteristic
29	C₂×C₄×C₁₉₁	—	2	2	no	—
30	C₂×C₄×C₁₉₁	—	2	2	no	—
31	C₂×C₄×C₁₉₁	—	4	2	no	—
32	C₂×C₄×C₁₉₁	—	4	4	no	—
33	C₂×C₄×C₁₉₁	C₄⋊C₄	1	4	yes	—
34	C₂×C₄×C₁₉₁	C₂²×C₄	1	1	yes	characteristic
35	C₂×C₄×C₁₉₁	C₂×Q₈	1	1	yes	characteristic
36	C₂×C₄×C₁₉₁	#41	1	2	yes	—
37	C₂×C₄×C₁₉₁	#41	1	2	yes	—
38	C₂²×C₁₉₁	C₂×C₄⋊C₄	1	1	yes	characteristic
39	C₂²×C₁₉₁	#124	1	1	yes	characteristic
40	C₂²×C₁₉₁	#124	1	1	yes	characteristic
41	C₂²×C₁₉₁	AGL(1,ℤ/8)	1	2	yes	—
42	C₂²×C₁₉₁	#137	1	2	yes	—
43	C₄×C₁₉₁	—	2	4	no	—
44	C₄×C₁₉₁	—	8	2	no	—
45	C₄×C₁₉₁	#117	1	2	yes	—
46	C₄×C₁₉₁	#125	1	2	yes	—
47	Dic₁₉₁	—	8	2	no	—
48	C₂×C₁₉₁	#382	1	1	yes	characteristic
49	C₂×C₁₉₁	#428	1	2	yes	—
50	C₂×C₁₉₁	#489	1	2	yes	—
51	C₂×C₁₉₁	#490	1	1	yes	characteristic
52	C₂×C₁₉₁	#491	1	1	yes	characteristic
53	C₁₉₁	#1836	1	1	yes	characteristic, complemented, Sylow
54	#1836	—	191	1	no	maximal, Sylow
55	C₂×#97	—	191	1	no	—
56	#384	D₁₉₁	1	1	yes	characteristic
57	#382	—	191	1	no	—
58	#340	—	191	4	no	—
59	C₂²×C₈	—	191	2	no	—
60	C₂×C₄⋊C₄	—	2	2	no	—
61	C₂×C₄⋊C₄	—	191	2	no	—
62	C₂×C₄⋊C₄	C₂×D₁₉₁	1	2	yes	—
63	C₂×C₄²	C₂×D₁₉₁	1	1	yes	characteristic
64	#97	—	191	4	no	—
65	#95	—	382	2	no	—
66	C₂²×C₄	—	2	2	no	—
67	C₂²×C₄	—	382	2	no	—
68	C₂²×C₄	C₂²⋊D₁₉₁	1	2	yes	—
69	C₂²×C₄	C₂²×D₁₉₁	1	1	yes	characteristic
70	C₂×C₈	—	382	2	no	—
71	C₂×C₈	—	382	2	no	—
72	C₂×C₈	—	191	4	no	—
73	C₄⋊C₄	—	2	4	no	—
74	C₄⋊C₄	—	4	4	no	—
75	C₄⋊C₄	—	382	4	no	—
76	C₄²	C₄×D₁₉₁	1	2	yes	—
77	C₄²	Dic₃₈₂	1	2	yes	—
78	C₂³	C₂×C₂²⋊D₁₉₁	1	1	yes	characteristic
79	C₂×C₄	—	2	2	no	—
80	C₂×C₄	—	2	2	no	—
81	C₂×C₄	—	4	4	no	—
82	C₂×C₄	—	4	2	no	—
83	C₂×C₄	—	764	2	no	—
84	C₂×C₄	—	764	2	no	—
85	C₂×C₄	—	764	2	no	—
86	C₂×C₄	Dic₁₉₁⋊C₄	1	4	yes	—
87	C₂×C₄	C₂×C₄×D₁₉₁	1	1	yes	characteristic
88	C₂×C₄	C₂×Dic₃₈₂	1	1	yes	characteristic
89	C₂×C₄	#15690286	1	2	yes	—
90	C₂×C₄	#15690288	1	2	yes	—
91	C₈	—	382	4	no	—
92	C₂²	C₂×Dic₁₉₁⋊C₄	1	1	yes	characteristic
93	C₂²	#20259309	1	1	yes	characteristic
94	C₂²	#20259311	1	1	yes	characteristic
95	C₂²	#20259348	1	2	yes	—
96	C₂²	#20259353	1	2	yes	—
97	C₄	—	2	4	no	—
98	C₄	—	8	2	no	—
99	C₄	—	1528	2	no	—
100	C₄	#20259280	1	2	yes	—
101	C₄	#20259313	1	2	yes	—
102	C₂	#27132528	1	1	yes	characteristic
103	C₂	#27132703	1	2	yes	—
104	C₂	#27133025	1	2	yes	—
105	C₂	#27133027	1	1	yes	characteristic
106	C₂	#27133031	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#20259233

C₂

yes

maximal, characteristic

#27132529

C₂

yes

maximal, characteristic

#27132425

C₂

yes

maximal

#384×C₁₉₁

C₂

yes

maximal, characteristic

C₂²×C₁₉₁⋊C₈

C₂²

yes

—

C₂×C₄⋊Dic₁₉₁

C₂²

yes

—

#20259233

C₄

yes

complemented

C₁₉₁⋊#95

—

C₂×C₄⋊C₄×C₁₉₁

—

C₂×C₄⋊C₄×C₁₉₁

C₂²

yes

—

C₂×C₄²×C₁₉₁

C₂²

yes

characteristic

C₂²×Dic₁₉₁

—

C₂×C₁₉₁⋊C₈

—

C₂×C₁₉₁⋊C₈

—

C₂×C₁₉₁⋊C₈

C₂×C₄

yes

—

C₄⋊Dic₁₉₁

—

C₂²×C₄×C₁₉₁

—

C₂²×C₄×C₁₉₁

D₄

yes

—

C₂²×C₄×C₁₉₁

C₂³

yes

characteristic

C₄⋊C₄×C₁₉₁

—

C₄⋊C₄×C₁₉₁

—

C₄²×C₁₉₁

C₂×C₄

yes

—

C₄²×C₁₉₁

Q₈

yes

—

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

C₄⋊C₄

yes

—

C₂×C₄×C₁₉₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₉₁

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₉₁

#41

yes

—

C₂×C₄×C₁₉₁

#41

yes

—

C₂²×C₁₉₁

C₂×C₄⋊C₄

yes

C₂²×C₁₉₁

yes

C₂²×C₁₉₁

yes

C₂²×C₁₉₁

yes

—

C₂²×C₁₉₁

#137

yes

—

C₄×C₁₉₁

—

C₄×C₁₉₁

—

C₄×C₁₉₁

#117

yes

—

C₄×C₁₉₁

#125

yes

—

Dic₁₉₁

—

C₂×C₁₉₁

#382

yes

characteristic

C₂×C₁₉₁

#428

yes

—

C₂×C₁₉₁

#489

yes

—

C₂×C₁₉₁

yes

C₂×C₁₉₁

yes

C₁₉₁

yes

characteristic, complemented, Sylow

#1836

—

191

maximal, Sylow

C₂×#97

—

191

—

#384

D₁₉₁

yes

characteristic

#382

—

191

—

#340

—

191

—

C₂²×C₈

—

191

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

191

—

C₂×C₄⋊C₄

C₂×D₁₉₁

yes

—

C₂×C₄²

C₂×D₁₉₁

yes

characteristic

#97

—

191

—

#95

—

382

—

C₂²×C₄

—

C₂²×C₄

—

382

—

C₂²×C₄

C₂²⋊D₁₉₁

yes

—

C₂²×C₄

C₂²×D₁₉₁

yes

characteristic

C₂×C₈

—

382

—

C₂×C₈

—

382

—

C₂×C₈

—

191

—

C₄⋊C₄

—

C₄⋊C₄

—

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

—

C₂×C₄

—

764

—

C₂×C₄

—

764

—

C₂×C₄

Dic₁₉₁⋊C₄

yes

—

C₂×C₄

C₂×C₄×D₁₉₁

yes

characteristic

C₂×C₄

C₂×Dic₃₈₂

yes

characteristic

C₂×C₄

#15690286

yes

—

C₂×C₄

#15690288

yes

—

C₈

—

382

—

C₂²

C₂×Dic₁₉₁⋊C₄

yes

C₂²

yes

C₂²

yes

C₂²

yes

—

C₂²

#20259353

yes

—

C₄

—

C₄

—

C₄

—

1528

—

100

C₄

#20259280

yes

—

101

C₄

#20259313

yes

—

102

C₂

#27132528

yes

characteristic

103

C₂

#27132703

yes

—

104

C₂

#27133025

yes

—

105

C₂

#27133027

yes

characteristic

106

C₂

#27133031

yes

characteristic

	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	#1836
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
7. #20259233	C₄	semidirect
53. C₁₉₁	#1836	semidirect

H ≅ G/N

type

7. #20259233

C₄

semidirect

53. C₁₉₁

#1836

semidirect

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

subgroup	index
C₂×#20259233	2
#27132529	2
#27132425	2
#384×C₁₉₁	2
#1836	191

subgroup

index

#384×C₁₉₁

191

Subgroups — #37572655

Conjugacy classes of subgroups

Complements

Maximal subgroups