Group #37572148 — subgroups

Subgroups — #37572148

Order 24448.

220 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 69 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	#27133061	C₂	1	1	yes	maximal, characteristic, complemented
2	#27133040	C₂	1	1	yes	maximal, characteristic
3	#27133036	C₂	1	2	yes	maximal, complemented
4	#27133010	C₂	1	2	yes	maximal, complemented
5	#322×C₁₉₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#15690276	—	2	4	no	—
7	C₂×D₁₅₂₈	—	2	2	no	—
8	#20259322	C₂²	1	2	yes	—
9	#20259319	C₂²	1	1	yes	characteristic
10	#20259319	C₂²	1	1	yes	characteristic
11	#20259254	—	2	2	no	—
12	#20259244	—	2	4	no	—
13	C₄×C₈×C₁₉₁	C₂²	1	2	yes	—
14	C₄×C₈×C₁₉₁	C₂²	1	1	yes	characteristic
15	C₂×Dic₃₈₂	—	2	2	no	—
16	C₂×D₇₆₄	—	2	1	no	—
17	C₂×D₇₆₄	—	2	1	no	—
18	C₂×D₇₆₄	—	2	2	no	—
19	C₂×D₇₆₄	—	2	2	no	—
20	#15690276	—	4	8	no	—
21	D₁₅₂₈	—	4	4	no	—
22	C₄⋊Dic₁₉₁	—	2	2	no	—
23	C₄⋊Dic₁₉₁	—	2	2	no	—
24	C₂×C₈×C₁₉₁	D₄	1	4	yes	—
25	C₂×C₈×C₁₉₁	D₄	1	2	yes	—
26	C₄²×C₁₉₁	C₂³	1	1	yes	characteristic
27	C₂²×D₁₉₁	—	4	1	no	—
28	C₂²×D₁₉₁	—	4	1	no	—
29	Dic₃₈₂	—	4	4	no	—
30	D₇₆₄	—	4	2	no	—
31	D₇₆₄	—	4	2	no	—
32	D₇₆₄	—	4	4	no	—
33	D₇₆₄	—	4	4	no	—
34	C₂×Dic₁₉₁	—	4	2	no	—
35	C₂×C₄×C₁₉₁	C₂×D₄	1	1	yes	characteristic
36	C₂×C₄×C₁₉₁	C₂×D₄	1	2	yes	—
37	C₈×C₁₉₁	—	2	4	no	—
38	C₈×C₁₉₁	SD₁₆	1	4	yes	complemented
39	C₂²×C₁₉₁	#127	1	1	yes	characteristic
40	C₂×D₁₉₁	—	8	1	no	—
41	C₂×D₁₉₁	—	8	1	no	—
42	C₂×D₁₉₁	—	8	2	no	—
43	C₂×D₁₉₁	—	8	2	no	—
44	C₄×C₁₉₁	C₂×SD₁₆	1	2	yes	—
45	C₄×C₁₉₁	AGL(1,ℤ/8)	1	4	yes	—
46	Dic₁₉₁	—	8	2	no	—
47	C₂×C₁₉₁	#492	1	1	yes	characteristic
48	C₂×C₁₉₁	#496	1	2	yes	—
49	D₁₉₁	—	16	1	no	—
50	D₁₉₁	—	16	1	no	—
51	C₁₉₁	#1715	1	1	yes	characteristic, complemented, Sylow
52	#1715	—	191	1	no	maximal, Sylow
53	#498	—	191	1	no	—
54	#493	—	191	1	no	—
55	#492	—	191	2	no	—
56	#486	—	191	2	no	—
57	#322	D₁₉₁	1	1	yes	characteristic, complemented
58	C₂×SD₁₆	—	382	4	no	—
59	C₂×D₈	—	382	2	no	—
60	C₄⋊Q₈	—	191	2	no	—
61	#127	—	191	1	no	—
62	#127	—	191	1	no	—
63	#106	—	382	2	no	—
64	#102	—	382	4	no	—
65	C₄×C₈	C₂×D₁₉₁	1	2	yes	—
66	C₄×C₈	C₂×D₁₉₁	1	1	yes	characteristic
67	C₂×Q₈	—	382	2	no	—
68	C₂×D₄	—	382	2	no	—
69	C₂×D₄	—	382	1	no	—
70	C₂×D₄	—	382	1	no	—
71	C₂×D₄	—	382	2	no	—
72	SD₁₆	—	764	8	no	—
73	D₈	—	764	4	no	—
74	C₂×C₈	D₇₆₄	1	4	yes	—
75	C₂×C₈	D₇₆₄	1	2	yes	—
76	C₄⋊C₄	—	382	2	no	—
77	C₄⋊C₄	—	382	2	no	—
78	C₄²	C₂²×D₁₉₁	1	1	yes	characteristic
79	C₂³	—	764	1	no	—
80	C₂³	—	764	1	no	—
81	Q₈	—	764	4	no	—
82	D₄	—	764	2	no	—
83	D₄	—	764	2	no	—
84	D₄	—	764	4	no	—
85	D₄	—	764	4	no	—
86	C₂×C₄	—	764	2	no	—
87	C₂×C₄	C₂×D₇₆₄	1	1	yes	characteristic
88	C₂×C₄	C₂×D₇₆₄	1	2	yes	—
89	C₈	—	2	4	no	—
90	C₈	#15690276	1	4	yes	complemented
91	C₂²	—	1528	1	no	—
92	C₂²	—	1528	1	no	—
93	C₂²	—	1528	2	no	—
94	C₂²	—	1528	2	no	—
95	C₂²	#20259319	1	1	yes	characteristic
96	C₄	—	1528	2	no	—
97	C₄	C₂×#15690276	1	2	yes	—
98	C₄	#20259350	1	4	yes	—
99	C₂	—	3056	1	no	—
100	C₂	—	3056	1	no	—
101	C₂	#27133036	1	1	yes	characteristic
102	C₂	#27133052	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#27133061

C₂

yes

maximal, characteristic, complemented

#27133040

C₂

yes

maximal, characteristic

#27133036

C₂

yes

maximal, complemented

#27133010

C₂

yes

maximal, complemented

#322×C₁₉₁

C₂

yes

maximal, characteristic, complemented

C₂×#15690276

—

C₂×D₁₅₂₈

—

#20259322

C₂²

yes

—

#20259319

C₂²

yes

characteristic

#20259319

C₂²

yes

characteristic

#20259254

—

#20259244

—

C₄×C₈×C₁₉₁

C₂²

yes

—

C₄×C₈×C₁₉₁

C₂²

yes

characteristic

C₂×Dic₃₈₂

—

C₂×D₇₆₄

—

C₂×D₇₆₄

—

C₂×D₇₆₄

—

C₂×D₇₆₄

—

#15690276

—

D₁₅₂₈

—

C₄⋊Dic₁₉₁

—

C₄⋊Dic₁₉₁

—

C₂×C₈×C₁₉₁

D₄

yes

—

C₂×C₈×C₁₉₁

D₄

yes

—

C₄²×C₁₉₁

C₂³

yes

characteristic

C₂²×D₁₉₁

—

C₂²×D₁₉₁

—

Dic₃₈₂

—

D₇₆₄

—

D₇₆₄

—

D₇₆₄

—

D₇₆₄

—

C₂×Dic₁₉₁

—

C₂×C₄×C₁₉₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₉₁

C₂×D₄

yes

—

C₈×C₁₉₁

—

C₈×C₁₉₁

SD₁₆

yes

complemented

C₂²×C₁₉₁

#127

yes

characteristic

C₂×D₁₉₁

—

C₂×D₁₉₁

—

C₂×D₁₉₁

—

C₂×D₁₉₁

—

C₄×C₁₉₁

C₂×SD₁₆

yes

—

C₄×C₁₉₁

AGL(1,ℤ/8)

yes

—

Dic₁₉₁

—

C₂×C₁₉₁

#492

yes

characteristic

C₂×C₁₉₁

#496

yes

—

D₁₉₁

—

D₁₉₁

—

C₁₉₁

#1715

yes

characteristic, complemented, Sylow

#1715

—

191

maximal, Sylow

#498

—

191

—

#493

—

191

—

#492

—

191

—

#486

—

191

—

#322

D₁₉₁

yes

characteristic, complemented

C₂×SD₁₆

—

382

—

C₂×D₈

—

382

—

C₄⋊Q₈

—

191

—

#127

—

191

—

#127

—

191

—

#106

—

382

—

#102

—

382

—

C₄×C₈

C₂×D₁₉₁

yes

—

C₄×C₈

C₂×D₁₉₁

yes

characteristic

C₂×Q₈

—

382

—

C₂×D₄

—

382

—

C₂×D₄

—

382

—

C₂×D₄

—

382

—

C₂×D₄

—

382

—

SD₁₆

—

764

—

D₈

—

764

—

C₂×C₈

D₇₆₄

yes

—

C₂×C₈

D₇₆₄

yes

—

C₄⋊C₄

—

382

—

C₄⋊C₄

—

382

—

C₄²

C₂²×D₁₉₁

yes

characteristic

C₂³

—

764

—

C₂³

—

764

—

Q₈

—

764

—

D₄

—

764

—

D₄

—

764

—

D₄

—

764

—

D₄

—

764

—

C₂×C₄

—

764

—

C₂×C₄

C₂×D₇₆₄

yes

characteristic

C₂×C₄

C₂×D₇₆₄

yes

—

C₈

—

C₈

#15690276

yes

complemented

C₂²

—

1528

—

C₂²

—

1528

—

C₂²

—

1528

—

C₂²

—

1528

—

C₂²

#20259319

yes

characteristic

C₄

—

1528

—

C₄

C₂×#15690276

yes

—

C₄

#20259350

yes

—

C₂

—

3056

—

100

C₂

—

3056

—

101

C₂

#27133036

yes

characteristic

102

C₂

#27133052

yes

—

	subgroup	G/H
center	C₂²	#20259319
commutator, cosocle	C₄²×C₁₉₁	C₂³
Frattini	C₄²	C₂²×D₁₉₁
Fitting	#322×C₁₉₁	C₂
socle	C₂²×C₁₉₁	#127
prime radical	#27133040	C₂
hypercenter	#322	D₁₉₁
hypocenter	C₁₉₁	#1715
LCS	G ▷ C₄²×C₁₉₁ ▷ C₂²×C₁₉₁ ▷ C₁₉₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #322

subgroup

G/H

center

C₂²

#20259319

commutator, cosocle

C₄²×C₁₉₁

C₂³

Frattini

C₄²

C₂²×D₁₉₁

Fitting

#322×C₁₉₁

C₂

C₂²×C₁₉₁

C₂

D₁₉₁

C₁₉₁

G ▷ C₄²×C₁₉₁ ▷ C₂²×C₁₉₁ ▷ C₁₉₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #322

N	H ≅ G/N	type
1. #27133061	C₂	semidirect
3. #27133036	C₂	semidirect
4. #27133010	C₂	semidirect
5. #322×C₁₉₁	C₂	semidirect
38. C₈×C₁₉₁	SD₁₆	semidirect
51. C₁₉₁	#1715	semidirect
57. #322	D₁₉₁	semidirect
90. C₈	#15690276	semidirect

H ≅ G/N

type

1. #27133061

C₂

semidirect

3. #27133036

C₂

semidirect

4. #27133010

C₂

semidirect

5. #322×C₁₉₁

C₂

semidirect

38. C₈×C₁₉₁

SD₁₆

semidirect

51. C₁₉₁

#1715

semidirect

57. #322

D₁₉₁

semidirect

90. C₈

#15690276

semidirect

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

subgroup	index
#27133061	2
#27133040	2
#27133036	2
#27133010	2
#322×C₁₉₁	2
#1715	191

subgroup

index

#322×C₁₉₁

191

Subgroups — #37572148

Conjugacy classes of subgroups

Complements

Maximal subgroups