Group #34098874 — subgroups

Subgroups — #34098874

Order 20096.

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

normal subgroups only

This group has 210 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	#23757297	C₂	1	2	yes	maximal
2	#23757019	C₂	1	4	yes	maximal, complemented
3	#533×C₁₅₇	C₂	1	1	yes	maximal, characteristic
4	C₂×#14578938	—	2	2	no	—
5	C₄⋊#11929434	—	2	2	no	—
6	C₄⋊#11929434	C₂²	1	4	yes	—
7	C₄⋊Q₈×C₁₅₇	C₂²	1	2	yes	—
8	#125×C₁₅₇	C₄	1	2	yes	—
9	C₂²⋊Q₈×C₁₅₇	—	2	2	no	—
10	C₄×Q₈×C₁₅₇	—	2	2	no	—
11	#117×C₁₅₇	C₄	1	2	yes	—
12	C₂×C₄²×C₁₅₇	C₂²	1	1	yes	characteristic
13	#14578938	—	4	4	no	—
14	C₂×#11929434	—	2	4	no	—
15	C₂×Q₈×C₁₅₇	—	2	2	no	—
16	C₂²×C₄×C₁₅₇	C₂×C₄	1	1	yes	characteristic
17	C₂²×C₄×C₁₅₇	D₄	1	2	yes	—
18	C₄⋊C₄×C₁₅₇	—	2	4	no	—
19	C₄⋊C₄×C₁₅₇	—	2	2	no	—
20	C₄⋊C₄×C₁₅₇	C₂×C₄	1	4	yes	—
21	C₂²⋊C₄×C₁₅₇	—	2	2	no	—
22	C₄²×C₁₅₇	—	2	2	no	—
23	C₄²×C₁₅₇	C₂×C₄	1	1	yes	characteristic
24	C₄²×C₁₅₇	D₄	1	2	yes	—
25	C₄²×C₁₅₇	C₂³	1	1	yes	characteristic
26	#11929434	—	4	4	no	—
27	C₂³×C₁₅₇	C₂²⋊C₄	1	1	yes	characteristic
28	Q₈×C₁₅₇	—	4	2	no	—
29	Q₈×C₁₅₇	—	4	2	no	—
30	C₂×C₄×C₁₅₇	—	2	2	no	—
31	C₂×C₄×C₁₅₇	—	2	2	no	—
32	C₂×C₄×C₁₅₇	—	2	2	no	—
33	C₂×C₄×C₁₅₇	—	2	4	no	—
34	C₂×C₄×C₁₅₇	C₂²⋊C₄	1	2	yes	—
35	C₂×C₄×C₁₅₇	C₂²⋊C₄	1	1	yes	characteristic
36	C₂×C₄×C₁₅₇	C₂²×C₄	1	1	yes	characteristic
37	C₂×C₄×C₁₅₇	C₂×D₄	1	2	yes	—
38	C₂²×C₁₅₇	—	2	1	no	—
39	C₂²×C₁₅₇	—	2	2	no	—
40	C₂²×C₁₅₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
41	C₄×C₁₅₇	—	2	1	no	—
42	C₄×C₁₅₇	—	2	2	no	—
43	C₄×C₁₅₇	—	4	4	no	—
44	C₄×C₁₅₇	#101	1	2	yes	—
45	C₄×C₁₅₇	AGL(1,ℤ/8)	1	2	yes	—
46	C₄×C₁₅₇	#137	1	2	yes	—
47	C₂×C₁₅₇	—	4	1	no	—
48	C₂×C₁₅₇	C₂×#101	1	1	yes	characteristic
49	C₂×C₁₅₇	#417	1	2	yes	—
50	C₁₅₇	#1561	1	1	yes	characteristic, complemented, Sylow
51	#1561	—	157	1	no	maximal, Sylow
52	#533	D₁₅₇	1	1	yes	characteristic
53	#423	—	157	2	no	—
54	#332	—	157	4	no	—
55	C₂×M₁₆	—	314	2	no	—
56	C₄⋊Q₈	C₂×D₁₅₇	1	2	yes	—
57	#125	Dic₁₅₇	1	2	yes	—
58	C₂²⋊Q₈	—	2	2	no	—
59	C₄×Q₈	—	2	2	no	—
60	#117	Dic₁₅₇	1	2	yes	—
61	C₂×C₄²	C₂×D₁₅₇	1	1	yes	characteristic
62	C₄⋊C₈	—	314	2	no	—
63	C₄⋊C₈	—	157	4	no	—
64	C₂×Q₈	—	2	2	no	—
65	C₂²×C₄	C₂×Dic₁₅₇	1	1	yes	characteristic
66	C₂²×C₄	C₂²⋊D₁₅₇	1	2	yes	—
67	M₁₆	—	628	4	no	—
68	C₂×C₈	—	314	4	no	—
69	C₄⋊C₄	—	2	2	no	—
70	C₄⋊C₄	—	2	4	no	—
71	C₄⋊C₄	C₂×Dic₁₅₇	1	4	yes	—
72	C₂²⋊C₄	—	2	2	no	—
73	C₄²	—	2	2	no	—
74	C₄²	C₂×Dic₁₅₇	1	1	yes	characteristic
75	C₄²	C₂²⋊D₁₅₇	1	2	yes	—
76	C₄²	C₂²×D₁₅₇	1	1	yes	characteristic
77	C₂³	C₂²⋊Dic₁₅₇	1	1	yes	characteristic
78	Q₈	—	4	2	no	—
79	Q₈	—	4	2	no	—
80	C₂×C₄	—	2	2	no	—
81	C₂×C₄	—	2	2	no	—
82	C₂×C₄	—	2	2	no	—
83	C₂×C₄	—	2	4	no	—
84	C₂×C₄	C₂²⋊Dic₁₅₇	1	2	yes	—
85	C₂×C₄	C₂²⋊Dic₁₅₇	1	1	yes	characteristic
86	C₂×C₄	C₂²×Dic₁₅₇	1	1	yes	characteristic
87	C₂×C₄	C₂×C₂²⋊D₁₅₇	1	2	yes	—
88	C₈	—	628	4	no	—
89	C₂²	—	2	1	no	—
90	C₂²	—	2	2	no	—
91	C₂²	C₂×C₂²⋊Dic₁₅₇	1	1	yes	characteristic
92	C₄	—	2	1	no	—
93	C₄	—	2	2	no	—
94	C₄	—	4	4	no	—
95	C₄	#19097690	1	2	yes	—
96	C₄	#19097796	1	2	yes	—
97	C₄	#19097801	1	2	yes	—
98	C₂	—	4	1	no	—
99	C₂	C₂×#19097690	1	1	yes	characteristic
100	C₂	#23757274	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#23757297

C₂

yes

maximal

#23757019

C₂

yes

maximal, complemented

#533×C₁₅₇

C₂

yes

maximal, characteristic

C₂×#14578938

—

C₄⋊#11929434

—

C₄⋊#11929434

C₂²

yes

—

C₄⋊Q₈×C₁₅₇

C₂²

yes

—

#125×C₁₅₇

C₄

yes

—

C₂²⋊Q₈×C₁₅₇

—

C₄×Q₈×C₁₅₇

—

#117×C₁₅₇

C₄

yes

—

C₂×C₄²×C₁₅₇

C₂²

yes

characteristic

#14578938

—

C₂×#11929434

—

C₂×Q₈×C₁₅₇

—

C₂²×C₄×C₁₅₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₅₇

D₄

yes

—

C₄⋊C₄×C₁₅₇

—

C₄⋊C₄×C₁₅₇

—

C₄⋊C₄×C₁₅₇

C₂×C₄

yes

—

C₂²⋊C₄×C₁₅₇

—

C₄²×C₁₅₇

—

C₄²×C₁₅₇

C₂×C₄

yes

characteristic

C₄²×C₁₅₇

D₄

yes

—

C₄²×C₁₅₇

C₂³

yes

characteristic

#11929434

—

C₂³×C₁₅₇

C₂²⋊C₄

yes

characteristic

Q₈×C₁₅₇

—

Q₈×C₁₅₇

—

C₂×C₄×C₁₅₇

—

C₂×C₄×C₁₅₇

—

C₂×C₄×C₁₅₇

—

C₂×C₄×C₁₅₇

—

C₂×C₄×C₁₅₇

C₂²⋊C₄

yes

—

C₂×C₄×C₁₅₇

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₅₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₅₇

C₂×D₄

yes

—

C₂²×C₁₅₇

—

C₂²×C₁₅₇

—

C₂²×C₁₅₇

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₁₅₇

—

C₄×C₁₅₇

—

C₄×C₁₅₇

—

C₄×C₁₅₇

#101

yes

—

C₄×C₁₅₇

AGL(1,ℤ/8)

yes

—

C₄×C₁₅₇

#137

yes

—

C₂×C₁₅₇

—

C₂×C₁₅₇

C₂×#101

yes

characteristic

C₂×C₁₅₇

#417

yes

—

C₁₅₇

#1561

yes

characteristic, complemented, Sylow

#1561

—

157

maximal, Sylow

#533

D₁₅₇

yes

characteristic

#423

—

157

—

#332

—

157

—

C₂×M₁₆

—

314

—

C₄⋊Q₈

C₂×D₁₅₇

yes

—

#125

Dic₁₅₇

yes

—

C₂²⋊Q₈

—

C₄×Q₈

—

#117

Dic₁₅₇

yes

—

C₂×C₄²

C₂×D₁₅₇

yes

characteristic

C₄⋊C₈

—

314

—

C₄⋊C₈

—

157

—

C₂×Q₈

—

C₂²×C₄

C₂×Dic₁₅₇

yes

characteristic

C₂²×C₄

C₂²⋊D₁₅₇

yes

—

M₁₆

—

628

—

C₂×C₈

—

314

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂×Dic₁₅₇

yes

—

C₂²⋊C₄

—

C₄²

—

C₄²

C₂×Dic₁₅₇

yes

characteristic

C₄²

C₂²⋊D₁₅₇

yes

—

C₄²

C₂²×D₁₅₇

yes

characteristic

C₂³

C₂²⋊Dic₁₅₇

yes

characteristic

Q₈

—

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂²⋊Dic₁₅₇

yes

—

C₂×C₄

C₂²⋊Dic₁₅₇

yes

characteristic

C₂×C₄

C₂²×Dic₁₅₇

yes

characteristic

C₂×C₄

C₂×C₂²⋊D₁₅₇

yes

—

C₈

—

628

—

C₂²

—

C₂²

—

C₂²

C₂×C₂²⋊Dic₁₅₇

yes

characteristic

C₄

—

C₄

—

C₄

—

C₄

#19097690

yes

—

C₄

#19097796

yes

—

C₄

#19097801

yes

—

C₂

—

C₂

C₂×#19097690

yes

characteristic

100

C₂

#23757274

yes

—

	subgroup	G/H
center	C₂²	C₂×C₂²⋊Dic₁₅₇
commutator	C₂×C₄×C₁₅₇	C₂²×C₄
Frattini	C₄²	C₂²×D₁₅₇
Fitting	#533×C₁₅₇	C₂
socle	C₂²×C₁₅₇	C₂×C₂²⋊C₄
prime radical	C₂³×C₁₅₇	C₂²⋊C₄
cosocle	C₄²×C₁₅₇	C₂³
hypercenter	#533	D₁₅₇
hypocenter	C₁₅₇	#1561
LCS	G ▷ C₂×C₄×C₁₅₇ ▷ C₂²×C₁₅₇ ▷ C₁₅₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #533

subgroup

G/H

center

C₂²

C₂×C₂²⋊Dic₁₅₇

commutator

C₂×C₄×C₁₅₇

C₂²×C₄

Frattini

C₄²

C₂²×D₁₅₇

Fitting

#533×C₁₅₇

C₂

socle

C₂²×C₁₅₇

C₂×C₂²⋊C₄

prime radical

C₂³×C₁₅₇

C₂²⋊C₄

cosocle

C₄²×C₁₅₇

C₂³

D₁₅₇

C₁₅₇

G ▷ C₂×C₄×C₁₅₇ ▷ C₂²×C₁₅₇ ▷ C₁₅₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #533

N	H ≅ G/N	type
2. #23757019	C₂	semidirect
50. C₁₅₇	#1561	semidirect

H ≅ G/N

type

2. #23757019

C₂

semidirect

50. C₁₅₇

#1561

semidirect

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

subgroup	index
#23757297	2
#23757019	2
#533×C₁₅₇	2
#1561	157

subgroup

index

#23757297

#23757019

#533×C₁₅₇

#1561

157

Subgroups — #34098874

Conjugacy classes of subgroups

Complements

Maximal subgroups