Group #34098955 — subgroups

Subgroups — #34098955

Order 20096.

160 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 160 conjugacy classes of subgroups — too many to draw. Only its 87 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	Q₈×#11929434	C₂	1	2	yes	maximal
2	#23757024	C₂	1	1	yes	maximal, characteristic
3	Q₈⋊#11929434	C₂	1	2	yes	maximal
4	#23756994	C₂	1	1	yes	maximal, characteristic
5	C₄×Q₁₆×C₁₅₇	C₂	1	1	yes	maximal, characteristic
6	C₄⋊#11929434	—	2	2	no	—
7	C₄⋊#11929434	C₂²	1	2	yes	—
8	C₄×#11929434	—	2	2	no	—
9	C₄×#11929434	C₂²	1	1	yes	characteristic
10	C₄×#11929434	C₂²	1	1	yes	characteristic
11	C₂×Q₁₆×C₁₅₇	C₄	1	1	yes	characteristic
12	C₄×Q₈×C₁₅₇	C₂²	1	2	yes	—
13	#107×C₁₅₇	C₄	1	1	yes	characteristic
14	#103×C₁₅₇	C₄	1	2	yes	—
15	C₄×C₈×C₁₅₇	C₂²	1	1	yes	characteristic
16	C₂×#11929434	—	2	2	no	—
17	C₂×#11929434	—	2	1	no	—
18	C₂×#11929434	D₄	1	2	yes	—
19	C₂×Q₈×C₁₅₇	C₂×C₄	1	2	yes	—
20	Q₁₆×C₁₅₇	C₈	1	4	yes	complemented
21	C₂×C₈×C₁₅₇	C₂×C₄	1	1	yes	characteristic
22	C₂×C₈×C₁₅₇	C₂×C₄	1	1	yes	characteristic
23	C₄⋊C₄×C₁₅₇	—	2	2	no	—
24	C₄⋊C₄×C₁₅₇	C₂×C₄	1	2	yes	—
25	C₄²×C₁₅₇	—	2	2	no	—
26	C₄²×C₁₅₇	C₂³	1	1	yes	characteristic
27	#11929434	—	2	2	no	—
28	#11929434	—	4	1	no	—
29	#11929434	—	4	2	no	—
30	Q₈×C₁₅₇	—	2	2	no	—
31	Q₈×C₁₅₇	C₂×C₈	1	4	yes	—
32	C₂×C₄×C₁₅₇	—	2	2	no	—
33	C₂×C₄×C₁₅₇	—	2	2	no	—
34	C₂×C₄×C₁₅₇	C₂²×C₄	1	1	yes	characteristic
35	C₂×C₄×C₁₅₇	C₂×D₄	1	1	yes	characteristic
36	C₂×C₄×C₁₅₇	#41	1	1	yes	characteristic
37	C₈×C₁₅₇	—	2	1	no	—
38	C₈×C₁₅₇	C₂×C₈	1	2	yes	—
39	C₂²×C₁₅₇	C₄×D₄	1	1	yes	characteristic
40	C₄×C₁₅₇	—	2	4	no	—
41	C₄×C₁₅₇	—	2	1	no	—
42	C₄×C₁₅₇	—	4	2	no	—
43	C₄×C₁₅₇	C₂²×C₈	1	1	yes	characteristic
44	C₄×C₁₅₇	#131	1	1	yes	characteristic
45	C₄×C₁₅₇	#137	1	2	yes	—
46	C₂×C₁₅₇	C₈×D₄	1	1	yes	characteristic
47	C₂×C₁₅₇	#441	1	1	yes	characteristic
48	C₂×C₁₅₇	#444	1	1	yes	characteristic
49	C₁₅₇	#1579	1	1	yes	characteristic, complemented, Sylow
50	#1579	—	157	1	no	maximal, Sylow
51	C₈×Q₈	—	157	2	no	—
52	C₄×Q₁₆	D₁₅₇	1	1	yes	characteristic
53	#334	—	157	1	no	—
54	Q₈⋊C₈	—	157	2	no	—
55	#322	—	157	1	no	—
56	C₂×Q₁₆	Dic₁₅₇	1	1	yes	characteristic
57	C₄×Q₈	C₂×D₁₅₇	1	2	yes	—
58	#107	Dic₁₅₇	1	1	yes	characteristic
59	C₄⋊C₈	—	314	2	no	—
60	C₄⋊C₈	—	157	2	no	—
61	#103	Dic₁₅₇	1	2	yes	—
62	C₄×C₈	—	314	2	no	—
63	C₄×C₈	—	157	1	no	—
64	C₄×C₈	—	157	1	no	—
65	C₄×C₈	C₂×D₁₅₇	1	1	yes	characteristic
66	C₂×Q₈	C₂×Dic₁₅₇	1	2	yes	—
67	Q₁₆	#11929434	1	4	yes	complemented
68	C₂×C₈	—	314	2	no	—
69	C₂×C₈	—	314	1	no	—
70	C₂×C₈	—	157	2	no	—
71	C₂×C₈	C₂×Dic₁₅₇	1	1	yes	characteristic
72	C₂×C₈	C₂×Dic₁₅₇	1	1	yes	characteristic
73	C₄⋊C₄	—	2	2	no	—
74	C₄⋊C₄	C₂×Dic₁₅₇	1	2	yes	—
75	C₄²	—	2	2	no	—
76	C₄²	C₂²×D₁₅₇	1	1	yes	characteristic
77	Q₈	—	2	2	no	—
78	Q₈	C₂×#11929434	1	4	yes	—
79	C₂×C₄	—	2	2	no	—
80	C₂×C₄	—	2	2	no	—
81	C₂×C₄	C₂²×Dic₁₅₇	1	1	yes	characteristic
82	C₂×C₄	D₄×D₁₅₇	1	1	yes	characteristic
83	C₂×C₄	#14578953	1	1	yes	characteristic
84	C₈	—	2	1	no	—
85	C₈	—	314	2	no	—
86	C₈	—	628	1	no	—
87	C₈	—	628	2	no	—
88	C₈	C₂×#11929434	1	2	yes	—
89	C₂²	D₄×Dic₁₅₇	1	1	yes	characteristic
90	C₄	—	2	4	no	—
91	C₄	—	2	1	no	—
92	C₄	—	4	2	no	—
93	C₄	C₂²×#11929434	1	1	yes	characteristic
94	C₄	#19097778	1	1	yes	characteristic
95	C₄	#19097803	1	2	yes	—
96	C₂	D₄×#11929434	1	1	yes	characteristic
97	C₂	#23757380	1	1	yes	characteristic
98	C₂	#23757395	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

Q₈×#11929434

C₂

yes

maximal

#23757024

C₂

yes

maximal, characteristic

Q₈⋊#11929434

C₂

yes

maximal

#23756994

C₂

yes

maximal, characteristic

C₄×Q₁₆×C₁₅₇

C₂

yes

maximal, characteristic

C₄⋊#11929434

—

C₄⋊#11929434

C₂²

yes

—

C₄×#11929434

—

C₄×#11929434

C₂²

yes

characteristic

C₄×#11929434

C₂²

yes

characteristic

C₂×Q₁₆×C₁₅₇

C₄

yes

characteristic

C₄×Q₈×C₁₅₇

C₂²

yes

—

#107×C₁₅₇

C₄

yes

characteristic

#103×C₁₅₇

C₄

yes

—

C₄×C₈×C₁₅₇

C₂²

yes

—

—

D₄

yes

—

C₂×Q₈×C₁₅₇

C₂×C₄

yes

—

Q₁₆×C₁₅₇

C₈

yes

complemented

C₂×C₈×C₁₅₇

C₂×C₄

yes

characteristic

C₂×C₈×C₁₅₇

C₂×C₄

yes

characteristic

C₄⋊C₄×C₁₅₇

—

C₄⋊C₄×C₁₅₇

C₂×C₄

yes

—

C₄²×C₁₅₇

—

C₄²×C₁₅₇

C₂³

yes

—

—

—

Q₈×C₁₅₇

—

Q₈×C₁₅₇

C₂×C₈

yes

—

C₂×C₄×C₁₅₇

—

C₂×C₄×C₁₅₇

—

C₂×C₄×C₁₅₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₅₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₅₇

#41

yes

characteristic

C₈×C₁₅₇

—

C₈×C₁₅₇

C₂×C₈

yes

—

C₂²×C₁₅₇

C₄×D₄

yes

characteristic

C₄×C₁₅₇

—

C₄×C₁₅₇

—

C₄×C₁₅₇

—

C₄×C₁₅₇

C₂²×C₈

yes

C₄×C₁₅₇

yes

C₄×C₁₅₇

yes

—

C₂×C₁₅₇

C₈×D₄

yes

C₂×C₁₅₇

yes

C₂×C₁₅₇

yes

C₁₅₇

yes

characteristic, complemented, Sylow

#1579

—

157

maximal, Sylow

C₈×Q₈

—

157

—

C₄×Q₁₆

D₁₅₇

yes

characteristic

#334

—

157

—

Q₈⋊C₈

—

157

—

#322

—

157

—

C₂×Q₁₆

Dic₁₅₇

yes

characteristic

C₄×Q₈

C₂×D₁₅₇

yes

—

#107

Dic₁₅₇

yes

characteristic

C₄⋊C₈

—

314

—

C₄⋊C₈

—

157

—

#103

Dic₁₅₇

yes

—

C₄×C₈

—

314

—

C₄×C₈

—

157

—

C₄×C₈

—

157

—

C₄×C₈

C₂×D₁₅₇

yes

characteristic

C₂×Q₈

C₂×Dic₁₅₇

yes

—

Q₁₆

#11929434

yes

complemented

C₂×C₈

—

314

—

C₂×C₈

—

314

—

C₂×C₈

—

157

—

C₂×C₈

C₂×Dic₁₅₇

yes

characteristic

C₂×C₈

C₂×Dic₁₅₇

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

C₂×Dic₁₅₇

yes

—

C₄²

—

C₄²

C₂²×D₁₅₇

yes

characteristic

Q₈

—

Q₈

C₂×#11929434

yes

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂²×Dic₁₅₇

yes

characteristic

C₂×C₄

D₄×D₁₅₇

yes

characteristic

C₂×C₄

#14578953

yes

characteristic

C₈

—

C₈

—

314

—

C₈

—

628

—

C₈

—

628

—

C₈

C₂×#11929434

yes

—

C₂²

D₄×Dic₁₅₇

yes

characteristic

C₄

—

C₄

—

C₄

—

C₄

yes

C₄

yes

C₄

yes

—

C₂

yes

C₂

yes

C₂

yes

	subgroup	G/H
center	C₂×C₄	D₄×D₁₅₇
commutator	C₄×C₁₅₇	C₂²×C₈
Frattini	C₄²	C₂²×D₁₅₇
Fitting	C₄×Q₁₆×C₁₅₇	C₂
socle, prime radical	C₂²×C₁₅₇	C₄×D₄
cosocle	C₄²×C₁₅₇	C₂³
hypercenter	C₄×Q₁₆	D₁₅₇
hypocenter	C₁₅₇	#1579
LCS	G ▷ C₄×C₁₅₇ ▷ C₂×C₁₅₇ ▷ C₁₅₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×Q₁₆

subgroup

G/H

center

C₂×C₄

D₄×D₁₅₇

commutator

C₄×C₁₅₇

C₂²×C₈

Frattini

C₄²

C₂²×D₁₅₇

Fitting

C₄×Q₁₆×C₁₅₇

C₂

socle, prime radical

C₂²×C₁₅₇

C₄×D₄

cosocle

C₄²×C₁₅₇

C₂³

hypercenter

C₄×Q₁₆

D₁₅₇

hypocenter

C₁₅₇

#1579

LCS

G ▷ C₄×C₁₅₇ ▷ C₂×C₁₅₇ ▷ C₁₅₇

UCS

C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×Q₁₆

N	H ≅ G/N	type
20. Q₁₆×C₁₅₇	C₈	semidirect
49. C₁₅₇	#1579	semidirect
67. Q₁₆	#11929434	semidirect

H ≅ G/N

type

20. Q₁₆×C₁₅₇

C₈

semidirect

49. C₁₅₇

#1579

semidirect

67. Q₁₆

#11929434

semidirect

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

subgroup	index
Q₈×#11929434	2
#23757024	2
Q₈⋊#11929434	2
#23756994	2
C₄×Q₁₆×C₁₅₇	2
#1579	157

subgroup

index

C₄×Q₁₆×C₁₅₇

157

Subgroups — #34098955

Conjugacy classes of subgroups

Complements

Maximal subgroups