Group #34099276 — subgroups

Subgroups — #34099276

Order 20096.

206 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 206 conjugacy classes of subgroups — too many to draw. Only its 65 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	#23757652	C₂	1	1	yes	maximal, characteristic
2	#23757568	C₂	1	2	yes	maximal
3	#23757022	C₂	1	1	yes	maximal, characteristic
4	Q₈⋊#11929434	C₂	1	2	yes	maximal
5	Q₈²×C₁₅₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#14578944	—	2	2	no	—
7	#19097771	C₂²	1	1	yes	characteristic
8	#19097704	—	2	2	no	—
9	C₄⋊#11929434	C₂²	1	2	yes	—
10	#19097695	—	2	2	no	—
11	C₄×#11929434	C₂²	1	1	yes	characteristic
12	C₄⋊Q₈×C₁₅₇	—	2	2	no	—
13	C₄⋊Q₈×C₁₅₇	—	2	2	no	—
14	C₄⋊Q₈×C₁₅₇	C₂²	1	1	yes	characteristic
15	C₄×Q₈×C₁₅₇	—	2	2	no	—
16	C₄×Q₈×C₁₅₇	C₂²	1	2	yes	—
17	C₂×Dic₃₁₄	—	2	1	no	—
18	#14578944	—	4	4	no	—
19	C₂×#11929434	—	2	1	no	—
20	C₂×#11929434	—	2	2	no	—
21	C₄⋊Dic₁₅₇	—	2	2	no	—
22	C₂×Q₈×C₁₅₇	—	2	2	no	—
23	C₂×Q₈×C₁₅₇	D₄	1	2	yes	—
24	C₂×Q₈×C₁₅₇	D₄	1	2	yes	—
25	C₄⋊C₄×C₁₅₇	—	2	4	no	—
26	C₄⋊C₄×C₁₅₇	—	2	2	no	—
27	C₄⋊C₄×C₁₅₇	—	2	2	no	—
28	C₄⋊C₄×C₁₅₇	D₄	1	2	yes	—
29	C₄²×C₁₅₇	—	2	2	no	—
30	C₄²×C₁₅₇	—	2	2	no	—
31	C₄²×C₁₅₇	C₂³	1	1	yes	characteristic
32	Dic₃₁₄	—	4	2	no	—
33	C₂×Dic₁₅₇	—	4	1	no	—
34	#11929434	—	2	2	no	—
35	#11929434	—	4	2	no	—
36	Q₈×C₁₅₇	—	2	2	no	—
37	Q₈×C₁₅₇	—	4	2	no	—
38	Q₈×C₁₅₇	—	8	2	no	—
39	Q₈×C₁₅₇	Q₁₆	1	4	yes	complemented
40	C₂×C₄×C₁₅₇	—	2	2	no	—
41	C₂×C₄×C₁₅₇	—	2	2	no	—
42	C₂×C₄×C₁₅₇	—	2	2	no	—
43	C₂×C₄×C₁₅₇	C₂×D₄	1	1	yes	characteristic
44	C₂×C₄×C₁₅₇	C₂×D₄	1	2	yes	—
45	C₂²×C₁₅₇	#120	1	1	yes	characteristic
46	C₄×C₁₅₇	—	2	4	no	—
47	C₄×C₁₅₇	—	2	1	no	—
48	C₄×C₁₅₇	—	4	2	no	—
49	C₄×C₁₅₇	—	4	2	no	—
50	C₄×C₁₅₇	C₂×Q₁₆	1	2	yes	—
51	C₄×C₁₅₇	#137	1	2	yes	—
52	Dic₁₅₇	—	8	1	no	—
53	C₂×C₁₅₇	#451	1	2	yes	—
54	C₂×C₁₅₇	#453	1	1	yes	characteristic
55	C₁₅₇	#1633	1	1	yes	characteristic, complemented, Sylow
56	#1633	—	157	1	no	maximal, Sylow
57	Q₈²	D₁₅₇	1	1	yes	characteristic
58	#494	—	157	1	no	—
59	#477	—	157	2	no	—
60	#333	—	157	1	no	—
61	Q₈⋊C₈	—	157	2	no	—
62	C₂×Q₁₆	—	314	2	no	—
63	C₄⋊Q₈	—	2	2	no	—
64	C₄⋊Q₈	—	2	2	no	—
65	C₄⋊Q₈	—	157	1	no	—
66	C₄⋊Q₈	C₂×D₁₅₇	1	1	yes	characteristic
67	C₄×Q₈	—	2	2	no	—
68	C₄×Q₈	C₂×D₁₅₇	1	2	yes	—
69	#107	—	314	2	no	—
70	C₄⋊C₈	—	157	2	no	—
71	#103	—	314	2	no	—
72	C₄×C₈	—	157	1	no	—
73	C₂×Q₈	—	2	2	no	—
74	C₂×Q₈	—	314	1	no	—
75	C₂×Q₈	C₂²⋊D₁₅₇	1	2	yes	—
76	C₂×Q₈	C₂²⋊D₁₅₇	1	2	yes	—
77	Q₁₆	—	628	4	no	—
78	C₂×C₈	—	314	1	no	—
79	C₂×C₈	—	314	2	no	—
80	C₄⋊C₄	—	2	4	no	—
81	C₄⋊C₄	—	2	2	no	—
82	C₄⋊C₄	—	2	2	no	—
83	C₄⋊C₄	—	314	2	no	—
84	C₄⋊C₄	C₂²⋊D₁₅₇	1	2	yes	—
85	C₄²	—	2	2	no	—
86	C₄²	—	2	2	no	—
87	C₄²	C₂²×D₁₅₇	1	1	yes	characteristic
88	Q₈	—	2	2	no	—
89	Q₈	—	4	2	no	—
90	Q₈	—	8	2	no	—
91	Q₈	—	628	2	no	—
92	Q₈	#14578944	1	4	yes	complemented
93	C₂×C₄	—	2	2	no	—
94	C₂×C₄	—	2	2	no	—
95	C₂×C₄	—	2	2	no	—
96	C₂×C₄	—	628	1	no	—
97	C₂×C₄	C₂×C₂²⋊D₁₅₇	1	1	yes	characteristic
98	C₂×C₄	C₂×C₂²⋊D₁₅₇	1	2	yes	—
99	C₈	—	314	2	no	—
100	C₈	—	628	2	no	—
101	C₂²	#19097740	1	1	yes	characteristic
102	C₄	—	2	4	no	—
103	C₄	—	2	1	no	—
104	C₄	—	4	2	no	—
105	C₄	—	4	2	no	—
106	C₄	—	1256	1	no	—
107	C₄	C₂×#14578944	1	2	yes	—
108	C₄	#19097802	1	2	yes	—
109	C₂	#23757429	1	2	yes	—
110	C₂	#23757441	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#23757652

C₂

yes

maximal, characteristic

#23757568

C₂

yes

maximal

#23757022

C₂

yes

maximal, characteristic

Q₈⋊#11929434

C₂

yes

maximal

Q₈²×C₁₅₇

C₂

yes

maximal, characteristic

C₂×#14578944

—

#19097771

C₂²

yes

characteristic

#19097704

—

C₄⋊#11929434

C₂²

yes

—

#19097695

—

C₄×#11929434

C₂²

yes

characteristic

C₄⋊Q₈×C₁₅₇

—

C₄⋊Q₈×C₁₅₇

—

C₄⋊Q₈×C₁₅₇

C₂²

yes

characteristic

C₄×Q₈×C₁₅₇

—

C₄×Q₈×C₁₅₇

C₂²

yes

—

C₂×Dic₃₁₄

—

#14578944

—

C₂×#11929434

—

C₂×#11929434

—

C₄⋊Dic₁₅₇

—

C₂×Q₈×C₁₅₇

—

C₂×Q₈×C₁₅₇

D₄

yes

—

C₂×Q₈×C₁₅₇

D₄

yes

—

C₄⋊C₄×C₁₅₇

—

C₄⋊C₄×C₁₅₇

—

C₄⋊C₄×C₁₅₇

—

C₄⋊C₄×C₁₅₇

D₄

yes

—

C₄²×C₁₅₇

—

C₄²×C₁₅₇

—

C₄²×C₁₅₇

C₂³

yes

characteristic

Dic₃₁₄

—

C₂×Dic₁₅₇

—

#11929434

—

#11929434

—

Q₈×C₁₅₇

—

Q₈×C₁₅₇

—

Q₈×C₁₅₇

—

Q₈×C₁₅₇

Q₁₆

yes

complemented

C₂×C₄×C₁₅₇

—

C₂×C₄×C₁₅₇

—

C₂×C₄×C₁₅₇

—

C₂×C₄×C₁₅₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₅₇

C₂×D₄

yes

—

C₂²×C₁₅₇

#120

yes

characteristic

C₄×C₁₅₇

—

C₄×C₁₅₇

—

C₄×C₁₅₇

—

C₄×C₁₅₇

—

C₄×C₁₅₇

C₂×Q₁₆

yes

—

C₄×C₁₅₇

#137

yes

—

Dic₁₅₇

—

C₂×C₁₅₇

#451

yes

—

C₂×C₁₅₇

#453

yes

characteristic

C₁₅₇

#1633

yes

characteristic, complemented, Sylow

#1633

—

157

maximal, Sylow

Q₈²

D₁₅₇

yes

characteristic

#494

—

157

—

#477

—

157

—

#333

—

157

—

Q₈⋊C₈

—

157

—

C₂×Q₁₆

—

314

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

157

—

C₄⋊Q₈

C₂×D₁₅₇

yes

characteristic

C₄×Q₈

—

C₄×Q₈

C₂×D₁₅₇

yes

—

#107

—

314

—

C₄⋊C₈

—

157

—

#103

—

314

—

C₄×C₈

—

157

—

C₂×Q₈

—

C₂×Q₈

—

314

—

C₂×Q₈

C₂²⋊D₁₅₇

yes

—

C₂×Q₈

C₂²⋊D₁₅₇

yes

—

Q₁₆

—

628

—

C₂×C₈

—

314

—

C₂×C₈

—

314

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

314

—

C₄⋊C₄

C₂²⋊D₁₅₇

yes

—

C₄²

—

C₄²

—

C₄²

C₂²×D₁₅₇

yes

characteristic

Q₈

—

Q₈

—

Q₈

—

Q₈

—

628

—

Q₈

#14578944

yes

complemented

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

628

—

C₂×C₄

C₂×C₂²⋊D₁₅₇

yes

characteristic

C₂×C₄

C₂×C₂²⋊D₁₅₇

yes

—

C₈

—

314

—

100

C₈

—

628

—

101

C₂²

#19097740

yes

characteristic

102

C₄

—

103

C₄

—

104

C₄

—

105

C₄

—

106

C₄

—

1256

—

107

C₄

C₂×#14578944

yes

—

108

C₄

#19097802

yes

—

109

C₂

#23757429

yes

—

110

C₂

#23757441

yes

characteristic

	subgroup	G/H
center	C₂²	#19097740
commutator, cosocle	C₄²×C₁₅₇	C₂³
Frattini	C₄²	C₂²×D₁₅₇
Fitting	Q₈²×C₁₅₇	C₂
socle, prime radical	C₂²×C₁₅₇	#120
hypercenter	Q₈²	D₁₅₇
hypocenter	C₁₅₇	#1633
LCS	G ▷ C₄²×C₁₅₇ ▷ C₂²×C₁₅₇ ▷ C₁₅₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ Q₈²

subgroup

G/H

center

C₂²

#19097740

commutator, cosocle

C₄²×C₁₅₇

C₂³

Frattini

C₄²

C₂²×D₁₅₇

Fitting

Q₈²×C₁₅₇

C₂

socle, prime radical

C₂²×C₁₅₇

#120

hypercenter

Q₈²

D₁₅₇

hypocenter

C₁₅₇

#1633

LCS

G ▷ C₄²×C₁₅₇ ▷ C₂²×C₁₅₇ ▷ C₁₅₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ Q₈²

N	H ≅ G/N	type
39. Q₈×C₁₅₇	Q₁₆	semidirect
55. C₁₅₇	#1633	semidirect
92. Q₈	#14578944	semidirect

H ≅ G/N

type

39. Q₈×C₁₅₇

Q₁₆

semidirect

55. C₁₅₇

#1633

semidirect

92. Q₈

#14578944

semidirect

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

subgroup	index
#23757652	2
#23757568	2
#23757022	2
Q₈⋊#11929434	2
Q₈²×C₁₅₇	2
#1633	157

subgroup

index

Q₈²×C₁₅₇

157

Subgroups — #34099276

Conjugacy classes of subgroups

Complements

Maximal subgroups