Group #34101130 — subgroups

Subgroups — #34101130

Order 20096.

228 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 97 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	1	yes	maximal, characteristic
2	C₄×#14578938	C₂	1	1	yes	maximal, characteristic
3	#23757042	C₂	1	4	yes	maximal
4	C₂×#125×C₁₅₇	C₂	1	1	yes	maximal, characteristic
5	C₂×#14578938	C₂²	1	2	yes	—
6	C₂×#14578938	C₂²	1	2	yes	—
7	C₄⋊#11929434	—	2	2	no	—
8	#19097682	—	2	1	no	—
9	C₄×#11929434	—	2	1	no	—
10	#125×C₁₅₇	—	2	1	no	—
11	#125×C₁₅₇	—	2	1	no	—
12	#125×C₁₅₇	—	2	2	no	—
13	C₂×C₄⋊C₄×C₁₅₇	C₄	1	4	yes	—
14	C₂×C₄⋊C₄×C₁₅₇	C₂²	1	2	yes	—
15	C₂×C₄²×C₁₅₇	C₂²	1	1	yes	characteristic
16	#14578938	—	2	2	no	—
17	#14578938	—	2	4	no	—
18	#14578938	Q₈	1	4	yes	—
19	C₂×#11929434	—	2	2	no	—
20	C₂×#11929434	—	2	2	no	—
21	C₂²×C₄×C₁₅₇	C₂×C₄	1	4	yes	—
22	C₂²×C₄×C₁₅₇	C₂×C₄	1	2	yes	—
23	C₂²×C₄×C₁₅₇	C₂³	1	1	yes	characteristic
24	C₄⋊C₄×C₁₅₇	—	2	4	no	—
25	C₄⋊C₄×C₁₅₇	—	2	4	no	—
26	C₄⋊C₄×C₁₅₇	—	2	2	no	—
27	C₄⋊C₄×C₁₅₇	—	2	2	no	—
28	C₄²×C₁₅₇	D₄	1	1	yes	characteristic
29	C₄²×C₁₅₇	D₄	1	1	yes	characteristic
30	C₄²×C₁₅₇	D₄	1	2	yes	—
31	#11929434	—	2	4	no	—
32	#11929434	—	4	2	no	—
33	C₂³×C₁₅₇	C₂²×C₄	1	1	yes	characteristic
34	C₂×C₄×C₁₅₇	—	2	4	no	—
35	C₂×C₄×C₁₅₇	—	4	2	no	—
36	C₂×C₄×C₁₅₇	—	4	4	no	—
37	C₂×C₄×C₁₅₇	C₂²⋊C₄	1	2	yes	—
38	C₂×C₄×C₁₅₇	C₂²⋊C₄	1	2	yes	—
39	C₂×C₄×C₁₅₇	C₂×D₄	1	1	yes	characteristic
40	C₂×C₄×C₁₅₇	C₂×D₄	1	1	yes	characteristic
41	C₂×C₄×C₁₅₇	C₂×Q₈	1	2	yes	—
42	C₂×C₄×C₁₅₇	#41	1	2	yes	—
43	C₂²×C₁₅₇	—	2	2	no	—
44	C₂²×C₁₅₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
45	C₂²×C₁₅₇	C₄×Q₈	1	2	yes	—
46	C₄×C₁₅₇	—	2	2	no	—
47	C₄×C₁₅₇	—	2	2	no	—
48	C₄×C₁₅₇	—	4	4	no	—
49	C₄×C₁₅₇	C₂²⋊Q₈	1	2	yes	—
50	C₄×C₁₅₇	#124	1	1	yes	characteristic
51	C₄×C₁₅₇	C₄⋊Q₈	1	1	yes	characteristic
52	C₂×C₁₅₇	—	2	2	no	—
53	C₂×C₁₅₇	#391	1	1	yes	characteristic
54	C₂×C₁₅₇	#413	1	2	yes	—
55	C₁₅₇	#1998	1	1	yes	characteristic, complemented, Sylow
56	#1998	—	157	1	no	maximal, Sylow
57	C₂×#125	D₁₅₇	1	1	yes	characteristic
58	#423	—	157	1	no	—
59	C₄×M₁₆	—	157	1	no	—
60	#343	—	157	4	no	—
61	C₂×M₁₆	—	157	2	no	—
62	C₂×M₁₆	—	157	2	no	—
63	#125	—	2	1	no	—
64	#125	—	2	1	no	—
65	#125	—	2	2	no	—
66	C₂×C₄⋊C₄	Dic₁₅₇	1	4	yes	—
67	C₂×C₄⋊C₄	C₂×D₁₅₇	1	2	yes	—
68	C₂×C₄²	C₂×D₁₅₇	1	1	yes	characteristic
69	C₄⋊C₈	—	314	2	no	—
70	#97	—	314	1	no	—
71	C₄×C₈	—	314	1	no	—
72	C₂²×C₄	C₂×Dic₁₅₇	1	4	yes	—
73	C₂²×C₄	C₂×Dic₁₅₇	1	2	yes	—
74	C₂²×C₄	C₂²×D₁₅₇	1	1	yes	characteristic
75	M₁₆	—	314	2	no	—
76	M₁₆	—	314	4	no	—
77	M₁₆	—	157	4	no	—
78	C₂×C₈	—	314	2	no	—
79	C₂×C₈	—	314	2	no	—
80	C₄⋊C₄	—	2	2	no	—
81	C₄⋊C₄	—	2	2	no	—
82	C₄⋊C₄	—	2	4	no	—
83	C₄⋊C₄	—	2	4	no	—
84	C₄²	C₂²⋊D₁₅₇	1	1	yes	characteristic
85	C₄²	C₂²⋊D₁₅₇	1	1	yes	characteristic
86	C₄²	C₂²⋊D₁₅₇	1	2	yes	—
87	C₂³	C₂²×Dic₁₅₇	1	1	yes	characteristic
88	C₂×C₄	—	2	4	no	—
89	C₂×C₄	—	4	2	no	—
90	C₂×C₄	—	4	4	no	—
91	C₂×C₄	C₂²⋊Dic₁₅₇	1	2	yes	—
92	C₂×C₄	C₂²⋊Dic₁₅₇	1	2	yes	—
93	C₂×C₄	C₂×C₂²⋊D₁₅₇	1	1	yes	characteristic
94	C₂×C₄	C₂×C₂²⋊D₁₅₇	1	1	yes	characteristic
95	C₂×C₄	Q₈×D₁₅₇	1	2	yes	—
96	C₂×C₄	#14578955	1	2	yes	—
97	C₈	—	314	4	no	—
98	C₈	—	628	2	no	—
99	C₂²	—	2	2	no	—
100	C₂²	C₂×C₂²⋊Dic₁₅₇	1	1	yes	characteristic
101	C₂²	Q₈×Dic₁₅₇	1	2	yes	—
102	C₄	—	2	2	no	—
103	C₄	—	2	2	no	—
104	C₄	—	4	4	no	—
105	C₄	#19097747	1	2	yes	—
106	C₄	#19097760	1	1	yes	characteristic
107	C₄	#19097769	1	1	yes	characteristic
108	C₂	—	2	2	no	—
109	C₂	#23757180	1	1	yes	characteristic
110	C₂	#23757257	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#23757297

C₂

yes

maximal, characteristic

C₄×#14578938

C₂

yes

maximal, characteristic

#23757042

C₂

yes

maximal

C₂×#125×C₁₅₇

C₂

yes

maximal, characteristic

C₂×#14578938

C₂²

yes

—

C₂×#14578938

C₂²

yes

—

C₄⋊#11929434

—

#19097682

—

C₄×#11929434

—

#125×C₁₅₇

—

#125×C₁₅₇

—

#125×C₁₅₇

—

C₂×C₄⋊C₄×C₁₅₇

C₄

yes

—

C₂×C₄⋊C₄×C₁₅₇

C₂²

yes

—

C₂×C₄²×C₁₅₇

C₂²

yes

—

—

Q₈

yes

—

C₂×#11929434

—

C₂×#11929434

—

C₂²×C₄×C₁₅₇

C₂×C₄

yes

—

C₂²×C₄×C₁₅₇

C₂×C₄

yes

—

C₂²×C₄×C₁₅₇

C₂³

yes

characteristic

C₄⋊C₄×C₁₅₇

—

C₄⋊C₄×C₁₅₇

—

C₄⋊C₄×C₁₅₇

—

C₄⋊C₄×C₁₅₇

—

C₄²×C₁₅₇

D₄

yes

characteristic

C₄²×C₁₅₇

D₄

yes

characteristic

C₄²×C₁₅₇

D₄

yes

—

#11929434

—

#11929434

—

C₂³×C₁₅₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₅₇

—

C₂×C₄×C₁₅₇

—

C₂×C₄×C₁₅₇

—

C₂×C₄×C₁₅₇

C₂²⋊C₄

yes

—

C₂×C₄×C₁₅₇

C₂²⋊C₄

yes

—

C₂×C₄×C₁₅₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₅₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₅₇

C₂×Q₈

yes

—

C₂×C₄×C₁₅₇

#41

yes

—

C₂²×C₁₅₇

—

C₂²×C₁₅₇

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₁₅₇

C₄×Q₈

yes

—

C₄×C₁₅₇

—

C₄×C₁₅₇

—

C₄×C₁₅₇

—

C₄×C₁₅₇

C₂²⋊Q₈

yes

—

C₄×C₁₅₇

#124

yes

characteristic

C₄×C₁₅₇

C₄⋊Q₈

yes

characteristic

C₂×C₁₅₇

—

C₂×C₁₅₇

#391

yes

characteristic

C₂×C₁₅₇

#413

yes

—

C₁₅₇

#1998

yes

characteristic, complemented, Sylow

#1998

—

157

maximal, Sylow

C₂×#125

D₁₅₇

yes

characteristic

#423

—

157

—

C₄×M₁₆

—

157

—

#343

—

157

—

C₂×M₁₆

—

157

—

C₂×M₁₆

—

157

—

#125

—

#125

—

#125

—

C₂×C₄⋊C₄

Dic₁₅₇

yes

—

C₂×C₄⋊C₄

C₂×D₁₅₇

yes

—

C₂×C₄²

C₂×D₁₅₇

yes

characteristic

C₄⋊C₈

—

314

—

#97

—

314

—

C₄×C₈

—

314

—

C₂²×C₄

C₂×Dic₁₅₇

yes

—

C₂²×C₄

C₂×Dic₁₅₇

yes

—

C₂²×C₄

C₂²×D₁₅₇

yes

characteristic

M₁₆

—

314

—

M₁₆

—

314

—

M₁₆

—

157

—

C₂×C₈

—

314

—

C₂×C₈

—

314

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

C₂²⋊D₁₅₇

yes

characteristic

C₄²

C₂²⋊D₁₅₇

yes

characteristic

C₄²

C₂²⋊D₁₅₇

yes

—

C₂³

C₂²×Dic₁₅₇

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂²⋊Dic₁₅₇

yes

—

C₂×C₄

C₂²⋊Dic₁₅₇

yes

—

C₂×C₄

C₂×C₂²⋊D₁₅₇

yes

characteristic

C₂×C₄

C₂×C₂²⋊D₁₅₇

yes

characteristic

C₂×C₄

Q₈×D₁₅₇

yes

—

C₂×C₄

#14578955

yes

—

C₈

—

314

—

C₈

—

628

—

C₂²

—

100

C₂²

C₂×C₂²⋊Dic₁₅₇

yes

characteristic

101

C₂²

Q₈×Dic₁₅₇

yes

—

102

C₄

—

103

C₄

—

104

C₄

—

105

C₄

#19097747

yes

—

106

C₄

#19097760

yes

characteristic

107

C₄

#19097769

yes

characteristic

108

C₂

—

109

C₂

#23757180

yes

characteristic

110

C₂

#23757257

yes

—

	subgroup	G/H
center	C₂²	C₂×C₂²⋊Dic₁₅₇
commutator, prime radical	C₂³×C₁₅₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₅₇
Fitting	C₂×#125×C₁₅₇	C₂
socle	C₂²×C₁₅₇	C₂×C₂²⋊C₄
cosocle	C₂²×C₄×C₁₅₇	C₂³
hypercenter	C₂×#125	D₁₅₇
hypocenter	C₁₅₇	#1998
LCS	G ▷ C₂³×C₁₅₇ ▷ C₂×C₁₅₇ ▷ C₁₅₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#125

subgroup

G/H

center

C₂²

C₂×C₂²⋊Dic₁₅₇

commutator, prime radical

C₂³×C₁₅₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₅₇

Fitting

C₂×#125×C₁₅₇

C₂

socle

C₂²×C₁₅₇

C₂×C₂²⋊C₄

cosocle

C₂²×C₄×C₁₅₇

C₂³

D₁₅₇

C₁₅₇

G ▷ C₂³×C₁₅₇ ▷ C₂×C₁₅₇ ▷ C₁₅₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#125

N	H ≅ G/N	type
55. C₁₅₇	#1998	semidirect

H ≅ G/N

type

55. C₁₅₇

#1998

semidirect

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

subgroup	index
#23757297	2
C₄×#14578938	2
#23757042	2
C₂×#125×C₁₅₇	2
#1998	157

subgroup

index

C₂×#125×C₁₅₇

157

Subgroups — #34101130

Conjugacy classes of subgroups

Complements

Maximal subgroups