Group #34098026 — subgroups

Subgroups — #34098026

Order 20096.

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

normal subgroups only

This group has 150 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	#23757326	C₂	1	1	yes	maximal, characteristic
2	#23757297	C₂	1	1	yes	maximal, characteristic
3	C₄×M₁₆×C₁₅₇	C₂	1	1	yes	maximal, characteristic
4	C₂×#14578938	C₄	1	2	yes	—
5	C₄⋊#11929434	—	2	1	no	—
6	C₄⋊#11929434	—	2	1	no	—
7	C₄⋊#11929434	C₄	1	2	yes	—
8	C₂²⋊#11929434	—	2	1	no	—
9	C₄×#11929434	—	2	1	no	—
10	C₂×M₁₆×C₁₅₇	—	2	1	no	—
11	C₂×C₄²×C₁₅₇	C₂²	1	1	yes	characteristic
12	#97×C₁₅₇	C₄	1	2	yes	—
13	C₄×C₈×C₁₅₇	—	2	1	no	—
14	#14578938	—	4	2	no	—
15	C₂×#11929434	—	2	2	no	—
16	C₂×#11929434	—	2	2	no	—
17	C₂²×C₄×C₁₅₇	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₄×C₁₅₇	D₄	1	1	yes	characteristic
19	C₂²×C₄×C₁₅₇	Q₈	1	1	yes	characteristic
20	M₁₆×C₁₅₇	—	2	4	no	—
21	C₂×C₈×C₁₅₇	—	2	2	no	—
22	C₄²×C₁₅₇	C₂×C₄	1	1	yes	characteristic
23	C₄²×C₁₅₇	C₂×C₄	1	1	yes	characteristic
24	C₄²×C₁₅₇	D₄	1	1	yes	characteristic
25	C₄²×C₁₅₇	D₄	1	1	yes	characteristic
26	#11929434	—	4	2	no	—
27	#11929434	—	4	2	no	—
28	C₂³×C₁₅₇	C₄⋊C₄	1	1	yes	characteristic
29	C₂×C₄×C₁₅₇	—	2	1	no	—
30	C₂×C₄×C₁₅₇	—	2	1	no	—
31	C₂×C₄×C₁₅₇	—	2	2	no	—
32	C₂×C₄×C₁₅₇	—	2	2	no	—
33	C₂×C₄×C₁₅₇	C₄²	1	1	yes	characteristic
34	C₂×C₄×C₁₅₇	C₂²⋊C₄	1	1	yes	characteristic
35	C₂×C₄×C₁₅₇	C₂²⋊C₄	1	1	yes	characteristic
36	C₂×C₄×C₁₅₇	C₂²⋊C₄	1	1	yes	characteristic
37	C₂×C₄×C₁₅₇	C₄⋊C₄	1	1	yes	characteristic
38	C₂×C₄×C₁₅₇	C₄⋊C₄	1	1	yes	characteristic
39	C₈×C₁₅₇	—	2	4	no	—
40	C₂²×C₁₅₇	—	2	1	no	—
41	C₂²×C₁₅₇	—	2	1	no	—
42	C₂²×C₁₅₇	—	2	1	no	—
43	C₂²×C₁₅₇	#95	1	1	yes	characteristic
44	C₄×C₁₅₇	—	2	1	no	—
45	C₄×C₁₅₇	—	2	1	no	—
46	C₄×C₁₅₇	—	2	1	no	—
47	C₄×C₁₅₇	#100	1	1	yes	characteristic
48	C₄×C₁₅₇	#101	1	1	yes	characteristic
49	C₄×C₁₅₇	#104	1	2	yes	—
50	C₄×C₁₅₇	#108	1	2	yes	—
51	C₂×C₁₅₇	—	4	1	no	—
52	C₂×C₁₅₇	#339	1	1	yes	characteristic
53	C₂×C₁₅₇	#341	1	1	yes	characteristic
54	C₂×C₁₅₇	#343	1	1	yes	characteristic
55	C₁₅₇	#1299	1	1	yes	characteristic, complemented, Sylow
56	#1299	—	157	1	no	maximal, Sylow
57	#431	—	157	1	no	—
58	#423	—	157	1	no	—
59	C₄×M₁₆	D₁₅₇	1	1	yes	characteristic
60	C₂×M₁₆	—	2	1	no	—
61	C₂×M₁₆	—	157	2	no	—
62	C₂×C₄²	C₂×D₁₅₇	1	1	yes	characteristic
63	C₄⋊C₈	—	314	1	no	—
64	C₄⋊C₈	—	314	1	no	—
65	C₄⋊C₈	—	157	2	no	—
66	C₂²⋊C₈	—	314	1	no	—
67	#97	Dic₁₅₇	1	2	yes	—
68	C₄×C₈	—	2	1	no	—
69	C₄×C₈	—	314	1	no	—
70	C₂²×C₄	C₄×D₁₅₇	1	1	yes	characteristic
71	C₂²×C₄	C₂²⋊D₁₅₇	1	1	yes	characteristic
72	C₂²×C₄	Dic₃₁₄	1	1	yes	characteristic
73	M₁₆	—	2	4	no	—
74	M₁₆	—	628	2	no	—
75	C₂×C₈	—	2	2	no	—
76	C₂×C₈	—	314	2	no	—
77	C₂×C₈	—	314	2	no	—
78	C₄²	C₄×D₁₅₇	1	1	yes	characteristic
79	C₄²	C₂×Dic₁₅₇	1	1	yes	characteristic
80	C₄²	C₂²⋊D₁₅₇	1	1	yes	characteristic
81	C₄²	D₆₂₈	1	1	yes	characteristic
82	C₂³	#14578933	1	1	yes	characteristic
83	C₂×C₄	—	2	1	no	—
84	C₂×C₄	—	2	1	no	—
85	C₂×C₄	—	2	2	no	—
86	C₂×C₄	—	2	2	no	—
87	C₂×C₄	C₄×Dic₁₅₇	1	1	yes	characteristic
88	C₂×C₄	#14578931	1	1	yes	characteristic
89	C₂×C₄	#14578931	1	1	yes	characteristic
90	C₂×C₄	C₂²⋊Dic₁₅₇	1	1	yes	characteristic
91	C₂×C₄	#14578933	1	1	yes	characteristic
92	C₂×C₄	C₄⋊Dic₁₅₇	1	1	yes	characteristic
93	C₈	—	2	4	no	—
94	C₈	—	628	2	no	—
95	C₈	—	628	2	no	—
96	C₂²	—	2	1	no	—
97	C₂²	—	2	1	no	—
98	C₂²	—	2	1	no	—
99	C₂²	#19097678	1	1	yes	characteristic
100	C₄	—	2	1	no	—
101	C₄	—	2	1	no	—
102	C₄	—	2	1	no	—
103	C₄	#19097687	1	1	yes	characteristic
104	C₄	#19097689	1	1	yes	characteristic
105	C₄	#19097699	1	2	yes	—
106	C₄	#19097706	1	2	yes	—
107	C₂	—	4	1	no	—
108	C₂	#23757033	1	1	yes	characteristic
109	C₂	#23757037	1	1	yes	characteristic
110	C₂	#23757041	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#23757326

C₂

yes

maximal, characteristic

#23757297

C₂

yes

maximal, characteristic

C₄×M₁₆×C₁₅₇

C₂

yes

maximal, characteristic

C₂×#14578938

C₄

yes

—

C₄⋊#11929434

—

C₄⋊#11929434

—

C₄⋊#11929434

C₄

yes

—

C₂²⋊#11929434

—

C₄×#11929434

—

C₂×M₁₆×C₁₅₇

—

C₂×C₄²×C₁₅₇

C₂²

yes

characteristic

#97×C₁₅₇

C₄

yes

—

C₄×C₈×C₁₅₇

—

#14578938

—

C₂×#11929434

—

C₂×#11929434

—

C₂²×C₄×C₁₅₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₅₇

D₄

yes

characteristic

C₂²×C₄×C₁₅₇

Q₈

yes

characteristic

M₁₆×C₁₅₇

—

C₂×C₈×C₁₅₇

—

C₄²×C₁₅₇

C₂×C₄

yes

characteristic

C₄²×C₁₅₇

C₂×C₄

yes

characteristic

C₄²×C₁₅₇

D₄

yes

characteristic

C₄²×C₁₅₇

D₄

yes

characteristic

#11929434

—

#11929434

—

C₂³×C₁₅₇

C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₅₇

—

C₂×C₄×C₁₅₇

—

C₂×C₄×C₁₅₇

—

C₂×C₄×C₁₅₇

—

C₂×C₄×C₁₅₇

C₄²

yes

characteristic

C₂×C₄×C₁₅₇

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₅₇

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₅₇

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₅₇

C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₅₇

C₄⋊C₄

yes

characteristic

C₈×C₁₅₇

—

C₂²×C₁₅₇

—

C₂²×C₁₅₇

—

C₂²×C₁₅₇

—

C₂²×C₁₅₇

#95

yes

characteristic

C₄×C₁₅₇

—

C₄×C₁₅₇

—

C₄×C₁₅₇

—

C₄×C₁₅₇

yes

C₄×C₁₅₇

yes

C₄×C₁₅₇

yes

—

C₄×C₁₅₇

#108

yes

—

C₂×C₁₅₇

—

C₂×C₁₅₇

yes

C₂×C₁₅₇

yes

C₂×C₁₅₇

yes

C₁₅₇

yes

characteristic, complemented, Sylow

#1299

—

157

maximal, Sylow

#431

—

157

—

#423

—

157

—

C₄×M₁₆

D₁₅₇

yes

characteristic

C₂×M₁₆

—

C₂×M₁₆

—

157

—

C₂×C₄²

C₂×D₁₅₇

yes

characteristic

C₄⋊C₈

—

314

—

C₄⋊C₈

—

314

—

C₄⋊C₈

—

157

—

C₂²⋊C₈

—

314

—

#97

Dic₁₅₇

yes

—

C₄×C₈

—

C₄×C₈

—

314

—

C₂²×C₄

C₄×D₁₅₇

yes

characteristic

C₂²×C₄

C₂²⋊D₁₅₇

yes

characteristic

C₂²×C₄

Dic₃₁₄

yes

characteristic

M₁₆

—

M₁₆

—

628

—

C₂×C₈

—

C₂×C₈

—

314

—

C₂×C₈

—

314

—

C₄²

C₄×D₁₅₇

yes

characteristic

C₄²

C₂×Dic₁₅₇

yes

characteristic

C₄²

C₂²⋊D₁₅₇

yes

characteristic

C₄²

D₆₂₈

yes

characteristic

C₂³

#14578933

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₄×Dic₁₅₇

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

C₂²⋊Dic₁₅₇

yes

characteristic

C₂×C₄

#14578933

yes

characteristic

C₂×C₄

C₄⋊Dic₁₅₇

yes

characteristic

C₈

—

C₈

—

628

—

C₈

—

628

—

C₂²

—

C₂²

—

C₂²

—

C₂²

#19097678

yes

characteristic

100

C₄

—

101

C₄

—

102

C₄

—

103

C₄

#19097687

yes

characteristic

104

C₄

#19097689

yes

characteristic

105

C₄

#19097699

yes

—

106

C₄

#19097706

yes

—

107

C₂

—

108

C₂

#23757033

yes

characteristic

109

C₂

#23757037

yes

characteristic

110

C₂

#23757041

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#14578931
commutator	C₂×C₄×C₁₅₇	C₄²
Frattini	C₂×C₄²	C₂×D₁₅₇
Fitting	C₄×M₁₆×C₁₅₇	C₂
socle	C₂²×C₁₅₇	#95
prime radical	C₂³×C₁₅₇	C₄⋊C₄
cosocle	C₂×C₄²×C₁₅₇	C₂²
hypercenter	C₄×M₁₆	D₁₅₇
hypocenter	C₁₅₇	#1299
LCS	G ▷ C₂×C₄×C₁₅₇ ▷ C₂²×C₁₅₇ ▷ C₁₅₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄×M₁₆

subgroup

G/H

center

C₂×C₄

#14578931

commutator

C₂×C₄×C₁₅₇

C₄²

Frattini

C₂×C₄²

C₂×D₁₅₇

Fitting

C₄×M₁₆×C₁₅₇

C₂

socle

C₂²×C₁₅₇

#95

prime radical

C₂³×C₁₅₇

C₄⋊C₄

cosocle

C₂×C₄²×C₁₅₇

C₂²

hypercenter

C₄×M₁₆

D₁₅₇

hypocenter

C₁₅₇

#1299

LCS

G ▷ C₂×C₄×C₁₅₇ ▷ C₂²×C₁₅₇ ▷ C₁₅₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄×M₁₆

N	H ≅ G/N	type
55. C₁₅₇	#1299	semidirect

H ≅ G/N

type

55. C₁₅₇

#1299

semidirect

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

subgroup	index
#23757326	2
#23757297	2
C₄×M₁₆×C₁₅₇	2
#1299	157

subgroup

index

#23757326

#23757297

C₄×M₁₆×C₁₅₇

#1299

157

Subgroups — #34098026

Conjugacy classes of subgroups

Complements

Maximal subgroups