Group #34096052 — subgroups

Subgroups — #1694×C₁₅₇ #34096052

Order 20096.

152 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 152 conjugacy classes of subgroups — too many to draw. Only its 68 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	#500×C₁₅₇	C₂	1	1	yes	maximal, characteristic
2	#462×C₁₅₇	C₂	1	2	yes	maximal
3	#446×C₁₅₇	C₂	1	1	yes	maximal, characteristic
4	#334×C₁₅₇	C₂	1	1	yes	maximal, characteristic
5	#332×C₁₅₇	C₂	1	2	yes	maximal
6	C₂×Q₁₆×C₁₅₇	—	2	2	no	—
7	C₄⋊Q₈×C₁₅₇	C₂²	1	2	yes	—
8	C₄×Q₈×C₁₅₇	C₂²	1	1	yes	characteristic
9	#107×C₁₅₇	—	2	1	no	—
10	#107×C₁₅₇	—	2	1	no	—
11	C₄⋊C₈×C₁₅₇	—	2	1	no	—
12	C₄⋊C₈×C₁₅₇	C₂²	1	1	yes	characteristic
13	C₄⋊C₈×C₁₅₇	C₂²	1	2	yes	—
14	#103×C₁₅₇	—	2	2	no	—
15	#97×C₁₅₇	—	2	1	no	—
16	C₄×C₈×C₁₅₇	C₂²	1	1	yes	characteristic
17	C₂×Q₈×C₁₅₇	—	2	2	no	—
18	C₂×Q₈×C₁₅₇	D₄	1	1	yes	characteristic
19	Q₁₆×C₁₅₇	—	4	4	no	—
20	C₂×C₈×C₁₅₇	—	2	1	no	—
21	C₂×C₈×C₁₅₇	—	2	2	no	—
22	C₂×C₈×C₁₅₇	D₄	1	1	yes	characteristic
23	C₂×C₈×C₁₅₇	D₄	1	1	yes	characteristic
24	C₄⋊C₄×C₁₅₇	—	2	1	no	—
25	C₄⋊C₄×C₁₅₇	—	2	2	no	—
26	C₄⋊C₄×C₁₅₇	—	2	2	no	—
27	C₄⋊C₄×C₁₅₇	D₄	1	1	yes	characteristic
28	C₄²×C₁₅₇	—	2	1	no	—
29	C₄²×C₁₅₇	C₂³	1	1	yes	characteristic
30	Q₈×C₁₅₇	—	2	2	no	—
31	Q₈×C₁₅₇	—	4	2	no	—
32	Q₈×C₁₅₇	—	4	2	no	—
33	C₂×C₄×C₁₅₇	—	2	1	no	—
34	C₂×C₄×C₁₅₇	—	2	1	no	—
35	C₂×C₄×C₁₅₇	—	4	2	no	—
36	C₂×C₄×C₁₅₇	C₂×D₄	1	1	yes	characteristic
37	C₂×C₄×C₁₅₇	C₂×D₄	1	1	yes	characteristic
38	C₂×C₄×C₁₅₇	#41	1	1	yes	characteristic
39	C₈×C₁₅₇	—	2	1	no	—
40	C₈×C₁₅₇	—	4	2	no	—
41	C₈×C₁₅₇	—	4	1	no	—
42	C₈×C₁₅₇	Q₁₆	1	2	yes	complemented
43	C₂²×C₁₅₇	#121	1	1	yes	characteristic
44	C₄×C₁₅₇	—	2	1	no	—
45	C₄×C₁₅₇	—	4	1	no	—
46	C₄×C₁₅₇	—	4	1	no	—
47	C₄×C₁₅₇	—	8	2	no	—
48	C₄×C₁₅₇	C₂×Q₁₆	1	1	yes	characteristic
49	C₄×C₁₅₇	AGL(1,ℤ/8)	1	2	yes	—
50	C₄×C₁₅₇	#137	1	1	yes	characteristic
51	C₂×C₁₅₇	#462	1	1	yes	characteristic
52	C₂×C₁₅₇	#469	1	1	yes	characteristic
53	C₂×C₁₅₇	#473	1	1	yes	characteristic
54	C₁₅₇	#1694	1	1	yes	characteristic, direct product factor, Sylow
55	#1694	C₁₅₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
56	#500	C₂×C₁₅₇	1	1	yes	characteristic
57	#462	C₂×C₁₅₇	1	2	yes	—
58	#446	C₂×C₁₅₇	1	1	yes	characteristic
59	#334	C₂×C₁₅₇	1	1	yes	characteristic
60	#332	C₂×C₁₅₇	1	2	yes	—
61	C₂×Q₁₆	—	2	2	no	—
62	C₄⋊Q₈	C₂²×C₁₅₇	1	2	yes	—
63	C₄×Q₈	C₂²×C₁₅₇	1	1	yes	characteristic
64	#107	—	2	1	no	—
65	#107	—	2	1	no	—
66	C₄⋊C₈	—	2	1	no	—
67	C₄⋊C₈	C₂²×C₁₅₇	1	1	yes	characteristic
68	C₄⋊C₈	C₂²×C₁₅₇	1	2	yes	—
69	#103	—	2	2	no	—
70	#97	—	2	1	no	—
71	C₄×C₈	C₂²×C₁₅₇	1	1	yes	characteristic
72	C₂×Q₈	—	2	2	no	—
73	C₂×Q₈	D₄×C₁₅₇	1	1	yes	characteristic
74	Q₁₆	—	4	4	no	—
75	C₂×C₈	—	2	1	no	—
76	C₂×C₈	—	2	2	no	—
77	C₂×C₈	D₄×C₁₅₇	1	1	yes	characteristic
78	C₂×C₈	D₄×C₁₅₇	1	1	yes	characteristic
79	C₄⋊C₄	—	2	1	no	—
80	C₄⋊C₄	—	2	2	no	—
81	C₄⋊C₄	—	2	2	no	—
82	C₄⋊C₄	D₄×C₁₅₇	1	1	yes	characteristic
83	C₄²	—	2	1	no	—
84	C₄²	C₂³×C₁₅₇	1	1	yes	characteristic
85	Q₈	—	2	2	no	—
86	Q₈	—	4	2	no	—
87	Q₈	—	4	2	no	—
88	C₂×C₄	—	2	1	no	—
89	C₂×C₄	—	2	1	no	—
90	C₂×C₄	—	4	2	no	—
91	C₂×C₄	C₂×D₄×C₁₅₇	1	1	yes	characteristic
92	C₂×C₄	C₂×D₄×C₁₅₇	1	1	yes	characteristic
93	C₂×C₄	#41×C₁₅₇	1	1	yes	characteristic
94	C₈	—	2	1	no	—
95	C₈	—	4	2	no	—
96	C₈	—	4	1	no	—
97	C₈	Q₁₆×C₁₅₇	1	2	yes	complemented
98	C₂²	#121×C₁₅₇	1	1	yes	characteristic
99	C₄	—	2	1	no	—
100	C₄	—	4	1	no	—
101	C₄	—	4	1	no	—
102	C₄	—	8	2	no	—
103	C₄	C₂×Q₁₆×C₁₅₇	1	1	yes	characteristic
104	C₄	AGL(1,ℤ/8)×C₁₅₇	1	2	yes	—
105	C₄	#137×C₁₅₇	1	1	yes	characteristic
106	C₂	#462×C₁₅₇	1	1	yes	characteristic
107	C₂	#469×C₁₅₇	1	1	yes	characteristic
108	C₂	#473×C₁₅₇	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#500×C₁₅₇

C₂

yes

maximal, characteristic

#462×C₁₅₇

C₂

yes

maximal

#446×C₁₅₇

C₂

yes

maximal, characteristic

#334×C₁₅₇

C₂

yes

maximal, characteristic

#332×C₁₅₇

C₂

yes

maximal

C₂×Q₁₆×C₁₅₇

—

C₄⋊Q₈×C₁₅₇

C₂²

yes

—

C₄×Q₈×C₁₅₇

C₂²

yes

characteristic

#107×C₁₅₇

—

#107×C₁₅₇

—

C₄⋊C₈×C₁₅₇

—

C₄⋊C₈×C₁₅₇

C₂²

yes

characteristic

C₄⋊C₈×C₁₅₇

C₂²

yes

—

#103×C₁₅₇

—

#97×C₁₅₇

—

C₄×C₈×C₁₅₇

C₂²

yes

characteristic

C₂×Q₈×C₁₅₇

—

C₂×Q₈×C₁₅₇

D₄

yes

characteristic

Q₁₆×C₁₅₇

—

C₂×C₈×C₁₅₇

—

C₂×C₈×C₁₅₇

—

C₂×C₈×C₁₅₇

D₄

yes

characteristic

C₂×C₈×C₁₅₇

D₄

yes

characteristic

C₄⋊C₄×C₁₅₇

—

C₄⋊C₄×C₁₅₇

—

C₄⋊C₄×C₁₅₇

—

C₄⋊C₄×C₁₅₇

D₄

yes

characteristic

C₄²×C₁₅₇

—

C₄²×C₁₅₇

C₂³

yes

characteristic

Q₈×C₁₅₇

—

Q₈×C₁₅₇

—

Q₈×C₁₅₇

—

C₂×C₄×C₁₅₇

—

C₂×C₄×C₁₅₇

—

C₂×C₄×C₁₅₇

—

C₂×C₄×C₁₅₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₅₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₅₇

#41

yes

characteristic

C₈×C₁₅₇

—

C₈×C₁₅₇

—

C₈×C₁₅₇

—

C₈×C₁₅₇

Q₁₆

yes

complemented

C₂²×C₁₅₇

#121

yes

characteristic

C₄×C₁₅₇

—

C₄×C₁₅₇

—

C₄×C₁₅₇

—

C₄×C₁₅₇

—

C₄×C₁₅₇

C₂×Q₁₆

yes

characteristic

C₄×C₁₅₇

AGL(1,ℤ/8)

yes

—

C₄×C₁₅₇

yes

C₂×C₁₅₇

yes

C₂×C₁₅₇

yes

C₂×C₁₅₇

yes

C₁₅₇

yes

characteristic, direct product factor, Sylow

#1694

C₁₅₇

yes

maximal, characteristic, direct product factor, Sylow

#500

C₂×C₁₅₇

yes

characteristic

#462

C₂×C₁₅₇

yes

—

#446

C₂×C₁₅₇

yes

characteristic

#334

C₂×C₁₅₇

yes

characteristic

#332

C₂×C₁₅₇

yes

—

C₂×Q₁₆

—

C₄⋊Q₈

C₂²×C₁₅₇

yes

—

C₄×Q₈

C₂²×C₁₅₇

yes

characteristic

#107

—

#107

—

C₄⋊C₈

—

C₄⋊C₈

C₂²×C₁₅₇

yes

characteristic

C₄⋊C₈

C₂²×C₁₅₇

yes

—

#103

—

#97

—

C₄×C₈

C₂²×C₁₅₇

yes

characteristic

C₂×Q₈

—

C₂×Q₈

D₄×C₁₅₇

yes

characteristic

Q₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

D₄×C₁₅₇

yes

characteristic

C₂×C₈

D₄×C₁₅₇

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

D₄×C₁₅₇

yes

characteristic

C₄²

—

C₄²

C₂³×C₁₅₇

yes

characteristic

Q₈

—

Q₈

—

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂×D₄×C₁₅₇

yes

characteristic

C₂×C₄

C₂×D₄×C₁₅₇

yes

characteristic

C₂×C₄

#41×C₁₅₇

yes

characteristic

C₈

—

C₈

—

C₈

—

C₈

Q₁₆×C₁₅₇

yes

complemented

C₂²

#121×C₁₅₇

yes

characteristic

C₄

—

100

C₄

—

101

C₄

—

102

C₄

—

103

C₄

C₂×Q₁₆×C₁₅₇

yes

characteristic

104

C₄

AGL(1,ℤ/8)×C₁₅₇

yes

—

105

C₄

#137×C₁₅₇

yes

characteristic

106

C₂

#462×C₁₅₇

yes

characteristic

107

C₂

#469×C₁₅₇

yes

characteristic

108

C₂

#473×C₁₅₇

yes

characteristic

	subgroup	G/H
center, socle, prime radical	C₂²×C₁₅₇	#121
commutator, Frattini, cosocle	C₄²	C₂³×C₁₅₇
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₁₅₇ ◁ C₄²×C₁₅₇ ◁ G

subgroup

G/H

center, socle, prime radical

C₂²×C₁₅₇

#121

commutator, Frattini, cosocle

C₄²

C₂³×C₁₅₇

LCS

G ▷ C₄² ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂²×C₁₅₇ ◁ C₄²×C₁₅₇ ◁ G

N	H ≅ G/N	type
42. C₈×C₁₅₇	Q₁₆	semidirect
54. C₁₅₇	#1694	direct (N × H)
55. #1694	C₁₅₇	direct (N × H)
97. C₈	Q₁₆×C₁₅₇	semidirect

H ≅ G/N

type

42. C₈×C₁₅₇

Q₁₆

semidirect

54. C₁₅₇

#1694

direct (N × H)

55. #1694

C₁₅₇

direct (N × H)

97. C₈

Q₁₆×C₁₅₇

semidirect

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

subgroup	index
#500×C₁₅₇	2
#462×C₁₅₇	2
#446×C₁₅₇	2
#334×C₁₅₇	2
#332×C₁₅₇	2
#1694	157

subgroup

index

#500×C₁₅₇

#462×C₁₅₇

#446×C₁₅₇

#334×C₁₅₇

#332×C₁₅₇

#1694

157

Subgroups — #1694×C₁₅₇ #34096052

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #1694×C157 #34096052

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #1694×C₁₅₇ #34096052