Group #34115000 — subgroups

Subgroups — #34115000

Order 20096.

136 conjugacy classes of subgroups; only the 44 normal subgroups are shown.

normal subgroups only

This group has 136 conjugacy classes of subgroups — too many to draw. Only its 44 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	#23758296	C₂	1	1	yes	maximal, characteristic
2	D₁₅₇⋊(C₄⋊C₈)	C₂	1	1	yes	maximal, characteristic
3	#23757328	C₂	1	1	yes	maximal, characteristic
4	C₂×D₁₅₇⋊C₈	C₄	1	2	yes	—
5	#19097838	—	2	1	no	—
6	#19097838	—	2	1	no	—
7	C₄⋊#11929441	—	2	1	no	—
8	#19097828	—	2	1	no	—
9	#19097826	C₄	1	2	yes	—
10	C₄²×D₁₅₇	C₂²	1	1	yes	characteristic
11	#19097700	—	2	1	no	—
12	#19097683	—	2	1	no	—
13	#19097682	C₄	1	1	yes	characteristic
14	#97×C₁₅₇	C₄	1	1	yes	characteristic
15	D₁₅₇⋊C₈	—	4	2	no	—
16	C₂×#11929441	—	2	2	no	—
17	C₂×#11929441	—	2	2	no	—
18	C₂×C₄×D₁₅₇	C₂×C₄	1	1	yes	characteristic
19	C₂×C₄×D₁₅₇	D₄	1	1	yes	characteristic
20	C₂×C₄×D₁₅₇	Q₈	1	1	yes	characteristic
21	C₂×#11929434	—	2	1	no	—
22	C₄×Dic₁₅₇	C₂×C₄	1	1	yes	characteristic
23	C₄×Dic₁₅₇	D₄	1	1	yes	characteristic
24	C₄×Dic₁₅₇	D₄	1	1	yes	characteristic
25	C₂×C₈×C₁₅₇	—	2	1	no	—
26	C₄²×C₁₅₇	C₂×C₄	1	1	yes	characteristic
27	#11929441	—	4	2	no	—
28	#11929441	—	4	2	no	—
29	C₂²×D₁₅₇	C₄⋊C₄	1	1	yes	characteristic
30	C₂×Dic₁₅₇	C₂²⋊C₄	1	1	yes	characteristic
31	C₂×Dic₁₅₇	C₂²⋊C₄	1	1	yes	characteristic
32	C₂×Dic₁₅₇	C₄⋊C₄	1	1	yes	characteristic
33	C₄×D₁₅₇	—	2	2	no	—
34	C₄×D₁₅₇	—	2	1	no	—
35	C₄×D₁₅₇	—	2	1	no	—
36	C₄×D₁₅₇	—	4	1	no	—
37	#11929434	—	4	1	no	—
38	C₂×C₄×C₁₅₇	C₄²	1	1	yes	characteristic
39	C₂×C₄×C₁₅₇	C₂²⋊C₄	1	1	yes	characteristic
40	C₂×C₄×C₁₅₇	C₄⋊C₄	1	1	yes	characteristic
41	C₈×C₁₅₇	—	4	1	no	—
42	C₂²×C₁₅₇	#95	1	1	yes	characteristic
43	C₂×D₁₅₇	—	2	1	no	—
44	C₂×D₁₅₇	—	2	1	no	—
45	C₂×D₁₅₇	#108	1	2	yes	—
46	C₄×C₁₅₇	—	2	1	no	—
47	C₄×C₁₅₇	—	2	1	no	—
48	C₄×C₁₅₇	—	2	1	no	—
49	Dic₁₅₇	—	2	1	no	—
50	Dic₁₅₇	—	2	1	no	—
51	Dic₁₅₇	#100	1	1	yes	characteristic
52	Dic₁₅₇	#101	1	1	yes	characteristic
53	C₂×C₁₅₇	H(3,ℤ/4)	1	1	yes	characteristic
54	C₂×C₁₅₇	#341	1	1	yes	characteristic
55	C₂×C₁₅₇	#343	1	1	yes	characteristic
56	D₁₅₇	—	2	2	no	—
57	C₁₅₇	#1290	1	1	yes	characteristic, complemented, Sylow
58	#1290	—	157	1	no	maximal, Sylow
59	#432	—	157	1	no	—
60	#432	—	157	1	no	—
61	C₂×C₄⋊C₈	—	157	1	no	—
62	C₂²×C₈	—	157	2	no	—
63	C₂×C₄²	—	157	1	no	—
64	C₄⋊C₈	—	314	1	no	—
65	C₄⋊C₈	—	314	1	no	—
66	C₄⋊C₈	—	314	1	no	—
67	C₄⋊C₈	—	314	1	no	—
68	C₂²⋊C₈	—	314	1	no	—
69	C₂²⋊C₈	—	314	1	no	—
70	#97	—	157	2	no	—
71	#97	—	157	1	no	—
72	#97	#10597845	1	1	yes	characteristic
73	C₂²×C₄	—	157	1	no	—
74	C₂²×C₄	—	157	1	no	—
75	C₂²×C₄	—	157	1	no	—
76	C₂×C₈	—	2	1	no	—
77	C₂×C₈	—	314	2	no	—
78	C₂×C₈	—	314	2	no	—
79	C₂×C₈	—	314	1	no	—
80	C₂×C₈	—	628	2	no	—
81	C₄²	—	157	1	no	—
82	C₄²	—	157	1	no	—
83	C₄²	—	157	1	no	—
84	C₄²	C₂×#10597845	1	1	yes	characteristic
85	C₂³	—	157	1	no	—
86	C₂×C₄	—	314	2	no	—
87	C₂×C₄	—	314	1	no	—
88	C₂×C₄	—	314	1	no	—
89	C₂×C₄	—	628	1	no	—
90	C₂×C₄	—	157	1	no	—
91	C₂×C₄	—	157	1	no	—
92	C₂×C₄	—	157	1	no	—
93	C₂×C₄	C₄×#10597845	1	1	yes	characteristic
94	C₂×C₄	C₂²⋊#10597845	1	1	yes	characteristic
95	C₂×C₄	C₄⋊#10597845	1	1	yes	characteristic
96	C₈	—	4	1	no	—
97	C₈	—	628	1	no	—
98	C₈	—	628	2	no	—
99	C₈	—	628	2	no	—
100	C₂²	—	314	1	no	—
101	C₂²	—	314	1	no	—
102	C₂²	—	157	2	no	—
103	C₂²	(C₂×#10597845)⋊C₄	1	1	yes	characteristic
104	C₄	—	2	1	no	—
105	C₄	—	2	1	no	—
106	C₄	—	2	1	no	—
107	C₄	—	314	1	no	—
108	C₄	—	314	1	no	—
109	C₄	—	157	1	no	—
110	C₄	—	157	1	no	—
111	C₂	—	314	2	no	—
112	C₂	#23758140	1	1	yes	characteristic
113	C₂	#23758148	1	1	yes	characteristic
114	C₂	#23758152	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#23758296

C₂

yes

maximal, characteristic

D₁₅₇⋊(C₄⋊C₈)

C₂

yes

maximal, characteristic

#23757328

C₂

yes

maximal, characteristic

C₂×D₁₅₇⋊C₈

C₄

yes

—

—

—

—

—

C₄

yes

—

C₄²×D₁₅₇

C₂²

yes

—

—

C₄

yes

characteristic

#97×C₁₅₇

C₄

yes

characteristic

D₁₅₇⋊C₈

—

C₂×#11929441

—

C₂×#11929441

—

C₂×C₄×D₁₅₇

C₂×C₄

yes

characteristic

C₂×C₄×D₁₅₇

D₄

yes

characteristic

C₂×C₄×D₁₅₇

Q₈

yes

characteristic

C₂×#11929434

—

C₄×Dic₁₅₇

C₂×C₄

yes

characteristic

C₄×Dic₁₅₇

D₄

yes

characteristic

C₄×Dic₁₅₇

D₄

yes

characteristic

C₂×C₈×C₁₅₇

—

C₄²×C₁₅₇

C₂×C₄

yes

characteristic

#11929441

—

#11929441

—

C₂²×D₁₅₇

C₄⋊C₄

yes

characteristic

C₂×Dic₁₅₇

C₂²⋊C₄

yes

characteristic

C₂×Dic₁₅₇

C₂²⋊C₄

yes

characteristic

C₂×Dic₁₅₇

C₄⋊C₄

yes

characteristic

C₄×D₁₅₇

—

C₄×D₁₅₇

—

C₄×D₁₅₇

—

C₄×D₁₅₇

—

#11929434

—

C₂×C₄×C₁₅₇

C₄²

yes

characteristic

C₂×C₄×C₁₅₇

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₅₇

C₄⋊C₄

yes

characteristic

C₈×C₁₅₇

—

C₂²×C₁₅₇

#95

yes

characteristic

C₂×D₁₅₇

—

C₂×D₁₅₇

—

C₂×D₁₅₇

#108

yes

—

C₄×C₁₅₇

—

C₄×C₁₅₇

—

C₄×C₁₅₇

—

Dic₁₅₇

—

Dic₁₅₇

—

Dic₁₅₇

yes

Dic₁₅₇

yes

C₂×C₁₅₇

yes

C₂×C₁₅₇

yes

C₂×C₁₅₇

yes

D₁₅₇

—

C₁₅₇

#1290

yes

characteristic, complemented, Sylow

#1290

—

157

maximal, Sylow

#432

—

157

—

#432

—

157

—

C₂×C₄⋊C₈

—

157

—

C₂²×C₈

—

157

—

C₂×C₄²

—

157

—

C₄⋊C₈

—

314

—

C₄⋊C₈

—

314

—

C₄⋊C₈

—

314

—

C₄⋊C₈

—

314

—

C₂²⋊C₈

—

314

—

C₂²⋊C₈

—

314

—

#97

—

157

—

#97

—

157

—

#97

#10597845

yes

characteristic

C₂²×C₄

—

157

—

C₂²×C₄

—

157

—

C₂²×C₄

—

157

—

C₂×C₈

—

C₂×C₈

—

314

—

C₂×C₈

—

314

—

C₂×C₈

—

314

—

C₂×C₈

—

628

—

C₄²

—

157

—

C₄²

—

157

—

C₄²

—

157

—

C₄²

C₂×#10597845

yes

characteristic

C₂³

—

157

—

C₂×C₄

—

314

—

C₂×C₄

—

314

—

C₂×C₄

—

314

—

C₂×C₄

—

628

—

C₂×C₄

—

157

—

C₂×C₄

—

157

—

C₂×C₄

—

157

—

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

C₈

—

C₈

—

628

—

C₈

—

628

—

C₈

—

628

—

100

C₂²

—

314

—

101

C₂²

—

314

—

102

C₂²

—

157

—

103

C₂²

(C₂×#10597845)⋊C₄

yes

characteristic

104

C₄

—

105

C₄

—

106

C₄

—

107

C₄

—

314

—

108

C₄

—

314

—

109

C₄

—

157

—

110

C₄

—

157

—

111

C₂

—

314

—

112

C₂

#23758140

yes

characteristic

113

C₂

#23758148

yes

characteristic

114

C₂

#23758152

yes

characteristic

	subgroup	G/H
center	C₂²	(C₂×#10597845)⋊C₄
commutator	C₂×C₄×C₁₅₇	C₄²
Frattini	C₄²	C₂×#10597845
Fitting	#97×C₁₅₇	C₄
socle	C₂²×C₁₅₇	#95
prime radical	C₂²×D₁₅₇	C₄⋊C₄
cosocle	C₄²×D₁₅₇	C₂²
hypercenter	#97	#10597845
hypocenter	C₁₅₇	#1290
LCS	G ▷ C₂×C₄×C₁₅₇ ▷ C₂×C₁₅₇ ▷ C₁₅₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #97

subgroup

G/H

center

C₂²

(C₂×#10597845)⋊C₄

commutator

C₂×C₄×C₁₅₇

C₄²

Frattini

C₄²

C₂×#10597845

Fitting

#97×C₁₅₇

C₄

socle

C₂²×C₁₅₇

#95

prime radical

C₂²×D₁₅₇

C₄⋊C₄

cosocle

C₄²×D₁₅₇

C₂²

C₁₅₇

G ▷ C₂×C₄×C₁₅₇ ▷ C₂×C₁₅₇ ▷ C₁₅₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #97

N	H ≅ G/N	type
57. C₁₅₇	#1290	semidirect

H ≅ G/N

type

57. C₁₅₇

#1290

semidirect

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

subgroup	index
#23758296	2
D₁₅₇⋊(C₄⋊C₈)	2
#23757328	2
#1290	157

subgroup

index

#23758296

D₁₅₇⋊(C₄⋊C₈)

#23757328

#1290

157

Subgroups — #34115000

Conjugacy classes of subgroups

Complements

Maximal subgroups