Group #34114957 — subgroups

Subgroups — #34114957

Order 20096.

166 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 166 conjugacy classes of subgroups — too many to draw. Only its 54 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	#23758139	C₂	1	2	yes	maximal
2	#23757178	C₂	1	1	yes	maximal, characteristic
3	C₂²×#11929441	—	2	2	no	—
4	C₂²×Dic₃₁₄	C₄	1	1	yes	characteristic
5	C₂×C₄×Dic₁₅₇	C₂²	1	1	yes	characteristic
6	#19097678	—	2	2	no	—
7	C₂×C₄⋊C₄×C₁₅₇	C₄	1	1	yes	characteristic
8	C₂×#11929441	—	2	4	no	—
9	C₂×#11929441	—	4	2	no	—
10	C₂×#11929441	—	4	2	no	—
11	C₂×Dic₃₁₄	—	2	1	no	—
12	C₂×Dic₃₁₄	—	4	1	no	—
13	C₂×Dic₃₁₄	—	4	1	no	—
14	C₂×Dic₃₁₄	C₈	1	2	yes	complemented
15	C₂²×Dic₁₅₇	—	2	1	no	—
16	C₂²×Dic₁₅₇	D₄	1	2	yes	—
17	C₄×Dic₁₅₇	—	4	1	no	—
18	C₂²×C₄×C₁₅₇	—	2	1	no	—
19	C₂²×C₄×C₁₅₇	C₂×C₄	1	1	yes	characteristic
20	C₄⋊C₄×C₁₅₇	—	4	1	no	—
21	#11929441	—	4	4	no	—
22	Dic₃₁₄	—	4	2	no	—
23	Dic₃₁₄	—	4	2	no	—
24	C₂×Dic₁₅₇	—	2	2	no	—
25	C₂×Dic₁₅₇	—	2	2	no	—
26	C₂×Dic₁₅₇	—	2	2	no	—
27	C₂×Dic₁₅₇	—	4	1	no	—
28	C₂×Dic₁₅₇	—	4	1	no	—
29	C₂×Dic₁₅₇	SD₁₆	1	2	yes	—
30	C₂×Dic₁₅₇	Q₁₆	1	2	yes	—
31	C₂³×C₁₅₇	C₂²⋊C₄	1	1	yes	characteristic
32	C₂×C₄×C₁₅₇	—	2	1	no	—
33	C₂×C₄×C₁₅₇	—	2	1	no	—
34	C₂×C₄×C₁₅₇	—	4	1	no	—
35	C₂×C₄×C₁₅₇	—	4	2	no	—
36	C₂×C₄×C₁₅₇	C₂×C₈	1	1	yes	characteristic
37	C₂×C₄×C₁₅₇	M₁₆	1	1	yes	characteristic
38	C₂²×C₁₅₇	C₂²⋊C₈	1	1	yes	characteristic
39	C₂²×C₁₅₇	#99	1	1	yes	characteristic
40	C₂²×C₁₅₇	#100	1	1	yes	characteristic
41	C₂²×C₁₅₇	#103	1	2	yes	—
42	C₂²×C₁₅₇	#104	1	2	yes	—
43	C₄×C₁₅₇	—	2	1	no	—
44	C₄×C₁₅₇	—	2	1	no	—
45	C₄×C₁₅₇	—	8	1	no	—
46	Dic₁₅₇	—	2	2	no	—
47	Dic₁₅₇	—	2	2	no	—
48	Dic₁₅₇	—	4	2	no	—
49	C₂×C₁₅₇	#323	1	1	yes	characteristic
50	C₂×C₁₅₇	Q₈⋊C₈	1	2	yes	—
51	C₂×C₁₅₇	#328	1	2	yes	—
52	C₂×C₁₅₇	#329	1	1	yes	characteristic
53	C₂×C₁₅₇	#333	1	1	yes	characteristic
54	C₁₅₇	#1272	1	1	yes	characteristic, complemented, Sylow
55	#1272	—	157	1	no	maximal, Sylow
56	#391	—	157	1	no	—
57	#336	—	157	2	no	—
58	C₂²×Q₈	—	157	1	no	—
59	C₂²×C₈	—	314	2	no	—
60	C₂×C₄⋊C₄	#10597845	1	1	yes	characteristic
61	C₂×C₄²	—	157	1	no	—
62	#95	—	314	2	no	—
63	C₂×Q₈	—	314	1	no	—
64	C₂×Q₈	—	628	1	no	—
65	C₂×Q₈	—	628	1	no	—
66	C₂×Q₈	—	157	2	no	—
67	C₂²×C₄	—	2	1	no	—
68	C₂²×C₄	—	314	1	no	—
69	C₂²×C₄	—	157	2	no	—
70	C₂²×C₄	C₂×#10597845	1	1	yes	characteristic
71	C₂×C₈	—	314	4	no	—
72	C₂×C₈	—	628	2	no	—
73	C₂×C₈	—	628	2	no	—
74	C₄⋊C₄	—	4	1	no	—
75	C₄²	—	628	1	no	—
76	C₂³	C₂²⋊#10597845	1	1	yes	characteristic
77	Q₈	—	628	2	no	—
78	Q₈	—	628	2	no	—
79	C₂×C₄	—	2	1	no	—
80	C₂×C₄	—	2	1	no	—
81	C₂×C₄	—	4	1	no	—
82	C₂×C₄	—	4	2	no	—
83	C₂×C₄	—	314	2	no	—
84	C₂×C₄	—	314	2	no	—
85	C₂×C₄	—	314	2	no	—
86	C₂×C₄	—	628	1	no	—
87	C₂×C₄	—	628	1	no	—
88	C₂×C₄	—	157	2	no	—
89	C₂×C₄	—	157	2	no	—
90	C₂×C₄	D₁₅₇⋊C₈	1	1	yes	characteristic
91	C₂×C₄	#14578964	1	1	yes	characteristic
92	C₈	—	628	4	no	—
93	C₂²	#19097828	1	1	yes	characteristic
94	C₂²	#19097830	1	1	yes	characteristic
95	C₂²	#19097831	1	1	yes	characteristic
96	C₂²	#19097834	1	2	yes	—
97	C₂²	#19097835	1	2	yes	—
98	C₄	—	2	1	no	—
99	C₄	—	2	1	no	—
100	C₄	—	8	1	no	—
101	C₄	—	314	2	no	—
102	C₄	—	314	2	no	—
103	C₄	—	628	2	no	—
104	C₂	#23758117	1	1	yes	characteristic
105	C₂	#23758123	1	2	yes	—
106	C₂	#23758126	1	2	yes	—
107	C₂	#23758128	1	1	yes	characteristic
108	C₂	#23758133	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#23758139

C₂

yes

maximal

#23757178

C₂

yes

maximal, characteristic

C₂²×#11929441

—

C₂²×Dic₃₁₄

C₄

yes

characteristic

C₂×C₄×Dic₁₅₇

C₂²

yes

characteristic

#19097678

—

C₂×C₄⋊C₄×C₁₅₇

C₄

yes

—

—

—

C₂×Dic₃₁₄

—

C₂×Dic₃₁₄

—

C₂×Dic₃₁₄

—

C₂×Dic₃₁₄

C₈

yes

complemented

C₂²×Dic₁₅₇

—

C₂²×Dic₁₅₇

D₄

yes

—

C₄×Dic₁₅₇

—

C₂²×C₄×C₁₅₇

—

C₂²×C₄×C₁₅₇

C₂×C₄

yes

characteristic

C₄⋊C₄×C₁₅₇

—

#11929441

—

Dic₃₁₄

—

Dic₃₁₄

—

C₂×Dic₁₅₇

—

C₂×Dic₁₅₇

—

C₂×Dic₁₅₇

—

C₂×Dic₁₅₇

—

C₂×Dic₁₅₇

—

C₂×Dic₁₅₇

SD₁₆

yes

—

C₂×Dic₁₅₇

Q₁₆

yes

—

C₂³×C₁₅₇

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₅₇

—

C₂×C₄×C₁₅₇

—

C₂×C₄×C₁₅₇

—

C₂×C₄×C₁₅₇

—

C₂×C₄×C₁₅₇

C₂×C₈

yes

characteristic

C₂×C₄×C₁₅₇

M₁₆

yes

characteristic

C₂²×C₁₅₇

C₂²⋊C₈

yes

C₂²×C₁₅₇

yes

C₂²×C₁₅₇

yes

C₂²×C₁₅₇

yes

—

C₂²×C₁₅₇

#104

yes

—

C₄×C₁₅₇

—

C₄×C₁₅₇

—

C₄×C₁₅₇

—

Dic₁₅₇

—

Dic₁₅₇

—

Dic₁₅₇

—

C₂×C₁₅₇

#323

yes

characteristic

C₂×C₁₅₇

Q₈⋊C₈

yes

—

C₂×C₁₅₇

#328

yes

—

C₂×C₁₅₇

yes

C₂×C₁₅₇

yes

C₁₅₇

yes

characteristic, complemented, Sylow

#1272

—

157

maximal, Sylow

#391

—

157

—

#336

—

157

—

C₂²×Q₈

—

157

—

C₂²×C₈

—

314

—

C₂×C₄⋊C₄

#10597845

yes

characteristic

C₂×C₄²

—

157

—

#95

—

314

—

C₂×Q₈

—

314

—

C₂×Q₈

—

628

—

C₂×Q₈

—

628

—

C₂×Q₈

—

157

—

C₂²×C₄

—

C₂²×C₄

—

314

—

C₂²×C₄

—

157

—

C₂²×C₄

C₂×#10597845

yes

characteristic

C₂×C₈

—

314

—

C₂×C₈

—

628

—

C₂×C₈

—

628

—

C₄⋊C₄

—

C₄²

—

628

—

C₂³

C₂²⋊#10597845

yes

characteristic

Q₈

—

628

—

Q₈

—

628

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

314

—

C₂×C₄

—

314

—

C₂×C₄

—

314

—

C₂×C₄

—

628

—

C₂×C₄

—

628

—

C₂×C₄

—

157

—

C₂×C₄

—

157

—

C₂×C₄

D₁₅₇⋊C₈

yes

characteristic

C₂×C₄

#14578964

yes

characteristic

C₈

—

628

—

C₂²

yes

C₂²

yes

C₂²

yes

C₂²

yes

—

C₂²

#19097835

yes

—

C₄

—

C₄

—

100

C₄

—

101

C₄

—

314

—

102

C₄

—

314

—

103

C₄

—

628

—

104

C₂

#23758117

yes

characteristic

105

C₂

#23758123

yes

—

106

C₂

#23758126

yes

—

107

C₂

#23758128

yes

characteristic

108

C₂

#23758133

yes

characteristic

	subgroup	G/H
center	C₂³	C₂²⋊#10597845
commutator	C₂×C₄×C₁₅₇	C₂×C₈
Frattini	C₂²×C₄	C₂×#10597845
Fitting	C₂×C₄⋊C₄×C₁₅₇	C₄
socle, prime radical	C₂³×C₁₅₇	C₂²⋊C₄
cosocle	C₂×C₄×Dic₁₅₇	C₂²
hypercenter	C₂×C₄⋊C₄	#10597845
hypocenter	C₁₅₇	#1272
LCS	G ▷ C₂×C₄×C₁₅₇ ▷ C₂²×C₁₅₇ ▷ C₁₅₇
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×C₄⋊C₄

subgroup

G/H

center

C₂³

C₂²⋊#10597845

commutator

C₂×C₄×C₁₅₇

C₂×C₈

Frattini

C₂²×C₄

C₂×#10597845

Fitting

C₂×C₄⋊C₄×C₁₅₇

C₄

socle, prime radical

C₂³×C₁₅₇

C₂²⋊C₄

cosocle

C₂×C₄×Dic₁₅₇

C₂²

C₂×C₄⋊C₄

C₁₅₇

G ▷ C₂×C₄×C₁₅₇ ▷ C₂²×C₁₅₇ ▷ C₁₅₇

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×C₄⋊C₄

N	H ≅ G/N	type
14. C₂×Dic₃₁₄	C₈	semidirect
54. C₁₅₇	#1272	semidirect

H ≅ G/N

type

14. C₂×Dic₃₁₄

C₈

semidirect

54. C₁₅₇

#1272

semidirect

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

subgroup	index
#23758139	2
#23757178	2
#1272	157

subgroup

index

#23758139

#23757178

#1272

157

Subgroups — #34114957

Conjugacy classes of subgroups

Complements

Maximal subgroups