Group #34100885 — subgroups

Subgroups — #34100885

Order 20096.

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

normal subgroups only

This group has 208 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	C₂×#19097707	C₂	1	1	yes	maximal, characteristic
2	#23757300	C₂	1	2	yes	maximal
3	C₂×C₈×Dic₁₅₇	C₂	1	1	yes	maximal, characteristic
4	#23757032	C₂	1	2	yes	maximal
5	#427×C₁₅₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#14578938	C₂²	1	2	yes	—
7	C₂²×#11929434	C₂²	1	1	yes	characteristic
8	C₂×C₄×Dic₁₅₇	C₂²	1	1	yes	characteristic
9	#19097707	—	2	2	no	—
10	C₄⋊#11929434	—	2	4	no	—
11	C₈×Dic₁₅₇	—	2	2	no	—
12	C₂²×C₈×C₁₅₇	C₂²	1	1	yes	characteristic
13	#117×C₁₅₇	C₂²	1	2	yes	—
14	#107×C₁₅₇	C₄	1	2	yes	complemented
15	#106×C₁₅₇	C₄	1	2	yes	complemented
16	C₂²×Dic₁₅₇	—	2	1	no	—
17	#14578938	—	2	2	no	—
18	#14578938	—	2	2	no	—
19	C₂×#11929434	—	2	1	no	—
20	C₂×#11929434	—	2	2	no	—
21	C₂×#11929434	—	2	1	no	—
22	C₂×#11929434	D₄	1	2	yes	—
23	C₄×Dic₁₅₇	—	2	1	no	—
24	C₄×Dic₁₅₇	Q₈	1	2	yes	—
25	C₂²×C₄×C₁₅₇	C₂³	1	1	yes	characteristic
26	C₂×C₈×C₁₅₇	C₂×C₄	1	1	yes	characteristic
27	C₂×C₈×C₁₅₇	C₂×C₄	1	1	yes	characteristic
28	C₂×C₈×C₁₅₇	D₄	1	2	yes	—
29	C₂×C₈×C₁₅₇	Q₈	1	2	yes	—
30	C₄⋊C₄×C₁₅₇	C₂×C₄	1	4	yes	—
31	C₂²⋊C₄×C₁₅₇	—	2	2	no	—
32	C₄²×C₁₅₇	—	2	2	no	—
33	C₂×Dic₁₅₇	—	2	4	no	—
34	C₂×Dic₁₅₇	—	2	2	no	—
35	#11929434	—	2	2	no	—
36	#11929434	—	4	2	no	—
37	C₂³×C₁₅₇	#41	1	1	yes	characteristic
38	C₂×C₄×C₁₅₇	—	2	4	no	—
39	C₂×C₄×C₁₅₇	C₂²×C₄	1	1	yes	characteristic
40	C₂×C₄×C₁₅₇	C₂×D₄	1	1	yes	characteristic
41	C₂×C₄×C₁₅₇	C₂×D₄	1	1	yes	characteristic
42	C₂×C₄×C₁₅₇	C₂×Q₈	1	1	yes	characteristic
43	C₂×C₄×C₁₅₇	C₂×Q₈	1	1	yes	characteristic
44	C₂×C₄×C₁₅₇	#41	1	1	yes	characteristic
45	C₈×C₁₅₇	C₄⋊C₄	1	2	yes	—
46	C₈×C₁₅₇	C₄⋊C₄	1	2	yes	—
47	C₂²×C₁₅₇	—	2	2	no	—
48	C₂²×C₁₅₇	C₄×D₄	1	1	yes	characteristic
49	C₂²×C₁₅₇	C₂²⋊Q₈	1	1	yes	characteristic
50	C₂²×C₁₅₇	#125	1	1	yes	characteristic
51	C₄×C₁₅₇	—	4	4	no	—
52	C₄×C₁₅₇	C₂×C₄⋊C₄	1	1	yes	characteristic
53	C₄×C₁₅₇	C₄×Q₈	1	1	yes	characteristic
54	C₄×C₁₅₇	#121	1	1	yes	characteristic
55	C₄×C₁₅₇	C₄⋊Q₈	1	1	yes	characteristic
56	Dic₁₅₇	—	2	4	no	—
57	C₂×C₁₅₇	—	2	1	no	—
58	C₂×C₁₅₇	—	2	1	no	—
59	C₂×C₁₅₇	#389	1	1	yes	characteristic
60	C₂×C₁₅₇	#443	1	2	yes	—
61	C₁₅₇	#1945	1	1	yes	characteristic, complemented, Sylow
62	#1945	—	157	1	no	maximal, Sylow
63	C₂×#108	—	157	1	no	—
64	#427	D₁₅₇	1	1	yes	characteristic
65	#424	—	157	2	no	—
66	C₂×C₄×C₈	—	157	1	no	—
67	#339	—	157	2	no	—
68	C₂×M₁₆	—	157	2	no	—
69	C₂²×C₈	—	157	1	no	—
70	C₂²×C₈	C₂×D₁₅₇	1	1	yes	characteristic
71	#117	C₂×D₁₅₇	1	2	yes	—
72	C₂×C₄²	—	157	1	no	—
73	#108	—	314	2	no	—
74	#107	Dic₁₅₇	1	2	yes	complemented
75	#106	Dic₁₅₇	1	2	yes	complemented
76	C₄⋊C₈	—	314	4	no	—
77	C₄×C₈	—	314	2	no	—
78	C₂²×C₄	—	314	1	no	—
79	C₂²×C₄	C₂²×D₁₅₇	1	1	yes	characteristic
80	M₁₆	—	314	2	no	—
81	M₁₆	—	314	2	no	—
82	C₂×C₈	—	314	1	no	—
83	C₂×C₈	—	314	1	no	—
84	C₂×C₈	—	314	2	no	—
85	C₂×C₈	—	157	2	no	—
86	C₂×C₈	C₂×Dic₁₅₇	1	1	yes	characteristic
87	C₂×C₈	C₂×Dic₁₅₇	1	1	yes	characteristic
88	C₂×C₈	D₆₂₈	1	2	yes	—
89	C₂×C₈	Dic₃₁₄	1	2	yes	—
90	C₄⋊C₄	C₂×Dic₁₅₇	1	4	yes	—
91	C₂²⋊C₄	—	2	2	no	—
92	C₄²	—	2	2	no	—
93	C₄²	—	314	1	no	—
94	C₄²	—	157	2	no	—
95	C₂³	#14578953	1	1	yes	characteristic
96	C₂×C₄	—	2	4	no	—
97	C₂×C₄	—	314	4	no	—
98	C₂×C₄	—	314	2	no	—
99	C₂×C₄	C₂²×Dic₁₅₇	1	1	yes	characteristic
100	C₂×C₄	D₄×D₁₅₇	1	1	yes	characteristic
101	C₂×C₄	C₂×D₆₂₈	1	1	yes	characteristic
102	C₂×C₄	Q₈×D₁₅₇	1	1	yes	characteristic
103	C₂×C₄	C₂×Dic₃₁₄	1	1	yes	characteristic
104	C₂×C₄	#14578955	1	1	yes	characteristic
105	C₈	—	314	2	no	—
106	C₈	—	628	2	no	—
107	C₈	C₄⋊Dic₁₅₇	1	2	yes	—
108	C₈	C₄⋊Dic₁₅₇	1	2	yes	—
109	C₂²	—	2	2	no	—
110	C₂²	D₄×Dic₁₅₇	1	1	yes	characteristic
111	C₂²	#19097751	1	1	yes	characteristic
112	C₂²	#19097762	1	1	yes	characteristic
113	C₄	—	4	4	no	—
114	C₄	—	314	4	no	—
115	C₄	C₂×C₄⋊Dic₁₅₇	1	1	yes	characteristic
116	C₄	Q₈×Dic₁₅₇	1	1	yes	characteristic
117	C₄	#19097746	1	1	yes	characteristic
118	C₄	#19097770	1	1	yes	characteristic
119	C₂	—	2	1	no	—
120	C₂	—	2	1	no	—
121	C₂	#23757172	1	1	yes	characteristic
122	C₂	#23757390	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#19097707

C₂

yes

maximal, characteristic

#23757300

C₂

yes

maximal

C₂×C₈×Dic₁₅₇

C₂

yes

maximal, characteristic

#23757032

C₂

yes

maximal

#427×C₁₅₇

C₂

yes

maximal, characteristic

C₂×#14578938

C₂²

yes

—

C₂²×#11929434

C₂²

yes

characteristic

C₂×C₄×Dic₁₅₇

C₂²

yes

characteristic

#19097707

—

C₄⋊#11929434

—

C₈×Dic₁₅₇

—

C₂²×C₈×C₁₅₇

C₂²

yes

characteristic

#117×C₁₅₇

C₂²

yes

—

#107×C₁₅₇

C₄

yes

complemented

#106×C₁₅₇

C₄

yes

complemented

C₂²×Dic₁₅₇

—

—

—

—

—

—

D₄

yes

—

C₄×Dic₁₅₇

—

C₄×Dic₁₅₇

Q₈

yes

—

C₂²×C₄×C₁₅₇

C₂³

yes

characteristic

C₂×C₈×C₁₅₇

C₂×C₄

yes

characteristic

C₂×C₈×C₁₅₇

C₂×C₄

yes

characteristic

C₂×C₈×C₁₅₇

D₄

yes

—

C₂×C₈×C₁₅₇

Q₈

yes

—

C₄⋊C₄×C₁₅₇

C₂×C₄

yes

—

C₂²⋊C₄×C₁₅₇

—

C₄²×C₁₅₇

—

C₂×Dic₁₅₇

—

C₂×Dic₁₅₇

—

#11929434

—

#11929434

—

C₂³×C₁₅₇

#41

yes

characteristic

C₂×C₄×C₁₅₇

—

C₂×C₄×C₁₅₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₅₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₅₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₅₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₅₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₅₇

#41

yes

characteristic

C₈×C₁₅₇

C₄⋊C₄

yes

—

C₈×C₁₅₇

C₄⋊C₄

yes

—

C₂²×C₁₅₇

—

C₂²×C₁₅₇

C₄×D₄

yes

characteristic

C₂²×C₁₅₇

C₂²⋊Q₈

yes

characteristic

C₂²×C₁₅₇

#125

yes

characteristic

C₄×C₁₅₇

—

C₄×C₁₅₇

C₂×C₄⋊C₄

yes

characteristic

C₄×C₁₅₇

C₄×Q₈

yes

characteristic

C₄×C₁₅₇

#121

yes

characteristic

C₄×C₁₅₇

C₄⋊Q₈

yes

characteristic

Dic₁₅₇

—

C₂×C₁₅₇

—

C₂×C₁₅₇

—

C₂×C₁₅₇

#389

yes

characteristic

C₂×C₁₅₇

#443

yes

—

C₁₅₇

#1945

yes

characteristic, complemented, Sylow

#1945

—

157

maximal, Sylow

C₂×#108

—

157

—

#427

D₁₅₇

yes

characteristic

#424

—

157

—

C₂×C₄×C₈

—

157

—

#339

—

157

—

C₂×M₁₆

—

157

—

C₂²×C₈

—

157

—

C₂²×C₈

C₂×D₁₅₇

yes

characteristic

#117

C₂×D₁₅₇

yes

—

C₂×C₄²

—

157

—

#108

—

314

—

#107

Dic₁₅₇

yes

complemented

#106

Dic₁₅₇

yes

complemented

C₄⋊C₈

—

314

—

C₄×C₈

—

314

—

C₂²×C₄

—

314

—

C₂²×C₄

C₂²×D₁₅₇

yes

characteristic

M₁₆

—

314

—

M₁₆

—

314

—

C₂×C₈

—

314

—

C₂×C₈

—

314

—

C₂×C₈

—

314

—

C₂×C₈

—

157

—

C₂×C₈

C₂×Dic₁₅₇

yes

characteristic

C₂×C₈

C₂×Dic₁₅₇

yes

characteristic

C₂×C₈

D₆₂₈

yes

—

C₂×C₈

Dic₃₁₄

yes

—

C₄⋊C₄

C₂×Dic₁₅₇

yes

—

C₂²⋊C₄

—

C₄²

—

C₄²

—

314

—

C₄²

—

157

—

C₂³

#14578953

yes

characteristic

C₂×C₄

—

C₂×C₄

—

314

—

C₂×C₄

—

314

—

C₂×C₄

C₂²×Dic₁₅₇

yes

characteristic

100

C₂×C₄

D₄×D₁₅₇

yes

characteristic

101

C₂×C₄

C₂×D₆₂₈

yes

characteristic

102

C₂×C₄

Q₈×D₁₅₇

yes

characteristic

103

C₂×C₄

C₂×Dic₃₁₄

yes

characteristic

104

C₂×C₄

#14578955

yes

characteristic

105

C₈

—

314

—

106

C₈

—

628

—

107

C₈

C₄⋊Dic₁₅₇

yes

—

108

C₈

C₄⋊Dic₁₅₇

yes

—

109

C₂²

—

110

C₂²

D₄×Dic₁₅₇

yes

characteristic

111

C₂²

#19097751

yes

characteristic

112

C₂²

#19097762

yes

characteristic

113

C₄

—

114

C₄

—

314

—

115

C₄

C₂×C₄⋊Dic₁₅₇

yes

characteristic

116

C₄

Q₈×Dic₁₅₇

yes

characteristic

117

C₄

#19097746

yes

characteristic

118

C₄

#19097770

yes

characteristic

119

C₂

—

120

C₂

—

121

C₂

#23757172

yes

characteristic

122

C₂

#23757390

yes

—

	subgroup	G/H
center	C₂×C₄	D₄×D₁₅₇
commutator	C₂×C₄×C₁₅₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₅₇
Fitting	#427×C₁₅₇	C₂
socle	C₂²×C₁₅₇	C₄×D₄
prime radical	C₂³×C₁₅₇	#41
cosocle	C₂²×C₄×C₁₅₇	C₂³
hypercenter	#427	D₁₅₇
hypocenter	C₁₅₇	#1945
LCS	G ▷ C₂×C₄×C₁₅₇ ▷ C₂×C₁₅₇ ▷ C₁₅₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #427

subgroup

G/H

center

C₂×C₄

D₄×D₁₅₇

commutator

C₂×C₄×C₁₅₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₅₇

Fitting

#427×C₁₅₇

C₂

socle

C₂²×C₁₅₇

C₄×D₄

prime radical

C₂³×C₁₅₇

#41

cosocle

C₂²×C₄×C₁₅₇

C₂³

D₁₅₇

C₁₅₇

G ▷ C₂×C₄×C₁₅₇ ▷ C₂×C₁₅₇ ▷ C₁₅₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #427

N	H ≅ G/N	type
14. #107×C₁₅₇	C₄	semidirect
15. #106×C₁₅₇	C₄	semidirect
61. C₁₅₇	#1945	semidirect
74. #107	Dic₁₅₇	semidirect
75. #106	Dic₁₅₇	semidirect

H ≅ G/N

type

14. #107×C₁₅₇

C₄

semidirect

15. #106×C₁₅₇

C₄

semidirect

61. C₁₅₇

#1945

semidirect

74. #107

Dic₁₅₇

semidirect

75. #106

Dic₁₅₇

semidirect

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

subgroup	index
C₂×#19097707	2
#23757300	2
C₂×C₈×Dic₁₅₇	2
#23757032	2
#427×C₁₅₇	2
#1945	157

subgroup

index

C₂×C₈×Dic₁₅₇

#427×C₁₅₇

157

Subgroups — #34100885

Conjugacy classes of subgroups

Complements

Maximal subgroups