Group #34100937 — subgroups

Subgroups — #34100937

Order 20096.

332 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 332 conjugacy classes of subgroups — too many to draw. Only its 137 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×#19097771	C₂	1	1	yes	maximal, characteristic
2	C₂×#19097694	C₂	1	4	yes	maximal
3	C₂×C₄×#11929434	C₂	1	1	yes	maximal, characteristic
4	#384×C₁₅₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₃₁₄	C₂²	1	2	yes	—
6	C₂²×#11929434	C₂²	1	2	yes	—
7	#19097771	—	2	2	no	—
8	#19097771	C₄	1	4	yes	complemented
9	C₂×C₄⋊Dic₁₅₇	—	2	2	no	—
10	#19097694	—	2	8	no	—
11	C₄×#11929434	—	2	2	no	—
12	C₂×C₄⋊C₄×C₁₅₇	—	2	2	no	—
13	C₂×C₄⋊C₄×C₁₅₇	C₂²	1	2	yes	—
14	C₂×C₄²×C₁₅₇	C₂²	1	1	yes	characteristic
15	C₂×Dic₃₁₄	—	2	2	no	—
16	C₂×Dic₃₁₄	—	2	4	no	—
17	C₂×Dic₃₁₄	—	2	4	no	—
18	C₂×Dic₃₁₄	C₂×C₄	1	4	yes	—
19	C₂²×Dic₁₅₇	—	2	2	no	—
20	C₂×#11929434	—	2	2	no	—
21	C₂×#11929434	—	2	2	no	—
22	C₂×#11929434	D₄	1	4	yes	—
23	C₄⋊Dic₁₅₇	—	2	8	no	—
24	C₂²×C₄×C₁₅₇	—	2	2	no	—
25	C₂²×C₄×C₁₅₇	D₄	1	2	yes	—
26	C₂²×C₄×C₁₅₇	C₂³	1	1	yes	characteristic
27	C₄⋊C₄×C₁₅₇	—	2	4	no	—
28	C₄⋊C₄×C₁₅₇	—	4	4	no	—
29	C₄²×C₁₅₇	C₂×C₄	1	2	yes	—
30	C₄²×C₁₅₇	D₄	1	2	yes	—
31	Dic₃₁₄	—	2	8	no	—
32	Dic₃₁₄	—	4	4	no	—
33	C₂×Dic₁₅₇	—	2	4	no	—
34	C₂×Dic₁₅₇	—	4	2	no	—
35	C₂×Dic₁₅₇	—	4	2	no	—
36	#11929434	—	2	4	no	—
37	C₂³×C₁₅₇	C₂×D₄	1	1	yes	characteristic
38	C₂×C₄×C₁₅₇	—	4	2	no	—
39	C₂×C₄×C₁₅₇	—	4	4	no	—
40	C₂×C₄×C₁₅₇	C₂²⋊C₄	1	4	yes	—
41	C₂×C₄×C₁₅₇	SD₁₆	1	4	yes	—
42	C₂×C₄×C₁₅₇	Q₁₆	1	4	yes	—
43	C₂×C₄×C₁₅₇	C₂²×C₄	1	1	yes	characteristic
44	C₂×C₄×C₁₅₇	C₂×D₄	1	2	yes	—
45	C₂×C₄×C₁₅₇	C₂×D₄	1	1	yes	characteristic
46	C₂×C₄×C₁₅₇	#41	1	2	yes	—
47	C₂²×C₁₅₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
48	C₂²×C₁₅₇	#124	1	1	yes	characteristic
49	C₂²×C₁₅₇	#127	1	1	yes	characteristic
50	C₂²×C₁₅₇	C₂×SD₁₆	1	2	yes	—
51	C₂²×C₁₅₇	C₂×Q₁₆	1	2	yes	—
52	C₄×C₁₅₇	—	8	2	no	—
53	C₄×C₁₅₇	#103	1	8	yes	complemented
54	C₄×C₁₅₇	C₄×D₄	1	2	yes	—
55	C₄×C₁₅₇	#121	1	2	yes	—
56	Dic₁₅₇	—	4	4	no	—
57	C₂×C₁₅₇	#390	1	1	yes	characteristic
58	C₂×C₁₅₇	C₂×#103	1	2	yes	—
59	C₂×C₁₅₇	#487	1	2	yes	—
60	C₂×C₁₅₇	#492	1	1	yes	characteristic
61	C₂×C₁₅₇	#494	1	1	yes	characteristic
62	C₁₅₇	#1956	1	1	yes	characteristic, complemented, Sylow
63	#1956	—	157	1	no	maximal, Sylow
64	C₂×C₄⋊Q₈	—	157	1	no	—
65	C₂×#103	—	157	4	no	—
66	C₂×C₄×C₈	—	157	1	no	—
67	#384	D₁₅₇	1	1	yes	characteristic
68	C₂²×Q₈	—	157	2	no	—
69	C₂²×C₈	—	157	2	no	—
70	C₄⋊Q₈	—	314	2	no	—
71	C₄⋊Q₈	—	157	4	no	—
72	C₂×C₄⋊C₄	—	2	2	no	—
73	C₂×C₄⋊C₄	—	314	2	no	—
74	C₂×C₄⋊C₄	C₂×D₁₅₇	1	2	yes	—
75	C₂×C₄²	C₂×D₁₅₇	1	1	yes	characteristic
76	#103	—	314	8	no	—
77	C₄×C₈	—	314	2	no	—
78	C₂×Q₈	—	314	2	no	—
79	C₂×Q₈	—	314	4	no	—
80	C₂×Q₈	—	314	4	no	—
81	C₂×Q₈	—	157	4	no	—
82	C₂²×C₄	—	2	2	no	—
83	C₂²×C₄	—	314	2	no	—
84	C₂²×C₄	C₂²⋊D₁₅₇	1	2	yes	—
85	C₂²×C₄	C₂²×D₁₅₇	1	1	yes	characteristic
86	C₂×C₈	—	314	2	no	—
87	C₂×C₈	—	314	2	no	—
88	C₂×C₈	—	157	4	no	—
89	C₄⋊C₄	—	2	4	no	—
90	C₄⋊C₄	—	4	4	no	—
91	C₄⋊C₄	—	314	8	no	—
92	C₄²	C₄×D₁₅₇	1	2	yes	—
93	C₄²	D₆₂₈	1	2	yes	—
94	C₂³	C₂×C₂²⋊D₁₅₇	1	1	yes	characteristic
95	Q₈	—	314	8	no	—
96	Q₈	—	628	4	no	—
97	C₂×C₄	—	4	2	no	—
98	C₂×C₄	—	4	4	no	—
99	C₂×C₄	—	314	4	no	—
100	C₂×C₄	—	628	2	no	—
101	C₂×C₄	—	628	2	no	—
102	C₂×C₄	#14578931	1	4	yes	—
103	C₂×C₄	#14578942	1	4	yes	—
104	C₂×C₄	#14578944	1	4	yes	—
105	C₂×C₄	C₂×C₄×D₁₅₇	1	1	yes	characteristic
106	C₂×C₄	D₄×D₁₅₇	1	2	yes	—
107	C₂×C₄	C₂×D₆₂₈	1	1	yes	characteristic
108	C₂×C₄	#14578955	1	2	yes	—
109	C₈	—	314	4	no	—
110	C₂²	C₂×#14578931	1	1	yes	characteristic
111	C₂²	#19097760	1	1	yes	characteristic
112	C₂²	#19097767	1	1	yes	characteristic
113	C₂²	C₂×#14578942	1	2	yes	—
114	C₂²	C₂×#14578944	1	2	yes	—
115	C₄	—	8	2	no	—
116	C₄	—	628	4	no	—
117	C₄	#19097694	1	8	yes	complemented
118	C₄	#19097733	1	2	yes	—
119	C₄	#19097746	1	2	yes	—
120	C₂	#23757174	1	1	yes	characteristic
121	C₂	C₂×#19097694	1	2	yes	—
122	C₂	#23757625	1	2	yes	—
123	C₂	#23757645	1	1	yes	characteristic
124	C₂	#23757652	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#19097771

C₂

yes

maximal, characteristic

C₂×#19097694

C₂

yes

maximal

C₂×C₄×#11929434

C₂

yes

maximal, characteristic

#384×C₁₅₇

C₂

yes

maximal, characteristic

C₂²×Dic₃₁₄

C₂²

yes

—

C₂²×#11929434

C₂²

yes

—

#19097771

—

#19097771

C₄

yes

complemented

C₂×C₄⋊Dic₁₅₇

—

#19097694

—

C₄×#11929434

—

C₂×C₄⋊C₄×C₁₅₇

—

C₂×C₄⋊C₄×C₁₅₇

C₂²

yes

—

C₂×C₄²×C₁₅₇

C₂²

yes

characteristic

C₂×Dic₃₁₄

—

C₂×Dic₃₁₄

—

C₂×Dic₃₁₄

—

C₂×Dic₃₁₄

C₂×C₄

yes

—

C₂²×Dic₁₅₇

—

C₂×#11929434

—

C₂×#11929434

—

C₂×#11929434

D₄

yes

—

C₄⋊Dic₁₅₇

—

C₂²×C₄×C₁₅₇

—

C₂²×C₄×C₁₅₇

D₄

yes

—

C₂²×C₄×C₁₅₇

C₂³

yes

characteristic

C₄⋊C₄×C₁₅₇

—

C₄⋊C₄×C₁₅₇

—

C₄²×C₁₅₇

C₂×C₄

yes

—

C₄²×C₁₅₇

D₄

yes

—

Dic₃₁₄

—

Dic₃₁₄

—

C₂×Dic₁₅₇

—

C₂×Dic₁₅₇

—

C₂×Dic₁₅₇

—

#11929434

—

C₂³×C₁₅₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₅₇

—

C₂×C₄×C₁₅₇

—

C₂×C₄×C₁₅₇

C₂²⋊C₄

yes

—

C₂×C₄×C₁₅₇

SD₁₆

yes

—

C₂×C₄×C₁₅₇

Q₁₆

yes

—

C₂×C₄×C₁₅₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₅₇

C₂×D₄

yes

—

C₂×C₄×C₁₅₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₅₇

#41

yes

—

C₂²×C₁₅₇

C₂×C₂²⋊C₄

yes

C₂²×C₁₅₇

yes

C₂²×C₁₅₇

yes

C₂²×C₁₅₇

C₂×SD₁₆

yes

—

C₂²×C₁₅₇

C₂×Q₁₆

yes

—

C₄×C₁₅₇

—

C₄×C₁₅₇

#103

yes

complemented

C₄×C₁₅₇

C₄×D₄

yes

—

C₄×C₁₅₇

#121

yes

—

Dic₁₅₇

—

C₂×C₁₅₇

#390

yes

characteristic

C₂×C₁₅₇

C₂×#103

yes

—

C₂×C₁₅₇

#487

yes

—

C₂×C₁₅₇

yes

C₂×C₁₅₇

yes

C₁₅₇

yes

characteristic, complemented, Sylow

#1956

—

157

maximal, Sylow

C₂×C₄⋊Q₈

—

157

—

C₂×#103

—

157

—

C₂×C₄×C₈

—

157

—

#384

D₁₅₇

yes

characteristic

C₂²×Q₈

—

157

—

C₂²×C₈

—

157

—

C₄⋊Q₈

—

314

—

C₄⋊Q₈

—

157

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

314

—

C₂×C₄⋊C₄

C₂×D₁₅₇

yes

—

C₂×C₄²

C₂×D₁₅₇

yes

characteristic

#103

—

314

—

C₄×C₈

—

314

—

C₂×Q₈

—

314

—

C₂×Q₈

—

314

—

C₂×Q₈

—

314

—

C₂×Q₈

—

157

—

C₂²×C₄

—

C₂²×C₄

—

314

—

C₂²×C₄

C₂²⋊D₁₅₇

yes

—

C₂²×C₄

C₂²×D₁₅₇

yes

characteristic

C₂×C₈

—

314

—

C₂×C₈

—

314

—

C₂×C₈

—

157

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

314

—

C₄²

C₄×D₁₅₇

yes

—

C₄²

D₆₂₈

yes

—

C₂³

C₂×C₂²⋊D₁₅₇

yes

characteristic

Q₈

—

314

—

Q₈

—

628

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

314

—

100

C₂×C₄

—

628

—

101

C₂×C₄

—

628

—

102

C₂×C₄

#14578931

yes

—

103

C₂×C₄

#14578942

yes

—

104

C₂×C₄

#14578944

yes

—

105

C₂×C₄

C₂×C₄×D₁₅₇

yes

characteristic

106

C₂×C₄

D₄×D₁₅₇

yes

—

107

C₂×C₄

C₂×D₆₂₈

yes

characteristic

108

C₂×C₄

#14578955

yes

—

109

C₈

—

314

—

110

C₂²

C₂×#14578931

yes

characteristic

111

C₂²

#19097760

yes

characteristic

112

C₂²

#19097767

yes

characteristic

113

C₂²

C₂×#14578942

yes

—

114

C₂²

C₂×#14578944

yes

—

115

C₄

—

116

C₄

—

628

—

117

C₄

#19097694

yes

complemented

118

C₄

#19097733

yes

—

119

C₄

#19097746

yes

—

120

C₂

#23757174

yes

characteristic

121

C₂

C₂×#19097694

yes

—

122

C₂

#23757625

yes

—

123

C₂

#23757645

yes

characteristic

124

C₂

#23757652

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₁₅₇
commutator	C₂×C₄×C₁₅₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₅₇
Fitting	#384×C₁₅₇	C₂
socle, prime radical	C₂³×C₁₅₇	C₂×D₄
cosocle	C₂²×C₄×C₁₅₇	C₂³
hypercenter	#384	D₁₅₇
hypocenter	C₁₅₇	#1956
LCS	G ▷ C₂×C₄×C₁₅₇ ▷ C₂×C₁₅₇ ▷ C₁₅₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #384

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₁₅₇

commutator

C₂×C₄×C₁₅₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₅₇

Fitting

#384×C₁₅₇

C₂

socle, prime radical

C₂³×C₁₅₇

C₂×D₄

cosocle

C₂²×C₄×C₁₅₇

C₂³

D₁₅₇

C₁₅₇

G ▷ C₂×C₄×C₁₅₇ ▷ C₂×C₁₅₇ ▷ C₁₅₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #384

N	H ≅ G/N	type
8. #19097771	C₄	semidirect
53. C₄×C₁₅₇	#103	semidirect
62. C₁₅₇	#1956	semidirect
117. C₄	#19097694	semidirect

H ≅ G/N

type

8. #19097771

C₄

semidirect

53. C₄×C₁₅₇

#103

semidirect

62. C₁₅₇

#1956

semidirect

117. C₄

#19097694

semidirect

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

subgroup	index
C₂×#19097771	2
C₂×#19097694	2
C₂×C₄×#11929434	2
#384×C₁₅₇	2
#1956	157

subgroup

index

#384×C₁₅₇

157

Subgroups — #34100937

Conjugacy classes of subgroups

Complements

Maximal subgroups