Group #34099986 — subgroups

Subgroups — C₄×#19097700 #34099986

Order 20096.

316 conjugacy classes of subgroups; only the 181 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 181 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×#19097700	C₂	1	2	yes	maximal
2	C₂×C₄×#11929434	C₂	1	1	yes	maximal, characteristic
3	C₄²×Dic₁₅₇	C₂	1	1	yes	maximal, characteristic
4	#23757027	C₂	1	2	yes	maximal
5	C₂×C₄×C₈×C₁₅₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11929434	C₂²	1	2	yes	—
7	C₂×C₄×Dic₁₅₇	C₄	1	2	yes	—
8	C₂×C₄×Dic₁₅₇	C₄	1	2	yes	—
9	C₂×C₄×Dic₁₅₇	C₂²	1	1	yes	characteristic
10	C₂×C₄×Dic₁₅₇	C₂²	1	1	yes	characteristic
11	#19097700	C₄	1	8	yes	direct product factor
12	C₄×#11929434	—	2	2	no	—
13	C₂²×C₈×C₁₅₇	C₂²	1	2	yes	—
14	C₂×C₄²×C₁₅₇	C₂²	1	1	yes	characteristic
15	C₄×C₈×C₁₅₇	—	2	2	no	—
16	C₂²×Dic₁₅₇	C₂×C₄	1	1	yes	characteristic
17	C₂²×Dic₁₅₇	C₂×C₄	1	2	yes	—
18	C₂²×Dic₁₅₇	C₂×C₄	1	1	yes	characteristic
19	C₂×#11929434	—	2	2	no	—
20	C₂×#11929434	—	2	2	no	—
21	C₂×#11929434	C₂×C₄	1	4	yes	—
22	C₄×Dic₁₅₇	—	2	4	no	—
23	C₄×Dic₁₅₇	—	2	2	no	—
24	C₄×Dic₁₅₇	C₈	1	8	yes	complemented
25	C₄×Dic₁₅₇	C₂×C₄	1	4	yes	—
26	C₂²×C₄×C₁₅₇	C₂×C₄	1	2	yes	—
27	C₂²×C₄×C₁₅₇	C₂³	1	1	yes	characteristic
28	C₂×C₈×C₁₅₇	—	2	2	no	—
29	C₂×C₈×C₁₅₇	—	2	2	no	—
30	C₂×C₈×C₁₅₇	C₂×C₄	1	4	yes	—
31	C₄²×C₁₅₇	D₄	1	2	yes	—
32	C₄²×C₁₅₇	Q₈	1	2	yes	—
33	C₂×Dic₁₅₇	—	2	2	no	—
34	C₂×Dic₁₅₇	—	2	2	no	—
35	C₂×Dic₁₅₇	—	2	2	no	—
36	C₂×Dic₁₅₇	C₄²	1	2	yes	—
37	C₂×Dic₁₅₇	C₄²	1	2	yes	—
38	C₂×Dic₁₅₇	C₂×C₈	1	4	yes	—
39	C₂×Dic₁₅₇	C₂×C₈	1	4	yes	—
40	#11929434	—	2	4	no	—
41	C₂³×C₁₅₇	C₂²×C₄	1	1	yes	characteristic
42	C₂×C₄×C₁₅₇	C₄⋊C₄	1	4	yes	—
43	C₂×C₄×C₁₅₇	C₂×C₈	1	4	yes	—
44	C₂×C₄×C₁₅₇	M₁₆	1	4	yes	—
45	C₂×C₄×C₁₅₇	C₂²×C₄	1	2	yes	—
46	C₂×C₄×C₁₅₇	C₂×D₄	1	1	yes	characteristic
47	C₂×C₄×C₁₅₇	C₂×Q₈	1	1	yes	characteristic
48	C₂×C₄×C₁₅₇	#41	1	2	yes	—
49	C₈×C₁₅₇	—	2	4	no	—
50	C₂²×C₁₅₇	C₂×C₄²	1	1	yes	characteristic
51	C₂²×C₁₅₇	C₂×C₄⋊C₄	1	1	yes	characteristic
52	C₂²×C₁₅₇	#117	1	1	yes	characteristic
53	C₂²×C₁₅₇	C₂²×C₈	1	2	yes	—
54	C₂²×C₁₅₇	C₂×M₁₆	1	2	yes	—
55	C₄×C₁₅₇	C₄⋊C₈	1	8	yes	complemented
56	C₄×C₁₅₇	C₄×D₄	1	2	yes	—
57	C₄×C₁₅₇	C₄×Q₈	1	2	yes	—
58	Dic₁₅₇	—	2	2	no	—
59	Dic₁₅₇	—	2	4	no	—
60	Dic₁₅₇	C₄×C₈	1	4	yes	complemented
61	C₂×C₁₅₇	C₄×C₄⋊C₄	1	1	yes	characteristic
62	C₂×C₁₅₇	C₂×C₄×C₈	1	1	yes	characteristic
63	C₂×C₁₅₇	C₄×M₁₆	1	1	yes	characteristic
64	C₂×C₁₅₇	C₂×C₄⋊C₈	1	2	yes	—
65	C₂×C₁₅₇	#431	1	2	yes	—
66	C₁₅₇	C₄×C₄⋊C₈	1	1	yes	characteristic, complemented, Sylow
67	C₄×C₄⋊C₈	—	157	1	no	maximal, Sylow
68	C₂×C₄⋊C₈	—	157	2	no	—
69	C₂×C₄×C₈	—	157	1	no	—
70	C₂×C₄×C₈	D₁₅₇	1	1	yes	characteristic
71	C₄³	—	157	1	no	—
72	#336	—	157	2	no	—
73	C₂²×C₈	—	157	2	no	—
74	C₂²×C₈	C₂×D₁₅₇	1	2	yes	—
75	C₂×C₄²	—	157	2	no	—
76	C₂×C₄²	—	157	1	no	—
77	C₂×C₄²	—	157	2	no	—
78	C₂×C₄²	—	157	1	no	—
79	C₂×C₄²	C₂×D₁₅₇	1	1	yes	characteristic
80	C₄⋊C₈	—	157	8	no	—
81	C₄×C₈	—	2	2	no	—
82	C₄×C₈	—	314	2	no	—
83	C₂²×C₄	—	157	2	no	—
84	C₂²×C₄	—	157	1	no	—
85	C₂²×C₄	—	157	1	no	—
86	C₂²×C₄	C₄×D₁₅₇	1	2	yes	—
87	C₂²×C₄	C₂²×D₁₅₇	1	1	yes	characteristic
88	C₂×C₈	—	2	2	no	—
89	C₂×C₈	—	2	2	no	—
90	C₂×C₈	—	314	2	no	—
91	C₂×C₈	—	314	2	no	—
92	C₂×C₈	—	157	4	no	—
93	C₂×C₈	C₄×D₁₅₇	1	4	yes	—
94	C₄²	—	314	2	no	—
95	C₄²	—	314	4	no	—
96	C₄²	—	157	4	no	—
97	C₄²	—	157	8	no	—
98	C₄²	C₂²⋊D₁₅₇	1	2	yes	—
99	C₄²	Dic₃₁₄	1	2	yes	—
100	C₂³	C₂×C₄×D₁₅₇	1	1	yes	characteristic
101	C₂×C₄	—	314	2	no	—
102	C₂×C₄	—	314	2	no	—
103	C₂×C₄	—	314	2	no	—
104	C₂×C₄	—	157	2	no	—
105	C₂×C₄	—	157	4	no	—
106	C₂×C₄	—	157	2	no	—
107	C₂×C₄	—	157	4	no	—
108	C₂×C₄	#14578933	1	4	yes	—
109	C₂×C₄	C₈×D₁₅₇	1	4	yes	—
110	C₂×C₄	#14578937	1	4	yes	—
111	C₂×C₄	C₂×C₄×D₁₅₇	1	2	yes	—
112	C₂×C₄	C₂×C₂²⋊D₁₅₇	1	1	yes	characteristic
113	C₂×C₄	C₂×Dic₃₁₄	1	1	yes	characteristic
114	C₂×C₄	#14578954	1	2	yes	—
115	C₈	—	2	4	no	—
116	C₈	—	314	4	no	—
117	C₂²	C₄²×D₁₅₇	1	1	yes	characteristic
118	C₂²	C₂×#14578933	1	1	yes	characteristic
119	C₂²	#19097728	1	1	yes	characteristic
120	C₂²	C₂×C₈×D₁₅₇	1	2	yes	—
121	C₂²	C₂×#14578937	1	2	yes	—
122	C₄	—	314	4	no	—
123	C₄	—	314	2	no	—
124	C₄	—	157	4	no	—
125	C₄	#19097700	1	8	yes	direct product factor
126	C₄	C₄×C₂²⋊D₁₅₇	1	2	yes	—
127	C₄	C₄×Dic₃₁₄	1	2	yes	—
128	C₂	C₄×#14578933	1	1	yes	characteristic
129	C₂	C₄×C₈×D₁₅₇	1	1	yes	characteristic
130	C₂	C₄×#14578937	1	1	yes	characteristic
131	C₂	C₂×#19097700	1	2	yes	—
132	C₂	#23757324	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#19097700

C₂

yes

maximal

C₂×C₄×#11929434

C₂

yes

maximal, characteristic

C₄²×Dic₁₅₇

C₂

yes

maximal, characteristic

#23757027

C₂

yes

maximal

C₂×C₄×C₈×C₁₅₇

C₂

yes

maximal, characteristic

C₂²×#11929434

C₂²

yes

—

C₂×C₄×Dic₁₅₇

C₄

yes

—

C₂×C₄×Dic₁₅₇

C₄

yes

—

C₂×C₄×Dic₁₅₇

C₂²

yes

characteristic

C₂×C₄×Dic₁₅₇

C₂²

yes

characteristic

#19097700

C₄

yes

direct product factor

C₄×#11929434

—

C₂²×C₈×C₁₅₇

C₂²

yes

—

C₂×C₄²×C₁₅₇

C₂²

yes

characteristic

C₄×C₈×C₁₅₇

—

C₂²×Dic₁₅₇

C₂×C₄

yes

characteristic

C₂²×Dic₁₅₇

C₂×C₄

yes

—

C₂²×Dic₁₅₇

C₂×C₄

yes

—

—

C₂×C₄

yes

—

C₄×Dic₁₅₇

—

C₄×Dic₁₅₇

—

C₄×Dic₁₅₇

C₈

yes

complemented

C₄×Dic₁₅₇

C₂×C₄

yes

—

C₂²×C₄×C₁₅₇

C₂×C₄

yes

—

C₂²×C₄×C₁₅₇

C₂³

yes

characteristic

C₂×C₈×C₁₅₇

—

C₂×C₈×C₁₅₇

—

C₂×C₈×C₁₅₇

C₂×C₄

yes

—

C₄²×C₁₅₇

D₄

yes

—

C₄²×C₁₅₇

Q₈

yes

—

C₂×Dic₁₅₇

—

C₂×Dic₁₅₇

—

C₂×Dic₁₅₇

—

C₂×Dic₁₅₇

C₄²

yes

—

C₂×Dic₁₅₇

C₄²

yes

—

C₂×Dic₁₅₇

C₂×C₈

yes

—

C₂×Dic₁₅₇

C₂×C₈

yes

—

#11929434

—

C₂³×C₁₅₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₅₇

C₄⋊C₄

yes

—

C₂×C₄×C₁₅₇

C₂×C₈

yes

—

C₂×C₄×C₁₅₇

M₁₆

yes

—

C₂×C₄×C₁₅₇

C₂²×C₄

yes

—

C₂×C₄×C₁₅₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₅₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₅₇

#41

yes

—

C₈×C₁₅₇

—

C₂²×C₁₅₇

C₂×C₄²

yes

characteristic

C₂²×C₁₅₇

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₁₅₇

#117

yes

characteristic

C₂²×C₁₅₇

C₂²×C₈

yes

—

C₂²×C₁₅₇

C₂×M₁₆

yes

—

C₄×C₁₅₇

C₄⋊C₈

yes

complemented

C₄×C₁₅₇

C₄×D₄

yes

—

C₄×C₁₅₇

C₄×Q₈

yes

—

Dic₁₅₇

—

Dic₁₅₇

—

Dic₁₅₇

C₄×C₈

yes

complemented

C₂×C₁₅₇

C₄×C₄⋊C₄

yes

characteristic

C₂×C₁₅₇

C₂×C₄×C₈

yes

characteristic

C₂×C₁₅₇

C₄×M₁₆

yes

characteristic

C₂×C₁₅₇

C₂×C₄⋊C₈

yes

—

C₂×C₁₅₇

#431

yes

—

C₁₅₇

C₄×C₄⋊C₈

yes

characteristic, complemented, Sylow

C₄×C₄⋊C₈

—

157

maximal, Sylow

C₂×C₄⋊C₈

—

157

—

C₂×C₄×C₈

—

157

—

C₂×C₄×C₈

D₁₅₇

yes

characteristic

C₄³

—

157

—

#336

—

157

—

C₂²×C₈

—

157

—

C₂²×C₈

C₂×D₁₅₇

yes

—

C₂×C₄²

—

157

—

C₂×C₄²

—

157

—

C₂×C₄²

—

157

—

C₂×C₄²

—

157

—

C₂×C₄²

C₂×D₁₅₇

yes

characteristic

C₄⋊C₈

—

157

—

C₄×C₈

—

C₄×C₈

—

314

—

C₂²×C₄

—

157

—

C₂²×C₄

—

157

—

C₂²×C₄

—

157

—

C₂²×C₄

C₄×D₁₅₇

yes

—

C₂²×C₄

C₂²×D₁₅₇

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

314

—

C₂×C₈

—

314

—

C₂×C₈

—

157

—

C₂×C₈

C₄×D₁₅₇

yes

—

C₄²

—

314

—

C₄²

—

314

—

C₄²

—

157

—

C₄²

—

157

—

C₄²

C₂²⋊D₁₅₇

yes

—

C₄²

Dic₃₁₄

yes

—

100

C₂³

C₂×C₄×D₁₅₇

yes

characteristic

101

C₂×C₄

—

314

—

102

C₂×C₄

—

314

—

103

C₂×C₄

—

314

—

104

C₂×C₄

—

157

—

105

C₂×C₄

—

157

—

106

C₂×C₄

—

157

—

107

C₂×C₄

—

157

—

108

C₂×C₄

#14578933

yes

—

109

C₂×C₄

C₈×D₁₅₇

yes

—

110

C₂×C₄

#14578937

yes

—

111

C₂×C₄

C₂×C₄×D₁₅₇

yes

—

112

C₂×C₄

C₂×C₂²⋊D₁₅₇

yes

characteristic

113

C₂×C₄

C₂×Dic₃₁₄

yes

characteristic

114

C₂×C₄

#14578954

yes

—

115

C₈

—

116

C₈

—

314

—

117

C₂²

C₄²×D₁₅₇

yes

characteristic

118

C₂²

C₂×#14578933

yes

characteristic

119

C₂²

#19097728

yes

characteristic

120

C₂²

C₂×C₈×D₁₅₇

yes

—

121

C₂²

C₂×#14578937

yes

—

122

C₄

—

314

—

123

C₄

—

314

—

124

C₄

—

157

—

125

C₄

#19097700

yes

direct product factor

126

C₄

C₄×C₂²⋊D₁₅₇

yes

—

127

C₄

C₄×Dic₃₁₄

yes

—

128

C₂

C₄×#14578933

yes

characteristic

129

C₂

C₄×C₈×D₁₅₇

yes

characteristic

130

C₂

C₄×#14578937

yes

characteristic

131

C₂

C₂×#19097700

yes

—

132

C₂

#23757324

yes

—

	subgroup	G/H
center	C₂×C₄²	C₂×D₁₅₇
commutator	C₂×C₁₅₇	C₂×C₄×C₈
Frattini	C₂²×C₄	C₂²×D₁₅₇
Fitting	C₂×C₄×C₈×C₁₅₇	C₂
socle, prime radical	C₂³×C₁₅₇	C₂²×C₄
cosocle	C₂²×C₄×C₁₅₇	C₂³
hypercenter	C₂×C₄×C₈	D₁₅₇
hypocenter	C₁₅₇	C₄×C₄⋊C₈
LCS	G ▷ C₂×C₁₅₇ ▷ C₁₅₇
UCS	C₁ ◁ C₂×C₄² ◁ C₂×C₄×C₈

subgroup

G/H

center

C₂×C₄²

C₂×D₁₅₇

commutator

C₂×C₁₅₇

C₂×C₄×C₈

Frattini

C₂²×C₄

C₂²×D₁₅₇

Fitting

C₂×C₄×C₈×C₁₅₇

C₂

socle, prime radical

C₂³×C₁₅₇

C₂²×C₄

cosocle

C₂²×C₄×C₁₅₇

C₂³

hypercenter

C₂×C₄×C₈

D₁₅₇

hypocenter

C₁₅₇

C₄×C₄⋊C₈

LCS

G ▷ C₂×C₁₅₇ ▷ C₁₅₇

UCS

C₁ ◁ C₂×C₄² ◁ C₂×C₄×C₈

N	H ≅ G/N	type
11. #19097700	C₄	direct (N × H)
24. C₄×Dic₁₅₇	C₈	semidirect
55. C₄×C₁₅₇	C₄⋊C₈	semidirect
60. Dic₁₅₇	C₄×C₈	semidirect
66. C₁₅₇	C₄×C₄⋊C₈	semidirect
125. C₄	#19097700	direct (N × H)

H ≅ G/N

type

11. #19097700

C₄

direct (N × H)

24. C₄×Dic₁₅₇

C₈

semidirect

55. C₄×C₁₅₇

C₄⋊C₈

semidirect

60. Dic₁₅₇

C₄×C₈

semidirect

66. C₁₅₇

C₄×C₄⋊C₈

semidirect

125. C₄

#19097700

direct (N × H)

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

subgroup	index
C₂×#19097700	2
C₂×C₄×#11929434	2
C₄²×Dic₁₅₇	2
#23757027	2
C₂×C₄×C₈×C₁₅₇	2
C₄×C₄⋊C₈	157

subgroup

index

C₂×#19097700

C₂×C₄×#11929434

C₄²×Dic₁₅₇

#23757027

C₂×C₄×C₈×C₁₅₇

C₄×C₄⋊C₈

157

Subgroups — C₄×#19097700 #34099986

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C4×#19097700 #34099986

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#19097700 #34099986