Group #34100979 — subgroups

Subgroups — #34100979

Order 20096.

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

normal subgroups only

This group has 292 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×#19097771	C₂	1	1	yes	maximal, characteristic
2	C₂×#19097689	C₂	1	4	yes	maximal
3	C₄×#14578938	C₂	1	1	yes	maximal, characteristic
4	#423×C₁₅₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₃₁₄	C₂²	1	2	yes	—
6	C₂×#14578938	C₂²	1	2	yes	—
7	#19097771	—	2	2	no	—
8	#19097771	—	2	1	no	—
9	#19097771	—	2	1	no	—
10	C₂×C₄⋊Dic₁₅₇	C₄	1	4	yes	—
11	#19097689	—	2	8	no	—
12	#19097682	—	2	1	no	—
13	C₄×#11929434	—	2	1	no	—
14	C₂×M₁₆×C₁₅₇	C₂²	1	2	yes	—
15	C₂×C₄²×C₁₅₇	C₂²	1	1	yes	characteristic
16	C₄⋊C₈×C₁₅₇	—	2	2	no	—
17	C₂×Dic₃₁₄	—	2	4	no	—
18	C₂×Dic₃₁₄	—	2	2	no	—
19	C₂×Dic₃₁₄	—	2	2	no	—
20	C₂×Dic₃₁₄	—	2	4	no	—
21	C₂²×Dic₁₅₇	C₂×C₄	1	4	yes	—
22	#14578938	—	2	2	no	—
23	#14578938	D₄	1	4	yes	—
24	C₂×#11929434	—	2	2	no	—
25	C₄⋊Dic₁₅₇	—	2	4	no	—
26	C₄⋊Dic₁₅₇	—	2	4	no	—
27	C₂²×C₄×C₁₅₇	C₂×C₄	1	2	yes	—
28	C₂²×C₄×C₁₅₇	C₂³	1	1	yes	characteristic
29	M₁₆×C₁₅₇	—	2	4	no	—
30	C₂×C₈×C₁₅₇	—	2	2	no	—
31	C₄²×C₁₅₇	D₄	1	2	yes	—
32	C₄²×C₁₅₇	D₄	1	1	yes	characteristic
33	C₄²×C₁₅₇	D₄	1	1	yes	characteristic
34	Dic₃₁₄	—	4	4	no	—
35	Dic₃₁₄	—	4	2	no	—
36	Dic₃₁₄	—	4	2	no	—
37	C₂×Dic₁₅₇	—	2	4	no	—
38	C₂×Dic₁₅₇	—	2	8	no	—
39	#11929434	—	2	4	no	—
40	C₂³×C₁₅₇	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₄×C₁₅₇	—	2	4	no	—
42	C₂×C₄×C₁₅₇	C₂²⋊C₄	1	2	yes	—
43	C₂×C₄×C₁₅₇	C₂²⋊C₄	1	2	yes	—
44	C₂×C₄×C₁₅₇	C₂×D₄	1	2	yes	—
45	C₂×C₄×C₁₅₇	C₂×D₄	1	1	yes	characteristic
46	C₂×C₄×C₁₅₇	C₂×D₄	1	1	yes	characteristic
47	C₂×C₄×C₁₅₇	#41	1	2	yes	—
48	C₈×C₁₅₇	—	4	2	no	—
49	C₂²×C₁₅₇	—	2	2	no	—
50	C₂²×C₁₅₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
51	C₂²×C₁₅₇	C₄×D₄	1	2	yes	—
52	C₄×C₁₅₇	—	2	2	no	—
53	C₄×C₁₅₇	#101	1	4	yes	complemented
54	C₄×C₁₅₇	#121	1	2	yes	—
55	C₄×C₁₅₇	#124	1	1	yes	characteristic
56	C₄×C₁₅₇	#127	1	1	yes	characteristic
57	Dic₁₅₇	—	4	4	no	—
58	C₂×C₁₅₇	—	2	2	no	—
59	C₂×C₁₅₇	#390	1	1	yes	characteristic
60	C₂×C₁₅₇	C₂×#101	1	2	yes	—
61	C₁₅₇	#1966	1	1	yes	characteristic, complemented, Sylow
62	#1966	—	157	1	no	maximal, Sylow
63	C₂×C₄⋊Q₈	—	157	1	no	—
64	#423	D₁₅₇	1	1	yes	characteristic
65	C₂×#101	—	157	4	no	—
66	C₄×M₁₆	—	157	1	no	—
67	C₂²×Q₈	—	157	2	no	—
68	C₂×M₁₆	—	157	2	no	—
69	C₂×M₁₆	C₂×D₁₅₇	1	2	yes	—
70	C₄⋊Q₈	—	314	2	no	—
71	C₄⋊Q₈	—	314	1	no	—
72	C₄⋊Q₈	—	314	1	no	—
73	C₂×C₄⋊C₄	—	157	4	no	—
74	C₂×C₄²	C₂×D₁₅₇	1	1	yes	characteristic
75	C₄⋊C₈	—	2	2	no	—
76	#101	—	314	8	no	—
77	#97	—	314	1	no	—
78	C₄×C₈	—	314	1	no	—
79	C₂×Q₈	—	314	4	no	—
80	C₂×Q₈	—	314	2	no	—
81	C₂×Q₈	—	314	2	no	—
82	C₂×Q₈	—	314	4	no	—
83	C₂²×C₄	—	157	4	no	—
84	C₂²×C₄	C₄×D₁₅₇	1	2	yes	—
85	C₂²×C₄	C₂²×D₁₅₇	1	1	yes	characteristic
86	M₁₆	—	2	4	no	—
87	M₁₆	—	314	2	no	—
88	M₁₆	—	157	4	no	—
89	C₂×C₈	—	2	2	no	—
90	C₂×C₈	—	314	2	no	—
91	C₄⋊C₄	—	314	4	no	—
92	C₄⋊C₄	—	314	4	no	—
93	C₄²	C₂²⋊D₁₅₇	1	2	yes	—
94	C₄²	D₆₂₈	1	1	yes	characteristic
95	C₄²	D₆₂₈	1	1	yes	characteristic
96	C₂³	C₂×C₄×D₁₅₇	1	1	yes	characteristic
97	Q₈	—	628	4	no	—
98	Q₈	—	628	2	no	—
99	Q₈	—	628	2	no	—
100	C₂×C₄	—	2	4	no	—
101	C₂×C₄	—	314	4	no	—
102	C₂×C₄	—	314	8	no	—
103	C₂×C₄	#14578931	1	2	yes	—
104	C₂×C₄	#14578931	1	2	yes	—
105	C₂×C₄	D₄×D₁₅₇	1	2	yes	—
106	C₂×C₄	C₂×C₂²⋊D₁₅₇	1	1	yes	characteristic
107	C₂×C₄	C₂×D₆₂₈	1	1	yes	characteristic
108	C₂×C₄	#14578955	1	2	yes	—
109	C₈	—	4	2	no	—
110	C₈	—	314	4	no	—
111	C₂²	—	2	2	no	—
112	C₂²	C₂×#14578931	1	1	yes	characteristic
113	C₂²	#19097733	1	2	yes	—
114	C₄	—	2	2	no	—
115	C₄	—	628	4	no	—
116	C₄	#19097689	1	4	yes	complemented
117	C₄	#19097746	1	2	yes	—
118	C₄	#19097760	1	1	yes	characteristic
119	C₄	#19097767	1	1	yes	characteristic
120	C₂	—	2	2	no	—
121	C₂	#23757174	1	1	yes	characteristic
122	C₂	C₂×#19097689	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#19097771

C₂

yes

maximal, characteristic

C₂×#19097689

C₂

yes

maximal

C₄×#14578938

C₂

yes

maximal, characteristic

#423×C₁₅₇

C₂

yes

maximal, characteristic

C₂²×Dic₃₁₄

C₂²

yes

—

C₂×#14578938

C₂²

yes

—

#19097771

—

#19097771

—

#19097771

—

C₂×C₄⋊Dic₁₅₇

C₄

yes

—

#19097689

—

#19097682

—

C₄×#11929434

—

C₂×M₁₆×C₁₅₇

C₂²

yes

—

C₂×C₄²×C₁₅₇

C₂²

yes

characteristic

C₄⋊C₈×C₁₅₇

—

C₂×Dic₃₁₄

—

C₂×Dic₃₁₄

—

C₂×Dic₃₁₄

—

C₂×Dic₃₁₄

—

C₂²×Dic₁₅₇

C₂×C₄

yes

—

#14578938

—

#14578938

D₄

yes

—

C₂×#11929434

—

C₄⋊Dic₁₅₇

—

C₄⋊Dic₁₅₇

—

C₂²×C₄×C₁₅₇

C₂×C₄

yes

—

C₂²×C₄×C₁₅₇

C₂³

yes

characteristic

M₁₆×C₁₅₇

—

C₂×C₈×C₁₅₇

—

C₄²×C₁₅₇

D₄

yes

—

C₄²×C₁₅₇

D₄

yes

characteristic

C₄²×C₁₅₇

D₄

yes

characteristic

Dic₃₁₄

—

Dic₃₁₄

—

Dic₃₁₄

—

C₂×Dic₁₅₇

—

C₂×Dic₁₅₇

—

#11929434

—

C₂³×C₁₅₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₅₇

—

C₂×C₄×C₁₅₇

C₂²⋊C₄

yes

—

C₂×C₄×C₁₅₇

C₂²⋊C₄

yes

—

C₂×C₄×C₁₅₇

C₂×D₄

yes

—

C₂×C₄×C₁₅₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₅₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₅₇

#41

yes

—

C₈×C₁₅₇

—

C₂²×C₁₅₇

—

C₂²×C₁₅₇

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₁₅₇

C₄×D₄

yes

—

C₄×C₁₅₇

—

C₄×C₁₅₇

#101

yes

complemented

C₄×C₁₅₇

#121

yes

—

C₄×C₁₅₇

yes

C₄×C₁₅₇

yes

Dic₁₅₇

—

C₂×C₁₅₇

—

C₂×C₁₅₇

#390

yes

characteristic

C₂×C₁₅₇

C₂×#101

yes

—

C₁₅₇

#1966

yes

characteristic, complemented, Sylow

#1966

—

157

maximal, Sylow

C₂×C₄⋊Q₈

—

157

—

#423

D₁₅₇

yes

characteristic

C₂×#101

—

157

—

C₄×M₁₆

—

157

—

C₂²×Q₈

—

157

—

C₂×M₁₆

—

157

—

C₂×M₁₆

C₂×D₁₅₇

yes

—

C₄⋊Q₈

—

314

—

C₄⋊Q₈

—

314

—

C₄⋊Q₈

—

314

—

C₂×C₄⋊C₄

—

157

—

C₂×C₄²

C₂×D₁₅₇

yes

characteristic

C₄⋊C₈

—

#101

—

314

—

#97

—

314

—

C₄×C₈

—

314

—

C₂×Q₈

—

314

—

C₂×Q₈

—

314

—

C₂×Q₈

—

314

—

C₂×Q₈

—

314

—

C₂²×C₄

—

157

—

C₂²×C₄

C₄×D₁₅₇

yes

—

C₂²×C₄

C₂²×D₁₅₇

yes

characteristic

M₁₆

—

M₁₆

—

314

—

M₁₆

—

157

—

C₂×C₈

—

C₂×C₈

—

314

—

C₄⋊C₄

—

314

—

C₄⋊C₄

—

314

—

C₄²

C₂²⋊D₁₅₇

yes

—

C₄²

D₆₂₈

yes

characteristic

C₄²

D₆₂₈

yes

characteristic

C₂³

C₂×C₄×D₁₅₇

yes

characteristic

Q₈

—

628

—

Q₈

—

628

—

Q₈

—

628

—

100

C₂×C₄

—

101

C₂×C₄

—

314

—

102

C₂×C₄

—

314

—

103

C₂×C₄

#14578931

yes

—

104

C₂×C₄

#14578931

yes

—

105

C₂×C₄

D₄×D₁₅₇

yes

—

106

C₂×C₄

C₂×C₂²⋊D₁₅₇

yes

characteristic

107

C₂×C₄

C₂×D₆₂₈

yes

characteristic

108

C₂×C₄

#14578955

yes

—

109

C₈

—

110

C₈

—

314

—

111

C₂²

—

112

C₂²

C₂×#14578931

yes

characteristic

113

C₂²

#19097733

yes

—

114

C₄

—

115

C₄

—

628

—

116

C₄

#19097689

yes

complemented

117

C₄

#19097746

yes

—

118

C₄

#19097760

yes

characteristic

119

C₄

#19097767

yes

characteristic

120

C₂

—

121

C₂

#23757174

yes

characteristic

122

C₂

C₂×#19097689

yes

—

	subgroup	G/H
center	C₂²	C₂×#14578931
commutator, prime radical	C₂³×C₁₅₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₅₇
Fitting	#423×C₁₅₇	C₂
socle	C₂²×C₁₅₇	C₂×C₂²⋊C₄
cosocle	C₂²×C₄×C₁₅₇	C₂³
hypercenter	#423	D₁₅₇
hypocenter	C₁₅₇	#1966
LCS	G ▷ C₂³×C₁₅₇ ▷ C₂×C₁₅₇ ▷ C₁₅₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #423

subgroup

G/H

center

C₂²

C₂×#14578931

commutator, prime radical

C₂³×C₁₅₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₅₇

Fitting

#423×C₁₅₇

C₂

socle

C₂²×C₁₅₇

C₂×C₂²⋊C₄

cosocle

C₂²×C₄×C₁₅₇

C₂³

D₁₅₇

C₁₅₇

G ▷ C₂³×C₁₅₇ ▷ C₂×C₁₅₇ ▷ C₁₅₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #423

N	H ≅ G/N	type
53. C₄×C₁₅₇	#101	semidirect
61. C₁₅₇	#1966	semidirect
116. C₄	#19097689	semidirect

H ≅ G/N

type

53. C₄×C₁₅₇

#101

semidirect

61. C₁₅₇

#1966

semidirect

116. C₄

#19097689

semidirect

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

subgroup	index
C₂×#19097771	2
C₂×#19097689	2
C₄×#14578938	2
#423×C₁₅₇	2
#1966	157

subgroup

index

#423×C₁₅₇

157

Subgroups — #34100979

Conjugacy classes of subgroups

Complements

Maximal subgroups