Group #34100309 — subgroups

Subgroups — #34100309

Order 20096.

212 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 212 conjugacy classes of subgroups — too many to draw. Only its 101 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	C₂×#19097706	C₂	1	2	yes	maximal
3	#23757296	C₂	1	2	yes	maximal
4	C₂×#19097681	C₂	1	1	yes	maximal, characteristic
5	C₂×#108×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	#19097706	—	2	4	no	—
11	#19097700	—	2	4	no	—
12	#19097681	C₄	1	4	yes	—
13	C₂×M₁₆×C₁₅₇	C₂²	1	2	yes	—
14	C₂²×C₈×C₁₅₇	C₂²	1	1	yes	characteristic
15	#108×C₁₅₇	—	2	2	no	—
16	C₂²×Dic₁₅₇	Q₈	1	2	yes	—
17	#14578938	—	2	2	no	—
18	#14578938	—	2	2	no	—
19	C₂×#11929434	—	2	2	no	—
20	C₂×#11929434	C₂×C₄	1	2	yes	—
21	C₂×#11929434	D₄	1	2	yes	—
22	C₂×#11929434	Q₈	1	2	yes	—
23	C₄×Dic₁₅₇	C₂×C₄	1	2	yes	—
24	C₄×Dic₁₅₇	D₄	1	2	yes	—
25	C₂²×C₄×C₁₅₇	C₂³	1	1	yes	characteristic
26	M₁₆×C₁₅₇	—	2	2	no	—
27	M₁₆×C₁₅₇	—	2	2	no	—
28	C₂×C₈×C₁₅₇	—	2	2	no	—
29	C₂×C₈×C₁₅₇	C₂×C₄	1	2	yes	—
30	C₂×C₈×C₁₅₇	D₄	1	2	yes	—
31	C₂×C₈×C₁₅₇	Q₈	1	2	yes	—
32	C₂×Dic₁₅₇	—	4	2	no	—
33	C₂×Dic₁₅₇	C₄⋊C₄	1	4	yes	—
34	#11929434	—	4	2	no	—
35	#11929434	C₄⋊C₄	1	4	yes	—
36	C₂³×C₁₅₇	C₂×Q₈	1	1	yes	characteristic
37	C₂×C₄×C₁₅₇	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₄×C₁₅₇	C₂×D₄	1	1	yes	characteristic
39	C₂×C₄×C₁₅₇	C₂×D₄	1	1	yes	characteristic
40	C₂×C₄×C₁₅₇	C₂×D₄	1	1	yes	characteristic
41	C₂×C₄×C₁₅₇	C₂×Q₈	1	1	yes	characteristic
42	C₂×C₄×C₁₅₇	C₂×Q₈	1	1	yes	characteristic
43	C₈×C₁₅₇	—	4	2	no	—
44	C₈×C₁₅₇	C₄⋊C₄	1	4	yes	—
45	C₂²×C₁₅₇	—	2	2	no	—
46	C₂²×C₁₅₇	C₂×C₄⋊C₄	1	1	yes	characteristic
47	C₂²×C₁₅₇	C₄⋊Q₈	1	1	yes	characteristic
48	C₂²×C₁₅₇	C₄⋊Q₈	1	1	yes	characteristic
49	C₄×C₁₅₇	C₂×C₄⋊C₄	1	1	yes	characteristic
50	C₄×C₁₅₇	C₂×C₄⋊C₄	1	1	yes	characteristic
51	C₄×C₁₅₇	#127	1	1	yes	characteristic
52	C₄×C₁₅₇	C₄⋊Q₈	1	1	yes	characteristic
53	Dic₁₅₇	—	2	4	no	—
54	C₂×C₁₅₇	—	2	1	no	—
55	C₂×C₁₅₇	—	2	1	no	—
56	C₂×C₁₅₇	#384	1	1	yes	characteristic
57	C₂×C₁₅₇	#430	1	2	yes	—
58	C₁₅₇	#1849	1	1	yes	characteristic, complemented, Sylow
59	#1849	—	157	1	no	maximal, Sylow
60	C₂×#108	—	157	2	no	—
61	C₂×#108	—	157	1	no	—
62	C₂×#108	D₁₅₇	1	1	yes	characteristic
63	#423	—	157	2	no	—
64	C₂×#97	—	157	1	no	—
65	C₂×M₁₆	—	157	2	no	—
66	C₂×M₁₆	C₂×D₁₅₇	1	2	yes	—
67	C₂²×C₈	—	157	1	no	—
68	C₂²×C₈	C₂×D₁₅₇	1	1	yes	characteristic
69	C₂×C₄²	—	157	1	no	—
70	#108	—	2	2	no	—
71	#108	—	314	4	no	—
72	#108	—	314	2	no	—
73	C₄⋊C₈	—	314	4	no	—
74	#97	—	157	4	no	—
75	C₂²×C₄	—	157	2	no	—
76	C₂²×C₄	C₂²×D₁₅₇	1	1	yes	characteristic
77	M₁₆	—	2	2	no	—
78	M₁₆	—	2	2	no	—
79	M₁₆	—	314	2	no	—
80	M₁₆	—	314	2	no	—
81	C₂×C₈	—	2	2	no	—
82	C₂×C₈	—	314	2	no	—
83	C₂×C₈	—	157	2	no	—
84	C₂×C₈	—	157	2	no	—
85	C₂×C₈	—	157	2	no	—
86	C₂×C₈	C₄×D₁₅₇	1	2	yes	—
87	C₂×C₈	C₂²⋊D₁₅₇	1	2	yes	—
88	C₂×C₈	Dic₃₁₄	1	2	yes	—
89	C₄²	—	157	2	no	—
90	C₄²	—	157	2	no	—
91	C₂³	Q₈×D₁₅₇	1	1	yes	characteristic
92	C₂×C₄	—	628	2	no	—
93	C₂×C₄	—	157	4	no	—
94	C₂×C₄	C₂×C₄×D₁₅₇	1	1	yes	characteristic
95	C₂×C₄	D₄×D₁₅₇	1	1	yes	characteristic
96	C₂×C₄	D₄×D₁₅₇	1	1	yes	characteristic
97	C₂×C₄	C₂×C₂²⋊D₁₅₇	1	1	yes	characteristic
98	C₂×C₄	Q₈×D₁₅₇	1	1	yes	characteristic
99	C₂×C₄	C₂×Dic₃₁₄	1	1	yes	characteristic
100	C₈	—	4	2	no	—
101	C₈	—	628	2	no	—
102	C₈	—	157	4	no	—
103	C₈	#14578933	1	4	yes	—
104	C₂²	—	2	2	no	—
105	C₂²	C₄⋊C₄×D₁₅₇	1	1	yes	characteristic
106	C₂²	#19097769	1	1	yes	characteristic
107	C₂²	#19097770	1	1	yes	characteristic
108	C₄	—	314	4	no	—
109	C₄	C₄⋊C₄×D₁₅₇	1	1	yes	characteristic
110	C₄	C₂×#14578933	1	1	yes	characteristic
111	C₄	#19097767	1	1	yes	characteristic
112	C₄	#19097770	1	1	yes	characteristic
113	C₂	—	2	1	no	—
114	C₂	—	2	1	no	—
115	C₂	#23757142	1	1	yes	characteristic
116	C₂	#23757320	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#19097707

C₂

yes

maximal, characteristic

C₂×#19097706

C₂

yes

maximal

#23757296

C₂

yes

maximal

C₂×#19097681

C₂

yes

maximal, characteristic

C₂×#108×C₁₅₇

C₂

yes

maximal, characteristic

C₂×#14578938

C₂²

yes

—

C₂²×#11929434

C₂²

yes

characteristic

C₂×C₄×Dic₁₅₇

C₂²

yes

—

—

—

C₄

yes

—

C₂×M₁₆×C₁₅₇

C₂²

yes

—

C₂²×C₈×C₁₅₇

C₂²

yes

characteristic

#108×C₁₅₇

—

C₂²×Dic₁₅₇

Q₈

yes

—

—

—

—

C₂×C₄

yes

—

C₂×#11929434

D₄

yes

—

C₂×#11929434

Q₈

yes

—

C₄×Dic₁₅₇

C₂×C₄

yes

—

C₄×Dic₁₅₇

D₄

yes

—

C₂²×C₄×C₁₅₇

C₂³

yes

characteristic

M₁₆×C₁₅₇

—

M₁₆×C₁₅₇

—

C₂×C₈×C₁₅₇

—

C₂×C₈×C₁₅₇

C₂×C₄

yes

—

C₂×C₈×C₁₅₇

D₄

yes

—

C₂×C₈×C₁₅₇

Q₈

yes

—

C₂×Dic₁₅₇

—

C₂×Dic₁₅₇

C₄⋊C₄

yes

—

#11929434

—

#11929434

C₄⋊C₄

yes

—

C₂³×C₁₅₇

C₂×Q₈

yes

characteristic

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×D₄

yes

characteristic

C₂×C₄×C₁₅₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₅₇

C₂×Q₈

yes

characteristic

C₈×C₁₅₇

—

C₈×C₁₅₇

C₄⋊C₄

yes

—

C₂²×C₁₅₇

—

C₂²×C₁₅₇

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₁₅₇

C₄⋊Q₈

yes

characteristic

C₂²×C₁₅₇

C₄⋊Q₈

yes

characteristic

C₄×C₁₅₇

C₂×C₄⋊C₄

yes

characteristic

C₄×C₁₅₇

C₂×C₄⋊C₄

yes

characteristic

C₄×C₁₅₇

#127

yes

characteristic

C₄×C₁₅₇

C₄⋊Q₈

yes

characteristic

Dic₁₅₇

—

C₂×C₁₅₇

—

C₂×C₁₅₇

—

C₂×C₁₅₇

#384

yes

characteristic

C₂×C₁₅₇

#430

yes

—

C₁₅₇

#1849

yes

characteristic, complemented, Sylow

#1849

—

157

maximal, Sylow

C₂×#108

—

157

—

C₂×#108

—

157

—

C₂×#108

D₁₅₇

yes

characteristic

#423

—

157

—

C₂×#97

—

157

—

C₂×M₁₆

—

157

—

C₂×M₁₆

C₂×D₁₅₇

yes

—

C₂²×C₈

—

157

—

C₂²×C₈

C₂×D₁₅₇

yes

characteristic

C₂×C₄²

—

157

—

#108

—

#108

—

314

—

#108

—

314

—

C₄⋊C₈

—

314

—

#97

—

157

—

C₂²×C₄

—

157

—

C₂²×C₄

C₂²×D₁₅₇

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

314

—

M₁₆

—

314

—

C₂×C₈

—

C₂×C₈

—

314

—

C₂×C₈

—

157

—

C₂×C₈

—

157

—

C₂×C₈

—

157

—

C₂×C₈

C₄×D₁₅₇

yes

—

C₂×C₈

C₂²⋊D₁₅₇

yes

—

C₂×C₈

Dic₃₁₄

yes

—

C₄²

—

157

—

C₄²

—

157

—

C₂³

Q₈×D₁₅₇

yes

characteristic

C₂×C₄

—

628

—

C₂×C₄

—

157

—

C₂×C₄

C₂×C₄×D₁₅₇

yes

characteristic

C₂×C₄

D₄×D₁₅₇

yes

characteristic

C₂×C₄

D₄×D₁₅₇

yes

characteristic

C₂×C₄

C₂×C₂²⋊D₁₅₇

yes

characteristic

C₂×C₄

Q₈×D₁₅₇

yes

characteristic

C₂×C₄

C₂×Dic₃₁₄

yes

characteristic

100

C₈

—

101

C₈

—

628

—

102

C₈

—

157

—

103

C₈

#14578933

yes

—

104

C₂²

—

105

C₂²

C₄⋊C₄×D₁₅₇

yes

characteristic

106

C₂²

#19097769

yes

characteristic

107

C₂²

#19097770

yes

characteristic

108

C₄

—

314

—

109

C₄

C₄⋊C₄×D₁₅₇

yes

characteristic

110

C₄

C₂×#14578933

yes

characteristic

111

C₄

#19097767

yes

characteristic

112

C₄

#19097770

yes

characteristic

113

C₂

—

114

C₂

—

115

C₂

#23757142

yes

characteristic

116

C₂

#23757320

yes

—

	subgroup	G/H
center	C₂×C₄	D₄×D₁₅₇
commutator	C₂×C₄×C₁₅₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₅₇
Fitting	C₂×#108×C₁₅₇	C₂
socle	C₂²×C₁₅₇	C₂×C₄⋊C₄
prime radical	C₂³×C₁₅₇	C₂×Q₈
cosocle	C₂²×C₄×C₁₅₇	C₂³
hypercenter	C₂×#108	D₁₅₇
hypocenter	C₁₅₇	#1849
LCS	G ▷ C₂×C₄×C₁₅₇ ▷ C₂×C₁₅₇ ▷ C₁₅₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×#108

subgroup

G/H

center

C₂×C₄

D₄×D₁₅₇

commutator

C₂×C₄×C₁₅₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₅₇

Fitting

C₂×#108×C₁₅₇

C₂

socle

C₂²×C₁₅₇

C₂×C₄⋊C₄

prime radical

C₂³×C₁₅₇

C₂×Q₈

cosocle

C₂²×C₄×C₁₅₇

C₂³

D₁₅₇

C₁₅₇

G ▷ C₂×C₄×C₁₅₇ ▷ C₂×C₁₅₇ ▷ C₁₅₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×#108

N	H ≅ G/N	type
58. C₁₅₇	#1849	semidirect

H ≅ G/N

type

58. C₁₅₇

#1849

semidirect

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

subgroup	index
C₂×#19097707	2
C₂×#19097706	2
#23757296	2
C₂×#19097681	2
C₂×#108×C₁₅₇	2
#1849	157

subgroup

index

C₂×#108×C₁₅₇

157

Subgroups — #34100309

Conjugacy classes of subgroups

Complements

Maximal subgroups