Group #34098907 — subgroups

Subgroups — #34098907

Order 20096.

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

normal subgroups only

This group has 168 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	#23757297	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊#11929434	C₂	1	1	yes	maximal, characteristic
3	#23757024	C₂	1	4	yes	maximal, complemented
4	#431×C₁₅₇	C₂	1	1	yes	maximal, characteristic
5	C₂×#14578938	—	2	1	no	—
6	C₂²×#11929434	—	2	1	no	—
7	C₄⋊#11929434	—	2	1	no	—
8	C₄⋊#11929434	—	2	1	no	—
9	C₄⋊#11929434	C₂²	1	2	yes	—
10	C₄⋊#11929434	C₂²	1	2	yes	—
11	C₂×C₄²×C₁₅₇	C₂²	1	1	yes	characteristic
12	C₄⋊C₈×C₁₅₇	C₄	1	2	yes	—
13	C₂²⋊C₈×C₁₅₇	C₄	1	2	yes	—
14	C₄×C₈×C₁₅₇	C₂²	1	2	yes	—
15	#14578938	—	4	2	no	—
16	C₂×#11929434	—	2	2	no	—
17	C₂×#11929434	—	2	2	no	—
18	C₂×#11929434	—	4	2	no	—
19	C₂²×C₄×C₁₅₇	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₄×C₁₅₇	D₄	1	1	yes	characteristic
21	C₂²×C₄×C₁₅₇	Q₈	1	1	yes	characteristic
22	C₂×C₈×C₁₅₇	C₂×C₄	1	4	yes	—
23	C₄²×C₁₅₇	C₂×C₄	1	1	yes	characteristic
24	C₄²×C₁₅₇	D₄	1	1	yes	characteristic
25	C₄²×C₁₅₇	Q₈	1	1	yes	characteristic
26	C₄²×C₁₅₇	C₂³	1	1	yes	characteristic
27	#11929434	—	4	2	no	—
28	#11929434	—	4	2	no	—
29	C₂³×C₁₅₇	C₄⋊C₄	1	1	yes	characteristic
30	C₂×C₄×C₁₅₇	—	2	2	no	—
31	C₂×C₄×C₁₅₇	—	4	1	no	—
32	C₂×C₄×C₁₅₇	C₄⋊C₄	1	1	yes	characteristic
33	C₂×C₄×C₁₅₇	C₄⋊C₄	1	1	yes	characteristic
34	C₂×C₄×C₁₅₇	C₄⋊C₄	1	1	yes	characteristic
35	C₂×C₄×C₁₅₇	SD₁₆	1	2	yes	—
36	C₂×C₄×C₁₅₇	SD₁₆	1	2	yes	—
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×Q₈	1	1	yes	characteristic
40	C₈×C₁₅₇	—	2	4	no	—
41	C₂²×C₁₅₇	—	2	1	no	—
42	C₂²×C₁₅₇	—	2	1	no	—
43	C₂²×C₁₅₇	#106	1	2	yes	—
44	C₂²×C₁₅₇	C₂×C₄⋊C₄	1	1	yes	characteristic
45	C₄×C₁₅₇	—	2	1	no	—
46	C₄×C₁₅₇	—	2	1	no	—
47	C₄×C₁₅₇	—	2	1	no	—
48	C₄×C₁₅₇	#106	1	2	yes	—
49	C₄×C₁₅₇	#131	1	2	yes	—
50	C₄×C₁₅₇	C₂×SD₁₆	1	1	yes	characteristic
51	C₄×C₁₅₇	C₂×SD₁₆	1	1	yes	characteristic
52	C₂×C₁₅₇	—	2	2	no	—
53	C₂×C₁₅₇	#424	1	1	yes	characteristic
54	C₂×C₁₅₇	C₂×#106	1	1	yes	characteristic
55	C₂×C₁₅₇	#430	1	1	yes	characteristic
56	C₁₅₇	#1570	1	1	yes	characteristic, complemented, Sylow
57	#1570	—	157	1	no	maximal, Sylow
58	#431	D₁₅₇	1	1	yes	characteristic
59	#423	—	157	1	no	—
60	C₂×C₄⋊C₈	—	157	1	no	—
61	#334	—	157	4	no	—
62	C₂×M₁₆	—	314	1	no	—
63	C₂²×C₈	—	314	1	no	—
64	C₂×C₄²	C₂×D₁₅₇	1	1	yes	characteristic
65	C₄⋊C₈	—	314	1	no	—
66	C₄⋊C₈	—	314	1	no	—
67	C₄⋊C₈	—	157	2	no	—
68	C₄⋊C₈	—	157	2	no	—
69	C₄⋊C₈	Dic₁₅₇	1	2	yes	—
70	C₂²⋊C₈	Dic₁₅₇	1	2	yes	—
71	C₄×C₈	C₂×D₁₅₇	1	2	yes	—
72	C₂²×C₄	C₂×Dic₁₅₇	1	1	yes	characteristic
73	C₂²×C₄	D₆₂₈	1	1	yes	characteristic
74	C₂²×C₄	Dic₃₁₄	1	1	yes	characteristic
75	M₁₆	—	628	2	no	—
76	C₂×C₈	—	314	2	no	—
77	C₂×C₈	—	314	2	no	—
78	C₂×C₈	—	628	2	no	—
79	C₂×C₈	C₂×Dic₁₅₇	1	4	yes	—
80	C₄²	C₂×Dic₁₅₇	1	1	yes	characteristic
81	C₄²	D₆₂₈	1	1	yes	characteristic
82	C₄²	Dic₃₁₄	1	1	yes	characteristic
83	C₄²	C₂²×D₁₅₇	1	1	yes	characteristic
84	C₂³	C₄⋊Dic₁₅₇	1	1	yes	characteristic
85	C₂×C₄	—	2	2	no	—
86	C₂×C₄	—	4	1	no	—
87	C₂×C₄	C₄⋊Dic₁₅₇	1	1	yes	characteristic
88	C₂×C₄	C₄⋊Dic₁₅₇	1	1	yes	characteristic
89	C₂×C₄	C₄⋊Dic₁₅₇	1	1	yes	characteristic
90	C₂×C₄	#14578943	1	2	yes	—
91	C₂×C₄	#14578943	1	2	yes	—
92	C₂×C₄	C₂²×Dic₁₅₇	1	1	yes	characteristic
93	C₂×C₄	C₂×D₆₂₈	1	1	yes	characteristic
94	C₂×C₄	C₂×Dic₃₁₄	1	1	yes	characteristic
95	C₈	—	2	4	no	—
96	C₈	—	628	2	no	—
97	C₈	—	628	2	no	—
98	C₂²	—	2	1	no	—
99	C₂²	—	2	1	no	—
100	C₂²	#19097703	1	2	yes	—
101	C₂²	C₂×C₄⋊Dic₁₅₇	1	1	yes	characteristic
102	C₄	—	2	1	no	—
103	C₄	—	2	1	no	—
104	C₄	—	2	1	no	—
105	C₄	#19097703	1	2	yes	—
106	C₄	#19097778	1	2	yes	—
107	C₄	C₂×#14578943	1	1	yes	characteristic
108	C₄	C₂×#14578943	1	1	yes	characteristic
109	C₂	—	2	2	no	—
110	C₂	#23757300	1	1	yes	characteristic
111	C₂	C₂×#19097703	1	1	yes	characteristic
112	C₂	#23757322	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#23757297

C₂

yes

maximal, characteristic

C₂×C₄⋊#11929434

C₂

yes

maximal, characteristic

#23757024

C₂

yes

maximal, complemented

#431×C₁₅₇

C₂

yes

maximal, characteristic

—

—

—

—

C₂²

yes

—

C₄⋊#11929434

C₂²

yes

—

C₂×C₄²×C₁₅₇

C₂²

yes

characteristic

C₄⋊C₈×C₁₅₇

C₄

yes

—

C₂²⋊C₈×C₁₅₇

C₄

yes

—

C₄×C₈×C₁₅₇

C₂²

yes

—

—

—

—

—

C₂²×C₄×C₁₅₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₅₇

D₄

yes

characteristic

C₂²×C₄×C₁₅₇

Q₈

yes

characteristic

C₂×C₈×C₁₅₇

C₂×C₄

yes

—

C₄²×C₁₅₇

C₂×C₄

yes

characteristic

C₄²×C₁₅₇

D₄

yes

characteristic

C₄²×C₁₅₇

Q₈

yes

characteristic

C₄²×C₁₅₇

C₂³

yes

characteristic

#11929434

—

#11929434

—

C₂³×C₁₅₇

C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₅₇

—

C₂×C₄×C₁₅₇

—

C₂×C₄×C₁₅₇

C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₅₇

C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₅₇

C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₅₇

SD₁₆

yes

—

C₂×C₄×C₁₅₇

SD₁₆

yes

—

C₂×C₄×C₁₅₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₅₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₅₇

C₂×Q₈

yes

characteristic

C₈×C₁₅₇

—

C₂²×C₁₅₇

—

C₂²×C₁₅₇

—

C₂²×C₁₅₇

#106

yes

—

C₂²×C₁₅₇

C₂×C₄⋊C₄

yes

characteristic

C₄×C₁₅₇

—

C₄×C₁₅₇

—

C₄×C₁₅₇

—

C₄×C₁₅₇

#106

yes

—

C₄×C₁₅₇

#131

yes

—

C₄×C₁₅₇

C₂×SD₁₆

yes

characteristic

C₄×C₁₅₇

C₂×SD₁₆

yes

characteristic

C₂×C₁₅₇

—

C₂×C₁₅₇

yes

C₂×C₁₅₇

yes

C₂×C₁₅₇

yes

C₁₅₇

yes

characteristic, complemented, Sylow

#1570

—

157

maximal, Sylow

#431

D₁₅₇

yes

characteristic

#423

—

157

—

C₂×C₄⋊C₈

—

157

—

#334

—

157

—

C₂×M₁₆

—

314

—

C₂²×C₈

—

314

—

C₂×C₄²

C₂×D₁₅₇

yes

characteristic

C₄⋊C₈

—

314

—

C₄⋊C₈

—

314

—

C₄⋊C₈

—

157

—

C₄⋊C₈

—

157

—

C₄⋊C₈

Dic₁₅₇

yes

—

C₂²⋊C₈

Dic₁₅₇

yes

—

C₄×C₈

C₂×D₁₅₇

yes

—

C₂²×C₄

C₂×Dic₁₅₇

yes

characteristic

C₂²×C₄

D₆₂₈

yes

characteristic

C₂²×C₄

Dic₃₁₄

yes

characteristic

M₁₆

—

628

—

C₂×C₈

—

314

—

C₂×C₈

—

314

—

C₂×C₈

—

628

—

C₂×C₈

C₂×Dic₁₅₇

yes

—

C₄²

C₂×Dic₁₅₇

yes

characteristic

C₄²

D₆₂₈

yes

characteristic

C₄²

Dic₃₁₄

yes

characteristic

C₄²

C₂²×D₁₅₇

yes

characteristic

C₂³

C₄⋊Dic₁₅₇

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₄⋊Dic₁₅₇

yes

characteristic

C₂×C₄

C₄⋊Dic₁₅₇

yes

characteristic

C₂×C₄

C₄⋊Dic₁₅₇

yes

characteristic

C₂×C₄

#14578943

yes

—

C₂×C₄

#14578943

yes

—

C₂×C₄

C₂²×Dic₁₅₇

yes

characteristic

C₂×C₄

C₂×D₆₂₈

yes

characteristic

C₂×C₄

C₂×Dic₃₁₄

yes

characteristic

C₈

—

C₈

—

628

—

C₈

—

628

—

C₂²

—

C₂²

—

100

C₂²

#19097703

yes

—

101

C₂²

C₂×C₄⋊Dic₁₅₇

yes

characteristic

102

C₄

—

103

C₄

—

104

C₄

—

105

C₄

#19097703

yes

—

106

C₄

#19097778

yes

—

107

C₄

C₂×#14578943

yes

characteristic

108

C₄

C₂×#14578943

yes

characteristic

109

C₂

—

110

C₂

#23757300

yes

characteristic

111

C₂

C₂×#19097703

yes

characteristic

112

C₂

#23757322

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₆₂₈
commutator	C₂×C₄×C₁₅₇	C₂²×C₄
Frattini	C₄²	C₂²×D₁₅₇
Fitting	#431×C₁₅₇	C₂
socle	C₂²×C₁₅₇	C₂×C₄⋊C₄
prime radical	C₂³×C₁₅₇	C₄⋊C₄
cosocle	C₄²×C₁₅₇	C₂³
hypercenter	#431	D₁₅₇
hypocenter	C₁₅₇	#1570
LCS	G ▷ C₂×C₄×C₁₅₇ ▷ C₂×C₁₅₇ ▷ C₁₅₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #431

subgroup

G/H

center

C₂×C₄

C₂×D₆₂₈

commutator

C₂×C₄×C₁₅₇

C₂²×C₄

Frattini

C₄²

C₂²×D₁₅₇

Fitting

#431×C₁₅₇

C₂

socle

C₂²×C₁₅₇

C₂×C₄⋊C₄

prime radical

C₂³×C₁₅₇

C₄⋊C₄

cosocle

C₄²×C₁₅₇

C₂³

D₁₅₇

C₁₅₇

G ▷ C₂×C₄×C₁₅₇ ▷ C₂×C₁₅₇ ▷ C₁₅₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #431

N	H ≅ G/N	type
3. #23757024	C₂	semidirect
56. C₁₅₇	#1570	semidirect

H ≅ G/N

type

3. #23757024

C₂

semidirect

56. C₁₅₇

#1570

semidirect

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

subgroup	index
#23757297	2
C₂×C₄⋊#11929434	2
#23757024	2
#431×C₁₅₇	2
#1570	157

subgroup

index

#431×C₁₅₇

157

Subgroups — #34098907

Conjugacy classes of subgroups

Complements

Maximal subgroups