Group #34099732 — subgroups

Subgroups — #34099732

Order 20096.

180 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 180 conjugacy classes of subgroups — too many to draw. Only its 65 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	#23757676	C₂	1	1	yes	maximal, characteristic
2	#23757669	C₂	1	1	yes	maximal, characteristic
3	#23757669	C₂	1	1	yes	maximal, characteristic
4	#23757652	C₂	1	1	yes	maximal, characteristic
5	#23757624	C₂	1	1	yes	maximal, characteristic
6	C₈×#11929434	C₂	1	1	yes	maximal, characteristic
7	#487×C₁₅₇	C₂	1	1	yes	maximal, characteristic
8	C₂×#14578944	—	2	2	no	—
9	#19097771	C₂²	1	1	yes	characteristic
10	#19097771	C₂²	1	1	yes	characteristic
11	#19097704	—	2	2	no	—
12	#19097702	—	2	2	no	—
13	#19097702	—	2	2	no	—
14	#19097696	—	2	2	no	—
15	C₄×#11929434	C₂²	1	1	yes	characteristic
16	C₄×#11929434	C₂²	1	1	yes	characteristic
17	C₄⋊Q₈×C₁₅₇	C₂²	1	1	yes	characteristic
18	C₄⋊Q₈×C₁₅₇	C₂²	1	1	yes	characteristic
19	#103×C₁₅₇	—	2	2	no	—
20	C₄×C₈×C₁₅₇	C₂²	1	1	yes	characteristic
21	C₂×Dic₃₁₄	—	2	1	no	—
22	C₂×Dic₃₁₄	—	2	1	no	—
23	#14578944	—	4	4	no	—
24	C₂×#11929434	D₄	1	2	yes	—
25	C₂×#11929434	D₄	1	2	yes	—
26	C₄⋊Dic₁₅₇	—	2	1	no	—
27	C₄⋊Dic₁₅₇	—	2	1	no	—
28	C₄⋊Dic₁₅₇	—	2	1	no	—
29	C₄⋊Dic₁₅₇	—	2	1	no	—
30	C₂×Q₈×C₁₅₇	—	2	1	no	—
31	C₂×Q₈×C₁₅₇	—	2	1	no	—
32	C₂×C₈×C₁₅₇	D₄	1	2	yes	—
33	C₄⋊C₄×C₁₅₇	—	2	1	no	—
34	C₄⋊C₄×C₁₅₇	—	2	1	no	—
35	C₄⋊C₄×C₁₅₇	—	2	1	no	—
36	C₄⋊C₄×C₁₅₇	—	2	1	no	—
37	C₄²×C₁₅₇	C₂³	1	1	yes	characteristic
38	Dic₃₁₄	—	4	2	no	—
39	Dic₃₁₄	—	4	2	no	—
40	C₂×Dic₁₅₇	—	4	1	no	—
41	C₂×Dic₁₅₇	—	4	1	no	—
42	#11929434	SD₁₆	1	4	yes	—
43	#11929434	Q₁₆	1	4	yes	complemented
44	Q₈×C₁₅₇	—	4	2	no	—
45	Q₈×C₁₅₇	—	4	2	no	—
46	C₂×C₄×C₁₅₇	—	4	1	no	—
47	C₂×C₄×C₁₅₇	—	4	1	no	—
48	C₂×C₄×C₁₅₇	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₁₅₇	C₂×D₄	1	1	yes	characteristic
50	C₂×C₄×C₁₅₇	C₂×D₄	1	1	yes	characteristic
51	C₈×C₁₅₇	—	2	2	no	—
52	C₂²×C₁₅₇	#127	1	1	yes	characteristic
53	C₄×C₁₅₇	—	8	1	no	—
54	C₄×C₁₅₇	—	8	1	no	—
55	C₄×C₁₅₇	C₂×SD₁₆	1	2	yes	—
56	C₄×C₁₅₇	C₂×Q₁₆	1	2	yes	—
57	C₄×C₁₅₇	#135	1	2	yes	—
58	Dic₁₅₇	—	8	1	no	—
59	Dic₁₅₇	—	8	1	no	—
60	C₂×C₁₅₇	#492	1	1	yes	characteristic
61	C₂×C₁₅₇	#494	1	1	yes	characteristic
62	C₂×C₁₅₇	#495	1	1	yes	characteristic
63	C₁₅₇	#1707	1	1	yes	characteristic, complemented, Sylow
64	#1707	—	157	1	no	maximal, Sylow
65	#500	—	157	1	no	—
66	#498	—	157	1	no	—
67	#498	—	157	1	no	—
68	#494	—	157	1	no	—
69	#487	—	157	1	no	—
70	#487	D₁₅₇	1	1	yes	characteristic
71	C₈²	—	157	1	no	—
72	C₂×Q₁₆	—	314	2	no	—
73	C₄⋊Q₈	—	157	1	no	—
74	C₄⋊Q₈	—	157	1	no	—
75	C₄⋊Q₈	C₂×D₁₅₇	1	1	yes	characteristic
76	C₄⋊Q₈	C₂×D₁₅₇	1	1	yes	characteristic
77	#107	—	314	2	no	—
78	#106	—	314	2	no	—
79	#106	—	314	2	no	—
80	#103	—	2	2	no	—
81	#103	—	314	2	no	—
82	C₄×C₈	—	157	1	no	—
83	C₄×C₈	—	157	1	no	—
84	C₄×C₈	C₂×D₁₅₇	1	1	yes	characteristic
85	C₂×Q₈	—	2	1	no	—
86	C₂×Q₈	—	2	1	no	—
87	C₂×Q₈	—	314	1	no	—
88	C₂×Q₈	—	314	1	no	—
89	Q₁₆	—	628	4	no	—
90	C₂×C₈	—	157	2	no	—
91	C₂×C₈	—	157	2	no	—
92	C₂×C₈	C₂²⋊D₁₅₇	1	2	yes	—
93	C₄⋊C₄	—	2	1	no	—
94	C₄⋊C₄	—	2	1	no	—
95	C₄⋊C₄	—	2	1	no	—
96	C₄⋊C₄	—	2	1	no	—
97	C₄⋊C₄	—	314	1	no	—
98	C₄⋊C₄	—	314	1	no	—
99	C₄⋊C₄	—	314	1	no	—
100	C₄⋊C₄	—	314	1	no	—
101	C₄²	C₂²×D₁₅₇	1	1	yes	characteristic
102	Q₈	—	4	2	no	—
103	Q₈	—	4	2	no	—
104	Q₈	—	628	2	no	—
105	Q₈	—	628	2	no	—
106	C₂×C₄	—	4	1	no	—
107	C₂×C₄	—	4	1	no	—
108	C₂×C₄	—	628	1	no	—
109	C₂×C₄	—	628	1	no	—
110	C₂×C₄	D₄×D₁₅₇	1	1	yes	characteristic
111	C₂×C₄	D₄×D₁₅₇	1	1	yes	characteristic
112	C₂×C₄	C₂×C₂²⋊D₁₅₇	1	1	yes	characteristic
113	C₈	—	2	2	no	—
114	C₈	—	157	4	no	—
115	C₈	—	157	4	no	—
116	C₂²	#19097767	1	1	yes	characteristic
117	C₄	—	8	1	no	—
118	C₄	—	8	1	no	—
119	C₄	—	1256	1	no	—
120	C₄	—	1256	1	no	—
121	C₄	SD₁₆×D₁₅₇	1	2	yes	—
122	C₄	Q₁₆×D₁₅₇	1	2	yes	—
123	C₄	#19097791	1	2	yes	—
124	C₂	#23757644	1	1	yes	characteristic
125	C₂	#23757651	1	1	yes	characteristic
126	C₂	#23757655	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#23757676

C₂

yes

maximal, characteristic

#23757669

C₂

yes

maximal, characteristic

#23757669

C₂

yes

maximal, characteristic

#23757652

C₂

yes

maximal, characteristic

#23757624

C₂

yes

maximal, characteristic

C₈×#11929434

C₂

yes

maximal, characteristic

#487×C₁₅₇

C₂

yes

maximal, characteristic

C₂×#14578944

—

#19097771

C₂²

yes

characteristic

#19097771

C₂²

yes

—

—

—

—

C₂²

yes

characteristic

C₄×#11929434

C₂²

yes

characteristic

C₄⋊Q₈×C₁₅₇

C₂²

yes

characteristic

C₄⋊Q₈×C₁₅₇

C₂²

yes

characteristic

#103×C₁₅₇

—

C₄×C₈×C₁₅₇

C₂²

yes

characteristic

C₂×Dic₃₁₄

—

C₂×Dic₃₁₄

—

#14578944

—

C₂×#11929434

D₄

yes

—

C₂×#11929434

D₄

yes

—

C₄⋊Dic₁₅₇

—

C₄⋊Dic₁₅₇

—

C₄⋊Dic₁₅₇

—

C₄⋊Dic₁₅₇

—

C₂×Q₈×C₁₅₇

—

C₂×Q₈×C₁₅₇

—

C₂×C₈×C₁₅₇

D₄

yes

—

C₄⋊C₄×C₁₅₇

—

C₄⋊C₄×C₁₅₇

—

C₄⋊C₄×C₁₅₇

—

C₄⋊C₄×C₁₅₇

—

C₄²×C₁₅₇

C₂³

yes

characteristic

Dic₃₁₄

—

Dic₃₁₄

—

C₂×Dic₁₅₇

—

C₂×Dic₁₅₇

—

#11929434

SD₁₆

yes

—

#11929434

Q₁₆

yes

complemented

Q₈×C₁₅₇

—

Q₈×C₁₅₇

—

C₂×C₄×C₁₅₇

—

C₂×C₄×C₁₅₇

—

C₂×C₄×C₁₅₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₅₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₅₇

C₂×D₄

yes

characteristic

C₈×C₁₅₇

—

C₂²×C₁₅₇

#127

yes

characteristic

C₄×C₁₅₇

—

C₄×C₁₅₇

—

C₄×C₁₅₇

C₂×SD₁₆

yes

—

C₄×C₁₅₇

C₂×Q₁₆

yes

—

C₄×C₁₅₇

#135

yes

—

Dic₁₅₇

—

Dic₁₅₇

—

C₂×C₁₅₇

yes

C₂×C₁₅₇

yes

C₂×C₁₅₇

yes

C₁₅₇

yes

characteristic, complemented, Sylow

#1707

—

157

maximal, Sylow

#500

—

157

—

#498

—

157

—

#498

—

157

—

#494

—

157

—

#487

—

157

—

#487

D₁₅₇

yes

characteristic

C₈²

—

157

—

C₂×Q₁₆

—

314

—

C₄⋊Q₈

—

157

—

C₄⋊Q₈

—

157

—

C₄⋊Q₈

C₂×D₁₅₇

yes

characteristic

C₄⋊Q₈

C₂×D₁₅₇

yes

characteristic

#107

—

314

—

#106

—

314

—

#106

—

314

—

#103

—

#103

—

314

—

C₄×C₈

—

157

—

C₄×C₈

—

157

—

C₄×C₈

C₂×D₁₅₇

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

314

—

C₂×Q₈

—

314

—

Q₁₆

—

628

—

C₂×C₈

—

157

—

C₂×C₈

—

157

—

C₂×C₈

C₂²⋊D₁₅₇

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

314

—

C₄⋊C₄

—

314

—

C₄⋊C₄

—

314

—

100

C₄⋊C₄

—

314

—

101

C₄²

C₂²×D₁₅₇

yes

characteristic

102

Q₈

—

103

Q₈

—

104

Q₈

—

628

—

105

Q₈

—

628

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

628

—

109

C₂×C₄

—

628

—

110

C₂×C₄

D₄×D₁₅₇

yes

characteristic

111

C₂×C₄

D₄×D₁₅₇

yes

characteristic

112

C₂×C₄

C₂×C₂²⋊D₁₅₇

yes

characteristic

113

C₈

—

114

C₈

—

157

—

115

C₈

—

157

—

116

C₂²

#19097767

yes

characteristic

117

C₄

—

118

C₄

—

119

C₄

—

1256

—

120

C₄

—

1256

—

121

C₄

SD₁₆×D₁₅₇

yes

—

122

C₄

Q₁₆×D₁₅₇

yes

—

123

C₄

#19097791

yes

—

124

C₂

#23757644

yes

characteristic

125

C₂

#23757651

yes

characteristic

126

C₂

#23757655

yes

characteristic

	subgroup	G/H
center	C₂²	#19097767
commutator, cosocle	C₄²×C₁₅₇	C₂³
Frattini	C₄²	C₂²×D₁₅₇
Fitting	#487×C₁₅₇	C₂
socle, prime radical	C₂²×C₁₅₇	#127
hypercenter	#487	D₁₅₇
hypocenter	C₁₅₇	#1707
LCS	G ▷ C₄²×C₁₅₇ ▷ C₂²×C₁₅₇ ▷ C₁₅₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #487

subgroup

G/H

center

C₂²

#19097767

commutator, cosocle

C₄²×C₁₅₇

C₂³

Frattini

C₄²

C₂²×D₁₅₇

Fitting

#487×C₁₅₇

C₂

C₂²×C₁₅₇

D₁₅₇

C₁₅₇

G ▷ C₄²×C₁₅₇ ▷ C₂²×C₁₅₇ ▷ C₁₅₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #487

N	H ≅ G/N	type
43. #11929434	Q₁₆	semidirect
63. C₁₅₇	#1707	semidirect

H ≅ G/N

type

43. #11929434

Q₁₆

semidirect

63. C₁₅₇

#1707

semidirect

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

subgroup	index
#23757676	2
#23757669	2
#23757652	2
#23757624	2
C₈×#11929434	2
#487×C₁₅₇	2
#1707	157

subgroup

index

#487×C₁₅₇

157

Subgroups — #34099732

Conjugacy classes of subgroups

Complements

Maximal subgroups