Group #23757168 — subgroups

Subgroups — #23757168

Order 10048.

170 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 170 conjugacy classes of subgroups — too many to draw. Only its 77 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×#14578933	C₂	1	1	yes	maximal, characteristic
2	C₂×#14578933	C₂	1	1	yes	maximal, characteristic
3	C₂×#14578933	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄×Dic₁₅₇	C₂	1	1	yes	maximal, characteristic
5	#19097678	C₂	1	1	yes	maximal, characteristic
6	#19097678	C₂	1	1	yes	maximal, characteristic
7	C₂×C₄⋊C₄×C₁₅₇	C₂	1	1	yes	maximal, characteristic
8	C₂²×Dic₁₅₇	C₂²	1	1	yes	characteristic
9	C₂²×Dic₁₅₇	C₂²	1	1	yes	characteristic
10	C₂²×Dic₁₅₇	C₂²	1	1	yes	characteristic
11	C₂²×Dic₁₅₇	C₂²	1	1	yes	characteristic
12	#14578933	—	2	2	no	—
13	#14578933	—	2	2	no	—
14	#14578933	C₄	1	4	yes	complemented
15	C₄×Dic₁₅₇	—	2	2	no	—
16	C₂²×C₄×C₁₅₇	C₂²	1	1	yes	characteristic
17	C₂²×C₄×C₁₅₇	C₂²	1	1	yes	characteristic
18	C₂²×C₄×C₁₅₇	C₂²	1	1	yes	characteristic
19	C₄⋊C₄×C₁₅₇	—	2	2	no	—
20	C₂×Dic₁₅₇	—	2	1	no	—
21	C₂×Dic₁₅₇	—	2	1	no	—
22	C₂×Dic₁₅₇	—	2	1	no	—
23	C₂×Dic₁₅₇	—	2	1	no	—
24	C₂×Dic₁₅₇	—	2	1	no	—
25	C₂×Dic₁₅₇	—	2	1	no	—
26	C₂×Dic₁₅₇	—	2	1	no	—
27	C₂×Dic₁₅₇	C₂×C₄	1	2	yes	—
28	C₂×Dic₁₅₇	C₂×C₄	1	2	yes	—
29	C₂×Dic₁₅₇	D₄	1	2	yes	—
30	C₂×Dic₁₅₇	Q₈	1	2	yes	—
31	C₂×Dic₁₅₇	Q₈	1	2	yes	—
32	C₂³×C₁₅₇	C₂³	1	1	yes	characteristic
33	C₂×C₄×C₁₅₇	—	2	1	no	—
34	C₂×C₄×C₁₅₇	—	2	1	no	—
35	C₂×C₄×C₁₅₇	—	2	1	no	—
36	C₂×C₄×C₁₅₇	—	2	1	no	—
37	C₂×C₄×C₁₅₇	—	2	1	no	—
38	C₂×C₄×C₁₅₇	—	2	1	no	—
39	C₂×C₄×C₁₅₇	—	2	1	no	—
40	C₂×C₄×C₁₅₇	C₂×C₄	1	2	yes	—
41	C₂×C₄×C₁₅₇	D₄	1	2	yes	—
42	C₂²×C₁₅₇	C₂²×C₄	1	1	yes	characteristic
43	C₂²×C₁₅₇	C₂×D₄	1	1	yes	characteristic
44	C₂²×C₁₅₇	C₂×D₄	1	1	yes	characteristic
45	C₂²×C₁₅₇	C₂×Q₈	1	1	yes	characteristic
46	C₂²×C₁₅₇	C₂×Q₈	1	1	yes	characteristic
47	C₂²×C₁₅₇	#41	1	1	yes	characteristic
48	C₂²×C₁₅₇	#41	1	1	yes	characteristic
49	C₄×C₁₅₇	—	2	2	no	—
50	C₄×C₁₅₇	—	2	2	no	—
51	C₄×C₁₅₇	—	4	1	no	—
52	Dic₁₅₇	—	2	2	no	—
53	Dic₁₅₇	—	2	2	no	—
54	Dic₁₅₇	—	4	1	no	—
55	Dic₁₅₇	C₄⋊C₄	1	4	yes	complemented
56	C₂×C₁₅₇	C₂×C₄⋊C₄	1	1	yes	characteristic
57	C₂×C₁₅₇	C₄×D₄	1	1	yes	characteristic
58	C₂×C₁₅₇	C₄×Q₈	1	1	yes	characteristic
59	C₂×C₁₅₇	#121	1	1	yes	characteristic
60	C₂×C₁₅₇	C₂²⋊Q₈	1	1	yes	characteristic
61	C₂×C₁₅₇	#125	1	1	yes	characteristic
62	C₂×C₁₅₇	C₄⋊Q₈	1	1	yes	characteristic
63	C₁₅₇	#389	1	1	yes	characteristic, complemented, Sylow
64	#389	—	157	1	no	maximal, Sylow
65	C₂×C₄⋊C₄	—	157	1	no	—
66	C₂×C₄⋊C₄	—	157	1	no	—
67	C₂×C₄⋊C₄	—	157	1	no	—
68	C₂×C₄⋊C₄	D₁₅₇	1	1	yes	characteristic
69	C₂×C₄²	—	157	1	no	—
70	#95	—	157	1	no	—
71	#95	—	157	1	no	—
72	C₂²×C₄	—	157	1	no	—
73	C₂²×C₄	—	157	1	no	—
74	C₂²×C₄	—	157	1	no	—
75	C₂²×C₄	—	157	1	no	—
76	C₂²×C₄	C₂×D₁₅₇	1	1	yes	characteristic
77	C₂²×C₄	C₂×D₁₅₇	1	1	yes	characteristic
78	C₂²×C₄	C₂×D₁₅₇	1	1	yes	characteristic
79	C₄⋊C₄	—	2	2	no	—
80	C₄⋊C₄	—	314	2	no	—
81	C₄⋊C₄	—	314	2	no	—
82	C₄⋊C₄	—	157	4	no	—
83	C₄²	—	314	2	no	—
84	C₂³	C₂²×D₁₅₇	1	1	yes	characteristic
85	C₂×C₄	—	2	1	no	—
86	C₂×C₄	—	2	1	no	—
87	C₂×C₄	—	2	1	no	—
88	C₂×C₄	—	2	1	no	—
89	C₂×C₄	—	2	1	no	—
90	C₂×C₄	—	2	1	no	—
91	C₂×C₄	—	2	1	no	—
92	C₂×C₄	—	314	1	no	—
93	C₂×C₄	—	314	1	no	—
94	C₂×C₄	—	314	1	no	—
95	C₂×C₄	—	314	1	no	—
96	C₂×C₄	—	314	1	no	—
97	C₂×C₄	—	314	1	no	—
98	C₂×C₄	—	314	1	no	—
99	C₂×C₄	—	157	2	no	—
100	C₂×C₄	—	157	2	no	—
101	C₂×C₄	—	157	2	no	—
102	C₂×C₄	—	157	2	no	—
103	C₂×C₄	—	157	2	no	—
104	C₂×C₄	C₄×D₁₅₇	1	2	yes	—
105	C₂×C₄	C₂²⋊D₁₅₇	1	2	yes	—
106	C₂²	C₂×C₄×D₁₅₇	1	1	yes	characteristic
107	C₂²	D₄×D₁₅₇	1	1	yes	characteristic
108	C₂²	C₂×C₂²⋊D₁₅₇	1	1	yes	characteristic
109	C₂²	Q₈×D₁₅₇	1	1	yes	characteristic
110	C₂²	Q₈×D₁₅₇	1	1	yes	characteristic
111	C₂²	#14578953	1	1	yes	characteristic
112	C₂²	#14578954	1	1	yes	characteristic
113	C₄	—	2	2	no	—
114	C₄	—	2	2	no	—
115	C₄	—	4	1	no	—
116	C₄	—	314	2	no	—
117	C₄	—	314	2	no	—
118	C₄	—	628	1	no	—
119	C₄	—	157	4	no	—
120	C₂	C₄⋊C₄×D₁₅₇	1	1	yes	characteristic
121	C₂	C₄×C₂²⋊D₁₅₇	1	1	yes	characteristic
122	C₂	#19097736	1	1	yes	characteristic
123	C₂	#19097743	1	1	yes	characteristic
124	C₂	#19097749	1	1	yes	characteristic
125	C₂	#19097761	1	1	yes	characteristic
126	C₂	#19097769	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14578933

C₂

yes

maximal, characteristic

C₂×#14578933

C₂

yes

maximal, characteristic

C₂×#14578933

C₂

yes

maximal, characteristic

C₂×C₄×Dic₁₅₇

C₂

yes

maximal, characteristic

#19097678

C₂

yes

maximal, characteristic

#19097678

C₂

yes

maximal, characteristic

C₂×C₄⋊C₄×C₁₅₇

C₂

yes

maximal, characteristic

C₂²×Dic₁₅₇

C₂²

yes

characteristic

C₂²×Dic₁₅₇

C₂²

yes

characteristic

C₂²×Dic₁₅₇

C₂²

yes

characteristic

C₂²×Dic₁₅₇

C₂²

yes

—

—

C₄

yes

complemented

C₄×Dic₁₅₇

—

C₂²×C₄×C₁₅₇

C₂²

yes

characteristic

C₂²×C₄×C₁₅₇

C₂²

yes

characteristic

C₂²×C₄×C₁₅₇

C₂²

yes

characteristic

C₄⋊C₄×C₁₅₇

—

C₂×Dic₁₅₇

—

C₂×Dic₁₅₇

—

C₂×Dic₁₅₇

—

C₂×Dic₁₅₇

—

C₂×Dic₁₅₇

—

C₂×Dic₁₅₇

—

C₂×Dic₁₅₇

—

C₂×Dic₁₅₇

C₂×C₄

yes

—

C₂×Dic₁₅₇

C₂×C₄

yes

—

C₂×Dic₁₅₇

D₄

yes

—

C₂×Dic₁₅₇

Q₈

yes

—

C₂×Dic₁₅₇

Q₈

yes

—

C₂³×C₁₅₇

C₂³

yes

characteristic

C₂×C₄×C₁₅₇

—

C₂×C₄×C₁₅₇

—

C₂×C₄×C₁₅₇

—

C₂×C₄×C₁₅₇

—

C₂×C₄×C₁₅₇

—

C₂×C₄×C₁₅₇

—

C₂×C₄×C₁₅₇

—

C₂×C₄×C₁₅₇

C₂×C₄

yes

—

C₂×C₄×C₁₅₇

D₄

yes

—

C₂²×C₁₅₇

C₂²×C₄

yes

characteristic

C₂²×C₁₅₇

C₂×D₄

yes

characteristic

C₂²×C₁₅₇

C₂×D₄

yes

characteristic

C₂²×C₁₅₇

C₂×Q₈

yes

characteristic

C₂²×C₁₅₇

C₂×Q₈

yes

C₂²×C₁₅₇

yes

C₂²×C₁₅₇

yes

C₄×C₁₅₇

—

C₄×C₁₅₇

—

C₄×C₁₅₇

—

Dic₁₅₇

—

Dic₁₅₇

—

Dic₁₅₇

—

Dic₁₅₇

C₄⋊C₄

yes

complemented

C₂×C₁₅₇

C₂×C₄⋊C₄

yes

characteristic

C₂×C₁₅₇

C₄×D₄

yes

characteristic

C₂×C₁₅₇

C₄×Q₈

yes

characteristic

C₂×C₁₅₇

#121

yes

characteristic

C₂×C₁₅₇

C₂²⋊Q₈

yes

characteristic

C₂×C₁₅₇

#125

yes

characteristic

C₂×C₁₅₇

C₄⋊Q₈

yes

characteristic

C₁₅₇

#389

yes

characteristic, complemented, Sylow

#389

—

157

maximal, Sylow

C₂×C₄⋊C₄

—

157

—

C₂×C₄⋊C₄

—

157

—

C₂×C₄⋊C₄

—

157

—

C₂×C₄⋊C₄

D₁₅₇

yes

characteristic

C₂×C₄²

—

157

—

#95

—

157

—

#95

—

157

—

C₂²×C₄

—

157

—

C₂²×C₄

—

157

—

C₂²×C₄

—

157

—

C₂²×C₄

—

157

—

C₂²×C₄

C₂×D₁₅₇

yes

characteristic

C₂²×C₄

C₂×D₁₅₇

yes

characteristic

C₂²×C₄

C₂×D₁₅₇

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

314

—

C₄⋊C₄

—

314

—

C₄⋊C₄

—

157

—

C₄²

—

314

—

C₂³

C₂²×D₁₅₇

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

314

—

C₂×C₄

—

314

—

C₂×C₄

—

314

—

C₂×C₄

—

314

—

C₂×C₄

—

314

—

C₂×C₄

—

314

—

C₂×C₄

—

314

—

C₂×C₄

—

157

—

100

C₂×C₄

—

157

—

101

C₂×C₄

—

157

—

102

C₂×C₄

—

157

—

103

C₂×C₄

—

157

—

104

C₂×C₄

C₄×D₁₅₇

yes

—

105

C₂×C₄

C₂²⋊D₁₅₇

yes

—

106

C₂²

C₂×C₄×D₁₅₇

yes

characteristic

107

C₂²

D₄×D₁₅₇

yes

characteristic

108

C₂²

C₂×C₂²⋊D₁₅₇

yes

characteristic

109

C₂²

Q₈×D₁₅₇

yes

characteristic

110

C₂²

Q₈×D₁₅₇

yes

characteristic

111

C₂²

#14578953

yes

characteristic

112

C₂²

#14578954

yes

characteristic

113

C₄

—

114

C₄

—

115

C₄

—

116

C₄

—

314

—

117

C₄

—

314

—

118

C₄

—

628

—

119

C₄

—

157

—

120

C₂

C₄⋊C₄×D₁₅₇

yes

characteristic

121

C₂

C₄×C₂²⋊D₁₅₇

yes

characteristic

122

C₂

#19097736

yes

characteristic

123

C₂

#19097743

yes

characteristic

124

C₂

#19097749

yes

characteristic

125

C₂

#19097761

yes

characteristic

126

C₂

#19097769

yes

characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂²×D₁₅₇
commutator	C₂²×C₁₅₇	C₂²×C₄
Fitting	C₂×C₄⋊C₄×C₁₅₇	C₂
socle, prime radical, cosocle	C₂³×C₁₅₇	C₂³
hypercenter	C₂×C₄⋊C₄	D₁₅₇
hypocenter	C₁₅₇	#389
LCS	G ▷ C₂²×C₁₅₇ ▷ C₁₅₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄⋊C₄

subgroup

G/H

center, Frattini

C₂³

C₂²×D₁₅₇

commutator

C₂²×C₁₅₇

C₂²×C₄

Fitting

C₂×C₄⋊C₄×C₁₅₇

C₂

socle, prime radical, cosocle

C₂³×C₁₅₇

C₂³

hypercenter

C₂×C₄⋊C₄

D₁₅₇

C₁₅₇

G ▷ C₂²×C₁₅₇ ▷ C₁₅₇

C₁ ◁ C₂³ ◁ C₂×C₄⋊C₄

N	H ≅ G/N	type
14. #14578933	C₄	semidirect
55. Dic₁₅₇	C₄⋊C₄	semidirect
63. C₁₅₇	#389	semidirect

H ≅ G/N

type

14. #14578933

C₄

semidirect

55. Dic₁₅₇

C₄⋊C₄

semidirect

63. C₁₅₇

#389

semidirect

Subgroups — #23757168

Conjugacy classes of subgroups

Complements