Group #23757413 — subgroups

Subgroups — #23757413

Order 10048.

158 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 158 conjugacy classes of subgroups — too many to draw. Only its 45 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×Q₈×D₁₅₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#14578943	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#14578941	C₂	1	1	yes	maximal, characteristic, complemented
4	#19097746	C₂	1	1	yes	maximal, characteristic
5	#19097694	C₂	1	1	yes	maximal, characteristic, complemented
6	#19097683	C₂	1	1	yes	maximal, characteristic, complemented
7	#103×C₁₅₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×Dic₃₁₄	—	2	1	no	—
9	C₂×Dic₃₁₄	C₂²	1	1	yes	characteristic
10	Q₈×D₁₅₇	—	2	2	no	—
11	Q₈×D₁₅₇	—	2	2	no	—
12	C₂×D₆₂₈	—	2	1	no	—
13	C₂×D₆₂₈	C₂²	1	1	yes	characteristic
14	C₂×C₄×D₁₅₇	—	2	1	no	—
15	C₂×C₄×D₁₅₇	C₂²	1	1	yes	characteristic
16	#14578943	—	2	2	no	—
17	#14578941	—	2	2	no	—
18	C₂×#11929434	C₂²	1	1	yes	characteristic
19	#14578931	—	2	1	no	—
20	C₂×Q₈×C₁₅₇	C₂²	1	1	yes	characteristic
21	C₂×C₈×C₁₅₇	C₂²	1	1	yes	characteristic
22	C₄⋊C₄×C₁₅₇	C₂²	1	1	yes	characteristic
23	C₂²×D₁₅₇	—	2	1	no	—
24	C₂²×D₁₅₇	D₄	1	1	yes	characteristic
25	Dic₃₁₄	—	2	4	no	—
26	Dic₃₁₄	—	2	1	no	—
27	Dic₃₁₄	D₄	1	2	yes	complemented
28	D₆₂₈	—	2	1	no	—
29	D₆₂₈	—	2	1	no	—
30	D₆₂₈	—	4	2	no	—
31	C₂×Dic₁₅₇	—	2	1	no	—
32	C₂×Dic₁₅₇	D₄	1	1	yes	characteristic
33	C₄×D₁₅₇	—	2	4	no	—
34	C₄×D₁₅₇	—	2	1	no	—
35	C₄×D₁₅₇	—	2	1	no	—
36	#11929434	—	2	1	no	—
37	Q₈×C₁₅₇	—	2	1	no	—
38	Q₈×C₁₅₇	D₄	1	2	yes	complemented
39	C₂×C₄×C₁₅₇	—	2	1	no	—
40	C₂×C₄×C₁₅₇	—	2	1	no	—
41	C₂×C₄×C₁₅₇	C₂³	1	1	yes	characteristic
42	C₈×C₁₅₇	—	2	1	no	—
43	C₂²×C₁₅₇	C₂×D₄	1	1	yes	characteristic
44	C₂×D₁₅₇	—	2	1	no	—
45	C₂×D₁₅₇	—	2	1	no	—
46	C₂×D₁₅₇	—	4	1	no	—
47	C₂×D₁₅₇	—	4	1	no	—
48	C₂×D₁₅₇	—	4	1	no	—
49	C₂×D₁₅₇	SD₁₆	1	2	yes	complemented
50	C₄×C₁₅₇	—	2	2	no	—
51	C₄×C₁₅₇	—	4	1	no	—
52	C₄×C₁₅₇	C₂×D₄	1	1	yes	characteristic
53	C₄×C₁₅₇	C₂×D₄	1	1	yes	characteristic
54	Dic₁₅₇	—	2	2	no	—
55	Dic₁₅₇	—	2	1	no	—
56	C₂×C₁₅₇	#120	1	1	yes	characteristic
57	C₂×C₁₅₇	C₂×SD₁₆	1	1	yes	characteristic
58	C₂×C₁₅₇	#137	1	1	yes	characteristic
59	D₁₅₇	—	2	2	no	—
60	D₁₅₇	—	8	1	no	—
61	C₁₅₇	#448	1	1	yes	characteristic, complemented, Sylow
62	#448	—	157	1	no	maximal, Sylow
63	C₂²×Q₈	—	157	1	no	—
64	C₂×SD₁₆	—	157	1	no	—
65	C₂×SD₁₆	—	157	1	no	—
66	#121	—	157	1	no	—
67	#103	—	157	1	no	—
68	#103	D₁₅₇	1	1	yes	characteristic, complemented
69	C₂²⋊C₈	—	157	1	no	—
70	C₂×Q₈	—	314	1	no	—
71	C₂×Q₈	—	314	2	no	—
72	C₂×Q₈	—	314	2	no	—
73	C₂×Q₈	—	157	1	no	—
74	C₂×Q₈	C₂×D₁₅₇	1	1	yes	characteristic
75	C₂×D₄	—	314	1	no	—
76	C₂×D₄	—	157	1	no	—
77	C₂²×C₄	—	314	1	no	—
78	C₂²×C₄	—	157	1	no	—
79	SD₁₆	—	314	2	no	—
80	SD₁₆	—	314	2	no	—
81	C₂×C₈	—	157	1	no	—
82	C₂×C₈	C₂×D₁₅₇	1	1	yes	characteristic
83	C₄⋊C₄	C₂×D₁₅₇	1	1	yes	characteristic
84	C₂²⋊C₄	—	314	1	no	—
85	C₂³	—	314	1	no	—
86	C₂³	—	157	1	no	—
87	Q₈	—	2	1	no	—
88	Q₈	—	314	4	no	—
89	Q₈	—	314	1	no	—
90	Q₈	—	157	2	no	—
91	Q₈	D₆₂₈	1	2	yes	complemented
92	D₄	—	314	1	no	—
93	D₄	—	314	1	no	—
94	D₄	—	628	2	no	—
95	C₂×C₄	—	2	1	no	—
96	C₂×C₄	—	2	1	no	—
97	C₂×C₄	—	314	4	no	—
98	C₂×C₄	—	314	1	no	—
99	C₂×C₄	—	314	1	no	—
100	C₂×C₄	—	314	1	no	—
101	C₂×C₄	—	157	1	no	—
102	C₂×C₄	C₂²×D₁₅₇	1	1	yes	characteristic
103	C₈	—	2	1	no	—
104	C₈	—	314	1	no	—
105	C₂²	—	314	1	no	—
106	C₂²	—	314	1	no	—
107	C₂²	—	628	1	no	—
108	C₂²	—	628	1	no	—
109	C₂²	—	628	1	no	—
110	C₂²	—	157	2	no	—
111	C₂²	D₄×D₁₅₇	1	1	yes	characteristic
112	C₄	—	2	2	no	—
113	C₄	—	4	1	no	—
114	C₄	—	314	2	no	—
115	C₄	—	314	1	no	—
116	C₄	D₄×D₁₅₇	1	1	yes	characteristic
117	C₄	C₂×D₆₂₈	1	1	yes	characteristic
118	C₂	—	314	2	no	—
119	C₂	—	1256	1	no	—
120	C₂	#19097741	1	1	yes	characteristic
121	C₂	SD₁₆×D₁₅₇	1	1	yes	characteristic
122	C₂	#19097803	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×Q₈×D₁₅₇

C₂

yes

maximal, characteristic, complemented

C₂×#14578943

C₂

yes

maximal, characteristic, complemented

C₂×#14578941

C₂

yes

maximal, characteristic, complemented

#19097746

C₂

yes

maximal, characteristic

#19097694

C₂

yes

maximal, characteristic, complemented

#19097683

C₂

yes

maximal, characteristic, complemented

#103×C₁₅₇

C₂

yes

maximal, characteristic, complemented

C₂×Dic₃₁₄

—

C₂×Dic₃₁₄

C₂²

yes

characteristic

Q₈×D₁₅₇

—

Q₈×D₁₅₇

—

C₂×D₆₂₈

—

C₂×D₆₂₈

C₂²

yes

characteristic

C₂×C₄×D₁₅₇

—

C₂×C₄×D₁₅₇

C₂²

yes

—

—

C₂²

yes

characteristic

#14578931

—

C₂×Q₈×C₁₅₇

C₂²

yes

characteristic

C₂×C₈×C₁₅₇

C₂²

yes

characteristic

C₄⋊C₄×C₁₅₇

C₂²

yes

characteristic

C₂²×D₁₅₇

—

C₂²×D₁₅₇

D₄

yes

characteristic

Dic₃₁₄

—

Dic₃₁₄

—

Dic₃₁₄

D₄

yes

complemented

D₆₂₈

—

D₆₂₈

—

D₆₂₈

—

C₂×Dic₁₅₇

—

C₂×Dic₁₅₇

D₄

yes

characteristic

C₄×D₁₅₇

—

C₄×D₁₅₇

—

C₄×D₁₅₇

—

#11929434

—

Q₈×C₁₅₇

—

Q₈×C₁₅₇

D₄

yes

complemented

C₂×C₄×C₁₅₇

—

C₂×C₄×C₁₅₇

—

C₂×C₄×C₁₅₇

C₂³

yes

characteristic

C₈×C₁₅₇

—

C₂²×C₁₅₇

C₂×D₄

yes

characteristic

C₂×D₁₅₇

—

C₂×D₁₅₇

—

C₂×D₁₅₇

—

C₂×D₁₅₇

—

C₂×D₁₅₇

—

C₂×D₁₅₇

SD₁₆

yes

complemented

C₄×C₁₅₇

—

C₄×C₁₅₇

—

C₄×C₁₅₇

C₂×D₄

yes

characteristic

C₄×C₁₅₇

C₂×D₄

yes

characteristic

Dic₁₅₇

—

Dic₁₅₇

—

C₂×C₁₅₇

#120

yes

characteristic

C₂×C₁₅₇

C₂×SD₁₆

yes

characteristic

C₂×C₁₅₇

#137

yes

characteristic

D₁₅₇

—

D₁₅₇

—

C₁₅₇

#448

yes

characteristic, complemented, Sylow

#448

—

157

maximal, Sylow

C₂²×Q₈

—

157

—

C₂×SD₁₆

—

157

—

C₂×SD₁₆

—

157

—

#121

—

157

—

#103

—

157

—

#103

D₁₅₇

yes

characteristic, complemented

C₂²⋊C₈

—

157

—

C₂×Q₈

—

314

—

C₂×Q₈

—

314

—

C₂×Q₈

—

314

—

C₂×Q₈

—

157

—

C₂×Q₈

C₂×D₁₅₇

yes

characteristic

C₂×D₄

—

314

—

C₂×D₄

—

157

—

C₂²×C₄

—

314

—

C₂²×C₄

—

157

—

SD₁₆

—

314

—

SD₁₆

—

314

—

C₂×C₈

—

157

—

C₂×C₈

C₂×D₁₅₇

yes

characteristic

C₄⋊C₄

C₂×D₁₅₇

yes

characteristic

C₂²⋊C₄

—

314

—

C₂³

—

314

—

C₂³

—

157

—

Q₈

—

Q₈

—

314

—

Q₈

—

314

—

Q₈

—

157

—

Q₈

D₆₂₈

yes

complemented

D₄

—

314

—

D₄

—

314

—

D₄

—

628

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

314

—

C₂×C₄

—

314

—

C₂×C₄

—

314

—

100

C₂×C₄

—

314

—

101

C₂×C₄

—

157

—

102

C₂×C₄

C₂²×D₁₅₇

yes

characteristic

103

C₈

—

104

C₈

—

314

—

105

C₂²

—

314

—

106

C₂²

—

314

—

107

C₂²

—

628

—

108

C₂²

—

628

—

109

C₂²

—

628

—

110

C₂²

—

157

—

111

C₂²

D₄×D₁₅₇

yes

characteristic

112

C₄

—

113

C₄

—

114

C₄

—

314

—

115

C₄

—

314

—

116

C₄

D₄×D₁₅₇

yes

characteristic

117

C₄

C₂×D₆₂₈

yes

characteristic

118

C₂

—

314

—

119

C₂

—

1256

—

120

C₂

#19097741

yes

characteristic

121

C₂

SD₁₆×D₁₅₇

yes

characteristic

122

C₂

#19097803

yes

characteristic

	subgroup	G/H
center	C₂²	D₄×D₁₅₇
commutator, cosocle	C₂×C₄×C₁₅₇	C₂³
Frattini	C₂×C₄	C₂²×D₁₅₇
Fitting	#103×C₁₅₇	C₂
socle	C₂²×C₁₅₇	C₂×D₄
prime radical	#19097746	C₂
hypercenter	#103	D₁₅₇
hypocenter	C₁₅₇	#448
LCS	G ▷ C₂×C₄×C₁₅₇ ▷ C₂×C₁₅₇ ▷ C₁₅₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #103

subgroup

G/H

center

C₂²

D₄×D₁₅₇

commutator, cosocle

C₂×C₄×C₁₅₇

C₂³

Frattini

C₂×C₄

C₂²×D₁₅₇

Fitting

#103×C₁₅₇

C₂

socle

C₂²×C₁₅₇

C₂×D₄

C₂

D₁₅₇

C₁₅₇

G ▷ C₂×C₄×C₁₅₇ ▷ C₂×C₁₅₇ ▷ C₁₅₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ #103

N	H ≅ G/N	type
1. C₂×Q₈×D₁₅₇	C₂	semidirect
2. C₂×#14578943	C₂	semidirect
3. C₂×#14578941	C₂	semidirect
5. #19097694	C₂	semidirect
6. #19097683	C₂	semidirect
7. #103×C₁₅₇	C₂	semidirect
27. Dic₃₁₄	D₄	semidirect
38. Q₈×C₁₅₇	D₄	semidirect
49. C₂×D₁₅₇	SD₁₆	semidirect
61. C₁₅₇	#448	semidirect
68. #103	D₁₅₇	semidirect
91. Q₈	D₆₂₈	semidirect

H ≅ G/N

type

1. C₂×Q₈×D₁₅₇

C₂

semidirect

2. C₂×#14578943

C₂

semidirect

3. C₂×#14578941

C₂

semidirect

5. #19097694

C₂

semidirect

6. #19097683

C₂

semidirect

7. #103×C₁₅₇

C₂

semidirect

27. Dic₃₁₄

D₄

semidirect

38. Q₈×C₁₅₇

D₄

semidirect

49. C₂×D₁₅₇

SD₁₆

semidirect

61. C₁₅₇

#448

semidirect

68. #103

D₁₅₇

semidirect

91. Q₈

D₆₂₈

semidirect

Subgroups — #23757413

Conjugacy classes of subgroups

Complements