Group #23757848 — subgroups

Subgroups — #23757848

Order 10048.

270 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 270 conjugacy classes of subgroups — too many to draw. Only its 103 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×#14578955	C₂	1	1	yes	maximal, characteristic
2	C₂×Q₈×D₁₅₇	C₂	1	1	yes	maximal, characteristic, complemented
3	#19097769	C₂	1	1	yes	maximal, characteristic, complemented
4	#19097760	C₂	1	1	yes	maximal, characteristic, complemented
5	#19097759	C₂	1	1	yes	maximal, characteristic, complemented
6	#19097756	C₂	1	4	yes	maximal, complemented
7	#19097749	C₂	1	4	yes	maximal, complemented
8	#19097728	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄⋊Q₈×C₁₅₇	C₂	1	1	yes	maximal, characteristic, complemented
10	#14578955	—	2	4	no	—
11	C₂×Dic₃₁₄	C₂²	1	2	yes	—
12	C₂×Dic₃₁₄	C₂²	1	1	yes	characteristic
13	Q₈×D₁₅₇	—	2	4	no	—
14	C₂×D₆₂₈	C₂²	1	2	yes	—
15	C₂×D₆₂₈	C₂²	1	1	yes	characteristic
16	C₂×C₄×D₁₅₇	C₂²	1	2	yes	—
17	C₂×C₄×D₁₅₇	C₂²	1	2	yes	—
18	C₂×C₄×D₁₅₇	C₂²	1	1	yes	characteristic
19	#14578933	C₂²	1	2	yes	—
20	#14578933	C₂²	1	4	yes	—
21	#14578931	C₂²	1	2	yes	—
22	#14578931	C₂²	1	4	yes	—
23	#14578931	C₂²	1	4	yes	—
24	C₄×Dic₁₅₇	C₂²	1	1	yes	characteristic
25	C₂×Q₈×C₁₅₇	C₂²	1	1	yes	characteristic
26	C₂×Q₈×C₁₅₇	C₂²	1	1	yes	characteristic
27	C₄⋊C₄×C₁₅₇	C₂²	1	4	yes	—
28	C₄²×C₁₅₇	C₂²	1	1	yes	characteristic
29	C₂²×D₁₅₇	C₂³	1	2	yes	—
30	C₂²×D₁₅₇	C₂³	1	1	yes	characteristic
31	Dic₃₁₄	—	2	2	no	—
32	Dic₃₁₄	—	2	4	no	—
33	D₆₂₈	—	2	4	no	—
34	D₆₂₈	—	2	2	no	—
35	C₂×Dic₁₅₇	C₂³	1	2	yes	—
36	C₂×Dic₁₅₇	C₂³	1	2	yes	—
37	C₂×Dic₁₅₇	C₂³	1	1	yes	characteristic
38	C₄×D₁₅₇	—	2	4	no	—
39	C₄×D₁₅₇	—	2	4	no	—
40	C₄×D₁₅₇	D₄	1	4	yes	—
41	Q₈×C₁₅₇	—	2	2	no	—
42	Q₈×C₁₅₇	—	2	2	no	—
43	C₂×C₄×C₁₅₇	C₂³	1	2	yes	—
44	C₂×C₄×C₁₅₇	C₂³	1	2	yes	—
45	C₂×C₄×C₁₅₇	C₂³	1	2	yes	—
46	C₂×C₄×C₁₅₇	C₂³	1	1	yes	characteristic
47	C₂²×C₁₅₇	C₂⁴	1	1	yes	characteristic
48	C₂×D₁₅₇	—	2	2	no	—
49	C₂×D₁₅₇	—	2	2	no	—
50	C₂×D₁₅₇	—	2	4	no	—
51	C₂×D₁₅₇	C₂×D₄	1	2	yes	—
52	C₄×C₁₅₇	—	2	2	no	—
53	C₄×C₁₅₇	—	2	2	no	—
54	C₄×C₁₅₇	—	2	2	no	—
55	C₄×C₁₅₇	C₂×D₄	1	2	yes	—
56	Dic₁₅₇	—	2	2	no	—
57	Dic₁₅₇	—	2	2	no	—
58	Dic₁₅₇	C₂×D₄	1	2	yes	—
59	C₂×C₁₅₇	C₂²×D₄	1	1	yes	characteristic
60	C₂×C₁₅₇	2₋¹⁺⁴	1	2	yes	—
61	D₁₅₇	—	2	2	no	—
62	D₁₅₇	—	4	2	no	—
63	C₁₅₇	#536	1	1	yes	characteristic, complemented, Sylow
64	#536	—	157	1	no	maximal, Sylow
65	C₂×#41	—	157	1	no	—
66	C₂²×Q₈	—	157	1	no	—
67	C₄⋊Q₈	—	157	1	no	—
68	C₄⋊Q₈	D₁₅₇	1	1	yes	characteristic, complemented
69	#124	—	157	1	no	—
70	#124	—	157	1	no	—
71	#123	—	157	4	no	—
72	C₂²⋊Q₈	—	157	4	no	—
73	#117	—	157	1	no	—
74	#41	—	314	4	no	—
75	C₂×Q₈	—	314	4	no	—
76	C₂×Q₈	—	157	2	no	—
77	C₂×Q₈	—	157	1	no	—
78	C₂×Q₈	C₂×D₁₅₇	1	1	yes	characteristic
79	C₂×Q₈	C₂×D₁₅₇	1	1	yes	characteristic
80	C₂×D₄	—	157	2	no	—
81	C₂×D₄	—	157	1	no	—
82	C₂²×C₄	—	157	2	no	—
83	C₂²×C₄	—	157	2	no	—
84	C₂²×C₄	—	157	1	no	—
85	C₄⋊C₄	—	157	2	no	—
86	C₄⋊C₄	—	157	4	no	—
87	C₄⋊C₄	C₂×D₁₅₇	1	4	yes	—
88	C₂²⋊C₄	—	157	2	no	—
89	C₂²⋊C₄	—	157	4	no	—
90	C₂²⋊C₄	—	157	4	no	—
91	C₄²	—	157	1	no	—
92	C₄²	C₂×D₁₅₇	1	1	yes	characteristic
93	C₂³	—	157	2	no	—
94	C₂³	—	157	1	no	—
95	Q₈	—	2	2	no	—
96	Q₈	—	2	2	no	—
97	Q₈	—	314	4	no	—
98	Q₈	—	314	2	no	—
99	D₄	—	314	4	no	—
100	D₄	—	314	2	no	—
101	C₂×C₄	—	314	4	no	—
102	C₂×C₄	—	314	4	no	—
103	C₂×C₄	—	157	2	no	—
104	C₂×C₄	—	157	2	no	—
105	C₂×C₄	—	157	1	no	—
106	C₂×C₄	—	157	4	no	—
107	C₂×C₄	C₂²×D₁₅₇	1	2	yes	—
108	C₂×C₄	C₂²×D₁₅₇	1	2	yes	—
109	C₂×C₄	C₂²×D₁₅₇	1	2	yes	—
110	C₂×C₄	C₂²×D₁₅₇	1	1	yes	characteristic
111	C₂²	—	314	2	no	—
112	C₂²	—	314	2	no	—
113	C₂²	—	314	4	no	—
114	C₂²	—	157	2	no	—
115	C₂²	C₂³×D₁₅₇	1	1	yes	characteristic
116	C₄	—	2	2	no	—
117	C₄	—	2	2	no	—
118	C₄	—	2	2	no	—
119	C₄	—	314	2	no	—
120	C₄	—	314	2	no	—
121	C₄	—	157	2	no	—
122	C₄	D₄×D₁₅₇	1	2	yes	—
123	C₂	—	314	2	no	—
124	C₂	—	628	2	no	—
125	C₂	C₂×D₄×D₁₅₇	1	1	yes	characteristic
126	C₂	#19097819	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14578955

C₂

yes

maximal, characteristic

C₂×Q₈×D₁₅₇

C₂

yes

maximal, characteristic, complemented

#19097769

C₂

yes

maximal, characteristic, complemented

#19097760

C₂

yes

maximal, characteristic, complemented

#19097759

C₂

yes

maximal, characteristic, complemented

#19097756

C₂

yes

maximal, complemented

#19097749

C₂

yes

maximal, complemented

#19097728

C₂

yes

maximal, characteristic, complemented

C₄⋊Q₈×C₁₅₇

C₂

yes

maximal, characteristic, complemented

#14578955

—

C₂×Dic₃₁₄

C₂²

yes

—

C₂×Dic₃₁₄

C₂²

yes

characteristic

Q₈×D₁₅₇

—

C₂×D₆₂₈

C₂²

yes

—

C₂×D₆₂₈

C₂²

yes

characteristic

C₂×C₄×D₁₅₇

C₂²

yes

—

C₂×C₄×D₁₅₇

C₂²

yes

—

C₂×C₄×D₁₅₇

C₂²

yes

characteristic

#14578933

C₂²

yes

—

#14578933

C₂²

yes

—

#14578931

C₂²

yes

—

#14578931

C₂²

yes

—

#14578931

C₂²

yes

—

C₄×Dic₁₅₇

C₂²

yes

characteristic

C₂×Q₈×C₁₅₇

C₂²

yes

characteristic

C₂×Q₈×C₁₅₇

C₂²

yes

characteristic

C₄⋊C₄×C₁₅₇

C₂²

yes

—

C₄²×C₁₅₇

C₂²

yes

characteristic

C₂²×D₁₅₇

C₂³

yes

—

C₂²×D₁₅₇

C₂³

yes

characteristic

Dic₃₁₄

—

Dic₃₁₄

—

D₆₂₈

—

D₆₂₈

—

C₂×Dic₁₅₇

C₂³

yes

—

C₂×Dic₁₅₇

C₂³

yes

—

C₂×Dic₁₅₇

C₂³

yes

characteristic

C₄×D₁₅₇

—

C₄×D₁₅₇

—

C₄×D₁₅₇

D₄

yes

—

Q₈×C₁₅₇

—

Q₈×C₁₅₇

—

C₂×C₄×C₁₅₇

C₂³

yes

—

C₂×C₄×C₁₅₇

C₂³

yes

—

C₂×C₄×C₁₅₇

C₂³

yes

—

C₂×C₄×C₁₅₇

C₂³

yes

characteristic

C₂²×C₁₅₇

C₂⁴

yes

characteristic

C₂×D₁₅₇

—

C₂×D₁₅₇

—

C₂×D₁₅₇

—

C₂×D₁₅₇

C₂×D₄

yes

—

C₄×C₁₅₇

—

C₄×C₁₅₇

—

C₄×C₁₅₇

—

C₄×C₁₅₇

C₂×D₄

yes

—

Dic₁₅₇

—

Dic₁₅₇

—

Dic₁₅₇

C₂×D₄

yes

—

C₂×C₁₅₇

C₂²×D₄

yes

characteristic

C₂×C₁₅₇

2₋¹⁺⁴

yes

—

D₁₅₇

—

D₁₅₇

—

C₁₅₇

#536

yes

characteristic, complemented, Sylow

#536

—

157

maximal, Sylow

C₂×#41

—

157

—

C₂²×Q₈

—

157

—

C₄⋊Q₈

—

157

—

C₄⋊Q₈

D₁₅₇

yes

characteristic, complemented

#124

—

157

—

#124

—

157

—

#123

—

157

—

C₂²⋊Q₈

—

157

—

#117

—

157

—

#41

—

314

—

C₂×Q₈

—

314

—

C₂×Q₈

—

157

—

C₂×Q₈

—

157

—

C₂×Q₈

C₂×D₁₅₇

yes

characteristic

C₂×Q₈

C₂×D₁₅₇

yes

characteristic

C₂×D₄

—

157

—

C₂×D₄

—

157

—

C₂²×C₄

—

157

—

C₂²×C₄

—

157

—

C₂²×C₄

—

157

—

C₄⋊C₄

—

157

—

C₄⋊C₄

—

157

—

C₄⋊C₄

C₂×D₁₅₇

yes

—

C₂²⋊C₄

—

157

—

C₂²⋊C₄

—

157

—

C₂²⋊C₄

—

157

—

C₄²

—

157

—

C₄²

C₂×D₁₅₇

yes

characteristic

C₂³

—

157

—

C₂³

—

157

—

Q₈

—

Q₈

—

Q₈

—

314

—

Q₈

—

314

—

D₄

—

314

—

100

D₄

—

314

—

101

C₂×C₄

—

314

—

102

C₂×C₄

—

314

—

103

C₂×C₄

—

157

—

104

C₂×C₄

—

157

—

105

C₂×C₄

—

157

—

106

C₂×C₄

—

157

—

107

C₂×C₄

C₂²×D₁₅₇

yes

—

108

C₂×C₄

C₂²×D₁₅₇

yes

—

109

C₂×C₄

C₂²×D₁₅₇

yes

—

110

C₂×C₄

C₂²×D₁₅₇

yes

characteristic

111

C₂²

—

314

—

112

C₂²

—

314

—

113

C₂²

—

314

—

114

C₂²

—

157

—

115

C₂²

C₂³×D₁₅₇

yes

characteristic

116

C₄

—

117

C₄

—

118

C₄

—

119

C₄

—

314

—

120

C₄

—

314

—

121

C₄

—

157

—

122

C₄

D₄×D₁₅₇

yes

—

123

C₂

—

314

—

124

C₂

—

628

—

125

C₂

C₂×D₄×D₁₅₇

yes

characteristic

126

C₂

#19097819

yes

—

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₁₅₇
commutator, socle, cosocle	C₂²×C₁₅₇	C₂⁴
Fitting	C₄⋊Q₈×C₁₅₇	C₂
prime radical	C₂×#14578955	C₂
hypercenter	C₄⋊Q₈	D₁₅₇
hypocenter	C₁₅₇	#536
LCS	G ▷ C₂²×C₁₅₇ ▷ C₁₅₇
UCS	C₁ ◁ C₂² ◁ C₄⋊Q₈

subgroup

G/H

center, Frattini

C₂²

C₂³×D₁₅₇

commutator, socle, cosocle

C₂²×C₁₅₇

C₂⁴

Fitting

C₄⋊Q₈×C₁₅₇

C₂

prime radical

C₂×#14578955

C₂

hypercenter

C₄⋊Q₈

D₁₅₇

C₁₅₇

G ▷ C₂²×C₁₅₇ ▷ C₁₅₇

C₁ ◁ C₂² ◁ C₄⋊Q₈

N	H ≅ G/N	type
2. C₂×Q₈×D₁₅₇	C₂	semidirect
3. #19097769	C₂	semidirect
4. #19097760	C₂	semidirect
5. #19097759	C₂	semidirect
6. #19097756	C₂	semidirect
7. #19097749	C₂	semidirect
8. #19097728	C₂	semidirect
9. C₄⋊Q₈×C₁₅₇	C₂	semidirect
63. C₁₅₇	#536	semidirect
68. C₄⋊Q₈	D₁₅₇	semidirect

H ≅ G/N

type

2. C₂×Q₈×D₁₅₇

C₂

semidirect

3. #19097769

C₂

semidirect

4. #19097760

C₂

semidirect

5. #19097759

C₂

semidirect

6. #19097756

C₂

semidirect

7. #19097749

C₂

semidirect

8. #19097728

C₂

semidirect

9. C₄⋊Q₈×C₁₅₇

C₂

semidirect

63. C₁₅₇

#536

semidirect

68. C₄⋊Q₈

D₁₅₇

semidirect

Subgroups — #23757848

Conjugacy classes of subgroups

Complements