Group #23757232 — subgroups

Subgroups — C₂²⋊C₈×D₁₅₇ #23757232

Order 10048.

202 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 202 conjugacy classes of subgroups — too many to draw. Only its 73 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	C₂²×C₄×D₁₅₇	C₂	1	1	yes	maximal, characteristic
2	C₂×C₈×D₁₅₇	C₂	1	2	yes	maximal, complemented
3	C₂²⋊#11929434	C₂	1	1	yes	maximal, characteristic, complemented
4	#19097683	C₂	1	2	yes	maximal, complemented
5	C₂²⋊C₈×C₁₅₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂³×D₁₅₇	C₄	1	1	yes	characteristic
7	C₂²×Dic₁₅₇	C₄	1	1	yes	characteristic
8	C₂×C₄×D₁₅₇	—	2	4	no	—
9	C₂×C₄×D₁₅₇	C₄	1	1	yes	characteristic
10	C₂×C₄×D₁₅₇	C₄	1	1	yes	characteristic
11	C₂×C₄×D₁₅₇	C₂²	1	1	yes	characteristic
12	C₂×C₄×D₁₅₇	C₂²	1	1	yes	characteristic
13	C₂×#11929434	C₂²	1	2	yes	complemented
14	C₈×D₁₅₇	—	2	4	no	—
15	C₂²×C₄×C₁₅₇	C₂²	1	1	yes	characteristic
16	C₂×C₈×C₁₅₇	C₂²	1	2	yes	complemented
17	C₂²×D₁₅₇	—	2	2	no	—
18	C₂²×D₁₅₇	—	2	2	no	—
19	C₂²×D₁₅₇	C₈	1	4	yes	complemented
20	C₂²×D₁₅₇	C₂×C₄	1	1	yes	characteristic
21	C₂²×D₁₅₇	C₂×C₄	1	1	yes	characteristic
22	C₂×Dic₁₅₇	—	2	2	no	—
23	C₂×Dic₁₅₇	C₂×C₄	1	1	yes	characteristic
24	C₂×Dic₁₅₇	C₂×C₄	1	1	yes	characteristic
25	C₄×D₁₅₇	—	2	2	no	—
26	C₄×D₁₅₇	—	2	2	no	—
27	C₄×D₁₅₇	—	2	2	no	—
28	C₄×D₁₅₇	D₄	1	4	yes	—
29	#11929434	—	2	2	no	—
30	C₂³×C₁₅₇	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄×C₁₅₇	—	2	2	no	—
32	C₂×C₄×C₁₅₇	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄×C₁₅₇	C₂³	1	1	yes	characteristic
34	C₈×C₁₅₇	—	2	2	no	—
35	C₂²×C₁₅₇	—	2	1	no	—
36	C₂²×C₁₅₇	—	2	1	no	—
37	C₂²×C₁₅₇	C₂×C₈	1	2	yes	complemented
38	C₂²×C₁₅₇	C₂²×C₄	1	1	yes	characteristic
39	C₂×D₁₅₇	—	2	1	no	—
40	C₂×D₁₅₇	—	2	1	no	—
41	C₂×D₁₅₇	—	2	8	no	—
42	C₂×D₁₅₇	C₂²⋊C₄	1	2	yes	—
43	C₂×D₁₅₇	C₂×C₈	1	2	yes	—
44	C₂×D₁₅₇	C₂×C₈	1	2	yes	complemented
45	C₂×D₁₅₇	M₁₆	1	2	yes	—
46	C₄×C₁₅₇	—	2	1	no	—
47	C₄×C₁₅₇	C₂×D₄	1	2	yes	—
48	Dic₁₅₇	—	2	1	no	—
49	Dic₁₅₇	C₂²⋊C₄	1	2	yes	—
50	C₂×C₁₅₇	—	2	2	no	—
51	C₂×C₁₅₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
52	C₂×C₁₅₇	C₂²×C₈	1	1	yes	characteristic
53	C₂×C₁₅₇	C₂×M₁₆	1	1	yes	characteristic
54	D₁₅₇	—	2	2	no	—
55	D₁₅₇	C₂²⋊C₈	1	4	yes	direct product factor
56	C₁₅₇	C₂×C₂²⋊C₈	1	1	yes	characteristic, complemented, Sylow
57	C₂×C₂²⋊C₈	—	157	1	no	maximal, Sylow
58	C₂³×C₄	—	157	1	no	—
59	C₂²×C₈	—	157	2	no	—
60	C₂²⋊C₈	—	157	1	no	—
61	C₂²⋊C₈	—	157	2	no	—
62	C₂²⋊C₈	D₁₅₇	1	1	yes	characteristic, direct product factor
63	C₂⁴	—	157	1	no	—
64	C₂²×C₄	—	314	4	no	—
65	C₂²×C₄	—	157	1	no	—
66	C₂²×C₄	—	157	1	no	—
67	C₂²×C₄	—	157	1	no	—
68	C₂²×C₄	—	157	1	no	—
69	C₂²×C₄	—	157	1	no	—
70	C₂²×C₄	C₂×D₁₅₇	1	1	yes	characteristic
71	C₂×C₈	—	314	4	no	—
72	C₂×C₈	—	157	2	no	—
73	C₂×C₈	C₂×D₁₅₇	1	2	yes	complemented
74	C₂³	—	314	2	no	—
75	C₂³	—	314	2	no	—
76	C₂³	—	157	1	no	—
77	C₂³	—	157	4	no	—
78	C₂³	—	157	1	no	—
79	C₂³	C₄×D₁₅₇	1	1	yes	characteristic
80	C₂×C₄	—	2	2	no	—
81	C₂×C₄	—	314	2	no	—
82	C₂×C₄	—	314	2	no	—
83	C₂×C₄	—	314	2	no	—
84	C₂×C₄	—	314	2	no	—
85	C₂×C₄	—	157	1	no	—
86	C₂×C₄	—	157	1	no	—
87	C₂×C₄	—	157	4	no	—
88	C₂×C₄	C₄×D₁₅₇	1	1	yes	characteristic
89	C₂×C₄	C₂²×D₁₅₇	1	1	yes	characteristic
90	C₈	—	2	2	no	—
91	C₈	—	314	2	no	—
92	C₂²	—	2	1	no	—
93	C₂²	—	2	1	no	—
94	C₂²	—	314	8	no	—
95	C₂²	—	314	1	no	—
96	C₂²	—	314	1	no	—
97	C₂²	—	157	2	no	—
98	C₂²	—	157	2	no	—
99	C₂²	—	157	2	no	—
100	C₂²	—	157	2	no	—
101	C₂²	C₈×D₁₅₇	1	2	yes	complemented
102	C₂²	C₂×C₄×D₁₅₇	1	1	yes	characteristic
103	C₄	—	2	1	no	—
104	C₄	—	314	1	no	—
105	C₄	—	157	2	no	—
106	C₄	D₄×D₁₅₇	1	2	yes	—
107	C₂	—	2	2	no	—
108	C₂	—	314	2	no	—
109	C₂	—	157	4	no	—
110	C₂	C₂²⋊C₄×D₁₅₇	1	1	yes	characteristic
111	C₂	C₂×C₈×D₁₅₇	1	1	yes	characteristic
112	C₂	M₁₆×D₁₅₇	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×C₄×D₁₅₇

C₂

yes

maximal, characteristic

C₂×C₈×D₁₅₇

C₂

yes

maximal, complemented

C₂²⋊#11929434

C₂

yes

maximal, characteristic, complemented

#19097683

C₂

yes

maximal, complemented

C₂²⋊C₈×C₁₅₇

C₂

yes

maximal, characteristic, complemented

C₂³×D₁₅₇

C₄

yes

characteristic

C₂²×Dic₁₅₇

C₄

yes

characteristic

C₂×C₄×D₁₅₇

—

C₂×C₄×D₁₅₇

C₄

yes

characteristic

C₂×C₄×D₁₅₇

C₄

yes

characteristic

C₂×C₄×D₁₅₇

C₂²

yes

characteristic

C₂×C₄×D₁₅₇

C₂²

yes

characteristic

C₂×#11929434

C₂²

yes

complemented

C₈×D₁₅₇

—

C₂²×C₄×C₁₅₇

C₂²

yes

characteristic

C₂×C₈×C₁₅₇

C₂²

yes

complemented

C₂²×D₁₅₇

—

C₂²×D₁₅₇

—

C₂²×D₁₅₇

C₈

yes

complemented

C₂²×D₁₅₇

C₂×C₄

yes

characteristic

C₂²×D₁₅₇

C₂×C₄

yes

characteristic

C₂×Dic₁₅₇

—

C₂×Dic₁₅₇

C₂×C₄

yes

characteristic

C₂×Dic₁₅₇

C₂×C₄

yes

characteristic

C₄×D₁₅₇

—

C₄×D₁₅₇

—

C₄×D₁₅₇

—

C₄×D₁₅₇

D₄

yes

—

#11929434

—

C₂³×C₁₅₇

C₂×C₄

yes

characteristic

C₂×C₄×C₁₅₇

—

C₂×C₄×C₁₅₇

C₂×C₄

yes

characteristic

C₂×C₄×C₁₅₇

C₂³

yes

characteristic

C₈×C₁₅₇

—

C₂²×C₁₅₇

—

C₂²×C₁₅₇

—

C₂²×C₁₅₇

C₂×C₈

yes

complemented

C₂²×C₁₅₇

C₂²×C₄

yes

characteristic

C₂×D₁₅₇

—

C₂×D₁₅₇

—

C₂×D₁₅₇

—

C₂×D₁₅₇

C₂²⋊C₄

yes

—

C₂×D₁₅₇

C₂×C₈

yes

—

C₂×D₁₅₇

C₂×C₈

yes

complemented

C₂×D₁₅₇

M₁₆

yes

—

C₄×C₁₅₇

—

C₄×C₁₅₇

C₂×D₄

yes

—

Dic₁₅₇

—

Dic₁₅₇

C₂²⋊C₄

yes

—

C₂×C₁₅₇

—

C₂×C₁₅₇

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₁₅₇

C₂²×C₈

yes

characteristic

C₂×C₁₅₇

C₂×M₁₆

yes

characteristic

D₁₅₇

—

D₁₅₇

C₂²⋊C₈

yes

direct product factor

C₁₅₇

C₂×C₂²⋊C₈

yes

characteristic, complemented, Sylow

C₂×C₂²⋊C₈

—

157

maximal, Sylow

C₂³×C₄

—

157

—

C₂²×C₈

—

157

—

C₂²⋊C₈

—

157

—

C₂²⋊C₈

—

157

—

C₂²⋊C₈

D₁₅₇

yes

characteristic, direct product factor

C₂⁴

—

157

—

C₂²×C₄

—

314

—

C₂²×C₄

—

157

—

C₂²×C₄

—

157

—

C₂²×C₄

—

157

—

C₂²×C₄

—

157

—

C₂²×C₄

—

157

—

C₂²×C₄

C₂×D₁₅₇

yes

characteristic

C₂×C₈

—

314

—

C₂×C₈

—

157

—

C₂×C₈

C₂×D₁₅₇

yes

complemented

C₂³

—

314

—

C₂³

—

314

—

C₂³

—

157

—

C₂³

—

157

—

C₂³

—

157

—

C₂³

C₄×D₁₅₇

yes

characteristic

C₂×C₄

—

C₂×C₄

—

314

—

C₂×C₄

—

314

—

C₂×C₄

—

314

—

C₂×C₄

—

314

—

C₂×C₄

—

157

—

C₂×C₄

—

157

—

C₂×C₄

—

157

—

C₂×C₄

C₄×D₁₅₇

yes

characteristic

C₂×C₄

C₂²×D₁₅₇

yes

characteristic

C₈

—

C₈

—

314

—

C₂²

—

C₂²

—

C₂²

—

314

—

C₂²

—

314

—

C₂²

—

314

—

C₂²

—

157

—

C₂²

—

157

—

C₂²

—

157

—

100

C₂²

—

157

—

101

C₂²

C₈×D₁₅₇

yes

complemented

102

C₂²

C₂×C₄×D₁₅₇

yes

characteristic

103

C₄

—

104

C₄

—

314

—

105

C₄

—

157

—

106

C₄

D₄×D₁₅₇

yes

—

107

C₂

—

108

C₂

—

314

—

109

C₂

—

157

—

110

C₂

C₂²⋊C₄×D₁₅₇

yes

characteristic

111

C₂

C₂×C₈×D₁₅₇

yes

characteristic

112

C₂

M₁₆×D₁₅₇

yes

characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×D₁₅₇
commutator	C₂×C₁₅₇	C₂²×C₈
Fitting	C₂²⋊C₈×C₁₅₇	C₂
socle	C₂²×C₁₅₇	C₂²×C₄
prime radical	C₂³×D₁₅₇	C₄
cosocle	C₂×C₄×C₁₅₇	C₂³
hypercenter	C₂²⋊C₈	D₁₅₇
hypocenter	C₁₅₇	C₂×C₂²⋊C₈
LCS	G ▷ C₂×C₁₅₇ ▷ C₁₅₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²⋊C₈

subgroup

G/H

center, Frattini

C₂×C₄

C₂²×D₁₅₇

commutator

C₂×C₁₅₇

C₂²×C₈

Fitting

C₂²⋊C₈×C₁₅₇

C₂

socle

C₂²×C₁₅₇

C₂²×C₄

prime radical

C₂³×D₁₅₇

C₄

cosocle

C₂×C₄×C₁₅₇

C₂³

hypercenter

C₂²⋊C₈

D₁₅₇

hypocenter

C₁₅₇

C₂×C₂²⋊C₈

LCS

G ▷ C₂×C₁₅₇ ▷ C₁₅₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂²⋊C₈

N	H ≅ G/N	type
2. C₂×C₈×D₁₅₇	C₂	semidirect
3. C₂²⋊#11929434	C₂	semidirect
4. #19097683	C₂	semidirect
5. C₂²⋊C₈×C₁₅₇	C₂	semidirect
13. C₂×#11929434	C₂²	semidirect
16. C₂×C₈×C₁₅₇	C₂²	semidirect
19. C₂²×D₁₅₇	C₈	semidirect
37. C₂²×C₁₅₇	C₂×C₈	semidirect
44. C₂×D₁₅₇	C₂×C₈	semidirect
55. D₁₅₇	C₂²⋊C₈	direct (N × H)
56. C₁₅₇	C₂×C₂²⋊C₈	semidirect
62. C₂²⋊C₈	D₁₅₇	direct (N × H)
73. C₂×C₈	C₂×D₁₅₇	semidirect
101. C₂²	C₈×D₁₅₇	semidirect

H ≅ G/N

type

2. C₂×C₈×D₁₅₇

C₂

semidirect

3. C₂²⋊#11929434

C₂

semidirect

4. #19097683

C₂

semidirect

5. C₂²⋊C₈×C₁₅₇

C₂

semidirect

13. C₂×#11929434

C₂²

semidirect

16. C₂×C₈×C₁₅₇

C₂²

semidirect

19. C₂²×D₁₅₇

C₈

semidirect

37. C₂²×C₁₅₇

C₂×C₈

semidirect

44. C₂×D₁₅₇

C₂×C₈

semidirect

55. D₁₅₇

C₂²⋊C₈

direct (N × H)

56. C₁₅₇

C₂×C₂²⋊C₈

semidirect

62. C₂²⋊C₈

D₁₅₇

direct (N × H)

73. C₂×C₈

C₂×D₁₅₇

semidirect

101. C₂²

C₈×D₁₅₇

semidirect

Subgroups — C₂²⋊C₈×D₁₅₇ #23757232

Conjugacy classes of subgroups

Complements

Subgroups — C22⋊C8×D157 #23757232

Conjugacy classes of subgroups

Complements

Subgroups — C₂²⋊C₈×D₁₅₇ #23757232