Group #23757901 — subgroups

Subgroups — #23757901

Order 10048.

206 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 206 conjugacy classes of subgroups — too many to draw. Only its 111 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	#19097771	C₂	1	2	yes	maximal, complemented
2	#19097770	C₂	1	2	yes	maximal, complemented
3	#19097763	C₂	1	2	yes	maximal, complemented
4	#19097762	C₂	1	2	yes	maximal, complemented
5	#19097748	C₂	1	4	yes	maximal
6	#19097730	C₂	1	1	yes	maximal, characteristic
7	C₂×C₄⋊Dic₁₅₇	C₂	1	1	yes	maximal, characteristic
8	#117×C₁₅₇	C₂	1	1	yes	maximal, characteristic
9	C₂×Dic₃₁₄	C₂²	1	4	yes	—
10	C₂²×Dic₁₅₇	C₂²	1	2	yes	—
11	C₄⋊Dic₁₅₇	C₂²	1	2	yes	—
12	C₄⋊Dic₁₅₇	C₂²	1	2	yes	—
13	C₄⋊Dic₁₅₇	C₂²	1	1	yes	characteristic
14	C₄⋊Dic₁₅₇	C₂²	1	1	yes	characteristic
15	C₄⋊Dic₁₅₇	C₂²	1	4	yes	—
16	#14578933	C₂²	1	4	yes	—
17	#14578933	C₂²	1	4	yes	—
18	C₂²⋊Dic₁₅₇	C₂²	1	2	yes	—
19	C₄×Dic₁₅₇	C₂²	1	2	yes	—
20	C₂²×C₄×C₁₅₇	C₂²	1	1	yes	characteristic
21	C₄⋊C₄×C₁₅₇	C₂²	1	1	yes	characteristic
22	C₄⋊C₄×C₁₅₇	C₂²	1	1	yes	characteristic
23	C₂²⋊C₄×C₁₅₇	C₂²	1	2	yes	—
24	C₄²×C₁₅₇	C₂²	1	2	yes	—
25	Dic₃₁₄	—	4	4	no	—
26	C₂×Dic₁₅₇	—	4	2	no	—
27	C₂×Dic₁₅₇	C₂³	1	4	yes	—
28	C₂×Dic₁₅₇	C₂³	1	2	yes	—
29	C₂×Dic₁₅₇	C₂³	1	2	yes	—
30	C₂³×C₁₅₇	C₂³	1	1	yes	characteristic
31	C₂×C₄×C₁₅₇	Q₈	1	4	yes	complemented
32	C₂×C₄×C₁₅₇	C₂³	1	2	yes	—
33	C₂×C₄×C₁₅₇	C₂³	1	2	yes	—
34	C₂×C₄×C₁₅₇	C₂³	1	1	yes	characteristic
35	C₂×C₄×C₁₅₇	C₂³	1	1	yes	characteristic
36	C₂²×C₁₅₇	—	2	1	no	—
37	C₂²×C₁₅₇	—	2	1	no	—
38	C₂²×C₁₅₇	C₂×Q₈	1	2	yes	—
39	C₂²×C₁₅₇	C₂⁴	1	1	yes	characteristic
40	C₄×C₁₅₇	—	2	2	no	—
41	C₄×C₁₅₇	—	2	2	no	—
42	C₄×C₁₅₇	C₂×Q₈	1	2	yes	—
43	C₄×C₁₅₇	C₂×Q₈	1	2	yes	—
44	Dic₁₅₇	—	2	4	no	—
45	Dic₁₅₇	—	2	2	no	—
46	Dic₁₅₇	—	2	2	no	—
47	C₂×C₁₅₇	—	2	2	no	—
48	C₂×C₁₅₇	C₂²×Q₈	1	1	yes	characteristic
49	C₂×C₁₅₇	H(5,𝔽₂)	1	1	yes	characteristic
50	C₂×C₁₅₇	2₋¹⁺⁴	1	1	yes	characteristic
51	C₁₅₇	#544	1	1	yes	characteristic, complemented, Sylow
52	#544	—	157	1	no	maximal, Sylow
53	C₄⋊Q₈	—	157	2	no	—
54	C₄⋊Q₈	—	157	2	no	—
55	#125	—	157	2	no	—
56	#125	—	157	2	no	—
57	C₂²⋊Q₈	—	157	4	no	—
58	#117	—	157	1	no	—
59	#117	D₁₅₇	1	1	yes	characteristic
60	C₂×C₄⋊C₄	—	157	1	no	—
61	C₂×Q₈	—	157	4	no	—
62	C₂²×C₄	—	157	2	no	—
63	C₂²×C₄	C₂×D₁₅₇	1	1	yes	characteristic
64	C₄⋊C₄	—	157	2	no	—
65	C₄⋊C₄	—	157	2	no	—
66	C₄⋊C₄	—	157	1	no	—
67	C₄⋊C₄	—	157	1	no	—
68	C₄⋊C₄	—	157	4	no	—
69	C₄⋊C₄	—	157	4	no	—
70	C₄⋊C₄	—	157	4	no	—
71	C₄⋊C₄	C₂×D₁₅₇	1	1	yes	characteristic
72	C₄⋊C₄	C₂×D₁₅₇	1	1	yes	characteristic
73	C₂²⋊C₄	—	157	2	no	—
74	C₂²⋊C₄	C₂×D₁₅₇	1	2	yes	—
75	C₄²	—	157	2	no	—
76	C₄²	C₂×D₁₅₇	1	2	yes	—
77	C₂³	C₂²×D₁₅₇	1	1	yes	characteristic
78	Q₈	—	628	4	no	—
79	C₂×C₄	—	628	2	no	—
80	C₂×C₄	—	157	2	no	—
81	C₂×C₄	—	157	2	no	—
82	C₂×C₄	—	157	4	no	—
83	C₂×C₄	Dic₃₁₄	1	4	yes	complemented
84	C₂×C₄	C₂²×D₁₅₇	1	2	yes	—
85	C₂×C₄	C₂²×D₁₅₇	1	2	yes	—
86	C₂×C₄	C₂²×D₁₅₇	1	1	yes	characteristic
87	C₂×C₄	C₂²×D₁₅₇	1	1	yes	characteristic
88	C₂²	—	2	1	no	—
89	C₂²	—	2	1	no	—
90	C₂²	C₂×Dic₃₁₄	1	2	yes	—
91	C₂²	C₂³×D₁₅₇	1	1	yes	characteristic
92	C₄	—	2	2	no	—
93	C₄	—	2	2	no	—
94	C₄	—	314	4	no	—
95	C₄	—	314	2	no	—
96	C₄	—	314	2	no	—
97	C₄	C₂×Dic₃₁₄	1	2	yes	—
98	C₄	C₂×Dic₃₁₄	1	2	yes	—
99	C₂	—	2	2	no	—
100	C₂	C₂²×Dic₃₁₄	1	1	yes	characteristic
101	C₂	#19097817	1	1	yes	characteristic
102	C₂	#19097818	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19097771

C₂

yes

maximal, complemented

#19097770

C₂

yes

maximal, complemented

#19097763

C₂

yes

maximal, complemented

#19097762

C₂

yes

maximal, complemented

#19097748

C₂

yes

maximal

#19097730

C₂

yes

maximal, characteristic

C₂×C₄⋊Dic₁₅₇

C₂

yes

maximal, characteristic

#117×C₁₅₇

C₂

yes

maximal, characteristic

C₂×Dic₃₁₄

C₂²

yes

—

C₂²×Dic₁₅₇

C₂²

yes

—

C₄⋊Dic₁₅₇

C₂²

yes

—

C₄⋊Dic₁₅₇

C₂²

yes

—

C₄⋊Dic₁₅₇

C₂²

yes

characteristic

C₄⋊Dic₁₅₇

C₂²

yes

characteristic

C₄⋊Dic₁₅₇

C₂²

yes

—

#14578933

C₂²

yes

—

#14578933

C₂²

yes

—

C₂²⋊Dic₁₅₇

C₂²

yes

—

C₄×Dic₁₅₇

C₂²

yes

—

C₂²×C₄×C₁₅₇

C₂²

yes

characteristic

C₄⋊C₄×C₁₅₇

C₂²

yes

characteristic

C₄⋊C₄×C₁₅₇

C₂²

yes

characteristic

C₂²⋊C₄×C₁₅₇

C₂²

yes

—

C₄²×C₁₅₇

C₂²

yes

—

Dic₃₁₄

—

C₂×Dic₁₅₇

—

C₂×Dic₁₅₇

C₂³

yes

—

C₂×Dic₁₅₇

C₂³

yes

—

C₂×Dic₁₅₇

C₂³

yes

—

C₂³×C₁₅₇

C₂³

yes

characteristic

C₂×C₄×C₁₅₇

Q₈

yes

complemented

C₂×C₄×C₁₅₇

C₂³

yes

—

C₂×C₄×C₁₅₇

C₂³

yes

—

C₂×C₄×C₁₅₇

C₂³

yes

characteristic

C₂×C₄×C₁₅₇

C₂³

yes

characteristic

C₂²×C₁₅₇

—

C₂²×C₁₅₇

—

C₂²×C₁₅₇

C₂×Q₈

yes

—

C₂²×C₁₅₇

C₂⁴

yes

characteristic

C₄×C₁₅₇

—

C₄×C₁₅₇

—

C₄×C₁₅₇

C₂×Q₈

yes

—

C₄×C₁₅₇

C₂×Q₈

yes

—

Dic₁₅₇

—

Dic₁₅₇

—

Dic₁₅₇

—

C₂×C₁₅₇

—

C₂×C₁₅₇

C₂²×Q₈

yes

characteristic

C₂×C₁₅₇

H(5,𝔽₂)

yes

characteristic

C₂×C₁₅₇

2₋¹⁺⁴

yes

characteristic

C₁₅₇

#544

yes

characteristic, complemented, Sylow

#544

—

157

maximal, Sylow

C₄⋊Q₈

—

157

—

C₄⋊Q₈

—

157

—

#125

—

157

—

#125

—

157

—

C₂²⋊Q₈

—

157

—

#117

—

157

—

#117

D₁₅₇

yes

characteristic

C₂×C₄⋊C₄

—

157

—

C₂×Q₈

—

157

—

C₂²×C₄

—

157

—

C₂²×C₄

C₂×D₁₅₇

yes

characteristic

C₄⋊C₄

—

157

—

C₄⋊C₄

—

157

—

C₄⋊C₄

—

157

—

C₄⋊C₄

—

157

—

C₄⋊C₄

—

157

—

C₄⋊C₄

—

157

—

C₄⋊C₄

—

157

—

C₄⋊C₄

C₂×D₁₅₇

yes

characteristic

C₄⋊C₄

C₂×D₁₅₇

yes

characteristic

C₂²⋊C₄

—

157

—

C₂²⋊C₄

C₂×D₁₅₇

yes

—

C₄²

—

157

—

C₄²

C₂×D₁₅₇

yes

—

C₂³

C₂²×D₁₅₇

yes

characteristic

Q₈

—

628

—

C₂×C₄

—

628

—

C₂×C₄

—

157

—

C₂×C₄

—

157

—

C₂×C₄

—

157

—

C₂×C₄

Dic₃₁₄

yes

complemented

C₂×C₄

C₂²×D₁₅₇

yes

—

C₂×C₄

C₂²×D₁₅₇

yes

—

C₂×C₄

C₂²×D₁₅₇

yes

characteristic

C₂×C₄

C₂²×D₁₅₇

yes

characteristic

C₂²

—

C₂²

—

C₂²

C₂×Dic₃₁₄

yes

—

C₂²

C₂³×D₁₅₇

yes

characteristic

C₄

—

C₄

—

C₄

—

314

—

C₄

—

314

—

C₄

—

314

—

C₄

C₂×Dic₃₁₄

yes

—

C₄

C₂×Dic₃₁₄

yes

—

C₂

—

100

C₂

C₂²×Dic₃₁₄

yes

characteristic

101

C₂

#19097817

yes

characteristic

102

C₂

#19097818

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₁₅₇
commutator, socle, cosocle	C₂²×C₁₅₇	C₂⁴
Fitting	#117×C₁₅₇	C₂
prime radical	C₂³×C₁₅₇	C₂³
hypercenter	#117	D₁₅₇
hypocenter	C₁₅₇	#544
LCS	G ▷ C₂²×C₁₅₇ ▷ C₁₅₇
UCS	C₁ ◁ C₂² ◁ #117

subgroup

G/H

center, Frattini

C₂²

C₂³×D₁₅₇

commutator, socle, cosocle

C₂²×C₁₅₇

C₂⁴

Fitting

#117×C₁₅₇

C₂

prime radical

C₂³×C₁₅₇

C₂³

D₁₅₇

C₁₅₇

G ▷ C₂²×C₁₅₇ ▷ C₁₅₇

C₁ ◁ C₂² ◁ #117

N	H ≅ G/N	type
1. #19097771	C₂	semidirect
2. #19097770	C₂	semidirect
3. #19097763	C₂	semidirect
4. #19097762	C₂	semidirect
31. C₂×C₄×C₁₅₇	Q₈	semidirect
51. C₁₅₇	#544	semidirect
83. C₂×C₄	Dic₃₁₄	semidirect

H ≅ G/N

type

1. #19097771

C₂

semidirect

2. #19097770

C₂

semidirect

3. #19097763

C₂

semidirect

4. #19097762

C₂

semidirect

31. C₂×C₄×C₁₅₇

Q₈

semidirect

51. C₁₅₇

#544

semidirect

83. C₂×C₄

Dic₃₁₄

semidirect

Subgroups — #23757901

Conjugacy classes of subgroups

Complements