Group #23757980 — subgroups

Subgroups — #23757980

Order 10048.

228 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 115 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	1	yes	maximal, characteristic, complemented
2	#19097762	C₂	1	2	yes	maximal, complemented
3	#19097750	C₂	1	2	yes	maximal, complemented
4	#19097748	C₂	1	4	yes	maximal, complemented
5	C₄×Dic₃₁₄	C₂	1	1	yes	maximal, characteristic, complemented
6	D₄×Dic₁₅₇	C₂	1	2	yes	maximal
7	C₂×C₄⋊Dic₁₅₇	C₂	1	2	yes	maximal, complemented
8	C₄×D₄×C₁₅₇	C₂	1	1	yes	maximal, characteristic
9	C₂×Dic₃₁₄	C₂²	1	1	yes	characteristic
10	C₂×Dic₃₁₄	C₂²	1	2	yes	—
11	C₂²×Dic₁₅₇	C₂²	1	4	yes	—
12	C₄⋊Dic₁₅₇	C₂²	1	1	yes	characteristic
13	C₄⋊Dic₁₅₇	C₂²	1	4	yes	—
14	C₄⋊Dic₁₅₇	C₂²	1	2	yes	—
15	C₄⋊Dic₁₅₇	C₂²	1	2	yes	—
16	#14578933	C₂²	1	2	yes	—
17	#14578933	C₂²	1	4	yes	—
18	C₂²⋊Dic₁₅₇	C₂²	1	4	yes	—
19	C₄×Dic₁₅₇	C₂²	1	2	yes	—
20	C₂×D₄×C₁₅₇	C₂²	1	1	yes	characteristic
21	C₂²×C₄×C₁₅₇	C₂²	1	2	yes	—
22	C₄⋊C₄×C₁₅₇	C₂²	1	1	yes	characteristic
23	C₂²⋊C₄×C₁₅₇	C₂²	1	2	yes	—
24	C₄²×C₁₅₇	C₂²	1	1	yes	characteristic
25	Dic₃₁₄	—	2	2	no	—
26	Dic₃₁₄	—	4	2	no	—
27	C₂×Dic₁₅₇	—	4	4	no	—
28	C₂×Dic₁₅₇	C₂³	1	2	yes	—
29	C₂×Dic₁₅₇	C₂³	1	4	yes	—
30	C₂×Dic₁₅₇	C₂³	1	2	yes	—
31	C₂³×C₁₅₇	C₂³	1	2	yes	—
32	D₄×C₁₅₇	Q₈	1	4	yes	complemented
33	C₂×C₄×C₁₅₇	—	2	4	no	—
34	C₂×C₄×C₁₅₇	C₂³	1	1	yes	characteristic
35	C₂×C₄×C₁₅₇	C₂³	1	2	yes	—
36	C₂×C₄×C₁₅₇	C₂³	1	1	yes	characteristic
37	C₂×C₄×C₁₅₇	C₂³	1	1	yes	characteristic
38	C₂²×C₁₅₇	—	2	2	no	—
39	C₂²×C₁₅₇	—	2	2	no	—
40	C₂²×C₁₅₇	C₂×Q₈	1	4	yes	—
41	C₂²×C₁₅₇	C₂⁴	1	1	yes	characteristic
42	C₄×C₁₅₇	—	2	2	no	—
43	C₄×C₁₅₇	—	2	1	no	—
44	C₄×C₁₅₇	C₂×Q₈	1	2	yes	—
45	C₄×C₁₅₇	#41	1	2	yes	—
46	Dic₁₅₇	—	2	2	no	—
47	Dic₁₅₇	—	2	2	no	—
48	Dic₁₅₇	—	2	4	no	—
49	C₂×C₁₅₇	—	2	4	no	—
50	C₂×C₁₅₇	C₂²×Q₈	1	1	yes	characteristic
51	C₂×C₁₅₇	C₂×#41	1	1	yes	characteristic
52	C₂×C₁₅₇	H(5,𝔽₂)	1	1	yes	characteristic
53	C₁₅₇	#554	1	1	yes	characteristic, complemented, Sylow
54	#554	—	157	1	no	maximal, Sylow
55	C₄⋊Q₈	—	157	1	no	—
56	#125	—	157	2	no	—
57	C₂²⋊Q₈	—	157	2	no	—
58	C₂²⋊Q₈	—	157	4	no	—
59	C₄×Q₈	—	157	1	no	—
60	C₄×D₄	—	157	2	no	—
61	C₄×D₄	D₁₅₇	1	1	yes	characteristic
62	C₂×C₄⋊C₄	—	157	2	no	—
63	C₂×Q₈	—	157	1	no	—
64	C₂×Q₈	—	157	2	no	—
65	C₂×D₄	C₂×D₁₅₇	1	1	yes	characteristic
66	C₂²×C₄	—	157	4	no	—
67	C₂²×C₄	C₂×D₁₅₇	1	2	yes	—
68	C₄⋊C₄	—	157	1	no	—
69	C₄⋊C₄	—	157	4	no	—
70	C₄⋊C₄	—	157	2	no	—
71	C₄⋊C₄	—	157	2	no	—
72	C₄⋊C₄	—	157	2	no	—
73	C₄⋊C₄	—	157	4	no	—
74	C₄⋊C₄	C₂×D₁₅₇	1	1	yes	characteristic
75	C₂²⋊C₄	—	157	4	no	—
76	C₂²⋊C₄	C₂×D₁₅₇	1	2	yes	—
77	C₄²	—	157	2	no	—
78	C₄²	C₂×D₁₅₇	1	1	yes	characteristic
79	C₂³	C₂²×D₁₅₇	1	2	yes	—
80	Q₈	—	314	2	no	—
81	Q₈	—	628	2	no	—
82	D₄	Dic₃₁₄	1	4	yes	complemented
83	C₂×C₄	—	2	4	no	—
84	C₂×C₄	—	628	4	no	—
85	C₂×C₄	—	157	2	no	—
86	C₂×C₄	—	157	2	no	—
87	C₂×C₄	—	157	4	no	—
88	C₂×C₄	C₂²×D₁₅₇	1	1	yes	characteristic
89	C₂×C₄	C₂²×D₁₅₇	1	2	yes	—
90	C₂×C₄	C₂²×D₁₅₇	1	1	yes	characteristic
91	C₂×C₄	C₂²×D₁₅₇	1	1	yes	characteristic
92	C₂²	—	2	2	no	—
93	C₂²	—	2	2	no	—
94	C₂²	C₂×Dic₃₁₄	1	4	yes	—
95	C₂²	C₂³×D₁₅₇	1	1	yes	characteristic
96	C₄	—	2	2	no	—
97	C₄	—	2	1	no	—
98	C₄	—	314	2	no	—
99	C₄	—	314	2	no	—
100	C₄	—	314	4	no	—
101	C₄	C₂×Dic₃₁₄	1	2	yes	—
102	C₄	#14578953	1	2	yes	—
103	C₂	—	2	4	no	—
104	C₂	C₂²×Dic₃₁₄	1	1	yes	characteristic
105	C₂	C₂×#14578953	1	1	yes	characteristic
106	C₂	#19097817	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19097771

C₂

yes

maximal, characteristic, complemented

#19097762

C₂

yes

maximal, complemented

#19097750

C₂

yes

maximal, complemented

#19097748

C₂

yes

maximal, complemented

C₄×Dic₃₁₄

C₂

yes

maximal, characteristic, complemented

D₄×Dic₁₅₇

C₂

yes

maximal

C₂×C₄⋊Dic₁₅₇

C₂

yes

maximal, complemented

C₄×D₄×C₁₅₇

C₂

yes

maximal, characteristic

C₂×Dic₃₁₄

C₂²

yes

characteristic

C₂×Dic₃₁₄

C₂²

yes

—

C₂²×Dic₁₅₇

C₂²

yes

—

C₄⋊Dic₁₅₇

C₂²

yes

characteristic

C₄⋊Dic₁₅₇

C₂²

yes

—

C₄⋊Dic₁₅₇

C₂²

yes

—

C₄⋊Dic₁₅₇

C₂²

yes

—

#14578933

C₂²

yes

—

#14578933

C₂²

yes

—

C₂²⋊Dic₁₅₇

C₂²

yes

—

C₄×Dic₁₅₇

C₂²

yes

—

C₂×D₄×C₁₅₇

C₂²

yes

characteristic

C₂²×C₄×C₁₅₇

C₂²

yes

—

C₄⋊C₄×C₁₅₇

C₂²

yes

characteristic

C₂²⋊C₄×C₁₅₇

C₂²

yes

—

C₄²×C₁₅₇

C₂²

yes

characteristic

Dic₃₁₄

—

Dic₃₁₄

—

C₂×Dic₁₅₇

—

C₂×Dic₁₅₇

C₂³

yes

—

C₂×Dic₁₅₇

C₂³

yes

—

C₂×Dic₁₅₇

C₂³

yes

—

C₂³×C₁₅₇

C₂³

yes

—

D₄×C₁₅₇

Q₈

yes

complemented

C₂×C₄×C₁₅₇

—

C₂×C₄×C₁₅₇

C₂³

yes

characteristic

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

#41

yes

—

Dic₁₅₇

—

Dic₁₅₇

—

Dic₁₅₇

—

C₂×C₁₅₇

—

C₂×C₁₅₇

C₂²×Q₈

yes

characteristic

C₂×C₁₅₇

C₂×#41

yes

C₂×C₁₅₇

yes

C₁₅₇

yes

characteristic, complemented, Sylow

#554

—

157

maximal, Sylow

C₄⋊Q₈

—

157

—

#125

—

157

—

C₂²⋊Q₈

—

157

—

C₂²⋊Q₈

—

157

—

C₄×Q₈

—

157

—

C₄×D₄

—

157

—

C₄×D₄

D₁₅₇

yes

characteristic

C₂×C₄⋊C₄

—

157

—

C₂×Q₈

—

157

—

C₂×Q₈

—

157

—

C₂×D₄

C₂×D₁₅₇

yes

characteristic

C₂²×C₄

—

157

—

C₂²×C₄

C₂×D₁₅₇

yes

—

C₄⋊C₄

—

157

—

C₄⋊C₄

—

157

—

C₄⋊C₄

—

157

—

C₄⋊C₄

—

157

—

C₄⋊C₄

—

157

—

C₄⋊C₄

—

157

—

C₄⋊C₄

C₂×D₁₅₇

yes

characteristic

C₂²⋊C₄

—

157

—

C₂²⋊C₄

C₂×D₁₅₇

yes

—

C₄²

—

157

—

C₄²

C₂×D₁₅₇

yes

characteristic

C₂³

C₂²×D₁₅₇

yes

—

Q₈

—

314

—

Q₈

—

628

—

D₄

Dic₃₁₄

yes

complemented

C₂×C₄

—

C₂×C₄

—

628

—

C₂×C₄

—

157

—

C₂×C₄

—

157

—

C₂×C₄

—

157

—

C₂×C₄

C₂²×D₁₅₇

yes

characteristic

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

—

100

C₄

—

314

—

101

C₄

C₂×Dic₃₁₄

yes

—

102

C₄

#14578953

yes

—

103

C₂

—

104

C₂

C₂²×Dic₃₁₄

yes

characteristic

105

C₂

C₂×#14578953

yes

characteristic

106

C₂

#19097817

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₁₅₇
commutator, socle, cosocle	C₂²×C₁₅₇	C₂⁴
Fitting	C₄×D₄×C₁₅₇	C₂
prime radical	C₂×D₄×C₁₅₇	C₂²
hypercenter	C₄×D₄	D₁₅₇
hypocenter	C₁₅₇	#554
LCS	G ▷ C₂²×C₁₅₇ ▷ C₁₅₇
UCS	C₁ ◁ C₂² ◁ C₄×D₄

subgroup

G/H

center, Frattini

C₂²

C₂³×D₁₅₇

commutator, socle, cosocle

C₂²×C₁₅₇

C₂⁴

Fitting

C₄×D₄×C₁₅₇

C₂

prime radical

C₂×D₄×C₁₅₇

C₂²

hypercenter

C₄×D₄

D₁₅₇

C₁₅₇

G ▷ C₂²×C₁₅₇ ▷ C₁₅₇

C₁ ◁ C₂² ◁ C₄×D₄

N	H ≅ G/N	type
1. #19097771	C₂	semidirect
2. #19097762	C₂	semidirect
3. #19097750	C₂	semidirect
4. #19097748	C₂	semidirect
5. C₄×Dic₃₁₄	C₂	semidirect
7. C₂×C₄⋊Dic₁₅₇	C₂	semidirect
32. D₄×C₁₅₇	Q₈	semidirect
53. C₁₅₇	#554	semidirect
82. D₄	Dic₃₁₄	semidirect

H ≅ G/N

type

1. #19097771

C₂

semidirect

2. #19097762

C₂

semidirect

3. #19097750

C₂

semidirect

4. #19097748

C₂

semidirect

5. C₄×Dic₃₁₄

C₂

semidirect

7. C₂×C₄⋊Dic₁₅₇

C₂

semidirect

32. D₄×C₁₅₇

Q₈

semidirect

53. C₁₅₇

#554

semidirect

82. D₄

Dic₃₁₄

semidirect

Subgroups — #23757980

Conjugacy classes of subgroups

Complements