Group #23757768 — subgroups

Subgroups — C₂×#19097747 #23757768

Order 10048.

322 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 322 conjugacy classes of subgroups — too many to draw. Only its 135 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	#19097747	C₂	1	8	yes	maximal, direct product factor
3	C₂×C₄⋊Dic₁₅₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#14578933	C₂	1	2	yes	maximal, complemented
5	C₂×#14578931	C₂	1	2	yes	maximal
6	C₂²×Q₈×C₁₅₇	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂³×D₁₅₇	C₂²	1	1	yes	characteristic
8	C₂²×Dic₁₅₇	C₂²	1	2	yes	—
9	C₂²×Dic₁₅₇	C₂²	1	1	yes	characteristic
10	C₂×C₄×D₁₅₇	—	2	4	no	—
11	C₂×C₄×D₁₅₇	C₂²	1	4	yes	—
12	C₄⋊Dic₁₅₇	C₂²	1	4	yes	complemented
13	#14578933	C₂²	1	8	yes	complemented
14	#14578931	C₂²	1	8	yes	—
15	C₂×Q₈×C₁₅₇	—	2	4	no	—
16	C₂×Q₈×C₁₅₇	C₂²	1	4	yes	complemented
17	C₂²×C₄×C₁₅₇	C₂²	1	2	yes	—
18	C₂²×C₄×C₁₅₇	C₂²	1	1	yes	characteristic
19	C₂²×D₁₅₇	—	2	2	no	—
20	C₂²×D₁₅₇	—	2	2	no	—
21	C₂²×D₁₅₇	Q₈	1	4	yes	complemented
22	C₂²×D₁₅₇	C₂³	1	2	yes	—
23	C₂×Dic₁₅₇	—	2	2	no	—
24	C₂×Dic₁₅₇	—	2	4	no	—
25	C₂×Dic₁₅₇	C₂³	1	2	yes	—
26	C₂×Dic₁₅₇	C₂³	1	4	yes	—
27	C₄×D₁₅₇	—	2	8	no	—
28	C₂³×C₁₅₇	C₂³	1	1	yes	characteristic
29	Q₈×C₁₅₇	—	2	8	no	—
30	C₂×C₄×C₁₅₇	—	2	4	no	—
31	C₂×C₄×C₁₅₇	D₄	1	4	yes	—
32	C₂×C₄×C₁₅₇	C₂³	1	4	yes	—
33	C₂×C₄×C₁₅₇	C₂³	1	2	yes	—
34	C₂²×C₁₅₇	C₂×D₄	1	2	yes	—
35	C₂²×C₁₅₇	C₂×Q₈	1	2	yes	—
36	C₂²×C₁₅₇	#41	1	2	yes	—
37	C₂²×C₁₅₇	C₂⁴	1	1	yes	characteristic
38	C₂×D₁₅₇	—	2	8	no	—
39	C₂×D₁₅₇	—	2	2	no	—
40	C₂×D₁₅₇	—	2	2	no	—
41	C₂×D₁₅₇	C₂×Q₈	1	4	yes	complemented
42	C₄×C₁₅₇	—	2	4	no	—
43	C₄×C₁₅₇	C₂×D₄	1	4	yes	—
44	Dic₁₅₇	—	2	2	no	—
45	Dic₁₅₇	—	2	4	no	—
46	C₂×C₁₅₇	C₂²⋊Q₈	1	4	yes	complemented
47	C₂×C₁₅₇	C₂²×D₄	1	1	yes	characteristic
48	C₂×C₁₅₇	C₂²×Q₈	1	1	yes	characteristic
49	C₂×C₁₅₇	C₂×#41	1	1	yes	characteristic
50	D₁₅₇	—	2	4	no	—
51	C₁₅₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic, complemented, Sylow
52	C₂×C₂²⋊Q₈	—	157	1	no	maximal, Sylow
53	C₂²×Q₈	D₁₅₇	1	1	yes	characteristic, complemented
54	C₂³×C₄	—	157	1	no	—
55	C₂²⋊Q₈	—	157	8	no	—
56	C₂×C₄⋊C₄	—	157	2	no	—
57	C₂×C₄⋊C₄	—	157	1	no	—
58	C₂×C₂²⋊C₄	—	157	2	no	—
59	C₂⁴	—	157	1	no	—
60	C₂×Q₈	—	2	4	no	—
61	C₂×Q₈	C₂×D₁₅₇	1	4	yes	complemented
62	C₂²×C₄	—	314	4	no	—
63	C₂²×C₄	—	157	2	no	—
64	C₂²×C₄	—	157	1	no	—
65	C₂²×C₄	—	157	4	no	—
66	C₂²×C₄	C₂×D₁₅₇	1	2	yes	—
67	C₂²×C₄	C₂×D₁₅₇	1	1	yes	characteristic
68	C₄⋊C₄	—	157	8	no	—
69	C₄⋊C₄	—	157	4	no	—
70	C₂²⋊C₄	—	157	8	no	—
71	C₂³	—	314	2	no	—
72	C₂³	—	314	2	no	—
73	C₂³	—	157	4	no	—
74	C₂³	—	157	2	no	—
75	C₂³	C₂²×D₁₅₇	1	1	yes	characteristic
76	Q₈	—	2	8	no	—
77	C₂×C₄	—	2	4	no	—
78	C₂×C₄	—	314	2	no	—
79	C₂×C₄	—	314	4	no	—
80	C₂×C₄	—	314	8	no	—
81	C₂×C₄	—	157	4	no	—
82	C₂×C₄	—	157	2	no	—
83	C₂×C₄	C₂²⋊D₁₅₇	1	4	yes	—
84	C₂×C₄	C₂²×D₁₅₇	1	4	yes	—
85	C₂×C₄	C₂²×D₁₅₇	1	2	yes	—
86	C₂²	—	314	8	no	—
87	C₂²	—	314	2	no	—
88	C₂²	—	314	2	no	—
89	C₂²	—	157	4	no	—
90	C₂²	C₂×C₂²⋊D₁₅₇	1	2	yes	—
91	C₂²	Q₈×D₁₅₇	1	2	yes	—
92	C₂²	#14578955	1	2	yes	—
93	C₂²	C₂³×D₁₅₇	1	1	yes	characteristic
94	C₄	—	2	4	no	—
95	C₄	—	314	2	no	—
96	C₄	—	314	4	no	—
97	C₄	C₂×C₂²⋊D₁₅₇	1	4	yes	—
98	C₂	—	314	4	no	—
99	C₂	#19097747	1	4	yes	direct product factor
100	C₂	C₂²×C₂²⋊D₁₅₇	1	1	yes	characteristic
101	C₂	C₂×Q₈×D₁₅₇	1	1	yes	characteristic
102	C₂	C₂×#14578955	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×C₄×D₁₅₇

C₂

yes

maximal, characteristic

#19097747

C₂

yes

maximal, direct product factor

C₂×C₄⋊Dic₁₅₇

C₂

yes

maximal, characteristic, complemented

C₂×#14578933

C₂

yes

maximal, complemented

C₂×#14578931

C₂

yes

maximal

C₂²×Q₈×C₁₅₇

C₂

yes

maximal, characteristic, complemented

C₂³×D₁₅₇

C₂²

yes

characteristic

C₂²×Dic₁₅₇

C₂²

yes

—

C₂²×Dic₁₅₇

C₂²

yes

characteristic

C₂×C₄×D₁₅₇

—

C₂×C₄×D₁₅₇

C₂²

yes

—

C₄⋊Dic₁₅₇

C₂²

yes

complemented

#14578933

C₂²

yes

complemented

#14578931

C₂²

yes

—

C₂×Q₈×C₁₅₇

—

C₂×Q₈×C₁₅₇

C₂²

yes

complemented

C₂²×C₄×C₁₅₇

C₂²

yes

—

C₂²×C₄×C₁₅₇

C₂²

yes

characteristic

C₂²×D₁₅₇

—

C₂²×D₁₅₇

—

C₂²×D₁₅₇

Q₈

yes

complemented

C₂²×D₁₅₇

C₂³

yes

—

C₂×Dic₁₅₇

—

C₂×Dic₁₅₇

—

C₂×Dic₁₅₇

C₂³

yes

—

C₂×Dic₁₅₇

C₂³

yes

—

C₄×D₁₅₇

—

C₂³×C₁₅₇

C₂³

yes

characteristic

Q₈×C₁₅₇

—

C₂×C₄×C₁₅₇

—

C₂×C₄×C₁₅₇

D₄

yes

—

C₂×C₄×C₁₅₇

C₂³

yes

—

C₂×C₄×C₁₅₇

C₂³

yes

—

C₂²×C₁₅₇

C₂×D₄

yes

—

C₂²×C₁₅₇

C₂×Q₈

yes

—

C₂²×C₁₅₇

#41

yes

—

C₂²×C₁₅₇

C₂⁴

yes

characteristic

C₂×D₁₅₇

—

C₂×D₁₅₇

—

C₂×D₁₅₇

—

C₂×D₁₅₇

C₂×Q₈

yes

complemented

C₄×C₁₅₇

—

C₄×C₁₅₇

C₂×D₄

yes

—

Dic₁₅₇

—

Dic₁₅₇

—

C₂×C₁₅₇

C₂²⋊Q₈

yes

complemented

C₂×C₁₅₇

C₂²×D₄

yes

characteristic

C₂×C₁₅₇

C₂²×Q₈

yes

characteristic

C₂×C₁₅₇

C₂×#41

yes

characteristic

D₁₅₇

—

C₁₅₇

C₂×C₂²⋊Q₈

yes

characteristic, complemented, Sylow

C₂×C₂²⋊Q₈

—

157

maximal, Sylow

C₂²×Q₈

D₁₅₇

yes

characteristic, complemented

C₂³×C₄

—

157

—

C₂²⋊Q₈

—

157

—

C₂×C₄⋊C₄

—

157

—

C₂×C₄⋊C₄

—

157

—

C₂×C₂²⋊C₄

—

157

—

C₂⁴

—

157

—

C₂×Q₈

—

C₂×Q₈

C₂×D₁₅₇

yes

complemented

C₂²×C₄

—

314

—

C₂²×C₄

—

157

—

C₂²×C₄

—

157

—

C₂²×C₄

—

157

—

C₂²×C₄

C₂×D₁₅₇

yes

—

C₂²×C₄

C₂×D₁₅₇

yes

characteristic

C₄⋊C₄

—

157

—

C₄⋊C₄

—

157

—

C₂²⋊C₄

—

157

—

C₂³

—

314

—

C₂³

—

314

—

C₂³

—

157

—

C₂³

—

157

—

C₂³

C₂²×D₁₅₇

yes

characteristic

Q₈

—

C₂×C₄

—

C₂×C₄

—

314

—

C₂×C₄

—

314

—

C₂×C₄

—

314

—

C₂×C₄

—

157

—

C₂×C₄

—

157

—

C₂×C₄

C₂²⋊D₁₅₇

yes

—

C₂×C₄

C₂²×D₁₅₇

yes

—

C₂×C₄

C₂²×D₁₅₇

yes

—

C₂²

—

314

—

C₂²

—

314

—

C₂²

—

314

—

C₂²

—

157

—

C₂²

C₂×C₂²⋊D₁₅₇

yes

—

C₂²

Q₈×D₁₅₇

yes

—

C₂²

#14578955

yes

—

C₂²

C₂³×D₁₅₇

yes

characteristic

C₄

—

C₄

—

314

—

C₄

—

314

—

C₄

C₂×C₂²⋊D₁₅₇

yes

—

C₂

—

314

—

C₂

#19097747

yes

direct product factor

100

C₂

C₂²×C₂²⋊D₁₅₇

yes

characteristic

101

C₂

C₂×Q₈×D₁₅₇

yes

characteristic

102

C₂

C₂×#14578955

yes

characteristic

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

subgroup

G/H

center

C₂³

C₂²×D₁₅₇

commutator, cosocle

C₂²×C₁₅₇

C₂⁴

Frattini

C₂²

C₂³×D₁₅₇

Fitting

C₂²×Q₈×C₁₅₇

C₂

socle

C₂³×C₁₅₇

C₂³

prime radical

C₂³×D₁₅₇

C₂²

hypercenter

C₂²×Q₈

D₁₅₇

hypocenter

C₁₅₇

C₂×C₂²⋊Q₈

LCS

G ▷ C₂²×C₁₅₇ ▷ C₁₅₇

UCS

C₁ ◁ C₂³ ◁ C₂²×Q₈

N	H ≅ G/N	type
2. #19097747	C₂	direct (N × H)
3. C₂×C₄⋊Dic₁₅₇	C₂	semidirect
4. C₂×#14578933	C₂	semidirect
6. C₂²×Q₈×C₁₅₇	C₂	semidirect
12. C₄⋊Dic₁₅₇	C₂²	semidirect
13. #14578933	C₂²	semidirect
16. C₂×Q₈×C₁₅₇	C₂²	semidirect
21. C₂²×D₁₅₇	Q₈	semidirect
41. C₂×D₁₅₇	C₂×Q₈	semidirect
46. C₂×C₁₅₇	C₂²⋊Q₈	semidirect
51. C₁₅₇	C₂×C₂²⋊Q₈	semidirect
53. C₂²×Q₈	D₁₅₇	semidirect
61. C₂×Q₈	C₂×D₁₅₇	semidirect
99. C₂	#19097747	direct (N × H)

H ≅ G/N

type

2. #19097747

C₂

direct (N × H)

3. C₂×C₄⋊Dic₁₅₇

C₂

semidirect

4. C₂×#14578933

C₂

semidirect

6. C₂²×Q₈×C₁₅₇

C₂

semidirect

12. C₄⋊Dic₁₅₇

C₂²

semidirect

13. #14578933

C₂²

semidirect

16. C₂×Q₈×C₁₅₇

C₂²

semidirect

21. C₂²×D₁₅₇

Q₈

semidirect

41. C₂×D₁₅₇

C₂×Q₈

semidirect

46. C₂×C₁₅₇

C₂²⋊Q₈

semidirect

51. C₁₅₇

C₂×C₂²⋊Q₈

semidirect

53. C₂²×Q₈

D₁₅₇

semidirect

61. C₂×Q₈

C₂×D₁₅₇

semidirect

99. C₂

#19097747

direct (N × H)

Subgroups — C₂×#19097747 #23757768

Conjugacy classes of subgroups

Complements

Subgroups — C2×#19097747 #23757768

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#19097747 #23757768