Group #18010994 — subgroups

Subgroups — #18010994

Order 4736.

172 conjugacy classes of subgroups; only the 48 normal subgroups are shown.

normal subgroups only

This group has 172 conjugacy classes of subgroups — too many to draw. Only its 48 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	D₈×#3866	C₂	1	2	yes	maximal, complemented
2	#14558460	C₂	1	1	yes	maximal, characteristic, complemented
3	#14558445	C₂	1	2	yes	maximal, complemented
4	#14558420	C₂	1	1	yes	maximal, characteristic, complemented
5	D₁₆×D₃₇	C₂	1	1	yes	maximal, characteristic
6	D₄×#3866	—	2	2	no	—
7	D₃₇⋊C₁₆	C₂²	1	1	yes	characteristic
8	#11924605	C₂²	1	2	yes	—
9	#11924598	—	2	2	no	—
10	C₈×#3866	C₂²	1	1	yes	characteristic
11	D₈×D₃₇	C₂²	1	2	yes	—
12	D₅₉₂	C₄	1	1	yes	characteristic, complemented
13	#11924477	C₄	1	2	yes	complemented
14	C₁₆×D₃₇	C₂²	1	1	yes	characteristic
15	D₁₆×C₃₇	C₄	1	1	yes	characteristic, complemented
16	C₂²×#3866	—	4	2	no	—
17	D₄×D₃₇	—	2	2	no	—
18	C₂²⋊#3866	—	4	2	no	—
19	C₄⋊#3866	—	2	2	no	—
20	C₄×#3866	D₄	1	1	yes	characteristic
21	D₃₇⋊C₈	D₄	1	1	yes	characteristic
22	D₈×C₃₇	C₂×C₄	1	2	yes	complemented
23	#10596939	—	2	2	no	—
24	D₂₉₆	C₂×C₄	1	2	yes	complemented
25	C₈×D₃₇	C₂³	1	1	yes	characteristic
26	#10596927	—	2	1	no	—
27	C₁₆×C₃₇	C₂×C₄	1	1	yes	characteristic, complemented
28	#10596925	C₂×C₄	1	1	yes	characteristic, complemented
29	C₂²×D₃₇	—	4	2	no	—
30	C₂×#3866	—	4	2	no	—
31	C₂×#3866	—	8	4	no	—
32	C₂×#3866	D₈	1	2	yes	—
33	D₄×C₃₇	—	2	2	no	—
34	C₂²⋊D₃₇	—	4	2	no	—
35	D₁₄₈	—	2	2	no	—
36	C₄×D₃₇	C₂×D₄	1	1	yes	characteristic
37	#64438	—	2	1	no	—
38	C₈×C₃₇	C₂²×C₄	1	1	yes	characteristic
39	#64436	#41	1	1	yes	characteristic
40	C₂²×C₃₇	—	4	2	no	—
41	C₂×D₃₇	—	4	2	no	—
42	C₂×D₃₇	—	8	2	no	—
43	C₂×D₃₇	—	8	2	no	—
44	C₂×D₃₇	C₂×D₈	1	1	yes	characteristic
45	#3866	—	2	2	no	—
46	#3866	—	8	2	no	—
47	C₄×C₃₇	C₄×D₄	1	1	yes	characteristic
48	Dic₃₇	#135	1	1	yes	characteristic
49	#2177	—	37	1	no	maximal, Sylow
50	C₂×C₃₇	—	8	2	no	—
51	C₂×C₃₇	C₄×D₈	1	1	yes	characteristic
52	D₃₇	—	8	2	no	—
53	D₃₇	#509	1	2	yes	—
54	C₂×D₁₆	—	37	1	no	—
55	C₄×D₈	—	37	2	no	—
56	#367	—	37	1	no	—
57	#357	—	37	2	no	—
58	#346	—	37	1	no	—
59	C₃₇	#2177	1	1	yes	characteristic, complemented, Sylow
60	C₂×D₈	—	37	2	no	—
61	C₄×D₄	—	74	2	no	—
62	D₁₆	—	37	2	no	—
63	D₁₆	—	37	1	no	—
64	D₁₆	#3866	1	1	yes	characteristic, complemented
65	C₂×C₁₆	—	37	1	no	—
66	C₂×C₁₆	—	37	1	no	—
67	#107	—	37	2	no	—
68	#102	—	74	2	no	—
69	C₄×C₈	—	37	1	no	—
70	C₂×D₄	—	74	2	no	—
71	C₂²×C₄	—	148	2	no	—
72	D₈	—	37	2	no	—
73	D₈	—	74	2	no	—
74	D₈	C₂×#3866	1	2	yes	complemented
75	C₂×C₈	—	37	1	no	—
76	C₂×C₈	—	37	1	no	—
77	C₄⋊C₄	—	74	2	no	—
78	C₂²⋊C₄	—	148	2	no	—
79	C₄²	—	37	1	no	—
80	C₁₆	—	37	1	no	—
81	C₁₆	—	74	1	no	—
82	C₁₆	C₂×#3866	1	1	yes	characteristic, complemented
83	C₂³	—	148	2	no	—
84	D₄	—	2	2	no	—
85	D₄	—	74	2	no	—
86	D₄	—	148	2	no	—
87	C₂×C₄	—	37	1	no	—
88	C₂×C₄	—	37	2	no	—
89	C₂×C₄	—	296	4	no	—
90	C₂×C₄	—	148	2	no	—
91	C₈	—	37	1	no	—
92	C₈	—	74	1	no	—
93	C₈	C₂²×#3866	1	1	yes	characteristic
94	C₂²	—	4	2	no	—
95	C₂²	—	37	1	no	—
96	C₂²	—	296	2	no	—
97	C₂²	—	296	2	no	—
98	C₂²	—	148	2	no	—
99	C₄	—	37	1	no	—
100	C₄	—	296	2	no	—
101	C₄	—	74	2	no	—
102	C₄	D₄×#3866	1	1	yes	characteristic
103	C₂	—	8	2	no	—
104	C₂	—	37	2	no	—
105	C₂	—	296	2	no	—
106	C₂	D₈×#3866	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

D₈×#3866

C₂

yes

maximal, complemented

#14558460

C₂

yes

maximal, characteristic, complemented

#14558445

C₂

yes

maximal, complemented

#14558420

C₂

yes

maximal, characteristic, complemented

D₁₆×D₃₇

C₂

yes

maximal, characteristic

D₄×#3866

—

D₃₇⋊C₁₆

C₂²

yes

characteristic

#11924605

C₂²

yes

—

#11924598

—

C₈×#3866

C₂²

yes

characteristic

D₈×D₃₇

C₂²

yes

—

D₅₉₂

C₄

yes

characteristic, complemented

#11924477

C₄

yes

complemented

C₁₆×D₃₇

C₂²

yes

characteristic

D₁₆×C₃₇

C₄

yes

characteristic, complemented

C₂²×#3866

—

D₄×D₃₇

—

C₂²⋊#3866

—

C₄⋊#3866

—

C₄×#3866

D₄

yes

characteristic

D₃₇⋊C₈

D₄

yes

characteristic

D₈×C₃₇

C₂×C₄

yes

complemented

#10596939

—

D₂₉₆

C₂×C₄

yes

complemented

C₈×D₃₇

C₂³

yes

characteristic

#10596927

—

C₁₆×C₃₇

C₂×C₄

yes

characteristic, complemented

#10596925

C₂×C₄

yes

characteristic, complemented

C₂²×D₃₇

—

C₂×#3866

—

C₂×#3866

—

C₂×#3866

D₈

yes

—

D₄×C₃₇

—

C₂²⋊D₃₇

—

D₁₄₈

—

C₄×D₃₇

C₂×D₄

yes

characteristic

#64438

—

C₈×C₃₇

C₂²×C₄

yes

yes

C₂²×C₃₇

—

C₂×D₃₇

—

C₂×D₃₇

—

C₂×D₃₇

—

C₂×D₃₇

C₂×D₈

yes

characteristic

#3866

—

#3866

—

C₄×C₃₇

C₄×D₄

yes

Dic₃₇

yes

—

C₂×C₃₇

—

C₂×C₃₇

C₄×D₈

yes

characteristic

D₃₇

—

D₃₇

#509

yes

—

C₂×D₁₆

—

C₄×D₈

—

#367

—

#357

—

#346

—

C₃₇

#2177

yes

characteristic, complemented, Sylow

C₂×D₈

—

C₄×D₄

—

D₁₆

—

D₁₆

—

D₁₆

#3866

yes

characteristic, complemented

C₂×C₁₆

—

C₂×C₁₆

—

#107

—

#102

—

C₄×C₈

—

C₂×D₄

—

C₂²×C₄

—

148

—

D₈

—

D₈

—

D₈

C₂×#3866

yes

complemented

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₂²⋊C₄

—

148

—

C₄²

—

C₁₆

—

C₁₆

—

C₁₆

C₂×#3866

yes

characteristic, complemented

C₂³

—

148

—

D₄

—

D₄

—

D₄

—

148

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

296

—

C₂×C₄

—

148

—

C₈

—

C₈

—

C₈

C₂²×#3866

yes

characteristic

C₂²

—

C₂²

—

C₂²

—

296

—

C₂²

—

296

—

C₂²

—

148

—

C₄

—

100

C₄

—

296

—

101

C₄

—

102

C₄

D₄×#3866

yes

characteristic

103

C₂

—

104

C₂

—

105

C₂

—

296

—

106

C₂

D₈×#3866

yes

characteristic

	subgroup	G/H
center	C₂	D₈×#3866
commutator	C₈×C₃₇	C₂²×C₄
Frattini	C₈	C₂²×#3866
Fitting	D₁₆×C₃₇	C₄
socle	C₂×C₃₇	C₄×D₈
prime radical	D₁₆×D₃₇	C₂
cosocle	C₈×D₃₇	C₂³
hypercenter	D₁₆	#3866
hypocenter	C₃₇	#2177
LCS	G ▷ C₈×C₃₇ ▷ C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ D₁₆

subgroup

G/H

center

C₂

D₈×#3866

commutator

C₈×C₃₇

C₂²×C₄

Frattini

C₈

C₂²×#3866

Fitting

D₁₆×C₃₇

C₄

socle

C₂×C₃₇

C₄×D₈

prime radical

D₁₆×D₃₇

C₂

cosocle

C₈×D₃₇

C₂³

D₁₆

C₃₇

G ▷ C₈×C₃₇ ▷ C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇

UCS

C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ D₁₆

N	H ≅ G/N	type
1. D₈×#3866	C₂	semidirect
2. #14558460	C₂	semidirect
3. #14558445	C₂	semidirect
4. #14558420	C₂	semidirect
12. D₅₉₂	C₄	semidirect
13. #11924477	C₄	semidirect
15. D₁₆×C₃₇	C₄	semidirect
22. D₈×C₃₇	C₂×C₄	semidirect
24. D₂₉₆	C₂×C₄	semidirect
27. C₁₆×C₃₇	C₂×C₄	semidirect
28. #10596925	C₂×C₄	semidirect
59. C₃₇	#2177	semidirect
64. D₁₆	#3866	semidirect
74. D₈	C₂×#3866	semidirect
82. C₁₆	C₂×#3866	semidirect

H ≅ G/N

type

1. D₈×#3866

C₂

semidirect

2. #14558460

C₂

semidirect

3. #14558445

C₂

semidirect

4. #14558420

C₂

semidirect

12. D₅₉₂

C₄

semidirect

13. #11924477

C₄

semidirect

15. D₁₆×C₃₇

C₄

semidirect

22. D₈×C₃₇

C₂×C₄

semidirect

24. D₂₉₆

C₂×C₄

semidirect

27. C₁₆×C₃₇

C₂×C₄

semidirect

28. #10596925

C₂×C₄

semidirect

59. C₃₇

#2177

semidirect

64. D₁₆

#3866

semidirect

74. D₈

C₂×#3866

semidirect

82. C₁₆

C₂×#3866

semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
D₈×#3866	2
#14558460	2
#14558445	2
#14558420	2
D₁₆×D₃₇	2
#2177	37

subgroup

index

D₁₆×D₃₇

Subgroups — #18010994

Conjugacy classes of subgroups

Complements

Maximal subgroups