Group #18011592 — subgroups

Subgroups — #18011592

Order 4736.

1424 conjugacy classes of subgroups; only the 750 normal subgroups are shown.

normal subgroups only

This group has 1424 conjugacy classes of subgroups — too many to draw. Only its 750 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	#14558613	C₂	1	5	yes	maximal, complemented
2	#14558612	C₂	1	10	yes	maximal, complemented
3	C₂×#11924624	C₂	1	5	yes	maximal, complemented
4	D₃₇⋊#131	C₂	1	10	yes	maximal, complemented
5	#14558370	C₂	1	1	yes	maximal, characteristic
6	#11924624	C₂²	1	10	yes	complemented
7	#11924624	C₂²	1	10	yes	complemented
8	#11924624	C₂²	1	20	yes	complemented
9	#11924623	C₂²	1	10	yes	complemented
10	#11924623	C₂²	1	30	yes	complemented
11	C₂×#10596953	C₂²	1	15	yes	complemented
12	D₃₇⋊M₁₆	C₂²	1	30	yes	complemented
13	C₂×D₃₇⋊C₈	C₂²	1	15	yes	complemented
14	#11924584	C₄	1	5	yes	—
15	#11924582	C₄	1	10	yes	—
16	C₂×#10596966	C₂²	1	5	yes	—
17	#41×D₃₇	C₂²	1	10	yes	—
18	2₋¹⁺⁴×C₃₇	C₄	1	1	yes	characteristic
19	#41×C₃₇	C₂×C₄	1	10	yes	—
20	C₂×Q₈×C₃₇	C₂×C₄	1	5	yes	—
21	#10596966	C₂×C₄	1	20	yes	—
22	#10596966	C₂³	1	10	yes	—
23	Q₈×D₃₇	C₂×C₄	1	10	yes	—
24	#10596964	C₂³	1	10	yes	—
25	D₄×D₃₇	C₂×C₄	1	30	yes	—
26	#10596962	C₂×C₄	1	30	yes	—
27	C₂×D₁₄₈	C₂×C₄	1	15	yes	—
28	C₂×C₄×D₃₇	C₂³	1	15	yes	—
29	#10596957	C₂³	1	20	yes	—
30	C₂×#64438	C₂³	1	5	yes	—
31	C₂×#64438	C₂³	1	15	yes	—
32	#10596953	C₂³	1	10	yes	—
33	#10596953	C₂³	1	30	yes	—
34	D₃₇⋊C₈	C₂³	1	10	yes	—
35	D₃₇⋊C₈	C₂³	1	30	yes	—
36	C₂²×D₃₇	C₂²×C₄	1	15	yes	—
37	Q₈×C₃₇	C₂²×C₄	1	10	yes	—
38	D₄×C₃₇	C₂²×C₄	1	10	yes	—
39	C₂×C₄×C₃₇	C₂²×C₄	1	15	yes	—
40	C₂²⋊D₃₇	C₂²×C₄	1	20	yes	—
41	C₂×Dic₃₇	C₂⁴	1	5	yes	—
42	D₁₄₈	C₂²×C₄	1	30	yes	—
43	C₄×D₃₇	C₂²×C₄	1	30	yes	—
44	C₄×D₃₇	C₂⁴	1	10	yes	—
45	Dic₇₄	C₂²×C₄	1	10	yes	—
46	#64438	C₂⁴	1	1	yes	characteristic
47	#64438	C₂⁴	1	5	yes	—
48	#64438	C₂⁴	1	10	yes	—
49	C₂²×C₃₇	C₂³×C₄	1	5	yes	—
50	C₂×D₃₇	—	4	15	no	—
51	C₂×D₃₇	C₂³×C₄	1	10	yes	—
52	C₄×C₃₇	C₂³×C₄	1	10	yes	—
53	Dic₃₇	C₂³×C₄	1	5	yes	—
54	Dic₃₇	C₂⁵	1	1	yes	characteristic
55	#3573	—	37	1	no	maximal, Sylow
56	C₂×C₃₇	—	2	5	no	—
57	C₂×C₃₇	C₂⁴×C₄	1	1	yes	characteristic
58	D₃₇	—	2	10	no	—
59	#585	—	37	1	no	—
60	#568	—	37	10	no	—
61	#568	—	37	5	no	—
62	C₂×#131	—	37	10	no	—
63	C₂×#131	—	37	5	no	—
64	C₃₇	#3573	1	1	yes	characteristic, complemented, Sylow
65	2₋¹⁺⁴	—	37	5	no	—
66	2₋¹⁺⁴	#3866	1	1	yes	characteristic
67	H(5,𝔽₂)	—	37	10	no	—
68	C₂×#41	—	37	10	no	—
69	C₂×#41	—	37	5	no	—
70	#131	—	37	10	no	—
71	#131	—	37	10	no	—
72	#131	—	37	30	no	—
73	#131	—	37	10	no	—
74	#131	—	37	20	no	—
75	C₂×M₁₆	—	37	30	no	—
76	C₂×M₁₆	—	37	15	no	—
77	C₂²×C₈	—	37	15	no	—
78	#41	—	37	20	no	—
79	#41	—	37	10	no	—
80	#41	—	37	10	no	—
81	#41	—	37	30	no	—
82	#41	C₂×#3866	1	10	yes	—
83	C₂×Q₈	—	37	10	no	—
84	C₂×Q₈	C₂×#3866	1	5	yes	—
85	C₂×D₄	—	37	30	no	—
86	C₂×D₄	—	37	15	no	—
87	C₂²×C₄	—	37	15	no	—
88	M₁₆	—	37	10	no	—
89	M₁₆	—	37	30	no	—
90	M₁₆	—	37	20	no	—
91	C₂×C₈	—	37	10	no	—
92	C₂×C₈	—	37	30	no	—
93	C₂×C₈	—	37	5	no	—
94	C₂×C₈	—	37	15	no	—
95	C₂³	—	37	15	no	—
96	Q₈	—	37	10	no	—
97	Q₈	C₂²×#3866	1	10	yes	—
98	D₄	—	37	20	no	—
99	D₄	—	37	30	no	—
100	D₄	C₂²×#3866	1	10	yes	—
101	C₂×C₄	—	37	5	no	—
102	C₂×C₄	—	37	10	no	—
103	C₂×C₄	—	37	30	no	—
104	C₂×C₄	C₂²×#3866	1	15	yes	—
105	C₈	—	37	1	no	—
106	C₈	—	37	10	no	—
107	C₈	—	37	5	no	—
108	C₂²	—	37	10	no	—
109	C₂²	—	148	15	no	—
110	C₂²	C₂³×#3866	1	5	yes	—
111	C₄	—	37	1	no	—
112	C₄	—	37	5	no	—
113	C₄	C₂³×#3866	1	10	yes	—
114	C₂	—	2	5	no	—
115	C₂	—	74	10	no	—
116	C₂	C₂⁴×#3866	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14558613

C₂

yes

maximal, complemented

#14558612

C₂

yes

maximal, complemented

C₂×#11924624

C₂

yes

maximal, complemented

D₃₇⋊#131

C₂

yes

maximal, complemented

#14558370

C₂

yes

maximal, characteristic

#11924624

C₂²

yes

complemented

#11924624

C₂²

yes

complemented

#11924624

C₂²

yes

complemented

#11924623

C₂²

yes

complemented

#11924623

C₂²

yes

complemented

C₂×#10596953

C₂²

yes

complemented

D₃₇⋊M₁₆

C₂²

yes

complemented

C₂×D₃₇⋊C₈

C₂²

yes

complemented

#11924584

C₄

yes

—

#11924582

C₄

yes

—

C₂×#10596966

C₂²

yes

—

#41×D₃₇

C₂²

yes

—

2₋¹⁺⁴×C₃₇

C₄

yes

characteristic

#41×C₃₇

C₂×C₄

yes

—

C₂×Q₈×C₃₇

C₂×C₄

yes

—

#10596966

C₂×C₄

yes

—

#10596966

C₂³

yes

—

Q₈×D₃₇

C₂×C₄

yes

—

#10596964

C₂³

yes

—

D₄×D₃₇

C₂×C₄

yes

—

#10596962

C₂×C₄

yes

—

C₂×D₁₄₈

C₂×C₄

yes

—

C₂×C₄×D₃₇

C₂³

yes

—

#10596957

C₂³

yes

—

C₂×#64438

C₂³

yes

—

C₂×#64438

C₂³

yes

—

#10596953

C₂³

yes

—

#10596953

C₂³

yes

—

D₃₇⋊C₈

C₂³

yes

—

D₃₇⋊C₈

C₂³

yes

—

C₂²×D₃₇

C₂²×C₄

yes

—

Q₈×C₃₇

C₂²×C₄

yes

—

D₄×C₃₇

C₂²×C₄

yes

—

C₂×C₄×C₃₇

C₂²×C₄

yes

—

C₂²⋊D₃₇

C₂²×C₄

yes

—

C₂×Dic₃₇

C₂⁴

yes

—

D₁₄₈

C₂²×C₄

yes

—

C₄×D₃₇

C₂²×C₄

yes

—

C₄×D₃₇

C₂⁴

yes

—

Dic₇₄

C₂²×C₄

yes

—

#64438

C₂⁴

yes

characteristic

#64438

C₂⁴

yes

—

#64438

C₂⁴

yes

—

C₂²×C₃₇

C₂³×C₄

yes

—

C₂×D₃₇

—

C₂×D₃₇

C₂³×C₄

yes

—

C₄×C₃₇

C₂³×C₄

yes

—

Dic₃₇

C₂³×C₄

yes

—

Dic₃₇

C₂⁵

yes

characteristic

#3573

—

maximal, Sylow

C₂×C₃₇

—

C₂×C₃₇

C₂⁴×C₄

yes

characteristic

D₃₇

—

—

—

—

—

—

C₃₇

#3573

yes

characteristic, complemented, Sylow

2₋¹⁺⁴

—

2₋¹⁺⁴

#3866

yes

characteristic

H(5,𝔽₂)

—

C₂×#41

—

C₂×#41

—

—

—

—

—

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

—

—

—

—

—

yes

—

C₂×Q₈

—

C₂×Q₈

C₂×#3866

yes

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

—

M₁₆

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂³

—

Q₈

—

Q₈

C₂²×#3866

yes

—

D₄

—

D₄

—

100

D₄

C₂²×#3866

yes

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

C₂²×#3866

yes

—

105

C₈

—

106

C₈

—

107

C₈

—

108

C₂²

—

109

C₂²

—

148

—

110

C₂²

C₂³×#3866

yes

—

111

C₄

—

112

C₄

—

113

C₄

C₂³×#3866

yes

—

114

C₂

—

115

C₂

—

116

C₂

C₂⁴×#3866

yes

characteristic

	subgroup	G/H
center, Frattini	C₂	C₂⁴×#3866
commutator, socle	C₂×C₃₇	C₂⁴×C₄
Fitting	2₋¹⁺⁴×C₃₇	C₄
prime radical	#14558370	C₂
cosocle	Dic₃₇	C₂⁵
hypercenter	2₋¹⁺⁴	#3866
hypocenter	C₃₇	#3573
LCS	G ▷ C₂×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂ ◁ 2₋¹⁺⁴

subgroup

G/H

center, Frattini

C₂

C₂⁴×#3866

commutator, socle

C₂×C₃₇

C₂⁴×C₄

Fitting

2₋¹⁺⁴×C₃₇

C₄

prime radical

#14558370

C₂

cosocle

Dic₃₇

C₂⁵

2₋¹⁺⁴

C₃₇

G ▷ C₂×C₃₇ ▷ C₃₇

C₁ ◁ C₂ ◁ 2₋¹⁺⁴

N	H ≅ G/N	type
1. #14558613	C₂	semidirect
2. #14558612	C₂	semidirect
3. C₂×#11924624	C₂	semidirect
4. D₃₇⋊#131	C₂	semidirect
6. #11924624	C₂²	semidirect
7. #11924624	C₂²	semidirect
8. #11924624	C₂²	semidirect
9. #11924623	C₂²	semidirect
10. #11924623	C₂²	semidirect
11. C₂×#10596953	C₂²	semidirect
12. D₃₇⋊M₁₆	C₂²	semidirect
13. C₂×D₃₇⋊C₈	C₂²	semidirect
64. C₃₇	#3573	semidirect

H ≅ G/N

type

1. #14558613

C₂

semidirect

2. #14558612

C₂

semidirect

3. C₂×#11924624

C₂

semidirect

4. D₃₇⋊#131

C₂

semidirect

6. #11924624

C₂²

semidirect

7. #11924624

C₂²

semidirect

8. #11924624

C₂²

semidirect

9. #11924623

C₂²

semidirect

10. #11924623

C₂²

semidirect

11. C₂×#10596953

C₂²

semidirect

12. D₃₇⋊M₁₆

C₂²

semidirect

13. C₂×D₃₇⋊C₈

C₂²

semidirect

64. C₃₇

#3573

semidirect

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

subgroup	index
#14558613	2
#14558612	2
C₂×#11924624	2
D₃₇⋊#131	2
#14558370	2
#3573	37

subgroup

index

D₃₇⋊#131

Subgroups — #18011592

Conjugacy classes of subgroups

Complements

Maximal subgroups