Group #17995759 — subgroups

Subgroups — #17995759

Order 4736.

466 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 466 conjugacy classes of subgroups — too many to draw. Only its 71 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	#14557388	C₂	1	3	yes	maximal
2	#14557300	C₂	1	3	yes	maximal, complemented
3	C₃₇×#543	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂⁴×D₃₇	C₂²	1	1	yes	characteristic
5	C₂×C₂²⋊Dic₃₇	C₂²	1	3	yes	—
6	C₂×#10596938	—	2	6	no	—
7	C₂²⋊Q₈×C₃₇	—	2	6	no	—
8	C₂×C₂²⋊C₄×C₃₇	C₂²	1	3	yes	—
9	C₂⁴×C₃₇	C₂³	1	1	yes	characteristic
10	C₂³×D₃₇	—	2	3	no	—
11	C₂³×D₃₇	—	2	3	no	—
12	C₂³×D₃₇	—	4	4	no	—
13	C₂³×D₃₇	Q₈	1	2	yes	complemented
14	C₂×Q₈×C₃₇	—	2	2	no	—
15	C₂²×C₄×C₃₇	D₄	1	6	yes	—
16	C₂²×Dic₃₇	—	2	3	no	—
17	C₄⋊C₄×C₃₇	—	2	6	no	—
18	C₂²⋊C₄×C₃₇	—	2	6	no	—
19	C₂²⋊Dic₃₇	—	2	6	no	—
20	#10596938	—	4	12	no	—
21	C₂³×C₃₇	—	4	2	no	—
22	C₂³×C₃₇	C₂×D₄	1	3	yes	—
23	C₂³×C₃₇	C₂×Q₈	1	1	yes	characteristic
24	C₂³×C₃₇	#41	1	3	yes	—
25	C₂²×D₃₇	—	2	12	no	—
26	C₂²×D₃₇	—	2	6	no	—
27	C₂²×D₃₇	—	4	4	no	—
28	C₂²×D₃₇	—	4	12	no	—
29	C₂²×D₃₇	—	4	6	no	—
30	C₂²×D₃₇	—	4	1	no	—
31	C₂²×D₃₇	—	4	3	no	—
32	Q₈×C₃₇	—	8	2	no	—
33	C₂×C₄×C₃₇	—	2	6	no	—
34	C₂×C₄×C₃₇	—	4	6	no	—
35	C₂×Dic₃₇	—	4	6	no	—
36	C₂×Dic₃₇	—	4	3	no	—
37	C₂²×C₃₇	—	2	6	no	—
38	C₂²×C₃₇	—	4	1	no	—
39	C₂²×C₃₇	—	4	3	no	—
40	C₂²×C₃₇	#120	1	1	yes	characteristic
41	C₂²×C₃₇	C₂²⋊Q₈	1	3	yes	—
42	C₂²×C₃₇	#124	1	3	yes	—
43	C₂×D₃₇	—	2	12	no	—
44	C₂×D₃₇	—	4	12	no	—
45	C₂×D₃₇	—	4	6	no	—
46	C₂×D₃₇	—	4	2	no	—
47	C₂×D₃₇	—	4	4	no	—
48	C₄×C₃₇	—	4	6	no	—
49	Dic₃₇	—	8	3	no	—
50	#2029	—	37	1	no	maximal, Sylow
51	C₂×C₃₇	—	2	3	no	—
52	C₂×C₃₇	—	2	3	no	—
53	C₂×C₃₇	#393	1	1	yes	characteristic
54	C₂×C₃₇	B(4,𝔽₂)	1	2	yes	—
55	D₃₇	—	4	4	no	—
56	#543	D₃₇	1	1	yes	characteristic, complemented
57	#379	—	37	3	no	—
58	#342	—	37	3	no	—
59	C₃₇	#2029	1	1	yes	characteristic, complemented, Sylow
60	C₂⁵	—	37	1	no	—
61	C₂²⋊Q₈	—	2	6	no	—
62	C₂×C₂²⋊C₄	—	37	3	no	—
63	C₂×C₂²⋊C₄	—	74	6	no	—
64	C₂×C₂²⋊C₄	C₂×D₃₇	1	3	yes	—
65	C₂⁴	—	37	2	no	—
66	C₂⁴	—	74	3	no	—
67	C₂⁴	—	74	3	no	—
68	C₂⁴	—	148	4	no	—
69	C₂⁴	C₂²×D₃₇	1	1	yes	characteristic
70	C₂×Q₈	—	2	2	no	—
71	C₂²×C₄	—	74	3	no	—
72	C₂²×C₄	C₂²⋊D₃₇	1	6	yes	—
73	C₄⋊C₄	—	2	6	no	—
74	C₂²⋊C₄	—	2	6	no	—
75	C₂²⋊C₄	—	74	6	no	—
76	C₂²⋊C₄	—	148	12	no	—
77	C₂³	—	4	2	no	—
78	C₂³	—	74	12	no	—
79	C₂³	—	74	6	no	—
80	C₂³	—	148	6	no	—
81	C₂³	—	148	4	no	—
82	C₂³	—	148	12	no	—
83	C₂³	—	148	1	no	—
84	C₂³	—	148	3	no	—
85	C₂³	Q₈×D₃₇	1	1	yes	characteristic
86	C₂³	#10596966	1	3	yes	—
87	C₂³	C₂×C₂²⋊D₃₇	1	3	yes	—
88	Q₈	—	8	2	no	—
89	C₂×C₄	—	2	6	no	—
90	C₂×C₄	—	4	6	no	—
91	C₂×C₄	—	148	6	no	—
92	C₂×C₄	—	148	3	no	—
93	C₂²	—	2	6	no	—
94	C₂²	—	4	1	no	—
95	C₂²	—	4	3	no	—
96	C₂²	—	74	12	no	—
97	C₂²	—	148	12	no	—
98	C₂²	—	148	6	no	—
99	C₂²	—	148	2	no	—
100	C₂²	—	148	4	no	—
101	C₂²	#11924505	1	1	yes	characteristic
102	C₂²	#11924512	1	3	yes	—
103	C₂²	#11924525	1	3	yes	—
104	C₄	—	4	6	no	—
105	C₄	—	296	3	no	—
106	C₂	—	2	3	no	—
107	C₂	—	2	3	no	—
108	C₂	—	148	4	no	—
109	C₂	#14557444	1	1	yes	characteristic
110	C₂	#14557720	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#14557388

C₂

yes

maximal

#14557300

C₂

yes

maximal, complemented

C₃₇×#543

C₂

yes

maximal, characteristic, complemented

C₂⁴×D₃₇

C₂²

yes

characteristic

C₂×C₂²⋊Dic₃₇

C₂²

yes

—

C₂×#10596938

—

C₂²⋊Q₈×C₃₇

—

C₂×C₂²⋊C₄×C₃₇

C₂²

yes

—

C₂⁴×C₃₇

C₂³

yes

characteristic

C₂³×D₃₇

—

C₂³×D₃₇

—

C₂³×D₃₇

—

C₂³×D₃₇

Q₈

yes

complemented

C₂×Q₈×C₃₇

—

C₂²×C₄×C₃₇

D₄

yes

—

C₂²×Dic₃₇

—

C₄⋊C₄×C₃₇

—

C₂²⋊C₄×C₃₇

—

C₂²⋊Dic₃₇

—

#10596938

—

C₂³×C₃₇

—

C₂³×C₃₇

C₂×D₄

yes

—

C₂³×C₃₇

C₂×Q₈

yes

characteristic

C₂³×C₃₇

#41

yes

—

C₂²×D₃₇

—

C₂²×D₃₇

—

C₂²×D₃₇

—

C₂²×D₃₇

—

C₂²×D₃₇

—

C₂²×D₃₇

—

C₂²×D₃₇

—

Q₈×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

—

C₂×Dic₃₇

—

C₂×Dic₃₇

—

C₂²×C₃₇

—

C₂²×C₃₇

—

C₂²×C₃₇

—

C₂²×C₃₇

#120

yes

characteristic

C₂²×C₃₇

C₂²⋊Q₈

yes

—

C₂²×C₃₇

#124

yes

—

C₂×D₃₇

—

C₂×D₃₇

—

C₂×D₃₇

—

C₂×D₃₇

—

C₂×D₃₇

—

C₄×C₃₇

—

Dic₃₇

—

#2029

—

maximal, Sylow

C₂×C₃₇

—

C₂×C₃₇

—

C₂×C₃₇

#393

yes

characteristic

C₂×C₃₇

B(4,𝔽₂)

yes

—

D₃₇

—

#543

D₃₇

yes

characteristic, complemented

#379

—

#342

—

C₃₇

#2029

yes

characteristic, complemented, Sylow

C₂⁵

—

C₂²⋊Q₈

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

C₂×D₃₇

yes

—

C₂⁴

—

C₂⁴

—

C₂⁴

—

C₂⁴

—

148

—

C₂⁴

C₂²×D₃₇

yes

characteristic

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

C₂²⋊D₃₇

yes

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

148

—

C₂³

—

C₂³

—

C₂³

—

C₂³

—

148

—

C₂³

—

148

—

C₂³

—

148

—

C₂³

—

148

—

C₂³

—

148

—

C₂³

Q₈×D₃₇

yes

characteristic

C₂³

#10596966

yes

—

C₂³

C₂×C₂²⋊D₃₇

yes

—

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

148

—

C₂×C₄

—

148

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

148

—

C₂²

—

148

—

C₂²

—

148

—

100

C₂²

—

148

—

101

C₂²

#11924505

yes

characteristic

102

C₂²

#11924512

yes

—

103

C₂²

#11924525

yes

—

104

C₄

—

105

C₄

—

296

—

106

C₂

—

107

C₂

—

108

C₂

—

148

—

109

C₂

#14557444

yes

characteristic

110

C₂

#14557720

yes

—

	subgroup	G/H
center	C₂²	#11924505
commutator, cosocle	C₂⁴×C₃₇	C₂³
Frattini	C₂⁴	C₂²×D₃₇
Fitting	C₃₇×#543	C₂
socle	C₂²×C₃₇	#120
prime radical	C₂⁴×D₃₇	C₂²
hypercenter	#543	D₃₇
hypocenter	C₃₇	#2029
LCS	G ▷ C₂⁴×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ #543

subgroup

G/H

center

C₂²

#11924505

commutator, cosocle

C₂⁴×C₃₇

C₂³

Frattini

C₂⁴

C₂²×D₃₇

Fitting

C₃₇×#543

C₂

socle

C₂²×C₃₇

#120

prime radical

C₂⁴×D₃₇

C₂²

D₃₇

C₃₇

G ▷ C₂⁴×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇

UCS

C₁ ◁ C₂² ◁ C₂⁴ ◁ #543

N	H ≅ G/N	type
2. #14557300	C₂	semidirect
3. C₃₇×#543	C₂	semidirect
13. C₂³×D₃₇	Q₈	semidirect
56. #543	D₃₇	semidirect
59. C₃₇	#2029	semidirect

H ≅ G/N

type

2. #14557300

C₂

semidirect

3. C₃₇×#543

C₂

semidirect

13. C₂³×D₃₇

Q₈

semidirect

56. #543

D₃₇

semidirect

59. C₃₇

#2029

semidirect

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

subgroup	index
#14557388	2
#14557300	2
C₃₇×#543	2
#2029	37

subgroup

index

#14557388

#14557300

C₃₇×#543

#2029

Subgroups — #17995759

Conjugacy classes of subgroups

Complements

Maximal subgroups