Group #17995574 — subgroups

Subgroups — #17995574

Order 4736.

284 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 284 conjugacy classes of subgroups — too many to draw. Only its 105 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	#14557557	C₂	1	1	yes	maximal, characteristic
2	C₄²×Dic₃₇	C₂	1	1	yes	maximal, characteristic
3	#14557292	C₂	1	4	yes	maximal
4	C₃₇×#533	C₂	1	1	yes	maximal, characteristic
5	C₂×#10596934	C₂²	1	2	yes	—
6	C₂×C₄×Dic₃₇	—	2	2	no	—
7	C₂×C₄×Dic₃₇	C₂²	1	2	yes	—
8	C₄⋊#64436	—	2	2	no	—
9	C₄⋊Q₈×C₃₇	C₄	1	2	yes	complemented
10	#125×C₃₇	C₄	1	2	yes	complemented
11	C₂²⋊Q₈×C₃₇	—	2	2	no	—
12	C₄×Q₈×C₃₇	—	2	2	no	—
13	#117×C₃₇	C₂²	1	2	yes	—
14	C₂×C₄²×C₃₇	C₂²	1	1	yes	characteristic
15	C₂×Q₈×C₃₇	—	2	2	no	—
16	C₂²×C₄×C₃₇	D₄	1	2	yes	—
17	C₂²×C₄×C₃₇	C₂³	1	1	yes	characteristic
18	C₂²×Dic₃₇	—	2	2	no	—
19	C₄⋊C₄×C₃₇	—	2	4	no	—
20	C₄⋊C₄×C₃₇	—	2	2	no	—
21	C₄⋊C₄×C₃₇	C₂×C₄	1	4	yes	—
22	C₂²⋊C₄×C₃₇	—	2	2	no	—
23	C₄²×C₃₇	—	2	2	no	—
24	C₄²×C₃₇	C₂×C₄	1	1	yes	characteristic
25	C₄²×C₃₇	C₂×C₄	1	1	yes	characteristic
26	C₄²×C₃₇	D₄	1	2	yes	—
27	C₄×Dic₃₇	—	2	8	no	—
28	C₄×Dic₃₇	—	2	2	no	—
29	C₄×Dic₃₇	Q₈	1	4	yes	complemented
30	#10596934	—	2	4	no	—
31	C₂×#64436	—	2	2	no	—
32	C₂³×C₃₇	C₂×D₄	1	1	yes	characteristic
33	Q₈×C₃₇	—	4	4	no	—
34	C₂×C₄×C₃₇	—	2	4	no	—
35	C₂×C₄×C₃₇	—	2	4	no	—
36	C₂×C₄×C₃₇	C₂²⋊C₄	1	2	yes	—
37	C₂×C₄×C₃₇	C₂²⋊C₄	1	2	yes	—
38	C₂×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
39	C₂×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
40	C₂×C₄×C₃₇	C₂×Q₈	1	2	yes	—
41	C₂×C₄×C₃₇	#41	1	2	yes	—
42	C₂×Dic₃₇	—	2	8	no	—
43	C₂×Dic₃₇	—	2	4	no	—
44	#64436	—	4	2	no	—
45	C₂²×C₃₇	—	2	2	no	—
46	C₂²×C₃₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
47	C₂²×C₃₇	C₂²⋊Q₈	1	2	yes	—
48	C₄×C₃₇	—	2	2	no	—
49	C₄×C₃₇	—	4	4	no	—
50	C₄×C₃₇	#104	1	4	yes	—
51	C₄×C₃₇	C₄×Q₈	1	2	yes	—
52	C₄×C₃₇	#124	1	1	yes	characteristic
53	C₄×C₃₇	C₄⋊Q₈	1	1	yes	characteristic
54	Dic₃₇	—	2	8	no	—
55	#1994	—	37	1	no	maximal, Sylow
56	C₂×C₃₇	—	2	2	no	—
57	C₂×C₃₇	#391	1	1	yes	characteristic
58	C₂×C₃₇	C₂×#104	1	2	yes	—
59	#533	D₃₇	1	1	yes	characteristic
60	#423	—	37	1	no	—
61	C₄³	—	37	1	no	—
62	#339	—	37	4	no	—
63	C₃₇	#1994	1	1	yes	characteristic, complemented, Sylow
64	C₂×M₁₆	—	37	2	no	—
65	C₄⋊Q₈	Dic₃₇	1	2	yes	complemented
66	#125	Dic₃₇	1	2	yes	complemented
67	C₂²⋊Q₈	—	2	2	no	—
68	C₄×Q₈	—	2	2	no	—
69	#117	C₂×D₃₇	1	2	yes	—
70	C₂×C₄²	—	37	2	no	—
71	C₂×C₄²	—	74	2	no	—
72	C₂×C₄²	C₂×D₃₇	1	1	yes	characteristic
73	C₄⋊C₈	—	74	2	no	—
74	C₂×Q₈	—	2	2	no	—
75	C₂²×C₄	—	74	2	no	—
76	C₂²×C₄	C₂²⋊D₃₇	1	2	yes	—
77	C₂²×C₄	C₂²×D₃₇	1	1	yes	characteristic
78	M₁₆	—	74	4	no	—
79	C₂×C₈	—	74	2	no	—
80	C₄⋊C₄	—	2	4	no	—
81	C₄⋊C₄	—	2	2	no	—
82	C₄⋊C₄	C₂×Dic₃₇	1	4	yes	—
83	C₂²⋊C₄	—	2	2	no	—
84	C₄²	—	2	2	no	—
85	C₄²	—	37	4	no	—
86	C₄²	—	74	8	no	—
87	C₄²	—	74	2	no	—
88	C₄²	C₂×Dic₃₇	1	1	yes	characteristic
89	C₄²	C₂×Dic₃₇	1	1	yes	characteristic
90	C₄²	C₂²⋊D₃₇	1	2	yes	—
91	C₂³	C₂×C₂²⋊D₃₇	1	1	yes	characteristic
92	Q₈	—	4	4	no	—
93	C₂×C₄	—	2	4	no	—
94	C₂×C₄	—	2	4	no	—
95	C₂×C₄	—	74	8	no	—
96	C₂×C₄	—	74	4	no	—
97	C₂×C₄	C₂²⋊Dic₃₇	1	2	yes	—
98	C₂×C₄	C₂²⋊Dic₃₇	1	2	yes	—
99	C₂×C₄	Q₈×D₃₇	1	2	yes	—
100	C₂×C₄	#10596966	1	2	yes	—
101	C₂×C₄	C₂²×Dic₃₇	1	1	yes	characteristic
102	C₂×C₄	C₂×C₂²⋊D₃₇	1	1	yes	characteristic
103	C₈	—	148	2	no	—
104	C₂²	—	2	2	no	—
105	C₂²	C₂×C₂²⋊Dic₃₇	1	1	yes	characteristic
106	C₂²	#11924512	1	2	yes	—
107	C₄	—	2	2	no	—
108	C₄	—	4	4	no	—
109	C₄	—	74	8	no	—
110	C₄	#11924463	1	4	yes	—
111	C₄	Q₈×Dic₃₇	1	2	yes	—
112	C₄	#11924525	1	1	yes	characteristic
113	C₄	#11924534	1	1	yes	characteristic
114	C₂	—	2	2	no	—
115	C₂	#14557440	1	1	yes	characteristic
116	C₂	C₂×#11924463	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#14557557

C₂

yes

maximal, characteristic

C₄²×Dic₃₇

C₂

yes

maximal, characteristic

#14557292

C₂

yes

maximal

C₃₇×#533

C₂

yes

maximal, characteristic

C₂×#10596934

C₂²

yes

—

C₂×C₄×Dic₃₇

—

C₂×C₄×Dic₃₇

C₂²

yes

—

C₄⋊#64436

—

C₄⋊Q₈×C₃₇

C₄

yes

complemented

#125×C₃₇

C₄

yes

complemented

C₂²⋊Q₈×C₃₇

—

C₄×Q₈×C₃₇

—

#117×C₃₇

C₂²

yes

—

C₂×C₄²×C₃₇

C₂²

yes

characteristic

C₂×Q₈×C₃₇

—

C₂²×C₄×C₃₇

D₄

yes

—

C₂²×C₄×C₃₇

C₂³

yes

characteristic

C₂²×Dic₃₇

—

C₄⋊C₄×C₃₇

—

C₄⋊C₄×C₃₇

—

C₄⋊C₄×C₃₇

C₂×C₄

yes

—

C₂²⋊C₄×C₃₇

—

C₄²×C₃₇

—

C₄²×C₃₇

C₂×C₄

yes

characteristic

C₄²×C₃₇

C₂×C₄

yes

characteristic

C₄²×C₃₇

D₄

yes

—

C₄×Dic₃₇

—

C₄×Dic₃₇

—

C₄×Dic₃₇

Q₈

yes

complemented

#10596934

—

C₂×#64436

—

C₂³×C₃₇

C₂×D₄

yes

characteristic

Q₈×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

C₂²⋊C₄

yes

—

C₂×C₄×C₃₇

C₂²⋊C₄

yes

—

C₂×C₄×C₃₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₃₇

C₂×D₄

yes

characteristic

C₂×C₄×C₃₇

C₂×Q₈

yes

—

C₂×C₄×C₃₇

#41

yes

—

C₂×Dic₃₇

—

C₂×Dic₃₇

—

#64436

—

C₂²×C₃₇

—

C₂²×C₃₇

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₃₇

C₂²⋊Q₈

yes

—

C₄×C₃₇

—

C₄×C₃₇

—

C₄×C₃₇

#104

yes

—

C₄×C₃₇

C₄×Q₈

yes

—

C₄×C₃₇

#124

yes

characteristic

C₄×C₃₇

C₄⋊Q₈

yes

characteristic

Dic₃₇

—

#1994

—

maximal, Sylow

C₂×C₃₇

—

C₂×C₃₇

#391

yes

characteristic

C₂×C₃₇

C₂×#104

yes

—

#533

D₃₇

yes

characteristic

#423

—

C₄³

—

#339

—

C₃₇

#1994

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

C₄⋊Q₈

Dic₃₇

yes

complemented

#125

Dic₃₇

yes

complemented

C₂²⋊Q₈

—

C₄×Q₈

—

#117

C₂×D₃₇

yes

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

C₂×D₃₇

yes

characteristic

C₄⋊C₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

C₂²⋊D₃₇

yes

—

C₂²×C₄

C₂²×D₃₇

yes

characteristic

M₁₆

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂×Dic₃₇

yes

—

C₂²⋊C₄

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

C₄²

C₂×Dic₃₇

yes

characteristic

C₄²

C₂×Dic₃₇

yes

characteristic

C₄²

C₂²⋊D₃₇

yes

—

C₂³

C₂×C₂²⋊D₃₇

yes

characteristic

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂²⋊Dic₃₇

yes

—

C₂×C₄

C₂²⋊Dic₃₇

yes

—

C₂×C₄

Q₈×D₃₇

yes

—

100

C₂×C₄

#10596966

yes

—

101

C₂×C₄

C₂²×Dic₃₇

yes

characteristic

102

C₂×C₄

C₂×C₂²⋊D₃₇

yes

characteristic

103

C₈

—

148

—

104

C₂²

—

105

C₂²

C₂×C₂²⋊Dic₃₇

yes

characteristic

106

C₂²

#11924512

yes

—

107

C₄

—

108

C₄

—

109

C₄

—

110

C₄

#11924463

yes

—

111

C₄

Q₈×Dic₃₇

yes

—

112

C₄

#11924525

yes

characteristic

113

C₄

#11924534

yes

characteristic

114

C₂

—

115

C₂

#14557440

yes

characteristic

116

C₂

C₂×#11924463

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₃₇
commutator	C₂×C₄×C₃₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₃₇
Fitting	C₃₇×#533	C₂
socle	C₂²×C₃₇	C₂×C₂²⋊C₄
prime radical	C₂³×C₃₇	C₂×D₄
cosocle	C₂²×C₄×C₃₇	C₂³
hypercenter	#533	D₃₇
hypocenter	C₃₇	#1994
LCS	G ▷ C₂×C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #533

subgroup

G/H

center

C₂×C₄

C₂×C₂²⋊D₃₇

commutator

C₂×C₄×C₃₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₃₇

Fitting

C₃₇×#533

C₂

socle

C₂²×C₃₇

C₂×C₂²⋊C₄

prime radical

C₂³×C₃₇

C₂×D₄

cosocle

C₂²×C₄×C₃₇

C₂³

D₃₇

C₃₇

G ▷ C₂×C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #533

N	H ≅ G/N	type
9. C₄⋊Q₈×C₃₇	C₄	semidirect
10. #125×C₃₇	C₄	semidirect
29. C₄×Dic₃₇	Q₈	semidirect
63. C₃₇	#1994	semidirect
65. C₄⋊Q₈	Dic₃₇	semidirect
66. #125	Dic₃₇	semidirect

H ≅ G/N

type

9. C₄⋊Q₈×C₃₇

C₄

semidirect

10. #125×C₃₇

C₄

semidirect

29. C₄×Dic₃₇

Q₈

semidirect

63. C₃₇

#1994

semidirect

65. C₄⋊Q₈

Dic₃₇

semidirect

66. #125

Dic₃₇

semidirect

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

subgroup	index
#14557557	2
C₄²×Dic₃₇	2
#14557292	2
C₃₇×#533	2
#1994	37

subgroup

index

#14557557

C₄²×Dic₃₇

#14557292

C₃₇×#533

#1994

Subgroups — #17995574

Conjugacy classes of subgroups

Complements

Maximal subgroups