Group #17994313 — subgroups

Subgroups — #17994313

Order 4736.

424 conjugacy classes of subgroups; only the 205 normal subgroups are shown.

normal subgroups only

This group has 424 conjugacy classes of subgroups — too many to draw. Only its 205 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	C₂×#11924495	C₂	1	2	yes	maximal
2	#14557296	C₂	1	4	yes	maximal, complemented
3	C₂³×C₈×C₃₇	C₂	1	1	yes	maximal, characteristic
4	#11924495	—	2	4	no	—
5	#11924495	C₄	1	8	yes	complemented
6	C₂×C₄⋊Dic₃₇	C₂²	1	4	yes	—
7	C₂×C₂²⋊Dic₃₇	—	2	2	no	—
8	C₂×C₄×Dic₃₇	—	2	2	no	—
9	C₂³×C₄×C₃₇	C₂²	1	1	yes	characteristic
10	C₂²×C₈×C₃₇	—	2	4	no	—
11	C₂²×C₈×C₃₇	C₄	1	4	yes	complemented
12	C₂²×C₈×C₃₇	C₂²	1	2	yes	—
13	C₂⁴×C₃₇	D₄	1	1	yes	characteristic
14	C₂²×C₄×C₃₇	C₂×C₄	1	4	yes	—
15	C₂²×C₄×C₃₇	C₂×C₄	1	2	yes	—
16	C₂²×C₄×C₃₇	D₄	1	1	yes	characteristic
17	C₂²×C₄×C₃₇	D₄	1	4	yes	—
18	C₂²×C₄×C₃₇	Q₈	1	2	yes	—
19	C₂²×C₄×C₃₇	C₂³	1	1	yes	characteristic
20	C₂²×Dic₃₇	—	2	4	no	—
21	C₂×C₈×C₃₇	—	2	8	no	—
22	C₂×C₈×C₃₇	—	2	4	no	—
23	C₂×C₈×C₃₇	C₂×C₄	1	4	yes	—
24	C₂²⋊Dic₃₇	—	2	8	no	—
25	C₄⋊Dic₃₇	—	2	4	no	—
26	C₄⋊Dic₃₇	C₂×C₄	1	8	yes	—
27	C₄×Dic₃₇	—	2	8	no	—
28	C₂³×C₃₇	—	2	4	no	—
29	C₂³×C₃₇	C₂²⋊C₄	1	4	yes	—
30	C₂³×C₃₇	C₄⋊C₄	1	2	yes	—
31	C₂³×C₃₇	C₂×D₄	1	1	yes	characteristic
32	C₂×C₄×C₃₇	C₄²	1	4	yes	—
33	C₂×C₄×C₃₇	C₂²⋊C₄	1	4	yes	—
34	C₂×C₄×C₃₇	C₂²⋊C₄	1	4	yes	—
35	C₂×C₄×C₃₇	C₄⋊C₄	1	8	yes	—
36	C₂×C₄×C₃₇	C₄⋊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	2	yes	—
40	C₂×C₄×C₃₇	C₂×Q₈	1	1	yes	characteristic
41	C₂×Dic₃₇	—	2	8	no	—
42	C₂×Dic₃₇	—	4	8	no	—
43	C₈×C₃₇	—	2	4	no	—
44	C₂²×C₃₇	—	2	8	no	—
45	C₂²×C₃₇	—	2	4	no	—
46	C₂²×C₃₇	#95	1	4	yes	—
47	C₂²×C₃₇	C₂×C₂²⋊C₄	1	2	yes	—
48	C₂²×C₃₇	C₂×C₄⋊C₄	1	1	yes	characteristic
49	C₂²×C₃₇	#135	1	4	yes	—
50	C₄×C₃₇	#95	1	4	yes	—
51	C₄×C₃₇	C₂×C₄²	1	1	yes	characteristic
52	C₄×C₃₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
53	C₄×C₃₇	C₂×C₄⋊C₄	1	2	yes	—
54	Dic₃₇	—	4	8	no	—
55	#1735	—	37	1	no	maximal, Sylow
56	C₂×C₃₇	—	2	4	no	—
57	C₂×C₃₇	C₂×#95	1	1	yes	characteristic
58	C₂×C₃₇	#416	1	4	yes	—
59	C₂×C₃₇	#427	1	2	yes	—
60	C₂³×C₈	D₃₇	1	1	yes	characteristic
61	C₂×#117	—	37	2	no	—
62	#340	—	37	4	no	—
63	C₃₇	#1735	1	1	yes	characteristic, complemented, Sylow
64	C₂³×C₄	C₂×D₃₇	1	1	yes	characteristic
65	C₂²×C₈	—	2	4	no	—
66	C₂²×C₈	Dic₃₇	1	4	yes	complemented
67	C₂²×C₈	C₂×D₃₇	1	2	yes	—
68	#117	—	37	8	no	—
69	#117	—	74	4	no	—
70	C₂×C₄⋊C₄	—	37	4	no	—
71	C₂×C₂²⋊C₄	—	74	2	no	—
72	C₂×C₄²	—	74	2	no	—
73	C₂⁴	D₁₄₈	1	1	yes	characteristic
74	C₂²×C₄	—	74	4	no	—
75	C₂²×C₄	Dic₇₄	1	2	yes	—
76	C₂²×C₄	C₄×D₃₇	1	4	yes	—
77	C₂²×C₄	D₁₄₈	1	1	yes	characteristic
78	C₂²×C₄	C₂×Dic₃₇	1	2	yes	—
79	C₂²×C₄	C₂²⋊D₃₇	1	4	yes	—
80	C₂²×C₄	C₂²×D₃₇	1	1	yes	characteristic
81	C₂×C₈	—	2	8	no	—
82	C₂×C₈	—	2	4	no	—
83	C₂×C₈	C₂×Dic₃₇	1	4	yes	—
84	C₄⋊C₄	—	37	8	no	—
85	C₄⋊C₄	—	74	4	no	—
86	C₂²⋊C₄	—	74	8	no	—
87	C₄²	—	74	8	no	—
88	C₂³	—	2	4	no	—
89	C₂³	C₄⋊Dic₃₇	1	2	yes	—
90	C₂³	#10596938	1	4	yes	—
91	C₂³	C₂×D₁₄₈	1	1	yes	characteristic
92	C₂×C₄	—	74	8	no	—
93	C₂×C₄	—	148	8	no	—
94	C₂×C₄	C₄×Dic₃₇	1	4	yes	—
95	C₂×C₄	#10596936	1	8	yes	—
96	C₂×C₄	C₄⋊Dic₃₇	1	2	yes	—
97	C₂×C₄	#10596938	1	4	yes	—
98	C₂×C₄	C₂²⋊Dic₃₇	1	4	yes	—
99	C₂×C₄	C₂×Dic₇₄	1	1	yes	characteristic
100	C₂×C₄	C₂×C₄×D₃₇	1	2	yes	—
101	C₂×C₄	C₂²×Dic₃₇	1	1	yes	characteristic
102	C₂×C₄	C₂×C₂²⋊D₃₇	1	2	yes	—
103	C₈	—	2	4	no	—
104	C₂²	—	2	8	no	—
105	C₂²	—	2	4	no	—
106	C₂²	#11924443	1	4	yes	—
107	C₂²	C₂×#10596938	1	2	yes	—
108	C₂²	C₂×C₄⋊Dic₃₇	1	1	yes	characteristic
109	C₂²	#11924558	1	4	yes	—
110	C₄	—	148	8	no	—
111	C₄	#11924443	1	4	yes	—
112	C₄	C₂×C₄×Dic₃₇	1	1	yes	characteristic
113	C₄	C₂×C₂²⋊Dic₃₇	1	1	yes	characteristic
114	C₄	C₂×#10596936	1	2	yes	—
115	C₂	—	2	4	no	—
116	C₂	C₂×#11924443	1	1	yes	characteristic
117	C₂	#14557530	1	4	yes	—
118	C₂	#14557571	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#11924495

C₂

yes

maximal

#14557296

C₂

yes

maximal, complemented

C₂³×C₈×C₃₇

C₂

yes

maximal, characteristic

#11924495

—

#11924495

C₄

yes

complemented

C₂×C₄⋊Dic₃₇

C₂²

yes

—

C₂×C₂²⋊Dic₃₇

—

C₂×C₄×Dic₃₇

—

C₂³×C₄×C₃₇

C₂²

yes

characteristic

C₂²×C₈×C₃₇

—

C₂²×C₈×C₃₇

C₄

yes

complemented

C₂²×C₈×C₃₇

C₂²

yes

—

C₂⁴×C₃₇

D₄

yes

characteristic

C₂²×C₄×C₃₇

C₂×C₄

yes

—

C₂²×C₄×C₃₇

C₂×C₄

yes

—

C₂²×C₄×C₃₇

D₄

yes

characteristic

C₂²×C₄×C₃₇

D₄

yes

—

C₂²×C₄×C₃₇

Q₈

yes

—

C₂²×C₄×C₃₇

C₂³

yes

characteristic

C₂²×Dic₃₇

—

C₂×C₈×C₃₇

—

C₂×C₈×C₃₇

—

C₂×C₈×C₃₇

C₂×C₄

yes

—

C₂²⋊Dic₃₇

—

C₄⋊Dic₃₇

—

C₄⋊Dic₃₇

C₂×C₄

yes

—

C₄×Dic₃₇

—

C₂³×C₃₇

—

C₂³×C₃₇

C₂²⋊C₄

yes

—

C₂³×C₃₇

C₄⋊C₄

yes

—

C₂³×C₃₇

C₂×D₄

yes

characteristic

C₂×C₄×C₃₇

C₄²

yes

—

C₂×C₄×C₃₇

C₂²⋊C₄

yes

—

C₂×C₄×C₃₇

C₂²⋊C₄

yes

—

C₂×C₄×C₃₇

C₄⋊C₄

yes

—

C₂×C₄×C₃₇

C₄⋊C₄

yes

—

C₂×C₄×C₃₇

C₂²×C₄

yes

—

C₂×C₄×C₃₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₃₇

C₂×D₄

yes

—

C₂×C₄×C₃₇

C₂×Q₈

yes

characteristic

C₂×Dic₃₇

—

C₂×Dic₃₇

—

C₈×C₃₇

—

C₂²×C₃₇

—

C₂²×C₃₇

—

C₂²×C₃₇

#95

yes

—

C₂²×C₃₇

C₂×C₂²⋊C₄

yes

—

C₂²×C₃₇

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₃₇

#135

yes

—

C₄×C₃₇

#95

yes

—

C₄×C₃₇

C₂×C₄²

yes

characteristic

C₄×C₃₇

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₃₇

C₂×C₄⋊C₄

yes

—

Dic₃₇

—

#1735

—

maximal, Sylow

C₂×C₃₇

—

C₂×C₃₇

C₂×#95

yes

characteristic

C₂×C₃₇

#416

yes

—

C₂×C₃₇

#427

yes

—

C₂³×C₈

D₃₇

yes

characteristic

C₂×#117

—

#340

—

C₃₇

#1735

yes

characteristic, complemented, Sylow

C₂³×C₄

C₂×D₃₇

yes

characteristic

C₂²×C₈

—

C₂²×C₈

Dic₃₇

yes

complemented

C₂²×C₈

C₂×D₃₇

yes

—

#117

—

#117

—

C₂×C₄⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₄²

—

C₂⁴

D₁₄₈

yes

characteristic

C₂²×C₄

—

C₂²×C₄

Dic₇₄

yes

—

C₂²×C₄

C₄×D₃₇

yes

—

C₂²×C₄

D₁₄₈

yes

characteristic

C₂²×C₄

C₂×Dic₃₇

yes

—

C₂²×C₄

C₂²⋊D₃₇

yes

—

C₂²×C₄

C₂²×D₃₇

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂×Dic₃₇

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₂³

—

C₂³

C₄⋊Dic₃₇

yes

—

C₂³

#10596938

yes

—

C₂³

C₂×D₁₄₈

yes

characteristic

C₂×C₄

—

C₂×C₄

—

148

—

C₂×C₄

C₄×Dic₃₇

yes

—

C₂×C₄

#10596936

yes

—

C₂×C₄

C₄⋊Dic₃₇

yes

—

C₂×C₄

#10596938

yes

—

C₂×C₄

C₂²⋊Dic₃₇

yes

—

C₂×C₄

C₂×Dic₇₄

yes

characteristic

100

C₂×C₄

C₂×C₄×D₃₇

yes

—

101

C₂×C₄

C₂²×Dic₃₇

yes

characteristic

102

C₂×C₄

C₂×C₂²⋊D₃₇

yes

—

103

C₈

—

104

C₂²

—

105

C₂²

—

106

C₂²

#11924443

yes

—

107

C₂²

C₂×#10596938

yes

—

108

C₂²

C₂×C₄⋊Dic₃₇

yes

characteristic

109

C₂²

#11924558

yes

—

110

C₄

—

148

—

111

C₄

#11924443

yes

—

112

C₄

C₂×C₄×Dic₃₇

yes

characteristic

113

C₄

C₂×C₂²⋊Dic₃₇

yes

characteristic

114

C₄

C₂×#10596936

yes

—

115

C₂

—

116

C₂

C₂×#11924443

yes

characteristic

117

C₂

#14557530

yes

—

118

C₂

#14557571

yes

—

	subgroup	G/H
center	C₂²×C₄	D₁₄₈
commutator	C₄×C₃₇	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₃₇
Fitting	C₂³×C₈×C₃₇	C₂
socle	C₂³×C₃₇	C₂×D₄
prime radical	C₂⁴×C₃₇	D₄
cosocle	C₂²×C₄×C₃₇	C₂³
hypercenter	C₂³×C₈	D₃₇
hypocenter	C₃₇	#1735
LCS	G ▷ C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ C₂³×C₈

subgroup

G/H

center

C₂²×C₄

D₁₄₈

commutator

C₄×C₃₇

C₂×C₄²

Frattini

C₂²×C₄

C₂²×D₃₇

Fitting

C₂³×C₈×C₃₇

C₂

socle

C₂³×C₃₇

C₂×D₄

prime radical

C₂⁴×C₃₇

D₄

cosocle

C₂²×C₄×C₃₇

C₂³

hypercenter

C₂³×C₈

D₃₇

hypocenter

C₃₇

#1735

LCS

G ▷ C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇

UCS

C₁ ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ C₂³×C₈

N	H ≅ G/N	type
2. #14557296	C₂	semidirect
5. #11924495	C₄	semidirect
11. C₂²×C₈×C₃₇	C₄	semidirect
63. C₃₇	#1735	semidirect
66. C₂²×C₈	Dic₃₇	semidirect

H ≅ G/N

type

2. #14557296

C₂

semidirect

5. #11924495

C₄

semidirect

11. C₂²×C₈×C₃₇

C₄

semidirect

63. C₃₇

#1735

semidirect

66. C₂²×C₈

Dic₃₇

semidirect

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

subgroup	index
C₂×#11924495	2
#14557296	2
C₂³×C₈×C₃₇	2
#1735	37

subgroup

index

C₂×#11924495

#14557296

C₂³×C₈×C₃₇

#1735

Subgroups — #17994313

Conjugacy classes of subgroups

Complements

Maximal subgroups