Group #17995572 — subgroups

Subgroups — #17995572

Order 4736.

284 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 284 conjugacy classes of subgroups — too many to draw. Only its 97 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	#14558090	C₂	1	1	yes	maximal, characteristic
2	C₄²×Dic₃₇	C₂	1	1	yes	maximal, characteristic
3	#14557294	C₂	1	4	yes	maximal
4	C₃₇×#423	C₂	1	1	yes	maximal, characteristic
5	#11924536	C₄	1	2	yes	complemented
6	#11924528	C₄	1	2	yes	complemented
7	#11924515	—	2	2	no	—
8	C₄×Dic₇₄	—	2	2	no	—
9	#11924495	C₂²	1	2	yes	—
10	C₂×C₄×Dic₃₇	—	2	2	no	—
11	C₂×C₄×Dic₃₇	C₂²	1	2	yes	—
12	C₂×M₁₆×C₃₇	C₂²	1	2	yes	—
13	C₂×C₄²×C₃₇	C₂²	1	1	yes	characteristic
14	C₄⋊C₈×C₃₇	—	2	2	no	—
15	C₂²×C₄×C₃₇	D₄	1	2	yes	—
16	C₂²×C₄×C₃₇	C₂³	1	1	yes	characteristic
17	C₂²×Dic₃₇	—	2	2	no	—
18	C₂×Dic₇₄	—	2	2	no	—
19	M₁₆×C₃₇	—	2	4	no	—
20	C₂×C₈×C₃₇	—	2	2	no	—
21	C₄²×C₃₇	C₂×C₄	1	1	yes	characteristic
22	C₄²×C₃₇	C₂×C₄	1	1	yes	characteristic
23	C₄²×C₃₇	D₄	1	2	yes	—
24	C₂²⋊Dic₃₇	—	2	2	no	—
25	C₄⋊Dic₃₇	—	2	2	no	—
26	C₄⋊Dic₃₇	C₂×C₄	1	4	yes	—
27	#10596936	—	2	4	no	—
28	C₄×Dic₃₇	—	2	8	no	—
29	C₄×Dic₃₇	—	2	2	no	—
30	C₄×Dic₃₇	—	2	2	no	—
31	C₄×Dic₃₇	Q₈	1	4	yes	complemented
32	C₂³×C₃₇	C₂×D₄	1	1	yes	characteristic
33	C₂×C₄×C₃₇	—	2	4	no	—
34	C₂×C₄×C₃₇	C₂²⋊C₄	1	2	yes	—
35	C₂×C₄×C₃₇	C₂²⋊C₄	1	2	yes	—
36	C₂×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
37	C₂×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
38	C₂×C₄×C₃₇	C₂×Q₈	1	2	yes	—
39	C₂×C₄×C₃₇	#41	1	2	yes	—
40	C₂×Dic₃₇	—	2	8	no	—
41	C₂×Dic₃₇	—	2	4	no	—
42	C₂×Dic₃₇	—	2	4	no	—
43	Dic₇₄	—	4	4	no	—
44	C₈×C₃₇	—	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	#104	1	4	yes	—
50	C₄×C₃₇	C₄×Q₈	1	2	yes	—
51	C₄×C₃₇	#124	1	1	yes	characteristic
52	C₄×C₃₇	C₄⋊Q₈	1	1	yes	characteristic
53	Dic₃₇	—	2	8	no	—
54	Dic₃₇	—	4	4	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	—	37	1	no	—
60	#423	D₃₇	1	1	yes	characteristic
61	C₄³	—	37	1	no	—
62	#339	—	37	4	no	—
63	C₃₇	#1994	1	1	yes	characteristic, complemented, Sylow
64	C₂×M₁₆	C₂×D₃₇	1	2	yes	—
65	C₄⋊Q₈	—	37	2	no	—
66	#125	—	37	2	no	—
67	C₂²⋊Q₈	—	74	2	no	—
68	C₄×Q₈	—	74	2	no	—
69	#117	—	37	2	no	—
70	C₂×C₄²	—	37	2	no	—
71	C₂×C₄²	—	74	2	no	—
72	C₂×C₄²	C₂×D₃₇	1	1	yes	characteristic
73	C₄⋊C₈	—	2	2	no	—
74	C₂×Q₈	—	74	2	no	—
75	C₂²×C₄	—	74	2	no	—
76	C₂²×C₄	D₁₄₈	1	2	yes	—
77	C₂²×C₄	C₂²×D₃₇	1	1	yes	characteristic
78	M₁₆	—	2	4	no	—
79	C₂×C₈	—	2	2	no	—
80	C₄⋊C₄	—	37	4	no	—
81	C₄⋊C₄	—	74	4	no	—
82	C₄⋊C₄	—	74	2	no	—
83	C₂²⋊C₄	—	74	2	no	—
84	C₄²	—	37	4	no	—
85	C₄²	—	74	8	no	—
86	C₄²	—	74	2	no	—
87	C₄²	—	74	2	no	—
88	C₄²	C₄×D₃₇	1	1	yes	characteristic
89	C₄²	C₄×D₃₇	1	1	yes	characteristic
90	C₄²	C₂²⋊D₃₇	1	2	yes	—
91	C₂³	C₂×D₁₄₈	1	1	yes	characteristic
92	Q₈	—	148	4	no	—
93	C₂×C₄	—	2	4	no	—
94	C₂×C₄	—	74	8	no	—
95	C₂×C₄	—	74	4	no	—
96	C₂×C₄	—	74	4	no	—
97	C₂×C₄	#10596938	1	2	yes	—
98	C₂×C₄	#10596938	1	2	yes	—
99	C₂×C₄	C₂×C₄×D₃₇	1	1	yes	characteristic
100	C₂×C₄	#10596964	1	2	yes	—
101	C₂×C₄	Q₈×D₃₇	1	2	yes	—
102	C₂×C₄	C₂×C₂²⋊D₃₇	1	1	yes	characteristic
103	C₈	—	4	2	no	—
104	C₂²	—	2	2	no	—
105	C₂²	C₂×#10596938	1	1	yes	characteristic
106	C₂²	#11924516	1	2	yes	—
107	C₄	—	2	2	no	—
108	C₄	—	74	8	no	—
109	C₄	—	148	4	no	—
110	C₄	#11924462	1	4	yes	—
111	C₄	#11924501	1	2	yes	—
112	C₄	#11924523	1	1	yes	characteristic
113	C₄	#11924534	1	1	yes	characteristic
114	C₂	—	2	2	no	—
115	C₂	#14557438	1	1	yes	characteristic
116	C₂	C₂×#11924462	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#14558090

C₂

yes

maximal, characteristic

C₄²×Dic₃₇

C₂

yes

maximal, characteristic

#14557294

C₂

yes

maximal

C₃₇×#423

C₂

yes

maximal, characteristic

#11924536

C₄

yes

complemented

#11924528

C₄

yes

complemented

#11924515

—

C₄×Dic₇₄

—

#11924495

C₂²

yes

—

C₂×C₄×Dic₃₇

—

C₂×C₄×Dic₃₇

C₂²

yes

—

C₂×M₁₆×C₃₇

C₂²

yes

—

C₂×C₄²×C₃₇

C₂²

yes

characteristic

C₄⋊C₈×C₃₇

—

C₂²×C₄×C₃₇

D₄

yes

—

C₂²×C₄×C₃₇

C₂³

yes

characteristic

C₂²×Dic₃₇

—

C₂×Dic₇₄

—

M₁₆×C₃₇

—

C₂×C₈×C₃₇

—

C₄²×C₃₇

C₂×C₄

yes

characteristic

C₄²×C₃₇

C₂×C₄

yes

characteristic

C₄²×C₃₇

D₄

yes

—

C₂²⋊Dic₃₇

—

C₄⋊Dic₃₇

—

C₄⋊Dic₃₇

C₂×C₄

yes

—

#10596936

—

C₄×Dic₃₇

—

C₄×Dic₃₇

—

C₄×Dic₃₇

—

C₄×Dic₃₇

Q₈

yes

complemented

C₂³×C₃₇

C₂×D₄

yes

characteristic

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

—

C₂×Dic₃₇

—

Dic₇₄

—

C₈×C₃₇

—

C₂²×C₃₇

—

C₂²×C₃₇

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₃₇

C₂²⋊Q₈

yes

—

C₄×C₃₇

—

C₄×C₃₇

#104

yes

—

C₄×C₃₇

C₄×Q₈

yes

—

C₄×C₃₇

#124

yes

characteristic

C₄×C₃₇

C₄⋊Q₈

yes

characteristic

Dic₃₇

—

Dic₃₇

—

#1994

—

maximal, Sylow

C₂×C₃₇

—

C₂×C₃₇

#391

yes

characteristic

C₂×C₃₇

C₂×#104

yes

—

#533

—

#423

D₃₇

yes

characteristic

C₄³

—

#339

—

C₃₇

#1994

yes

characteristic, complemented, Sylow

C₂×M₁₆

C₂×D₃₇

yes

—

C₄⋊Q₈

—

#125

—

C₂²⋊Q₈

—

C₄×Q₈

—

#117

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

C₂×D₃₇

yes

characteristic

C₄⋊C₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

D₁₄₈

yes

—

C₂²×C₄

C₂²×D₃₇

yes

characteristic

M₁₆

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

C₄²

C₄×D₃₇

yes

characteristic

C₄²

C₄×D₃₇

yes

characteristic

C₄²

C₂²⋊D₃₇

yes

—

C₂³

C₂×D₁₄₈

yes

characteristic

Q₈

—

148

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

#10596938

yes

—

C₂×C₄

#10596938

yes

—

C₂×C₄

C₂×C₄×D₃₇

yes

characteristic

100

C₂×C₄

#10596964

yes

—

101

C₂×C₄

Q₈×D₃₇

yes

—

102

C₂×C₄

C₂×C₂²⋊D₃₇

yes

characteristic

103

C₈

—

104

C₂²

—

105

C₂²

C₂×#10596938

yes

characteristic

106

C₂²

#11924516

yes

—

107

C₄

—

108

C₄

—

109

C₄

—

148

—

110

C₄

#11924462

yes

—

111

C₄

#11924501

yes

—

112

C₄

#11924523

yes

characteristic

113

C₄

#11924534

yes

characteristic

114

C₂

—

115

C₂

#14557438

yes

characteristic

116

C₂

C₂×#11924462

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₃₇
commutator	C₂×C₄×C₃₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₃₇
Fitting	C₃₇×#423	C₂
socle	C₂²×C₃₇	C₂×C₂²⋊C₄
prime radical	C₂³×C₃₇	C₂×D₄
cosocle	C₂²×C₄×C₃₇	C₂³
hypercenter	#423	D₃₇
hypocenter	C₃₇	#1994
LCS	G ▷ C₂×C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #423

subgroup

G/H

center

C₂×C₄

C₂×C₂²⋊D₃₇

commutator

C₂×C₄×C₃₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₃₇

Fitting

C₃₇×#423

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 ◁ #423

N	H ≅ G/N	type
5. #11924536	C₄	semidirect
6. #11924528	C₄	semidirect
31. C₄×Dic₃₇	Q₈	semidirect
63. C₃₇	#1994	semidirect

H ≅ G/N

type

5. #11924536

C₄

semidirect

6. #11924528

C₄

semidirect

31. C₄×Dic₃₇

Q₈

semidirect

63. C₃₇

#1994

semidirect

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

subgroup	index
#14558090	2
C₄²×Dic₃₇	2
#14557294	2
C₃₇×#423	2
#1994	37

subgroup

index

#14558090

C₄²×Dic₃₇

#14557294

C₃₇×#423

#1994

Subgroups — #17995572

Conjugacy classes of subgroups

Complements

Maximal subgroups