Group #17995398 — subgroups

Subgroups — #17995398

Order 4736.

332 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 332 conjugacy classes of subgroups — too many to draw. Only its 141 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×#11924536	C₂	1	1	yes	maximal, characteristic
2	C₂×#11924461	C₂	1	4	yes	maximal
3	#14557403	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄×C₈×C₃₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₇₄	C₂²	1	2	yes	—
6	#11924536	—	2	2	no	—
7	#11924536	C₄	1	4	yes	complemented
8	C₂×C₄⋊Dic₃₇	—	2	2	no	—
9	C₂×C₄⋊Dic₃₇	—	2	2	no	—
10	C₂×C₄⋊Dic₃₇	C₂²	1	2	yes	—
11	#11924461	—	2	8	no	—
12	C₂²×C₈×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	C₂²×Dic₃₇	—	2	2	no	—
19	C₂×Dic₇₄	—	2	4	no	—
20	C₂×Dic₇₄	—	2	4	no	—
21	C₂×Dic₇₄	—	2	2	no	—
22	C₂×Dic₇₄	C₂×C₄	1	4	yes	—
23	C₂×C₈×C₃₇	—	2	2	no	—
24	C₂×C₈×C₃₇	—	2	2	no	—
25	C₂×C₈×C₃₇	D₄	1	4	yes	—
26	C₄²×C₃₇	C₂×C₄	1	2	yes	—
27	C₄²×C₃₇	D₄	1	2	yes	—
28	C₄⋊Dic₃₇	—	2	8	no	—
29	C₄⋊Dic₃₇	—	2	4	no	—
30	C₄⋊Dic₃₇	—	4	4	no	—
31	C₂³×C₃₇	C₂×D₄	1	1	yes	characteristic
32	C₂×C₄×C₃₇	C₂²⋊C₄	1	4	yes	—
33	C₂×C₄×C₃₇	SD₁₆	1	4	yes	—
34	C₂×C₄×C₃₇	Q₁₆	1	4	yes	—
35	C₂×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
36	C₂×C₄×C₃₇	C₂×D₄	1	2	yes	—
37	C₂×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
38	C₂×C₄×C₃₇	#41	1	2	yes	—
39	C₂×Dic₃₇	—	2	4	no	—
40	C₂×Dic₃₇	—	4	2	no	—
41	C₂×Dic₃₇	—	4	2	no	—
42	C₂×Dic₃₇	—	4	2	no	—
43	C₂×Dic₃₇	—	4	4	no	—
44	Dic₇₄	—	2	8	no	—
45	Dic₇₄	—	4	4	no	—
46	C₈×C₃₇	—	2	4	no	—
47	C₂²×C₃₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
48	C₂²×C₃₇	#124	1	1	yes	characteristic
49	C₂²×C₃₇	#127	1	1	yes	characteristic
50	C₂²×C₃₇	C₂×SD₁₆	1	2	yes	—
51	C₂²×C₃₇	C₂×Q₁₆	1	2	yes	—
52	C₄×C₃₇	#103	1	8	yes	complemented
53	C₄×C₃₇	C₄×D₄	1	2	yes	—
54	C₄×C₃₇	#121	1	2	yes	—
55	Dic₃₇	—	4	4	no	—
56	Dic₃₇	—	8	2	no	—
57	#1956	—	37	1	no	maximal, Sylow
58	C₂×C₃₇	#390	1	1	yes	characteristic
59	C₂×C₃₇	C₂×#103	1	2	yes	—
60	C₂×C₃₇	#487	1	2	yes	—
61	C₂×C₃₇	#492	1	1	yes	characteristic
62	C₂×C₃₇	#494	1	1	yes	characteristic
63	C₂×C₄⋊Q₈	—	37	1	no	—
64	C₂×#103	—	37	4	no	—
65	C₂×C₄×C₈	D₃₇	1	1	yes	characteristic
66	#384	—	37	1	no	—
67	C₃₇	#1956	1	1	yes	characteristic, complemented, Sylow
68	C₂²×Q₈	—	37	2	no	—
69	C₂²×C₈	C₂×D₃₇	1	2	yes	—
70	C₄⋊Q₈	—	37	4	no	—
71	C₄⋊Q₈	—	74	2	no	—
72	C₂×C₄⋊C₄	—	37	2	no	—
73	C₂×C₄⋊C₄	—	74	2	no	—
74	C₂×C₄⋊C₄	—	74	2	no	—
75	C₂×C₄²	C₂×D₃₇	1	1	yes	characteristic
76	#103	—	74	8	no	—
77	C₄×C₈	—	2	2	no	—
78	C₂×Q₈	—	37	4	no	—
79	C₂×Q₈	—	74	2	no	—
80	C₂×Q₈	—	74	4	no	—
81	C₂×Q₈	—	74	4	no	—
82	C₂²×C₄	—	74	2	no	—
83	C₂²×C₄	—	74	2	no	—
84	C₂²×C₄	D₁₄₈	1	2	yes	—
85	C₂²×C₄	C₂²×D₃₇	1	1	yes	characteristic
86	C₂×C₈	—	2	2	no	—
87	C₂×C₈	—	2	2	no	—
88	C₂×C₈	D₁₄₈	1	4	yes	—
89	C₄⋊C₄	—	74	8	no	—
90	C₄⋊C₄	—	74	4	no	—
91	C₄⋊C₄	—	148	4	no	—
92	C₄²	C₄×D₃₇	1	2	yes	—
93	C₄²	C₂²⋊D₃₇	1	2	yes	—
94	C₂³	C₂×D₁₄₈	1	1	yes	characteristic
95	Q₈	—	74	8	no	—
96	Q₈	—	148	4	no	—
97	C₂×C₄	—	74	4	no	—
98	C₂×C₄	—	148	2	no	—
99	C₂×C₄	—	148	4	no	—
100	C₂×C₄	—	148	2	no	—
101	C₂×C₄	—	148	2	no	—
102	C₂×C₄	#10596930	1	4	yes	—
103	C₂×C₄	Dic₁₄₈	1	4	yes	—
104	C₂×C₄	#10596938	1	4	yes	—
105	C₂×C₄	C₂×C₄×D₃₇	1	1	yes	characteristic
106	C₂×C₄	C₂×D₁₄₈	1	2	yes	—
107	C₂×C₄	#10596962	1	2	yes	—
108	C₂×C₄	C₂×C₂²⋊D₃₇	1	1	yes	characteristic
109	C₈	—	2	4	no	—
110	C₂²	C₂×#10596938	1	1	yes	characteristic
111	C₂²	#11924524	1	1	yes	characteristic
112	C₂²	#11924533	1	1	yes	characteristic
113	C₂²	C₂×#10596930	1	2	yes	—
114	C₂²	C₂×Dic₁₄₈	1	2	yes	—
115	C₄	—	296	2	no	—
116	C₄	—	148	4	no	—
117	C₄	#11924461	1	8	yes	complemented
118	C₄	C₄×D₁₄₈	1	2	yes	—
119	C₄	#11924510	1	2	yes	—
120	C₂	#14557435	1	1	yes	characteristic
121	C₂	C₂×#11924461	1	2	yes	—
122	C₂	#14557884	1	2	yes	—
123	C₂	#14557906	1	1	yes	characteristic
124	C₂	#14557913	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#11924536

C₂

yes

maximal, characteristic

C₂×#11924461

C₂

yes

maximal

#14557403

C₂

yes

maximal, characteristic

C₂×C₄×C₈×C₃₇

C₂

yes

maximal, characteristic

C₂²×Dic₇₄

C₂²

yes

—

#11924536

—

#11924536

C₄

yes

complemented

C₂×C₄⋊Dic₃₇

—

C₂×C₄⋊Dic₃₇

—

C₂×C₄⋊Dic₃₇

C₂²

yes

—

#11924461

—

C₂²×C₈×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

—

C₂²×Dic₃₇

—

C₂×Dic₇₄

—

C₂×Dic₇₄

—

C₂×Dic₇₄

—

C₂×Dic₇₄

C₂×C₄

yes

—

C₂×C₈×C₃₇

—

C₂×C₈×C₃₇

—

C₂×C₈×C₃₇

D₄

yes

—

C₄²×C₃₇

C₂×C₄

yes

—

C₄²×C₃₇

D₄

yes

—

C₄⋊Dic₃₇

—

C₄⋊Dic₃₇

—

C₄⋊Dic₃₇

—

C₂³×C₃₇

C₂×D₄

yes

characteristic

C₂×C₄×C₃₇

C₂²⋊C₄

yes

—

C₂×C₄×C₃₇

SD₁₆

yes

—

C₂×C₄×C₃₇

Q₁₆

yes

—

C₂×C₄×C₃₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₃₇

C₂×D₄

yes

—

C₂×C₄×C₃₇

C₂×D₄

yes

characteristic

C₂×C₄×C₃₇

#41

yes

—

C₂×Dic₃₇

—

C₂×Dic₃₇

—

C₂×Dic₃₇

—

C₂×Dic₃₇

—

C₂×Dic₃₇

—

Dic₇₄

—

Dic₇₄

—

C₈×C₃₇

—

C₂²×C₃₇

C₂×C₂²⋊C₄

yes

C₂²×C₃₇

yes

C₂²×C₃₇

yes

C₂²×C₃₇

C₂×SD₁₆

yes

—

C₂²×C₃₇

C₂×Q₁₆

yes

—

C₄×C₃₇

#103

yes

complemented

C₄×C₃₇

C₄×D₄

yes

—

C₄×C₃₇

#121

yes

—

Dic₃₇

—

Dic₃₇

—

—

C₂×C₃₇

yes

C₂×C₃₇

yes

—

C₂×C₃₇

#487

yes

—

C₂×C₃₇

yes

C₂×C₃₇

yes

C₂×C₄⋊Q₈

—

C₂×#103

—

C₂×C₄×C₈

D₃₇

yes

characteristic

#384

—

C₃₇

#1956

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₂²×C₈

C₂×D₃₇

yes

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂×D₃₇

yes

characteristic

#103

—

C₄×C₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

D₁₄₈

yes

—

C₂²×C₄

C₂²×D₃₇

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

D₁₄₈

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

148

—

C₄²

C₄×D₃₇

yes

—

C₄²

C₂²⋊D₃₇

yes

—

C₂³

C₂×D₁₄₈

yes

characteristic

Q₈

—

Q₈

—

148

—

C₂×C₄

—

C₂×C₄

—

148

—

C₂×C₄

—

148

—

100

C₂×C₄

—

148

—

101

C₂×C₄

—

148

—

102

C₂×C₄

#10596930

yes

—

103

C₂×C₄

Dic₁₄₈

yes

—

104

C₂×C₄

#10596938

yes

—

105

C₂×C₄

C₂×C₄×D₃₇

yes

characteristic

106

C₂×C₄

C₂×D₁₄₈

yes

—

107

C₂×C₄

#10596962

yes

—

108

C₂×C₄

C₂×C₂²⋊D₃₇

yes

characteristic

109

C₈

—

110

C₂²

C₂×#10596938

yes

characteristic

111

C₂²

#11924524

yes

characteristic

112

C₂²

#11924533

yes

characteristic

113

C₂²

C₂×#10596930

yes

—

114

C₂²

C₂×Dic₁₄₈

yes

—

115

C₄

—

296

—

116

C₄

—

148

—

117

C₄

#11924461

yes

complemented

118

C₄

C₄×D₁₄₈

yes

—

119

C₄

#11924510

yes

—

120

C₂

#14557435

yes

characteristic

121

C₂

C₂×#11924461

yes

—

122

C₂

#14557884

yes

—

123

C₂

#14557906

yes

characteristic

124

C₂

#14557913

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×D₁₄₈
commutator	C₂×C₄×C₃₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₃₇
Fitting	C₂×C₄×C₈×C₃₇	C₂
socle, prime radical	C₂³×C₃₇	C₂×D₄
cosocle	C₂²×C₄×C₃₇	C₂³
hypercenter	C₂×C₄×C₈	D₃₇
hypocenter	C₃₇	#1956
LCS	G ▷ C₂×C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈

subgroup

G/H

center

C₂³

C₂×D₁₄₈

commutator

C₂×C₄×C₃₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₃₇

Fitting

C₂×C₄×C₈×C₃₇

C₂

socle, prime radical

C₂³×C₃₇

C₂×D₄

cosocle

C₂²×C₄×C₃₇

C₂³

hypercenter

C₂×C₄×C₈

D₃₇

hypocenter

C₃₇

#1956

LCS

G ▷ C₂×C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈

N	H ≅ G/N	type
7. #11924536	C₄	semidirect
52. C₄×C₃₇	#103	semidirect
67. C₃₇	#1956	semidirect
117. C₄	#11924461	semidirect

H ≅ G/N

type

7. #11924536

C₄

semidirect

52. C₄×C₃₇

#103

semidirect

67. C₃₇

#1956

semidirect

117. C₄

#11924461

semidirect

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

subgroup	index
C₂×#11924536	2
C₂×#11924461	2
#14557403	2
C₂×C₄×C₈×C₃₇	2
#1956	37

subgroup

index

C₂×C₄×C₈×C₃₇

Subgroups — #17995398

Conjugacy classes of subgroups

Complements

Maximal subgroups