Group #17993656 — subgroups

Subgroups — #17993656

Order 4736.

240 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 53 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	#14558269	C₂	1	1	yes	maximal, characteristic, complemented
2	#14557909	C₂	1	1	yes	maximal, characteristic
3	#14557734	C₂	1	2	yes	maximal, complemented
4	#14557263	C₂	1	2	yes	maximal, complemented
5	C₃₇×#331	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#10596930	—	2	2	no	—
7	C₂×#10596939	—	2	2	no	—
8	#11924533	C₂²	1	1	yes	characteristic
9	#11924523	—	2	2	no	—
10	#11924517	—	2	2	no	—
11	C₄×Dic₇₄	C₂²	1	2	yes	—
12	D₄×Dic₃₇	—	2	2	no	—
13	#11924458	—	2	2	no	—
14	C₄×#64436	C₂²	1	1	yes	characteristic
15	#127×C₃₇	C₂²	1	1	yes	characteristic
16	C₄⋊C₈×C₃₇	C₂²	1	2	yes	—
17	C₂×D₄×C₃₇	—	2	2	no	—
18	C₂×D₄×C₃₇	D₄	1	2	yes	—
19	C₂²×Dic₃₇	—	2	2	no	—
20	C₂×D₁₄₈	—	2	1	no	—
21	C₂×D₁₄₈	—	2	2	no	—
22	C₂×Dic₇₄	D₄	1	2	yes	—
23	C₂×C₈×C₃₇	—	2	2	no	—
24	C₄²×C₃₇	C₂³	1	1	yes	characteristic
25	C₂²⋊Dic₃₇	—	2	4	no	—
26	C₂²⋊Dic₃₇	—	2	2	no	—
27	#10596939	—	4	4	no	—
28	C₄⋊Dic₃₇	D₄	1	2	yes	—
29	#10596936	—	2	2	no	—
30	C₄×Dic₃₇	—	2	2	no	—
31	C₂×#64436	—	2	1	no	—
32	#10596930	—	4	4	no	—
33	C₂³×C₃₇	—	2	2	no	—
34	C₂²×D₃₇	—	4	1	no	—
35	D₄×C₃₇	—	4	2	no	—
36	D₄×C₃₇	—	4	2	no	—
37	D₄×C₃₇	—	4	2	no	—
38	C₂×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
39	C₂×C₄×C₃₇	C₂×D₄	1	2	yes	—
40	C₂×Dic₃₇	—	2	2	no	—
41	C₂×Dic₃₇	—	2	2	no	—
42	C₂×Dic₃₇	—	8	2	no	—
43	D₁₄₈	—	4	2	no	—
44	D₁₄₈	—	4	2	no	—
45	D₁₄₈	—	4	2	no	—
46	Dic₇₄	—	2	2	no	—
47	Dic₇₄	—	2	2	no	—
48	C₈×C₃₇	—	4	2	no	—
49	#64436	—	2	2	no	—
50	C₂²×C₃₇	—	4	4	no	—
51	C₂²×C₃₇	—	4	2	no	—
52	C₂²×C₃₇	#120	1	1	yes	characteristic
53	C₂×D₃₇	—	8	2	no	—
54	C₂×D₃₇	—	8	1	no	—
55	C₄×C₃₇	—	2	1	no	—
56	C₄×C₃₇	#135	1	2	yes	—
57	C₄×C₃₇	AGL(1,ℤ/8)	1	2	yes	—
58	Dic₃₇	—	4	2	no	—
59	Dic₃₇	—	4	2	no	—
60	#1620	—	37	1	no	maximal, Sylow
61	C₂×C₃₇	—	8	2	no	—
62	C₂×C₃₇	#449	1	2	yes	—
63	C₂×C₃₇	#453	1	1	yes	characteristic
64	D₃₇	—	16	1	no	—
65	#559	—	37	1	no	—
66	#493	—	37	1	no	—
67	#460	—	37	2	no	—
68	#331	D₃₇	1	1	yes	characteristic, complemented
69	Q₈⋊C₈	—	37	2	no	—
70	C₃₇	#1620	1	1	yes	characteristic, complemented, Sylow
71	C₂×SD₁₆	—	74	2	no	—
72	C₂×D₈	—	74	2	no	—
73	#127	—	37	1	no	—
74	#127	C₂×D₃₇	1	1	yes	characteristic
75	#124	—	74	2	no	—
76	#123	—	74	2	no	—
77	C₄×Q₈	—	37	2	no	—
78	C₄×D₄	—	74	2	no	—
79	C₄⋊C₈	C₂×D₃₇	1	2	yes	—
80	#102	—	74	2	no	—
81	C₄×C₈	—	37	1	no	—
82	C₂×Q₈	—	37	2	no	—
83	C₂×D₄	—	2	2	no	—
84	C₂×D₄	—	74	1	no	—
85	C₂×D₄	—	74	2	no	—
86	C₂×D₄	D₁₄₈	1	2	yes	—
87	C₂²×C₄	—	74	2	no	—
88	SD₁₆	—	148	4	no	—
89	D₈	—	148	4	no	—
90	C₂×C₈	—	2	2	no	—
91	C₂×C₈	—	74	1	no	—
92	C₄⋊C₄	—	37	2	no	—
93	C₄⋊C₄	—	74	2	no	—
94	C₂²⋊C₄	—	74	4	no	—
95	C₂²⋊C₄	—	74	2	no	—
96	C₄²	—	74	2	no	—
97	C₄²	C₂²×D₃₇	1	1	yes	characteristic
98	C₂³	—	2	2	no	—
99	C₂³	—	148	1	no	—
100	Q₈	—	74	2	no	—
101	Q₈	—	74	2	no	—
102	D₄	—	4	2	no	—
103	D₄	—	4	2	no	—
104	D₄	—	4	2	no	—
105	D₄	—	148	2	no	—
106	D₄	—	148	2	no	—
107	D₄	—	148	2	no	—
108	C₂×C₄	—	296	2	no	—
109	C₂×C₄	—	74	2	no	—
110	C₂×C₄	—	74	2	no	—
111	C₂×C₄	C₂×D₁₄₈	1	1	yes	characteristic
112	C₂×C₄	D₄×D₃₇	1	2	yes	—
113	C₈	—	4	2	no	—
114	C₈	—	74	2	no	—
115	C₂²	—	4	2	no	—
116	C₂²	—	4	4	no	—
117	C₂²	—	296	1	no	—
118	C₂²	—	296	2	no	—
119	C₂²	#11924506	1	1	yes	characteristic
120	C₄	—	2	1	no	—
121	C₄	—	148	2	no	—
122	C₄	—	148	2	no	—
123	C₄	#11924557	1	2	yes	—
124	C₄	#11924560	1	2	yes	—
125	C₂	—	8	2	no	—
126	C₂	—	592	1	no	—
127	C₂	#14557679	1	2	yes	—
128	C₂	#14557703	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14558269

C₂

yes

maximal, characteristic, complemented

#14557909

C₂

yes

maximal, characteristic

#14557734

C₂

yes

maximal, complemented

#14557263

C₂

yes

maximal, complemented

C₃₇×#331

C₂

yes

maximal, characteristic, complemented

C₂×#10596930

—

C₂×#10596939

—

#11924533

C₂²

yes

characteristic

#11924523

—

#11924517

—

C₄×Dic₇₄

C₂²

yes

—

D₄×Dic₃₇

—

#11924458

—

C₄×#64436

C₂²

yes

characteristic

#127×C₃₇

C₂²

yes

characteristic

C₄⋊C₈×C₃₇

C₂²

yes

—

C₂×D₄×C₃₇

—

C₂×D₄×C₃₇

D₄

yes

—

C₂²×Dic₃₇

—

C₂×D₁₄₈

—

C₂×D₁₄₈

—

C₂×Dic₇₄

D₄

yes

—

C₂×C₈×C₃₇

—

C₄²×C₃₇

C₂³

yes

characteristic

C₂²⋊Dic₃₇

—

C₂²⋊Dic₃₇

—

#10596939

—

C₄⋊Dic₃₇

D₄

yes

—

#10596936

—

C₄×Dic₃₇

—

C₂×#64436

—

#10596930

—

C₂³×C₃₇

—

C₂²×D₃₇

—

D₄×C₃₇

—

D₄×C₃₇

—

D₄×C₃₇

—

C₂×C₄×C₃₇

C₂×D₄

yes

characteristic

C₂×C₄×C₃₇

C₂×D₄

yes

—

C₂×Dic₃₇

—

C₂×Dic₃₇

—

C₂×Dic₃₇

—

D₁₄₈

—

D₁₄₈

—

D₁₄₈

—

Dic₇₄

—

Dic₇₄

—

C₈×C₃₇

—

#64436

—

C₂²×C₃₇

—

C₂²×C₃₇

—

C₂²×C₃₇

#120

yes

characteristic

C₂×D₃₇

—

C₂×D₃₇

—

C₄×C₃₇

—

C₄×C₃₇

#135

yes

—

C₄×C₃₇

AGL(1,ℤ/8)

yes

—

Dic₃₇

—

Dic₃₇

—

#1620

—

maximal, Sylow

C₂×C₃₇

—

C₂×C₃₇

#449

yes

—

C₂×C₃₇

#453

yes

characteristic

D₃₇

—

—

—

—

D₃₇

yes

characteristic, complemented

Q₈⋊C₈

—

C₃₇

#1620

yes

characteristic, complemented, Sylow

C₂×SD₁₆

—

C₂×D₈

—

#127

—

#127

C₂×D₃₇

yes

characteristic

#124

—

#123

—

C₄×Q₈

—

C₄×D₄

—

C₄⋊C₈

C₂×D₃₇

yes

—

#102

—

C₄×C₈

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

D₁₄₈

yes

—

C₂²×C₄

—

SD₁₆

—

148

—

D₈

—

148

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

C₂²×D₃₇

yes

characteristic

C₂³

—

C₂³

—

148

—

100

Q₈

—

101

Q₈

—

102

D₄

—

103

D₄

—

104

D₄

—

105

D₄

—

148

—

106

D₄

—

148

—

107

D₄

—

148

—

108

C₂×C₄

—

296

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

C₂×D₁₄₈

yes

characteristic

112

C₂×C₄

D₄×D₃₇

yes

—

113

C₈

—

114

C₈

—

115

C₂²

—

116

C₂²

—

117

C₂²

—

296

—

118

C₂²

—

296

—

119

C₂²

#11924506

yes

characteristic

120

C₄

—

121

C₄

—

148

—

122

C₄

—

148

—

123

C₄

#11924557

yes

—

124

C₄

#11924560

yes

—

125

C₂

—

126

C₂

—

592

—

127

C₂

#14557679

yes

—

128

C₂

#14557703

yes

characteristic

	subgroup	G/H
center	C₂²	#11924506
commutator, cosocle	C₄²×C₃₇	C₂³
Frattini	C₄²	C₂²×D₃₇
Fitting	C₃₇×#331	C₂
socle	C₂²×C₃₇	#120
prime radical	#14557909	C₂
hypercenter	#331	D₃₇
hypocenter	C₃₇	#1620
LCS	G ▷ C₄²×C₃₇ ▷ C₂²×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #331

subgroup

G/H

center

C₂²

#11924506

commutator, cosocle

C₄²×C₃₇

C₂³

Frattini

C₄²

C₂²×D₃₇

Fitting

C₃₇×#331

C₂

C₂²×C₃₇

C₂

D₃₇

C₃₇

G ▷ C₄²×C₃₇ ▷ C₂²×C₃₇ ▷ C₃₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #331

N	H ≅ G/N	type
1. #14558269	C₂	semidirect
3. #14557734	C₂	semidirect
4. #14557263	C₂	semidirect
5. C₃₇×#331	C₂	semidirect
68. #331	D₃₇	semidirect
70. C₃₇	#1620	semidirect

H ≅ G/N

type

1. #14558269

C₂

semidirect

3. #14557734

C₂

semidirect

4. #14557263

C₂

semidirect

5. C₃₇×#331

C₂

semidirect

68. #331

D₃₇

semidirect

70. C₃₇

#1620

semidirect

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

subgroup	index
#14558269	2
#14557909	2
#14557734	2
#14557263	2
C₃₇×#331	2
#1620	37

subgroup

index

C₃₇×#331

Subgroups — #17993656

Conjugacy classes of subgroups

Complements

Maximal subgroups