Group #17993677 — subgroups

Subgroups — #17993677

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	#14558249	C₂	1	1	yes	maximal, characteristic, complemented
2	#14557725	C₂	1	2	yes	maximal, complemented
3	#14557282	C₂	1	1	yes	maximal, characteristic, complemented
4	#14557262	C₂	1	2	yes	maximal, complemented
5	C₃₇×#492	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#10596939	—	2	2	no	—
7	#11924525	—	2	2	no	—
8	#11924511	—	2	2	no	—
9	C₄×D₁₄₈	C₂²	1	2	yes	—
10	#11924494	—	2	2	no	—
11	C₄⋊#64436	C₂²	1	2	yes	—
12	#11924457	—	2	2	no	—
13	C₂×SD₁₆×C₃₇	—	2	2	no	—
14	C₄⋊Q₈×C₃₇	C₂²	1	1	yes	characteristic
15	#127×C₃₇	C₂²	1	1	yes	characteristic
16	C₄×C₈×C₃₇	C₂²	1	1	yes	characteristic
17	C₂×Q₈×C₃₇	D₄	1	2	yes	—
18	C₂×D₄×C₃₇	—	2	2	no	—
19	C₂×D₄×C₃₇	—	2	1	no	—
20	C₂×D₁₄₈	—	2	2	no	—
21	C₂×D₁₄₈	D₄	1	2	yes	—
22	C₂×C₄×D₃₇	—	2	2	no	—
23	SD₁₆×C₃₇	—	4	4	no	—
24	C₂×C₈×C₃₇	—	2	1	no	—
25	C₄⋊C₄×C₃₇	—	2	2	no	—
26	C₄²×C₃₇	C₂³	1	1	yes	characteristic
27	#10596939	—	4	4	no	—
28	#10596938	—	2	4	no	—
29	#10596938	—	2	2	no	—
30	C₄⋊Dic₃₇	D₄	1	2	yes	—
31	C₄×Dic₃₇	—	2	2	no	—
32	C₂×#64436	—	2	2	no	—
33	C₂³×C₃₇	—	4	1	no	—
34	C₂²×D₃₇	—	2	2	no	—
35	Q₈×C₃₇	—	4	2	no	—
36	D₄×C₃₇	—	4	2	no	—
37	D₄×C₃₇	—	4	2	no	—
38	D₄×C₃₇	—	4	2	no	—
39	C₂×C₄×C₃₇	—	2	2	no	—
40	C₂×C₄×C₃₇	C₂×D₄	1	2	yes	—
41	C₂×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
42	C₂×Dic₃₇	—	2	2	no	—
43	D₁₄₈	—	2	2	no	—
44	D₁₄₈	—	2	2	no	—
45	D₁₄₈	—	8	2	no	—
46	C₄×D₃₇	—	4	2	no	—
47	C₄×D₃₇	—	4	2	no	—
48	C₈×C₃₇	—	2	2	no	—
49	#64436	—	4	2	no	—
50	C₂²×C₃₇	—	8	2	no	—
51	C₂²×C₃₇	—	8	1	no	—
52	C₂²×C₃₇	#120	1	1	yes	characteristic
53	C₂×D₃₇	—	4	2	no	—
54	C₂×D₃₇	—	4	2	no	—
55	C₂×D₃₇	—	4	2	no	—
56	C₄×C₃₇	—	2	1	no	—
57	C₄×C₃₇	—	4	2	no	—
58	C₄×C₃₇	#135	1	2	yes	—
59	C₄×C₃₇	AGL(1,ℤ/8)	1	2	yes	—
60	Dic₃₇	—	4	2	no	—
61	#1624	—	37	1	no	maximal, Sylow
62	C₂×C₃₇	—	16	1	no	—
63	C₂×C₃₇	#449	1	2	yes	—
64	C₂×C₃₇	#453	1	1	yes	characteristic
65	D₃₇	—	8	2	no	—
66	#555	—	37	1	no	—
67	#492	D₃₇	1	1	yes	characteristic, complemented
68	#459	—	37	2	no	—
69	#333	—	37	1	no	—
70	D₄⋊C₈	—	37	2	no	—
71	C₃₇	#1624	1	1	yes	characteristic, complemented, Sylow
72	C₂×SD₁₆	—	2	2	no	—
73	C₂×D₈	—	74	2	no	—
74	C₄⋊Q₈	C₂×D₃₇	1	1	yes	characteristic
75	#127	C₂×D₃₇	1	1	yes	characteristic
76	#124	—	74	2	no	—
77	#121	—	74	2	no	—
78	C₄×D₄	—	37	2	no	—
79	#117	—	74	2	no	—
80	C₄⋊C₈	—	37	2	no	—
81	#102	—	74	2	no	—
82	C₄×C₈	C₂×D₃₇	1	1	yes	characteristic
83	C₂×Q₈	C₂²⋊D₃₇	1	2	yes	—
84	C₂×D₄	—	2	2	no	—
85	C₂×D₄	—	2	1	no	—
86	C₂×D₄	—	37	2	no	—
87	C₂×D₄	—	74	2	no	—
88	C₂²×C₄	—	74	2	no	—
89	SD₁₆	—	4	4	no	—
90	D₈	—	148	4	no	—
91	C₂×C₈	—	2	1	no	—
92	C₂×C₈	—	74	2	no	—
93	C₄⋊C₄	—	2	2	no	—
94	C₄⋊C₄	—	37	2	no	—
95	C₂²⋊C₄	—	74	4	no	—
96	C₂²⋊C₄	—	74	2	no	—
97	C₄²	—	74	2	no	—
98	C₄²	C₂²×D₃₇	1	1	yes	characteristic
99	C₂³	—	4	1	no	—
100	C₂³	—	74	2	no	—
101	Q₈	—	4	2	no	—
102	D₄	—	4	2	no	—
103	D₄	—	4	2	no	—
104	D₄	—	4	2	no	—
105	D₄	—	296	2	no	—
106	D₄	—	74	2	no	—
107	D₄	—	74	2	no	—
108	C₂×C₄	—	2	2	no	—
109	C₂×C₄	—	74	2	no	—
110	C₂×C₄	—	148	2	no	—
111	C₂×C₄	—	148	2	no	—
112	C₂×C₄	D₄×D₃₇	1	2	yes	—
113	C₂×C₄	C₂×C₂²⋊D₃₇	1	1	yes	characteristic
114	C₈	—	2	2	no	—
115	C₈	—	148	2	no	—
116	C₂²	—	8	1	no	—
117	C₂²	—	8	2	no	—
118	C₂²	—	148	2	no	—
119	C₂²	—	148	2	no	—
120	C₂²	—	148	2	no	—
121	C₂²	#11924504	1	1	yes	characteristic
122	C₄	—	2	1	no	—
123	C₄	—	4	2	no	—
124	C₄	—	148	2	no	—
125	C₄	#11924557	1	2	yes	—
126	C₄	#11924563	1	2	yes	—
127	C₂	—	16	1	no	—
128	C₂	—	296	2	no	—
129	C₂	#14557677	1	2	yes	—
130	C₂	#14557700	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14558249

C₂

yes

maximal, characteristic, complemented

#14557725

C₂

yes

maximal, complemented

#14557282

C₂

yes

maximal, characteristic, complemented

#14557262

C₂

yes

maximal, complemented

C₃₇×#492

C₂

yes

maximal, characteristic, complemented

C₂×#10596939

—

#11924525

—

#11924511

—

C₄×D₁₄₈

C₂²

yes

—

#11924494

—

C₄⋊#64436

C₂²

yes

—

#11924457

—

C₂×SD₁₆×C₃₇

—

C₄⋊Q₈×C₃₇

C₂²

yes

characteristic

#127×C₃₇

C₂²

yes

characteristic

C₄×C₈×C₃₇

C₂²

yes

characteristic

C₂×Q₈×C₃₇

D₄

yes

—

C₂×D₄×C₃₇

—

C₂×D₄×C₃₇

—

C₂×D₁₄₈

—

C₂×D₁₄₈

D₄

yes

—

C₂×C₄×D₃₇

—

SD₁₆×C₃₇

—

C₂×C₈×C₃₇

—

C₄⋊C₄×C₃₇

—

C₄²×C₃₇

C₂³

yes

—

—

—

C₄⋊Dic₃₇

D₄

yes

—

C₄×Dic₃₇

—

C₂×#64436

—

C₂³×C₃₇

—

C₂²×D₃₇

—

Q₈×C₃₇

—

D₄×C₃₇

—

D₄×C₃₇

—

D₄×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

C₂×D₄

yes

—

C₂×C₄×C₃₇

C₂×D₄

yes

characteristic

C₂×Dic₃₇

—

D₁₄₈

—

D₁₄₈

—

D₁₄₈

—

C₄×D₃₇

—

C₄×D₃₇

—

C₈×C₃₇

—

#64436

—

C₂²×C₃₇

—

C₂²×C₃₇

—

C₂²×C₃₇

#120

yes

characteristic

C₂×D₃₇

—

C₂×D₃₇

—

C₂×D₃₇

—

C₄×C₃₇

—

C₄×C₃₇

—

C₄×C₃₇

#135

yes

—

C₄×C₃₇

AGL(1,ℤ/8)

yes

—

Dic₃₇

—

#1624

—

maximal, Sylow

C₂×C₃₇

—

C₂×C₃₇

#449

yes

—

C₂×C₃₇

#453

yes

characteristic

D₃₇

—

#555

—

#492

D₃₇

yes

characteristic, complemented

#459

—

#333

—

D₄⋊C₈

—

C₃₇

#1624

yes

characteristic, complemented, Sylow

C₂×SD₁₆

—

C₂×D₈

—

C₄⋊Q₈

C₂×D₃₇

yes

characteristic

#127

C₂×D₃₇

yes

characteristic

#124

—

#121

—

C₄×D₄

—

#117

—

C₄⋊C₈

—

#102

—

C₄×C₈

C₂×D₃₇

yes

characteristic

C₂×Q₈

C₂²⋊D₃₇

yes

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

—

SD₁₆

—

D₈

—

148

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

C₂²×D₃₇

yes

characteristic

C₂³

—

100

C₂³

—

101

Q₈

—

102

D₄

—

103

D₄

—

104

D₄

—

105

D₄

—

296

—

106

D₄

—

107

D₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

148

—

111

C₂×C₄

—

148

—

112

C₂×C₄

D₄×D₃₇

yes

—

113

C₂×C₄

C₂×C₂²⋊D₃₇

yes

characteristic

114

C₈

—

115

C₈

—

148

—

116

C₂²

—

117

C₂²

—

118

C₂²

—

148

—

119

C₂²

—

148

—

120

C₂²

—

148

—

121

C₂²

#11924504

yes

characteristic

122

C₄

—

123

C₄

—

124

C₄

—

148

—

125

C₄

#11924557

yes

—

126

C₄

#11924563

yes

—

127

C₂

—

128

C₂

—

296

—

129

C₂

#14557677

yes

—

130

C₂

#14557700

yes

characteristic

	subgroup	G/H
center	C₂²	#11924504
commutator, cosocle	C₄²×C₃₇	C₂³
Frattini	C₄²	C₂²×D₃₇
Fitting	C₃₇×#492	C₂
socle	C₂²×C₃₇	#120
hypercenter	#492	D₃₇
hypocenter	C₃₇	#1624
LCS	G ▷ C₄²×C₃₇ ▷ C₂²×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #492

subgroup

G/H

center

C₂²

#11924504

commutator, cosocle

C₄²×C₃₇

C₂³

Frattini

C₄²

C₂²×D₃₇

Fitting

C₃₇×#492

C₂

C₂²×C₃₇

D₃₇

C₃₇

G ▷ C₄²×C₃₇ ▷ C₂²×C₃₇ ▷ C₃₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #492

N	H ≅ G/N	type
1. #14558249	C₂	semidirect
2. #14557725	C₂	semidirect
3. #14557282	C₂	semidirect
4. #14557262	C₂	semidirect
5. C₃₇×#492	C₂	semidirect
67. #492	D₃₇	semidirect
71. C₃₇	#1624	semidirect

H ≅ G/N

type

1. #14558249

C₂

semidirect

2. #14557725

C₂

semidirect

3. #14557282

C₂

semidirect

4. #14557262

C₂

semidirect

5. C₃₇×#492

C₂

semidirect

67. #492

D₃₇

semidirect

71. C₃₇

#1624

semidirect

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

subgroup	index
#14558249	2
#14557725	2
#14557282	2
#14557262	2
C₃₇×#492	2
#1624	37

subgroup

index

C₃₇×#492

Subgroups — #17993677

Conjugacy classes of subgroups

Complements

Maximal subgroups