Group #17997776 — subgroups

Subgroups — #17997776

Order 4736.

424 conjugacy classes of subgroups; only the 173 normal subgroups are shown.

normal subgroups only

This group has 424 conjugacy classes of subgroups — too many to draw. Only its 173 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	#14557437	C₂	1	4	yes	maximal
3	#14557418	C₂	1	4	yes	maximal
4	#14557416	C₂	1	4	yes	maximal
5	#14557401	C₂	1	1	yes	maximal, characteristic
6	C₃₇×#383	C₂	1	1	yes	maximal, characteristic
7	C₂²×Dic₇₄	C₂²	1	2	yes	—
8	#11924536	C₄	1	8	yes	complemented
9	C₂×C₄⋊Dic₃₇	C₂²	1	2	yes	—
10	C₂×C₄⋊Dic₃₇	C₂²	1	2	yes	—
11	C₂×#10596936	C₂²	1	4	yes	—
12	C₂×C₄×Dic₃₇	C₂²	1	4	yes	—
13	#11924443	C₂²	1	2	yes	—
14	#11924443	C₂²	1	2	yes	—
15	#11924443	C₂²	1	2	yes	—
16	#11924443	C₂²	1	8	yes	—
17	C₂×C₄²×C₃₇	C₂²	1	1	yes	characteristic
18	#95×C₃₇	C₂²	1	2	yes	—
19	#95×C₃₇	C₂²	1	2	yes	—
20	#95×C₃₇	C₂²	1	2	yes	—
21	C₂²×C₄×C₃₇	C₂³	1	4	yes	—
22	C₂²×C₄×C₃₇	C₂³	1	1	yes	characteristic
23	C₂²×C₄×C₃₇	C₂³	1	1	yes	characteristic
24	C₂²×C₄×C₃₇	C₂³	1	1	yes	characteristic
25	C₂²×Dic₃₇	C₂³	1	4	yes	—
26	C₂²×Dic₃₇	C₂³	1	4	yes	—
27	C₂×Dic₇₄	—	4	2	no	—
28	C₂×Dic₇₄	—	4	2	no	—
29	C₂×Dic₇₄	C₂×C₄	1	8	yes	—
30	C₄²×C₃₇	C₂×C₄	1	4	yes	—
31	C₄⋊Dic₃₇	C₂×C₄	1	8	yes	—
32	C₄⋊Dic₃₇	C₂×C₄	1	8	yes	—
33	#10596936	—	4	4	no	—
34	C₄×Dic₃₇	—	4	4	no	—
35	C₂³×C₃₇	C₂⁴	1	1	yes	characteristic
36	C₂×C₄×C₃₇	—	2	2	no	—
37	C₂×C₄×C₃₇	—	2	2	no	—
38	C₂×C₄×C₃₇	—	2	2	no	—
39	C₂×C₄×C₃₇	—	2	4	no	—
40	C₂×C₄×C₃₇	—	2	4	no	—
41	C₂×C₄×C₃₇	—	2	4	no	—
42	C₂×C₄×C₃₇	C₂²×C₄	1	2	yes	—
43	C₂×C₄×C₃₇	C₂²×C₄	1	2	yes	—
44	C₂×C₄×C₃₇	C₂²×C₄	1	2	yes	—
45	C₂×Dic₃₇	—	2	4	no	—
46	C₂×Dic₃₇	—	2	4	no	—
47	C₂×Dic₃₇	—	2	4	no	—
48	C₂×Dic₃₇	—	2	4	no	—
49	C₂×Dic₃₇	—	2	4	no	—
50	C₂×Dic₃₇	C₂²×C₄	1	8	yes	—
51	Dic₇₄	—	4	8	no	—
52	C₂²×C₃₇	C₂³×C₄	1	1	yes	characteristic
53	C₂²×C₃₇	H(5,𝔽₂)	1	2	yes	—
54	C₂²×C₃₇	2₋¹⁺⁴	1	2	yes	—
55	C₂²×C₃₇	2₋¹⁺⁴	1	2	yes	—
56	C₄×C₃₇	—	2	2	no	—
57	C₄×C₃₇	—	2	2	no	—
58	C₄×C₃₇	—	2	2	no	—
59	C₄×C₃₇	—	4	4	no	—
60	Dic₃₇	—	2	8	no	—
61	Dic₃₇	—	4	4	no	—
62	#2381	—	37	1	no	maximal, Sylow
63	C₂×C₃₇	#518	1	1	yes	characteristic
64	C₂×C₃₇	#519	1	1	yes	characteristic
65	C₂×C₃₇	#519	1	1	yes	characteristic
66	C₂×C₃₇	#563	1	4	yes	—
67	C₂×C₄⋊Q₈	—	37	1	no	—
68	#391	—	37	4	no	—
69	#387	—	37	4	no	—
70	#387	—	37	4	no	—
71	#383	—	37	1	no	—
72	#383	D₃₇	1	1	yes	characteristic
73	C₃₇	#2381	1	1	yes	characteristic, complemented, Sylow
74	C₂²×Q₈	—	37	2	no	—
75	C₄⋊Q₈	—	37	8	no	—
76	C₂×C₄⋊C₄	—	37	2	no	—
77	C₂×C₄⋊C₄	—	37	2	no	—
78	C₂×C₄⋊C₄	—	37	4	no	—
79	C₂×C₄²	—	37	4	no	—
80	C₂×C₄²	C₂×D₃₇	1	1	yes	characteristic
81	#95	—	37	2	no	—
82	#95	—	37	2	no	—
83	#95	—	37	2	no	—
84	#95	—	37	8	no	—
85	#95	C₂×D₃₇	1	2	yes	—
86	#95	C₂×D₃₇	1	2	yes	—
87	#95	C₂×D₃₇	1	2	yes	—
88	C₂×Q₈	—	37	8	no	—
89	C₂×Q₈	—	148	2	no	—
90	C₂×Q₈	—	148	2	no	—
91	C₂²×C₄	—	37	4	no	—
92	C₂²×C₄	—	37	4	no	—
93	C₂²×C₄	C₂²×D₃₇	1	4	yes	—
94	C₂²×C₄	C₂²×D₃₇	1	1	yes	characteristic
95	C₂²×C₄	C₂²×D₃₇	1	1	yes	characteristic
96	C₂²×C₄	C₂²×D₃₇	1	1	yes	characteristic
97	C₄⋊C₄	—	37	8	no	—
98	C₄⋊C₄	—	37	8	no	—
99	C₄⋊C₄	—	148	4	no	—
100	C₄²	—	148	4	no	—
101	C₄²	C₄×D₃₇	1	4	yes	—
102	C₂³	C₂³×D₃₇	1	1	yes	characteristic
103	Q₈	—	148	8	no	—
104	C₂×C₄	—	2	2	no	—
105	C₂×C₄	—	2	2	no	—
106	C₂×C₄	—	2	2	no	—
107	C₂×C₄	—	2	4	no	—
108	C₂×C₄	—	2	4	no	—
109	C₂×C₄	—	2	4	no	—
110	C₂×C₄	—	37	8	no	—
111	C₂×C₄	—	74	4	no	—
112	C₂×C₄	—	74	4	no	—
113	C₂×C₄	—	74	4	no	—
114	C₂×C₄	—	74	4	no	—
115	C₂×C₄	—	74	4	no	—
116	C₂×C₄	C₂×C₄×D₃₇	1	2	yes	—
117	C₂×C₄	C₂×C₄×D₃₇	1	2	yes	—
118	C₂×C₄	C₂×C₄×D₃₇	1	2	yes	—
119	C₂²	C₂²×C₄×D₃₇	1	1	yes	characteristic
120	C₂²	#11924582	1	2	yes	—
121	C₂²	#11924583	1	2	yes	—
122	C₂²	#11924583	1	2	yes	—
123	C₄	—	2	2	no	—
124	C₄	—	2	2	no	—
125	C₄	—	2	2	no	—
126	C₄	—	4	4	no	—
127	C₄	—	74	8	no	—
128	C₄	—	148	4	no	—
129	C₂	#14558007	1	1	yes	characteristic
130	C₂	#14558010	1	1	yes	characteristic
131	C₂	#14558010	1	1	yes	characteristic
132	C₂	#14558285	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#11924536

C₂

yes

maximal, characteristic

#14557437

C₂

yes

maximal

#14557418

C₂

yes

maximal

#14557416

C₂

yes

maximal

#14557401

C₂

yes

maximal, characteristic

C₃₇×#383

C₂

yes

maximal, characteristic

C₂²×Dic₇₄

C₂²

yes

—

#11924536

C₄

yes

complemented

C₂×C₄⋊Dic₃₇

C₂²

yes

—

C₂×C₄⋊Dic₃₇

C₂²

yes

—

C₂×#10596936

C₂²

yes

—

C₂×C₄×Dic₃₇

C₂²

yes

—

#11924443

C₂²

yes

—

#11924443

C₂²

yes

—

#11924443

C₂²

yes

—

#11924443

C₂²

yes

—

C₂×C₄²×C₃₇

C₂²

yes

characteristic

#95×C₃₇

C₂²

yes

—

#95×C₃₇

C₂²

yes

—

#95×C₃₇

C₂²

yes

—

C₂²×C₄×C₃₇

C₂³

yes

—

C₂²×C₄×C₃₇

C₂³

yes

characteristic

C₂²×C₄×C₃₇

C₂³

yes

characteristic

C₂²×C₄×C₃₇

C₂³

yes

characteristic

C₂²×Dic₃₇

C₂³

yes

—

C₂²×Dic₃₇

C₂³

yes

—

C₂×Dic₇₄

—

C₂×Dic₇₄

—

C₂×Dic₇₄

C₂×C₄

yes

—

C₄²×C₃₇

C₂×C₄

yes

—

C₄⋊Dic₃₇

C₂×C₄

yes

—

C₄⋊Dic₃₇

C₂×C₄

yes

—

#10596936

—

C₄×Dic₃₇

—

C₂³×C₃₇

C₂⁴

yes

characteristic

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

C₂²×C₄

yes

—

C₂×C₄×C₃₇

C₂²×C₄

yes

—

C₂×C₄×C₃₇

C₂²×C₄

yes

—

C₂×Dic₃₇

—

C₂×Dic₃₇

—

C₂×Dic₃₇

—

C₂×Dic₃₇

—

C₂×Dic₃₇

—

C₂×Dic₃₇

C₂²×C₄

yes

—

Dic₇₄

—

C₂²×C₃₇

C₂³×C₄

yes

characteristic

C₂²×C₃₇

H(5,𝔽₂)

yes

—

C₂²×C₃₇

2₋¹⁺⁴

yes

—

C₂²×C₃₇

2₋¹⁺⁴

yes

—

C₄×C₃₇

—

C₄×C₃₇

—

C₄×C₃₇

—

C₄×C₃₇

—

Dic₃₇

—

Dic₃₇

—

—

C₂×C₃₇

yes

C₂×C₃₇

yes

C₂×C₃₇

yes

C₂×C₃₇

yes

—

C₂×C₄⋊Q₈

—

—

—

—

—

D₃₇

yes

characteristic

C₃₇

#2381

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₄⋊Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

—

C₂×C₄²

C₂×D₃₇

yes

—

—

—

—

C₂×D₃₇

yes

—

#95

C₂×D₃₇

yes

—

#95

C₂×D₃₇

yes

—

C₂×Q₈

—

C₂×Q₈

—

148

—

C₂×Q₈

—

148

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₃₇

yes

—

C₂²×C₄

C₂²×D₃₇

yes

characteristic

C₂²×C₄

C₂²×D₃₇

yes

characteristic

C₂²×C₄

C₂²×D₃₇

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

148

—

100

C₄²

—

148

—

101

C₄²

C₄×D₃₇

yes

—

102

C₂³

C₂³×D₃₇

yes

characteristic

103

Q₈

—

148

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

C₂×C₄×D₃₇

yes

—

117

C₂×C₄

C₂×C₄×D₃₇

yes

—

118

C₂×C₄

C₂×C₄×D₃₇

yes

—

119

C₂²

C₂²×C₄×D₃₇

yes

characteristic

120

C₂²

#11924582

yes

—

121

C₂²

#11924583

yes

—

122

C₂²

#11924583

yes

—

123

C₄

—

124

C₄

—

125

C₄

—

126

C₄

—

127

C₄

—

128

C₄

—

148

—

129

C₂

#14558007

yes

characteristic

130

C₂

#14558010

yes

characteristic

131

C₂

#14558010

yes

characteristic

132

C₂

#14558285

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₃₇
commutator	C₂²×C₃₇	C₂³×C₄
Fitting	C₃₇×#383	C₂
socle, prime radical, cosocle	C₂³×C₃₇	C₂⁴
hypercenter	#383	D₃₇
hypocenter	C₃₇	#2381
LCS	G ▷ C₂²×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂³ ◁ #383

subgroup

G/H

center, Frattini

C₂³

C₂³×D₃₇

commutator

C₂²×C₃₇

C₂³×C₄

Fitting

C₃₇×#383

C₂

socle, prime radical, cosocle

C₂³×C₃₇

C₂⁴

D₃₇

C₃₇

G ▷ C₂²×C₃₇ ▷ C₃₇

C₁ ◁ C₂³ ◁ #383

N	H ≅ G/N	type
8. #11924536	C₄	semidirect
73. C₃₇	#2381	semidirect

H ≅ G/N

type

8. #11924536

C₄

semidirect

73. C₃₇

#2381

semidirect

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

subgroup	index
C₂×#11924536	2
#14557437	2
#14557418	2
#14557416	2
#14557401	2
C₃₇×#383	2
#2381	37

subgroup

index

C₃₇×#383

Subgroups — #17997776

Conjugacy classes of subgroups

Complements

Maximal subgroups