Group #17994488 — subgroups

Subgroups — #17994488

Order 4736.

280 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 280 conjugacy classes of subgroups — too many to draw. Only its 129 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×#11924469	C₂	1	2	yes	maximal
2	C₂×C₈×Dic₃₇	C₂	1	1	yes	maximal, characteristic
3	C₄⋊C₄×Dic₃₇	C₂	1	2	yes	maximal
4	#14557296	C₂	1	1	yes	maximal, characteristic
5	C₃₇×#340	C₂	1	1	yes	maximal, characteristic
6	C₂²×#64436	C₂²	1	1	yes	characteristic
7	C₂×C₄⋊Dic₃₇	C₂²	1	2	yes	—
8	C₂×C₄×Dic₃₇	—	2	2	no	—
9	C₂×C₄×Dic₃₇	C₂²	1	1	yes	characteristic
10	#11924469	C₄	1	8	yes	complemented
11	C₈×Dic₃₇	C₄	1	4	yes	complemented
12	#11924443	—	2	2	no	—
13	C₂²×C₈×C₃₇	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊C₄×C₃₇	C₂²	1	2	yes	—
15	C₂²×C₄×C₃₇	—	2	2	no	—
16	C₂²×C₄×C₃₇	C₂³	1	1	yes	characteristic
17	C₂²×Dic₃₇	—	2	2	no	—
18	C₂²×Dic₃₇	D₄	1	1	yes	characteristic
19	C₂²×Dic₃₇	D₄	1	1	yes	characteristic
20	C₂×C₈×C₃₇	—	2	2	no	—
21	C₂×C₈×C₃₇	C₂×C₄	1	2	yes	—
22	C₄⋊C₄×C₃₇	—	2	2	no	—
23	C₄⋊C₄×C₃₇	C₂×C₄	1	4	yes	—
24	C₄⋊Dic₃₇	—	2	2	no	—
25	C₄⋊Dic₃₇	C₂×C₄	1	4	yes	—
26	C₄×Dic₃₇	—	4	4	no	—
27	C₄×Dic₃₇	C₂×C₄	1	2	yes	—
28	C₄×Dic₃₇	Q₈	1	2	yes	—
29	C₂×#64436	C₂×C₄	1	2	yes	—
30	C₂×#64436	C₂×C₄	1	4	yes	—
31	C₂³×C₃₇	C₂×D₄	1	1	yes	characteristic
32	C₂×C₄×C₃₇	—	2	4	no	—
33	C₂×C₄×C₃₇	—	4	2	no	—
34	C₂×C₄×C₃₇	—	4	2	no	—
35	C₂×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
36	C₂×C₄×C₃₇	C₂²×C₄	1	2	yes	—
37	C₂×C₄×C₃₇	C₂×Q₈	1	1	yes	characteristic
38	C₂×C₄×C₃₇	#41	1	2	yes	—
39	C₂×Dic₃₇	—	2	2	no	—
40	C₂×Dic₃₇	—	2	4	no	—
41	C₂×Dic₃₇	—	4	2	no	—
42	C₂×Dic₃₇	—	4	2	no	—
43	C₂×Dic₃₇	C₄⋊C₄	1	2	yes	—
44	C₂×Dic₃₇	C₄⋊C₄	1	2	yes	—
45	C₂×Dic₃₇	D₈	1	2	yes	—
46	C₂×Dic₃₇	Q₁₆	1	2	yes	—
47	C₈×C₃₇	—	2	2	no	—
48	#64436	C₄²	1	4	yes	complemented
49	C₂²×C₃₇	C₂×C₄⋊C₄	1	1	yes	characteristic
50	C₂²×C₃₇	C₄×D₄	1	2	yes	—
51	C₂²×C₃₇	C₂×D₈	1	1	yes	characteristic
52	C₂²×C₃₇	C₂×Q₁₆	1	1	yes	characteristic
53	C₂²×C₃₇	#135	1	2	yes	—
54	C₄×C₃₇	—	4	4	no	—
55	C₄×C₃₇	C₂×C₄²	1	1	yes	characteristic
56	C₄×C₃₇	#117	1	1	yes	characteristic
57	C₄×C₃₇	C₄×Q₈	1	2	yes	—
58	Dic₃₇	—	2	2	no	—
59	Dic₃₇	—	4	4	no	—
60	Dic₃₇	#107	1	4	yes	complemented
61	C₄×#107	—	37	1	no	maximal, Sylow
62	C₂×C₃₇	C₄×C₄⋊C₄	1	1	yes	characteristic
63	C₂×C₃₇	C₂×#107	1	1	yes	characteristic
64	C₂×C₃₇	#427	1	1	yes	characteristic
65	C₂×C₃₇	C₄×D₈	1	2	yes	—
66	C₂×C₃₇	C₄×Q₁₆	1	2	yes	—
67	C₂×#107	—	37	2	no	—
68	C₂×C₄×C₈	—	37	1	no	—
69	C₄×C₄⋊C₄	—	37	2	no	—
70	#340	—	37	1	no	—
71	#340	D₃₇	1	1	yes	characteristic
72	C₃₇	C₄×#107	1	1	yes	characteristic, complemented, Sylow
73	C₂²×C₈	—	37	1	no	—
74	C₂²×C₈	C₂×D₃₇	1	1	yes	characteristic
75	C₂×C₄⋊C₄	—	37	2	no	—
76	C₂×C₄⋊C₄	C₂×D₃₇	1	2	yes	—
77	C₂×C₄²	—	37	1	no	—
78	C₂×C₄²	—	74	2	no	—
79	#107	—	37	8	no	—
80	C₄×C₈	—	37	4	no	—
81	#95	—	74	2	no	—
82	C₂²×C₄	—	2	2	no	—
83	C₂²×C₄	—	37	1	no	—
84	C₂²×C₄	—	37	1	no	—
85	C₂²×C₄	—	74	2	no	—
86	C₂²×C₄	C₂²×D₃₇	1	1	yes	characteristic
87	C₂×C₈	—	2	2	no	—
88	C₂×C₈	—	37	2	no	—
89	C₂×C₈	—	37	4	no	—
90	C₂×C₈	C₄×D₃₇	1	2	yes	—
91	C₄⋊C₄	—	2	2	no	—
92	C₄⋊C₄	—	37	4	no	—
93	C₄⋊C₄	—	74	2	no	—
94	C₄⋊C₄	C₄×D₃₇	1	4	yes	—
95	C₄²	—	37	2	no	—
96	C₄²	—	37	2	no	—
97	C₄²	—	148	4	no	—
98	C₂³	D₄×D₃₇	1	1	yes	characteristic
99	C₂×C₄	—	2	4	no	—
100	C₂×C₄	—	4	2	no	—
101	C₂×C₄	—	4	2	no	—
102	C₂×C₄	—	37	2	no	—
103	C₂×C₄	—	37	2	no	—
104	C₂×C₄	—	37	2	no	—
105	C₂×C₄	—	37	2	no	—
106	C₂×C₄	—	74	2	no	—
107	C₂×C₄	—	74	4	no	—
108	C₂×C₄	—	148	2	no	—
109	C₂×C₄	—	148	2	no	—
110	C₂×C₄	C₂×C₄×D₃₇	1	1	yes	characteristic
111	C₂×C₄	C₂×C₄×D₃₇	1	2	yes	—
112	C₂×C₄	#10596964	1	2	yes	—
113	C₂×C₄	Q₈×D₃₇	1	1	yes	characteristic
114	C₈	—	2	2	no	—
115	C₈	—	37	4	no	—
116	C₂²	C₄⋊C₄×D₃₇	1	1	yes	characteristic
117	C₂²	#11924496	1	2	yes	—
118	C₂²	D₈×D₃₇	1	1	yes	characteristic
119	C₂²	Q₁₆×D₃₇	1	1	yes	characteristic
120	C₂²	#11924557	1	2	yes	—
121	C₄	—	4	4	no	—
122	C₄	—	37	4	no	—
123	C₄	—	74	2	no	—
124	C₄	—	148	4	no	—
125	C₄	C₄²×D₃₇	1	1	yes	characteristic
126	C₄	#11924492	1	1	yes	characteristic
127	C₄	#11924501	1	2	yes	—
128	C₂	#14557384	1	1	yes	characteristic
129	C₂	#107×D₃₇	1	1	yes	characteristic
130	C₂	#14557568	1	1	yes	characteristic
131	C₂	#14557613	1	2	yes	—
132	C₂	#14557625	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#11924469

C₂

yes

maximal

C₂×C₈×Dic₃₇

C₂

yes

maximal, characteristic

C₄⋊C₄×Dic₃₇

C₂

yes

maximal

#14557296

C₂

yes

maximal, characteristic

C₃₇×#340

C₂

yes

maximal, characteristic

C₂²×#64436

C₂²

yes

characteristic

C₂×C₄⋊Dic₃₇

C₂²

yes

—

C₂×C₄×Dic₃₇

—

C₂×C₄×Dic₃₇

C₂²

yes

characteristic

#11924469

C₄

yes

complemented

C₈×Dic₃₇

C₄

yes

complemented

#11924443

—

C₂²×C₈×C₃₇

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₃₇

C₂²

yes

—

C₂²×C₄×C₃₇

—

C₂²×C₄×C₃₇

C₂³

yes

characteristic

C₂²×Dic₃₇

—

C₂²×Dic₃₇

D₄

yes

characteristic

C₂²×Dic₃₇

D₄

yes

characteristic

C₂×C₈×C₃₇

—

C₂×C₈×C₃₇

C₂×C₄

yes

—

C₄⋊C₄×C₃₇

—

C₄⋊C₄×C₃₇

C₂×C₄

yes

—

C₄⋊Dic₃₇

—

C₄⋊Dic₃₇

C₂×C₄

yes

—

C₄×Dic₃₇

—

C₄×Dic₃₇

C₂×C₄

yes

—

C₄×Dic₃₇

Q₈

yes

—

C₂×#64436

C₂×C₄

yes

—

C₂×#64436

C₂×C₄

yes

—

C₂³×C₃₇

C₂×D₄

yes

characteristic

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₃₇

C₂²×C₄

yes

—

C₂×C₄×C₃₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₃₇

#41

yes

—

C₂×Dic₃₇

—

C₂×Dic₃₇

—

C₂×Dic₃₇

—

C₂×Dic₃₇

—

C₂×Dic₃₇

C₄⋊C₄

yes

—

C₂×Dic₃₇

C₄⋊C₄

yes

—

C₂×Dic₃₇

D₈

yes

—

C₂×Dic₃₇

Q₁₆

yes

—

C₈×C₃₇

—

#64436

C₄²

yes

complemented

C₂²×C₃₇

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₃₇

C₄×D₄

yes

—

C₂²×C₃₇

C₂×D₈

yes

characteristic

C₂²×C₃₇

C₂×Q₁₆

yes

characteristic

C₂²×C₃₇

#135

yes

—

C₄×C₃₇

—

C₄×C₃₇

C₂×C₄²

yes

characteristic

C₄×C₃₇

#117

yes

characteristic

C₄×C₃₇

C₄×Q₈

yes

—

Dic₃₇

—

Dic₃₇

—

Dic₃₇

yes

—

C₂×C₃₇

C₄×C₄⋊C₄

yes

C₂×C₃₇

yes

C₂×C₃₇

yes

C₂×C₃₇

C₄×D₈

yes

—

C₂×C₃₇

C₄×Q₁₆

yes

—

C₂×#107

—

C₂×C₄×C₈

—

C₄×C₄⋊C₄

—

#340

—

#340

D₃₇

yes

characteristic

C₃₇

C₄×#107

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂²×C₈

C₂×D₃₇

yes

characteristic

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₃₇

yes

—

C₂×C₄²

—

C₂×C₄²

—

#107

—

C₄×C₈

—

#95

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₃₇

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₄×D₃₇

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₄×D₃₇

yes

—

C₄²

—

C₄²

—

C₄²

—

148

—

C₂³

D₄×D₃₇

yes

characteristic

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

148

—

109

C₂×C₄

—

148

—

110

C₂×C₄

C₂×C₄×D₃₇

yes

characteristic

111

C₂×C₄

C₂×C₄×D₃₇

yes

—

112

C₂×C₄

#10596964

yes

—

113

C₂×C₄

Q₈×D₃₇

yes

characteristic

114

C₈

—

115

C₈

—

116

C₂²

C₄⋊C₄×D₃₇

yes

characteristic

117

C₂²

#11924496

yes

—

118

C₂²

D₈×D₃₇

yes

characteristic

119

C₂²

Q₁₆×D₃₇

yes

characteristic

120

C₂²

#11924557

yes

—

121

C₄

—

122

C₄

—

123

C₄

—

124

C₄

—

148

—

125

C₄

C₄²×D₃₇

yes

characteristic

126

C₄

#11924492

yes

characteristic

127

C₄

#11924501

yes

—

128

C₂

#14557384

yes

characteristic

129

C₂

#107×D₃₇

yes

characteristic

130

C₂

#14557568

yes

characteristic

131

C₂

#14557613

yes

—

132

C₂

#14557625

yes

—

	subgroup	G/H
center	C₂³	D₄×D₃₇
commutator	C₄×C₃₇	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₃₇
Fitting	C₃₇×#340	C₂
socle, prime radical	C₂³×C₃₇	C₂×D₄
cosocle	C₂²×C₄×C₃₇	C₂³
hypercenter	#340	D₃₇
hypocenter	C₃₇	C₄×#107
LCS	G ▷ C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ #340

subgroup

G/H

center

C₂³

D₄×D₃₇

commutator

C₄×C₃₇

C₂×C₄²

Frattini

C₂²×C₄

C₂²×D₃₇

Fitting

C₃₇×#340

C₂

socle, prime radical

C₂³×C₃₇

C₂×D₄

cosocle

C₂²×C₄×C₃₇

C₂³

D₃₇

C₃₇

G ▷ C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ #340

N	H ≅ G/N	type
10. #11924469	C₄	semidirect
11. C₈×Dic₃₇	C₄	semidirect
48. #64436	C₄²	semidirect
60. Dic₃₇	#107	semidirect
72. C₃₇	C₄×#107	semidirect

H ≅ G/N

type

10. #11924469

C₄

semidirect

11. C₈×Dic₃₇

C₄

semidirect

48. #64436

C₄²

semidirect

60. Dic₃₇

#107

semidirect

72. C₃₇

C₄×#107

semidirect

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

subgroup	index
C₂×#11924469	2
C₂×C₈×Dic₃₇	2
C₄⋊C₄×Dic₃₇	2
#14557296	2
C₃₇×#340	2
C₄×#107	37

subgroup

index

C₂×#11924469

C₂×C₈×Dic₃₇

C₄⋊C₄×Dic₃₇

#14557296

C₃₇×#340

C₄×#107

Subgroups — #17994488

Conjugacy classes of subgroups

Complements

Maximal subgroups