Group #17992897 — subgroups

Subgroups — #17992897

Order 4736.

178 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 178 conjugacy classes of subgroups — too many to draw. Only its 89 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	#14557584	C₂	1	1	yes	maximal, characteristic
2	#14557584	C₂	1	1	yes	maximal, characteristic
3	#14557584	C₂	1	1	yes	maximal, characteristic
4	#14557255	C₂	1	1	yes	maximal, characteristic, complemented
5	#14557254	C₂	1	1	yes	maximal, characteristic, complemented
6	C₈×#64436	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₇×#322	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄²×D₃₇	C₂²	1	1	yes	characteristic
9	#11924465	C₄	1	2	yes	—
10	#11924465	C₄	1	2	yes	—
11	#11924465	C₄	1	2	yes	—
12	#11924448	C₄	1	2	yes	—
13	#11924448	C₄	1	2	yes	—
14	#11924448	C₄	1	2	yes	—
15	C₄×#64436	C₂²	1	1	yes	characteristic
16	C₄×#64436	C₂²	1	1	yes	characteristic
17	C₄×#64436	C₂²	1	1	yes	characteristic
18	C₄×C₈×C₃₇	C₂²	1	1	yes	characteristic
19	C₄×C₈×C₃₇	C₂²	1	1	yes	characteristic
20	C₄×C₈×C₃₇	C₂²	1	1	yes	characteristic
21	C₂×C₄×D₃₇	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄×D₃₇	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄×D₃₇	C₂×C₄	1	1	yes	characteristic
24	C₂×C₈×C₃₇	C₂×C₄	1	2	yes	—
25	C₂×C₈×C₃₇	C₂×C₄	1	2	yes	—
26	C₂×C₈×C₃₇	C₂×C₄	1	2	yes	—
27	C₄²×C₃₇	C₂³	1	1	yes	characteristic
28	C₄×Dic₃₇	C₂×C₄	1	1	yes	characteristic
29	C₄×Dic₃₇	C₂×C₄	1	1	yes	characteristic
30	C₄×Dic₃₇	C₂×C₄	1	1	yes	characteristic
31	C₂×#64436	C₂×C₄	1	2	yes	—
32	C₂×#64436	C₂×C₄	1	2	yes	—
33	C₂×#64436	C₂×C₄	1	2	yes	—
34	C₂²×D₃₇	C₄²	1	1	yes	characteristic
35	C₂×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
36	C₂×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
37	C₂×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
38	C₂×Dic₃₇	C₄²	1	1	yes	characteristic
39	C₂×Dic₃₇	C₄²	1	1	yes	characteristic
40	C₂×Dic₃₇	C₄²	1	1	yes	characteristic
41	C₄×D₃₇	—	2	2	no	—
42	C₄×D₃₇	—	2	2	no	—
43	C₄×D₃₇	—	2	2	no	—
44	C₈×C₃₇	—	2	2	no	—
45	C₈×C₃₇	—	2	2	no	—
46	C₈×C₃₇	—	2	2	no	—
47	#64436	—	2	2	no	—
48	#64436	—	2	2	no	—
49	#64436	—	2	2	no	—
50	C₂²×C₃₇	C₂×C₄²	1	1	yes	characteristic
51	C₂×D₃₇	—	2	1	no	—
52	C₂×D₃₇	—	2	1	no	—
53	C₂×D₃₇	—	2	1	no	—
54	C₄×C₃₇	#131	1	2	yes	—
55	C₄×C₃₇	#131	1	2	yes	—
56	C₄×C₃₇	#131	1	2	yes	—
57	Dic₃₇	—	2	1	no	—
58	Dic₃₇	—	2	1	no	—
59	Dic₃₇	—	2	1	no	—
60	#1454	—	37	1	no	maximal, Sylow
61	C₂×C₃₇	#405	1	1	yes	characteristic
62	C₂×C₃₇	#405	1	1	yes	characteristic
63	C₂×C₃₇	#405	1	1	yes	characteristic
64	D₃₇	—	4	1	no	—
65	#431	—	37	1	no	—
66	#431	—	37	1	no	—
67	#431	—	37	1	no	—
68	#322	—	37	1	no	—
69	#322	—	37	1	no	—
70	#322	D₃₇	1	1	yes	characteristic, complemented
71	C₈²	—	37	1	no	—
72	C₃₇	#1454	1	1	yes	characteristic, complemented, Sylow
73	C₂×C₄²	—	37	1	no	—
74	C₄⋊C₈	—	37	2	no	—
75	C₄⋊C₈	—	37	2	no	—
76	C₄⋊C₈	—	37	2	no	—
77	C₂²⋊C₈	—	37	2	no	—
78	C₂²⋊C₈	—	37	2	no	—
79	C₂²⋊C₈	—	37	2	no	—
80	C₄×C₈	—	37	1	no	—
81	C₄×C₈	—	37	1	no	—
82	C₄×C₈	—	37	1	no	—
83	C₄×C₈	C₂×D₃₇	1	1	yes	characteristic
84	C₄×C₈	C₂×D₃₇	1	1	yes	characteristic
85	C₄×C₈	C₂×D₃₇	1	1	yes	characteristic
86	C₂²×C₄	—	37	1	no	—
87	C₂²×C₄	—	37	1	no	—
88	C₂²×C₄	—	37	1	no	—
89	C₂×C₈	—	37	2	no	—
90	C₂×C₈	—	37	2	no	—
91	C₂×C₈	—	37	2	no	—
92	C₂×C₈	C₄×D₃₇	1	2	yes	—
93	C₂×C₈	C₄×D₃₇	1	2	yes	—
94	C₂×C₈	C₄×D₃₇	1	2	yes	—
95	C₄²	—	37	1	no	—
96	C₄²	—	37	1	no	—
97	C₄²	—	37	1	no	—
98	C₄²	C₂²×D₃₇	1	1	yes	characteristic
99	C₂³	—	37	1	no	—
100	C₂×C₄	—	37	1	no	—
101	C₂×C₄	—	37	1	no	—
102	C₂×C₄	—	37	1	no	—
103	C₂×C₄	—	74	2	no	—
104	C₂×C₄	—	74	2	no	—
105	C₂×C₄	—	74	2	no	—
106	C₂×C₄	C₂×C₄×D₃₇	1	1	yes	characteristic
107	C₂×C₄	C₂×C₄×D₃₇	1	1	yes	characteristic
108	C₂×C₄	C₂×C₄×D₃₇	1	1	yes	characteristic
109	C₈	—	2	2	no	—
110	C₈	—	2	2	no	—
111	C₈	—	2	2	no	—
112	C₈	—	74	2	no	—
113	C₈	—	74	2	no	—
114	C₈	—	74	2	no	—
115	C₂²	—	74	1	no	—
116	C₂²	—	74	1	no	—
117	C₂²	—	74	1	no	—
118	C₂²	C₄²×D₃₇	1	1	yes	characteristic
119	C₄	—	74	1	no	—
120	C₄	—	74	1	no	—
121	C₄	—	74	1	no	—
122	C₄	#11924542	1	2	yes	—
123	C₄	#11924544	1	2	yes	—
124	C₄	#11924544	1	2	yes	—
125	C₂	—	148	1	no	—
126	C₂	#14557487	1	1	yes	characteristic
127	C₂	#14557490	1	1	yes	characteristic
128	C₂	#14557490	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14557584

C₂

yes

maximal, characteristic

#14557584

C₂

yes

maximal, characteristic

#14557584

C₂

yes

maximal, characteristic

#14557255

C₂

yes

maximal, characteristic, complemented

#14557254

C₂

yes

maximal, characteristic, complemented

C₈×#64436

C₂

yes

maximal, characteristic, complemented

C₃₇×#322

C₂

yes

maximal, characteristic, complemented

C₄²×D₃₇

C₂²

yes

characteristic

#11924465

C₄

yes

—

#11924465

C₄

yes

—

#11924465

C₄

yes

—

#11924448

C₄

yes

—

#11924448

C₄

yes

—

#11924448

C₄

yes

—

C₄×#64436

C₂²

yes

characteristic

C₄×#64436

C₂²

yes

characteristic

C₄×#64436

C₂²

yes

characteristic

C₄×C₈×C₃₇

C₂²

yes

characteristic

C₄×C₈×C₃₇

C₂²

yes

characteristic

C₄×C₈×C₃₇

C₂²

yes

characteristic

C₂×C₄×D₃₇

C₂×C₄

yes

characteristic

C₂×C₄×D₃₇

C₂×C₄

yes

characteristic

C₂×C₄×D₃₇

C₂×C₄

yes

characteristic

C₂×C₈×C₃₇

C₂×C₄

yes

—

C₂×C₈×C₃₇

C₂×C₄

yes

—

C₂×C₈×C₃₇

C₂×C₄

yes

—

C₄²×C₃₇

C₂³

yes

characteristic

C₄×Dic₃₇

C₂×C₄

yes

characteristic

C₄×Dic₃₇

C₂×C₄

yes

characteristic

C₄×Dic₃₇

C₂×C₄

yes

characteristic

C₂×#64436

C₂×C₄

yes

—

C₂×#64436

C₂×C₄

yes

—

C₂×#64436

C₂×C₄

yes

—

C₂²×D₃₇

C₄²

yes

characteristic

C₂×C₄×C₃₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₃₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₃₇

C₂²×C₄

yes

characteristic

C₂×Dic₃₇

C₄²

yes

characteristic

C₂×Dic₃₇

C₄²

yes

characteristic

C₂×Dic₃₇

C₄²

yes

characteristic

C₄×D₃₇

—

C₄×D₃₇

—

C₄×D₃₇

—

C₈×C₃₇

—

C₈×C₃₇

—

C₈×C₃₇

—

#64436

—

#64436

—

#64436

—

C₂²×C₃₇

C₂×C₄²

yes

characteristic

C₂×D₃₇

—

C₂×D₃₇

—

C₂×D₃₇

—

C₄×C₃₇

#131

yes

—

C₄×C₃₇

#131

yes

—

C₄×C₃₇

#131

yes

—

Dic₃₇

—

Dic₃₇

—

Dic₃₇

—

—

C₂×C₃₇

yes

C₂×C₃₇

yes

C₂×C₃₇

yes

D₃₇

—

—

—

—

—

—

D₃₇

yes

characteristic, complemented

C₈²

—

C₃₇

#1454

yes

characteristic, complemented, Sylow

C₂×C₄²

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₂²⋊C₈

—

C₂²⋊C₈

—

C₂²⋊C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×D₃₇

yes

characteristic

C₄×C₈

C₂×D₃₇

yes

characteristic

C₄×C₈

C₂×D₃₇

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₄×D₃₇

yes

—

C₂×C₈

C₄×D₃₇

yes

—

C₂×C₈

C₄×D₃₇

yes

—

C₄²

—

C₄²

—

C₄²

—

C₄²

C₂²×D₃₇

yes

characteristic

C₂³

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

C₂×C₄×D₃₇

yes

characteristic

107

C₂×C₄

C₂×C₄×D₃₇

yes

characteristic

108

C₂×C₄

C₂×C₄×D₃₇

yes

characteristic

109

C₈

—

110

C₈

—

111

C₈

—

112

C₈

—

113

C₈

—

114

C₈

—

115

C₂²

—

116

C₂²

—

117

C₂²

—

118

C₂²

C₄²×D₃₇

yes

characteristic

119

C₄

—

120

C₄

—

121

C₄

—

122

C₄

#11924542

yes

—

123

C₄

#11924544

yes

—

124

C₄

#11924544

yes

—

125

C₂

—

148

—

126

C₂

#14557487

yes

characteristic

127

C₂

#14557490

yes

characteristic

128

C₂

#14557490

yes

characteristic

	subgroup	G/H
center, Frattini	C₄²	C₂²×D₃₇
commutator, socle	C₂²×C₃₇	C₂×C₄²
Fitting	C₃₇×#322	C₂
prime radical	C₂²×D₃₇	C₄²
cosocle	C₄²×C₃₇	C₂³
hypercenter	#322	D₃₇
hypocenter	C₃₇	#1454
LCS	G ▷ C₂²×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₄² ◁ #322

subgroup

G/H

center, Frattini

C₄²

C₂²×D₃₇

commutator, socle

C₂²×C₃₇

C₂×C₄²

Fitting

C₃₇×#322

C₂

prime radical

C₂²×D₃₇

C₄²

cosocle

C₄²×C₃₇

C₂³

D₃₇

C₃₇

G ▷ C₂²×C₃₇ ▷ C₃₇

C₁ ◁ C₄² ◁ #322

N	H ≅ G/N	type
4. #14557255	C₂	semidirect
5. #14557254	C₂	semidirect
6. C₈×#64436	C₂	semidirect
7. C₃₇×#322	C₂	semidirect
70. #322	D₃₇	semidirect
72. C₃₇	#1454	semidirect

H ≅ G/N

type

4. #14557255

C₂

semidirect

5. #14557254

C₂

semidirect

6. C₈×#64436

C₂

semidirect

7. C₃₇×#322

C₂

semidirect

70. #322

D₃₇

semidirect

72. C₃₇

#1454

semidirect

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

subgroup	index
#14557584	2
#14557255	2
#14557254	2
C₈×#64436	2
C₃₇×#322	2
#1454	37

subgroup

index

C₃₇×#322

Subgroups — #17992897

Conjugacy classes of subgroups

Complements

Maximal subgroups