Group #17993328 — subgroups

Subgroups — #17993328

Order 4736.

254 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 254 conjugacy classes of subgroups — too many to draw. Only its 67 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	#14558081	C₂	1	1	yes	maximal, characteristic
2	#14557555	C₂	1	2	yes	maximal, complemented
3	#14557281	C₂	1	2	yes	maximal, complemented
4	#14557277	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₇×#331	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#10596964	C₄	1	2	yes	—
7	C₂×#10596929	—	2	2	no	—
8	#11924536	C₂²	1	1	yes	characteristic
9	#11924523	C₄	1	2	yes	—
10	#11924507	—	2	2	no	—
11	D₄×Dic₃₇	—	2	2	no	—
12	C₄²×D₃₇	C₂²	1	1	yes	characteristic
13	C₄⋊#64436	C₂²	1	2	yes	—
14	#11924465	—	2	2	no	—
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	C₂×C₄	1	2	yes	—
19	C₂×C₂²⋊D₃₇	—	2	2	no	—
20	C₂²×Dic₃₇	—	2	2	no	—
21	#10596964	—	4	4	no	—
22	C₂×C₄×D₃₇	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄×D₃₇	D₄	1	2	yes	—
24	C₂×Dic₇₄	C₂×C₄	1	2	yes	—
25	C₂×C₈×C₃₇	—	2	2	no	—
26	C₄²×C₃₇	C₂³	1	1	yes	characteristic
27	C₂²⋊Dic₃₇	—	2	4	no	—
28	C₄⋊Dic₃₇	—	2	2	no	—
29	C₄×Dic₃₇	C₂×C₄	1	1	yes	characteristic
30	C₄×Dic₃₇	D₄	1	2	yes	—
31	C₂×#64436	—	2	2	no	—
32	#10596929	—	4	4	no	—
33	C₂³×C₃₇	—	2	2	no	—
34	C₂²×D₃₇	C₂²⋊C₄	1	1	yes	characteristic
35	D₄×C₃₇	—	4	2	no	—
36	D₄×C₃₇	—	4	2	no	—
37	D₄×C₃₇	—	4	2	no	—
38	C₂×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
39	C₂×C₄×C₃₇	C₂×D₄	1	2	yes	—
40	C₂²⋊D₃₇	—	4	4	no	—
41	C₂×Dic₃₇	—	2	2	no	—
42	C₂×Dic₃₇	—	4	4	no	—
43	C₂×Dic₃₇	C₂²⋊C₄	1	2	yes	—
44	C₂×Dic₃₇	C₂²⋊C₄	1	1	yes	characteristic
45	C₄×D₃₇	—	2	2	no	—
46	C₄×D₃₇	—	2	2	no	—
47	C₄×D₃₇	—	2	2	no	—
48	Dic₇₄	—	4	2	no	—
49	C₈×C₃₇	—	4	2	no	—
50	#64436	—	4	2	no	—
51	C₂²×C₃₇	—	4	2	no	—
52	C₂²×C₃₇	—	4	4	no	—
53	C₂²×C₃₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
54	C₂×D₃₇	—	2	1	no	—
55	C₂×D₃₇	—	2	2	no	—
56	C₄×C₃₇	—	2	1	no	—
57	C₄×C₃₇	AGL(1,ℤ/8)	1	2	yes	—
58	C₄×C₃₇	#137	1	2	yes	—
59	Dic₃₇	—	2	2	no	—
60	Dic₃₇	—	4	2	no	—
61	Dic₃₇	#100	1	2	yes	—
62	#1560	—	37	1	no	maximal, Sylow
63	C₂×C₃₇	—	8	2	no	—
64	C₂×C₃₇	C₂×#100	1	1	yes	characteristic
65	C₂×C₃₇	#417	1	2	yes	—
66	D₃₇	—	4	1	no	—
67	#532	—	37	1	no	—
68	#423	—	37	2	no	—
69	#333	—	37	2	no	—
70	#331	—	37	1	no	—
71	#331	D₃₇	1	1	yes	characteristic, complemented
72	C₃₇	#1560	1	1	yes	characteristic, complemented, Sylow
73	C₂×#41	—	37	2	no	—
74	C₂×M₁₆	—	74	2	no	—
75	C₄⋊Q₈	—	37	1	no	—
76	#127	C₂×D₃₇	1	1	yes	characteristic
77	#124	—	37	2	no	—
78	#121	—	74	2	no	—
79	C₄×D₄	—	74	2	no	—
80	C₂×C₄²	—	37	1	no	—
81	C₄⋊C₈	—	37	2	no	—
82	C₄⋊C₈	—	74	2	no	—
83	C₄⋊C₈	C₂×D₃₇	1	2	yes	—
84	#41	—	148	4	no	—
85	C₂×Q₈	—	37	2	no	—
86	C₂×D₄	—	2	2	no	—
87	C₂×D₄	—	74	2	no	—
88	C₂×D₄	C₄×D₃₇	1	2	yes	—
89	C₂²×C₄	—	37	1	no	—
90	C₂²×C₄	—	37	2	no	—
91	C₂²×C₄	—	74	2	no	—
92	M₁₆	—	148	4	no	—
93	C₂×C₈	—	2	2	no	—
94	C₂×C₈	—	74	2	no	—
95	C₄⋊C₄	—	74	2	no	—
96	C₂²⋊C₄	—	74	4	no	—
97	C₄²	—	37	1	no	—
98	C₄²	—	37	2	no	—
99	C₄²	C₂²×D₃₇	1	1	yes	characteristic
100	C₂³	—	2	2	no	—
101	C₂³	—	37	1	no	—
102	Q₈	—	148	2	no	—
103	D₄	—	4	2	no	—
104	D₄	—	4	2	no	—
105	D₄	—	4	2	no	—
106	D₄	—	148	4	no	—
107	C₂×C₄	—	37	1	no	—
108	C₂×C₄	—	37	2	no	—
109	C₂×C₄	—	74	2	no	—
110	C₂×C₄	—	74	2	no	—
111	C₂×C₄	—	74	2	no	—
112	C₂×C₄	—	74	2	no	—
113	C₂×C₄	—	148	4	no	—
114	C₂×C₄	C₂×C₄×D₃₇	1	1	yes	characteristic
115	C₂×C₄	D₄×D₃₇	1	2	yes	—
116	C₈	—	4	2	no	—
117	C₈	—	148	2	no	—
118	C₂²	—	4	2	no	—
119	C₂²	—	4	4	no	—
120	C₂²	—	74	1	no	—
121	C₂²	—	74	2	no	—
122	C₂²	C₂²⋊C₄×D₃₇	1	1	yes	characteristic
123	C₄	—	2	1	no	—
124	C₄	—	37	2	no	—
125	C₄	—	74	2	no	—
126	C₄	—	148	2	no	—
127	C₄	#11924560	1	2	yes	—
128	C₄	#11924565	1	2	yes	—
129	C₂	—	8	2	no	—
130	C₂	—	148	1	no	—
131	C₂	#100×D₃₇	1	1	yes	characteristic
132	C₂	#14557531	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#14558081

C₂

yes

maximal, characteristic

#14557555

C₂

yes

maximal, complemented

#14557281

C₂

yes

maximal, complemented

#14557277

C₂

yes

maximal, characteristic, complemented

C₃₇×#331

C₂

yes

maximal, characteristic, complemented

C₂×#10596964

C₄

yes

—

C₂×#10596929

—

#11924536

C₂²

yes

characteristic

#11924523

C₄

yes

—

#11924507

—

D₄×Dic₃₇

—

C₄²×D₃₇

C₂²

yes

characteristic

C₄⋊#64436

C₂²

yes

—

#11924465

—

#127×C₃₇

C₂²

yes

characteristic

C₄⋊C₈×C₃₇

C₂²

yes

—

C₂×D₄×C₃₇

—

C₂×D₄×C₃₇

C₂×C₄

yes

—

C₂×C₂²⋊D₃₇

—

C₂²×Dic₃₇

—

#10596964

—

C₂×C₄×D₃₇

C₂×C₄

yes

characteristic

C₂×C₄×D₃₇

D₄

yes

—

C₂×Dic₇₄

C₂×C₄

yes

—

C₂×C₈×C₃₇

—

C₄²×C₃₇

C₂³

yes

characteristic

C₂²⋊Dic₃₇

—

C₄⋊Dic₃₇

—

C₄×Dic₃₇

C₂×C₄

yes

characteristic

C₄×Dic₃₇

D₄

yes

—

C₂×#64436

—

#10596929

—

C₂³×C₃₇

—

C₂²×D₃₇

C₂²⋊C₄

yes

characteristic

D₄×C₃₇

—

D₄×C₃₇

—

D₄×C₃₇

—

C₂×C₄×C₃₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₃₇

C₂×D₄

yes

—

C₂²⋊D₃₇

—

C₂×Dic₃₇

—

C₂×Dic₃₇

—

C₂×Dic₃₇

C₂²⋊C₄

yes

—

C₂×Dic₃₇

C₂²⋊C₄

yes

characteristic

C₄×D₃₇

—

C₄×D₃₇

—

C₄×D₃₇

—

Dic₇₄

—

C₈×C₃₇

—

#64436

—

C₂²×C₃₇

—

C₂²×C₃₇

—

C₂²×C₃₇

C₂×C₂²⋊C₄

yes

characteristic

C₂×D₃₇

—

C₂×D₃₇

—

C₄×C₃₇

—

C₄×C₃₇

AGL(1,ℤ/8)

yes

—

C₄×C₃₇

#137

yes

—

Dic₃₇

—

Dic₃₇

—

Dic₃₇

#100

yes

—

#1560

—

maximal, Sylow

C₂×C₃₇

—

C₂×C₃₇

C₂×#100

yes

characteristic

C₂×C₃₇

#417

yes

—

D₃₇

—

—

—

—

—

D₃₇

yes

characteristic, complemented

C₃₇

#1560

yes

characteristic, complemented, Sylow

C₂×#41

—

C₂×M₁₆

—

C₄⋊Q₈

—

#127

C₂×D₃₇

yes

characteristic

#124

—

#121

—

C₄×D₄

—

C₂×C₄²

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

C₂×D₃₇

yes

—

#41

—

148

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

C₄×D₃₇

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

M₁₆

—

148

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

—

C₄²

C₂²×D₃₇

yes

characteristic

100

C₂³

—

101

C₂³

—

102

Q₈

—

148

—

103

D₄

—

104

D₄

—

105

D₄

—

106

D₄

—

148

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

148

—

114

C₂×C₄

C₂×C₄×D₃₇

yes

characteristic

115

C₂×C₄

D₄×D₃₇

yes

—

116

C₈

—

117

C₈

—

148

—

118

C₂²

—

119

C₂²

—

120

C₂²

—

121

C₂²

—

122

C₂²

C₂²⋊C₄×D₃₇

yes

characteristic

123

C₄

—

124

C₄

—

125

C₄

—

126

C₄

—

148

—

127

C₄

#11924560

yes

—

128

C₄

#11924565

yes

—

129

C₂

—

130

C₂

—

148

—

131

C₂

#100×D₃₇

yes

characteristic

132

C₂

#14557531

yes

—

	subgroup	G/H
center	C₂²	C₂²⋊C₄×D₃₇
commutator	C₂×C₄×C₃₇	C₂²×C₄
Frattini	C₄²	C₂²×D₃₇
Fitting	C₃₇×#331	C₂
socle	C₂²×C₃₇	C₂×C₂²⋊C₄
prime radical	#14558081	C₂
cosocle	C₄²×C₃₇	C₂³
hypercenter	#331	D₃₇
hypocenter	C₃₇	#1560
LCS	G ▷ C₂×C₄×C₃₇ ▷ C₂²×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #331

subgroup

G/H

center

C₂²

C₂²⋊C₄×D₃₇

commutator

C₂×C₄×C₃₇

C₂²×C₄

Frattini

C₄²

C₂²×D₃₇

Fitting

C₃₇×#331

C₂

socle

C₂²×C₃₇

C₂×C₂²⋊C₄

prime radical

#14558081

C₂

cosocle

C₄²×C₃₇

C₂³

D₃₇

C₃₇

G ▷ C₂×C₄×C₃₇ ▷ C₂²×C₃₇ ▷ C₃₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #331

N	H ≅ G/N	type
2. #14557555	C₂	semidirect
3. #14557281	C₂	semidirect
4. #14557277	C₂	semidirect
5. C₃₇×#331	C₂	semidirect
71. #331	D₃₇	semidirect
72. C₃₇	#1560	semidirect

H ≅ G/N

type

2. #14557555

C₂

semidirect

3. #14557281

C₂

semidirect

4. #14557277

C₂

semidirect

5. C₃₇×#331

C₂

semidirect

71. #331

D₃₇

semidirect

72. C₃₇

#1560

semidirect

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

subgroup	index
#14558081	2
#14557555	2
#14557281	2
#14557277	2
C₃₇×#331	2
#1560	37

subgroup

index

C₃₇×#331

Subgroups — #17993328

Conjugacy classes of subgroups

Complements

Maximal subgroups