Group #17993989 — subgroups

Subgroups — #17993989

Order 4736.

156 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 156 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	#14557936	C₂	1	1	yes	maximal, characteristic
2	#14557744	C₂	1	1	yes	maximal, characteristic
3	#14557740	C₂	1	1	yes	maximal, characteristic
4	#14557657	C₂	1	1	yes	maximal, characteristic
5	#14557285	C₂	1	1	yes	maximal, characteristic
6	#14557278	C₂	1	1	yes	maximal, characteristic
7	C₃₇×#332	C₂	1	1	yes	maximal, characteristic
8	C₂×Dic₁₄₈	—	2	1	no	—
9	C₂×#10596942	—	2	1	no	—
10	#11924536	C₂²	1	1	yes	characteristic
11	C₄×Dic₇₄	C₂²	1	1	yes	characteristic
12	#11924469	—	2	1	no	—
13	#11924469	—	2	1	no	—
14	C₄⋊#64436	C₂²	1	1	yes	characteristic
15	#11924465	—	2	1	no	—
16	#11924461	—	2	1	no	—
17	#11924460	—	2	1	no	—
18	C₄×#64436	C₂²	1	1	yes	characteristic
19	C₈×Dic₃₇	—	2	1	no	—
20	C₄⋊Q₈×C₃₇	C₂²	1	1	yes	characteristic
21	C₄⋊C₈×C₃₇	C₂²	1	1	yes	characteristic
22	C₄⋊C₈×C₃₇	C₂²	1	1	yes	characteristic
23	C₂×Q₈×C₃₇	—	2	1	no	—
24	C₂×Dic₇₄	—	2	1	no	—
25	C₂×Dic₇₄	D₄	1	1	yes	characteristic
26	C₂×C₈×C₃₇	—	2	1	no	—
27	C₂×C₈×C₃₇	—	2	1	no	—
28	C₄⋊C₄×C₃₇	—	2	1	no	—
29	C₄⋊C₄×C₃₇	—	2	1	no	—
30	C₄²×C₃₇	C₂³	1	1	yes	characteristic
31	#10596942	—	4	2	no	—
32	C₄⋊Dic₃₇	—	2	1	no	—
33	C₄⋊Dic₃₇	—	2	1	no	—
34	C₄⋊Dic₃₇	D₄	1	1	yes	characteristic
35	#10596936	—	2	1	no	—
36	C₄×Dic₃₇	—	2	1	no	—
37	C₂×#64436	—	2	1	no	—
38	C₂×#64436	D₄	1	1	yes	characteristic
39	C₂×#64436	D₄	1	1	yes	characteristic
40	Dic₁₄₈	—	4	2	no	—
41	Q₈×C₃₇	—	4	1	no	—
42	Q₈×C₃₇	—	4	1	no	—
43	C₂×C₄×C₃₇	—	4	1	no	—
44	C₂×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
45	C₂×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
46	C₂×C₄×C₃₇	#41	1	1	yes	characteristic
47	C₂×Dic₃₇	—	2	1	no	—
48	C₂×Dic₃₇	—	2	1	no	—
49	C₂×Dic₃₇	—	4	1	no	—
50	Dic₇₄	—	2	1	no	—
51	Dic₇₄	—	4	1	no	—
52	Dic₇₄	—	4	1	no	—
53	Dic₇₄	D₈	1	2	yes	—
54	C₈×C₃₇	—	4	1	no	—
55	C₈×C₃₇	—	4	1	no	—
56	#64436	—	2	1	no	—
57	#64436	—	4	1	no	—
58	#64436	Q₁₆	1	2	yes	complemented
59	C₂²×C₃₇	#121	1	1	yes	characteristic
60	C₄×C₃₇	—	2	1	no	—
61	C₄×C₃₇	—	8	1	no	—
62	C₄×C₃₇	C₂×D₈	1	1	yes	characteristic
63	C₄×C₃₇	C₂×Q₁₆	1	1	yes	characteristic
64	C₄×C₃₇	#135	1	1	yes	characteristic
65	C₄×C₃₇	#137	1	1	yes	characteristic
66	Dic₃₇	—	2	2	no	—
67	Dic₃₇	—	4	1	no	—
68	Dic₃₇	—	8	1	no	—
69	#1674	—	37	1	no	maximal, Sylow
70	C₂×C₃₇	#462	1	1	yes	characteristic
71	C₂×C₃₇	#466	1	1	yes	characteristic
72	C₂×C₃₇	#473	1	1	yes	characteristic
73	#500	—	37	1	no	—
74	#462	—	37	1	no	—
75	#462	—	37	1	no	—
76	C₈×Q₈	—	37	1	no	—
77	#335	—	37	1	no	—
78	#332	—	37	1	no	—
79	#332	D₃₇	1	1	yes	characteristic
80	C₃₇	#1674	1	1	yes	characteristic, complemented, Sylow
81	C₂×Q₁₆	—	74	1	no	—
82	C₂×Q₁₆	—	74	1	no	—
83	C₄⋊Q₈	—	37	1	no	—
84	C₄⋊Q₈	C₂×D₃₇	1	1	yes	characteristic
85	C₄×Q₈	—	37	1	no	—
86	#107	—	74	1	no	—
87	#107	—	74	1	no	—
88	C₄⋊C₈	—	37	1	no	—
89	C₄⋊C₈	—	74	1	no	—
90	C₄⋊C₈	C₂×D₃₇	1	1	yes	characteristic
91	C₄⋊C₈	C₂×D₃₇	1	1	yes	characteristic
92	#103	—	74	1	no	—
93	#103	—	74	1	no	—
94	C₄×C₈	—	37	1	no	—
95	C₄×C₈	—	74	1	no	—
96	C₂×Q₈	—	2	1	no	—
97	C₂×Q₈	—	37	1	no	—
98	C₂×Q₈	—	74	1	no	—
99	Q₁₆	—	148	2	no	—
100	Q₁₆	—	148	2	no	—
101	C₂×C₈	—	2	1	no	—
102	C₂×C₈	—	2	1	no	—
103	C₂×C₈	—	37	1	no	—
104	C₂×C₈	—	37	1	no	—
105	C₂×C₈	—	74	1	no	—
106	C₄⋊C₄	—	2	1	no	—
107	C₄⋊C₄	—	2	1	no	—
108	C₄⋊C₄	—	37	1	no	—
109	C₄⋊C₄	—	74	1	no	—
110	C₄⋊C₄	—	74	1	no	—
111	C₄⋊C₄	—	74	1	no	—
112	C₄²	—	74	1	no	—
113	C₄²	C₂²×D₃₇	1	1	yes	characteristic
114	Q₈	—	4	1	no	—
115	Q₈	—	4	1	no	—
116	Q₈	—	37	2	no	—
117	Q₈	—	74	1	no	—
118	Q₈	—	148	1	no	—
119	Q₈	—	148	1	no	—
120	C₂×C₄	—	4	1	no	—
121	C₂×C₄	—	74	1	no	—
122	C₂×C₄	—	74	1	no	—
123	C₂×C₄	—	148	1	no	—
124	C₂×C₄	#10596962	1	1	yes	characteristic
125	C₂×C₄	D₄×D₃₇	1	1	yes	characteristic
126	C₂×C₄	D₄×D₃₇	1	1	yes	characteristic
127	C₈	—	4	1	no	—
128	C₈	—	4	1	no	—
129	C₈	—	37	2	no	—
130	C₈	—	74	1	no	—
131	C₈	—	148	1	no	—
132	C₂²	#11924509	1	1	yes	characteristic
133	C₄	—	2	1	no	—
134	C₄	—	8	1	no	—
135	C₄	—	296	1	no	—
136	C₄	—	74	2	no	—
137	C₄	—	148	1	no	—
138	C₄	D₈×D₃₇	1	1	yes	characteristic
139	C₄	Q₁₆×D₃₇	1	1	yes	characteristic
140	C₄	#11924557	1	1	yes	characteristic
141	C₄	#11924565	1	1	yes	characteristic
142	C₂	#14557742	1	1	yes	characteristic
143	C₂	#14557768	1	1	yes	characteristic
144	C₂	#14557807	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14557936

C₂

yes

maximal, characteristic

#14557744

C₂

yes

maximal, characteristic

#14557740

C₂

yes

maximal, characteristic

#14557657

C₂

yes

maximal, characteristic

#14557285

C₂

yes

maximal, characteristic

#14557278

C₂

yes

maximal, characteristic

C₃₇×#332

C₂

yes

maximal, characteristic

C₂×Dic₁₄₈

—

C₂×#10596942

—

#11924536

C₂²

yes

characteristic

C₄×Dic₇₄

C₂²

yes

—

—

C₂²

yes

—

—

—

C₂²

yes

characteristic

C₈×Dic₃₇

—

C₄⋊Q₈×C₃₇

C₂²

yes

characteristic

C₄⋊C₈×C₃₇

C₂²

yes

characteristic

C₄⋊C₈×C₃₇

C₂²

yes

characteristic

C₂×Q₈×C₃₇

—

C₂×Dic₇₄

—

C₂×Dic₇₄

D₄

yes

characteristic

C₂×C₈×C₃₇

—

C₂×C₈×C₃₇

—

C₄⋊C₄×C₃₇

—

C₄⋊C₄×C₃₇

—

C₄²×C₃₇

C₂³

yes

characteristic

#10596942

—

C₄⋊Dic₃₇

—

C₄⋊Dic₃₇

—

C₄⋊Dic₃₇

D₄

yes

characteristic

#10596936

—

C₄×Dic₃₇

—

C₂×#64436

—

C₂×#64436

D₄

yes

characteristic

C₂×#64436

D₄

yes

characteristic

Dic₁₄₈

—

Q₈×C₃₇

—

Q₈×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

C₂×D₄

yes

characteristic

C₂×C₄×C₃₇

C₂×D₄

yes

characteristic

C₂×C₄×C₃₇

#41

yes

characteristic

C₂×Dic₃₇

—

C₂×Dic₃₇

—

C₂×Dic₃₇

—

Dic₇₄

—

Dic₇₄

—

Dic₇₄

—

Dic₇₄

D₈

yes

—

C₈×C₃₇

—

C₈×C₃₇

—

#64436

—

#64436

—

#64436

Q₁₆

yes

complemented

C₂²×C₃₇

#121

yes

characteristic

C₄×C₃₇

—

C₄×C₃₇

—

C₄×C₃₇

C₂×D₈

yes

characteristic

C₄×C₃₇

C₂×Q₁₆

yes

C₄×C₃₇

yes

C₄×C₃₇

yes

Dic₃₇

—

Dic₃₇

—

Dic₃₇

—

—

C₂×C₃₇

yes

C₂×C₃₇

yes

C₂×C₃₇

yes

—

—

—

C₈×Q₈

—

#335

—

#332

—

#332

D₃₇

yes

characteristic

C₃₇

#1674

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

C₂×Q₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₃₇

yes

characteristic

C₄×Q₈

—

#107

—

#107

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

C₂×D₃₇

yes

characteristic

C₄⋊C₈

C₂×D₃₇

yes

characteristic

#103

—

#103

—

C₄×C₈

—

C₄×C₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

Q₁₆

—

148

—

100

Q₁₆

—

148

—

101

C₂×C₈

—

102

C₂×C₈

—

103

C₂×C₈

—

104

C₂×C₈

—

105

C₂×C₈

—

106

C₄⋊C₄

—

107

C₄⋊C₄

—

108

C₄⋊C₄

—

109

C₄⋊C₄

—

110

C₄⋊C₄

—

111

C₄⋊C₄

—

112

C₄²

—

113

C₄²

C₂²×D₃₇

yes

characteristic

114

Q₈

—

115

Q₈

—

116

Q₈

—

117

Q₈

—

118

Q₈

—

148

—

119

Q₈

—

148

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

148

—

124

C₂×C₄

#10596962

yes

characteristic

125

C₂×C₄

D₄×D₃₇

yes

characteristic

126

C₂×C₄

D₄×D₃₇

yes

characteristic

127

C₈

—

128

C₈

—

129

C₈

—

130

C₈

—

131

C₈

—

148

—

132

C₂²

#11924509

yes

characteristic

133

C₄

—

134

C₄

—

135

C₄

—

296

—

136

C₄

—

137

C₄

—

148

—

138

C₄

D₈×D₃₇

yes

characteristic

139

C₄

Q₁₆×D₃₇

yes

characteristic

140

C₄

#11924557

yes

characteristic

141

C₄

#11924565

yes

characteristic

142

C₂

#14557742

yes

characteristic

143

C₂

#14557768

yes

characteristic

144

C₂

#14557807

yes

characteristic

	subgroup	G/H
center	C₂²	#11924509
commutator, cosocle	C₄²×C₃₇	C₂³
Frattini	C₄²	C₂²×D₃₇
Fitting	C₃₇×#332	C₂
socle, prime radical	C₂²×C₃₇	#121
hypercenter	#332	D₃₇
hypocenter	C₃₇	#1674
LCS	G ▷ C₄²×C₃₇ ▷ C₂²×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #332

subgroup

G/H

center

C₂²

#11924509

commutator, cosocle

C₄²×C₃₇

C₂³

Frattini

C₄²

C₂²×D₃₇

Fitting

C₃₇×#332

C₂

C₂²×C₃₇

D₃₇

C₃₇

G ▷ C₄²×C₃₇ ▷ C₂²×C₃₇ ▷ C₃₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #332

N	H ≅ G/N	type
58. #64436	Q₁₆	semidirect
80. C₃₇	#1674	semidirect

H ≅ G/N

type

58. #64436

Q₁₆

semidirect

80. C₃₇

#1674

semidirect

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

subgroup	index
#14557936	2
#14557744	2
#14557740	2
#14557657	2
#14557285	2
#14557278	2
C₃₇×#332	2
#1674	37

subgroup

index

C₃₇×#332

Subgroups — #17993989

Conjugacy classes of subgroups

Complements

Maximal subgroups