Group #17994028 — subgroups

Subgroups — #17994028

Order 4736.

150 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 150 conjugacy classes of subgroups — too many to draw. Only its 55 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	#14557885	C₂	1	1	yes	maximal, characteristic
2	#14557818	C₂	1	1	yes	maximal, characteristic
3	Q₈×#64436	C₂	1	1	yes	maximal, characteristic
4	#14557283	C₂	1	1	yes	maximal, characteristic
5	#14557279	C₂	1	1	yes	maximal, characteristic
6	#14557278	C₂	1	1	yes	maximal, characteristic
7	C₃₇×#462	C₂	1	1	yes	maximal, characteristic
8	#11924536	C₂²	1	1	yes	characteristic
9	#11924467	—	2	1	no	—
10	C₄⋊#64436	—	2	1	no	—
11	C₄⋊#64436	C₂²	1	1	yes	characteristic
12	C₄⋊#64436	C₂²	1	1	yes	characteristic
13	#11924460	—	2	1	no	—
14	#11924459	—	2	1	no	—
15	#11924459	—	2	1	no	—
16	C₄×#64436	—	2	1	no	—
17	C₄×#64436	C₂²	1	1	yes	characteristic
18	C₂×Q₁₆×C₃₇	—	2	1	no	—
19	C₄⋊Q₈×C₃₇	C₂²	1	1	yes	characteristic
20	C₄×Q₈×C₃₇	C₂²	1	1	yes	characteristic
21	C₄⋊C₈×C₃₇	C₂²	1	1	yes	characteristic
22	#103×C₃₇	—	2	1	no	—
23	C₂×Q₈×C₃₇	—	2	1	no	—
24	C₂×Q₈×C₃₇	D₄	1	1	yes	characteristic
25	C₂×Dic₇₄	—	2	1	no	—
26	Q₁₆×C₃₇	—	4	2	no	—
27	C₂×C₈×C₃₇	—	2	1	no	—
28	C₄⋊C₄×C₃₇	—	2	1	no	—
29	C₄⋊C₄×C₃₇	—	2	1	no	—
30	C₄⋊C₄×C₃₇	—	2	1	no	—
31	C₄⋊C₄×C₃₇	D₄	1	1	yes	characteristic
32	C₄²×C₃₇	—	2	1	no	—
33	C₄²×C₃₇	C₂³	1	1	yes	characteristic
34	C₄⋊Dic₃₇	—	2	1	no	—
35	C₄⋊Dic₃₇	—	2	1	no	—
36	C₂×#64436	—	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	Q₈×C₃₇	—	2	1	no	—
41	Q₈×C₃₇	—	4	1	no	—
42	Q₈×C₃₇	—	4	1	no	—
43	Q₈×C₃₇	SD₁₆	1	2	yes	—
44	C₂×C₄×C₃₇	—	2	1	no	—
45	C₂×C₄×C₃₇	—	2	1	no	—
46	C₂×C₄×C₃₇	—	4	1	no	—
47	C₂×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
48	C₂×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₃₇	#41	1	1	yes	characteristic
50	C₂×Dic₃₇	—	4	1	no	—
51	Dic₇₄	—	4	1	no	—
52	Dic₇₄	—	4	1	no	—
53	C₈×C₃₇	—	4	1	no	—
54	#64436	—	2	1	no	—
55	#64436	—	2	1	no	—
56	#64436	—	4	1	no	—
57	#64436	—	4	1	no	—
58	C₂²×C₃₇	#121	1	1	yes	characteristic
59	C₄×C₃₇	—	2	2	no	—
60	C₄×C₃₇	—	2	1	no	—
61	C₄×C₃₇	—	4	1	no	—
62	C₄×C₃₇	—	8	1	no	—
63	C₄×C₃₇	C₂×SD₁₆	1	1	yes	characteristic
64	C₄×C₃₇	#135	1	1	yes	characteristic
65	C₄×C₃₇	#135	1	1	yes	characteristic
66	C₄×C₃₇	#137	1	1	yes	characteristic
67	Dic₃₇	—	8	1	no	—
68	#1680	—	37	1	no	maximal, Sylow
69	C₂×C₃₇	#464	1	1	yes	characteristic
70	C₂×C₃₇	#465	1	1	yes	characteristic
71	C₂×C₃₇	#473	1	1	yes	characteristic
72	#487	—	37	1	no	—
73	#475	—	37	1	no	—
74	#462	D₃₇	1	1	yes	characteristic
75	C₈×Q₈	—	37	1	no	—
76	#334	—	37	1	no	—
77	#332	—	37	1	no	—
78	#332	—	37	1	no	—
79	C₃₇	#1680	1	1	yes	characteristic, complemented, Sylow
80	C₂×Q₁₆	—	2	1	no	—
81	C₄⋊Q₈	—	37	1	no	—
82	C₄⋊Q₈	C₂×D₃₇	1	1	yes	characteristic
83	C₄×Q₈	C₂×D₃₇	1	1	yes	characteristic
84	#106	—	74	1	no	—
85	C₄⋊C₈	—	37	1	no	—
86	C₄⋊C₈	—	37	1	no	—
87	C₄⋊C₈	—	74	1	no	—
88	C₄⋊C₈	C₂×D₃₇	1	1	yes	characteristic
89	#103	—	2	1	no	—
90	#103	—	74	1	no	—
91	#103	—	74	1	no	—
92	#103	—	74	1	no	—
93	C₄×C₈	—	37	1	no	—
94	C₄×C₈	—	74	1	no	—
95	C₂×Q₈	—	2	1	no	—
96	C₂×Q₈	—	74	1	no	—
97	C₂×Q₈	C₂²⋊D₃₇	1	1	yes	characteristic
98	Q₁₆	—	4	2	no	—
99	C₂×C₈	—	2	1	no	—
100	C₂×C₈	—	37	1	no	—
101	C₂×C₈	—	37	1	no	—
102	C₂×C₈	—	74	1	no	—
103	C₂×C₈	—	74	1	no	—
104	C₄⋊C₄	—	2	1	no	—
105	C₄⋊C₄	—	2	1	no	—
106	C₄⋊C₄	—	2	1	no	—
107	C₄⋊C₄	—	74	1	no	—
108	C₄⋊C₄	—	74	1	no	—
109	C₄⋊C₄	C₂²⋊D₃₇	1	1	yes	characteristic
110	C₄²	—	2	1	no	—
111	C₄²	C₂²×D₃₇	1	1	yes	characteristic
112	Q₈	—	2	1	no	—
113	Q₈	—	4	1	no	—
114	Q₈	—	4	1	no	—
115	Q₈	—	148	1	no	—
116	Q₈	—	148	1	no	—
117	Q₈	#10596940	1	2	yes	—
118	C₂×C₄	—	2	1	no	—
119	C₂×C₄	—	2	1	no	—
120	C₂×C₄	—	4	1	no	—
121	C₂×C₄	—	148	1	no	—
122	C₂×C₄	D₄×D₃₇	1	1	yes	characteristic
123	C₂×C₄	#10596964	1	1	yes	characteristic
124	C₂×C₄	C₂×C₂²⋊D₃₇	1	1	yes	characteristic
125	C₈	—	4	1	no	—
126	C₈	—	74	1	no	—
127	C₈	—	74	1	no	—
128	C₈	—	148	1	no	—
129	C₈	—	148	1	no	—
130	C₂²	#11924508	1	1	yes	characteristic
131	C₄	—	2	2	no	—
132	C₄	—	2	1	no	—
133	C₄	—	4	1	no	—
134	C₄	—	8	1	no	—
135	C₄	—	296	1	no	—
136	C₄	C₂×#10596940	1	1	yes	characteristic
137	C₄	#11924556	1	1	yes	characteristic
138	C₄	#11924559	1	1	yes	characteristic
139	C₄	#11924568	1	1	yes	characteristic
140	C₂	#14557757	1	1	yes	characteristic
141	C₂	#14557760	1	1	yes	characteristic
142	C₂	#14557806	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14557885

C₂

yes

maximal, characteristic

#14557818

C₂

yes

maximal, characteristic

Q₈×#64436

C₂

yes

maximal, characteristic

#14557283

C₂

yes

maximal, characteristic

#14557279

C₂

yes

maximal, characteristic

#14557278

C₂

yes

maximal, characteristic

C₃₇×#462

C₂

yes

maximal, characteristic

#11924536

C₂²

yes

—

—

C₂²

yes

characteristic

C₄⋊#64436

C₂²

yes

—

—

—

—

C₂²

yes

characteristic

C₂×Q₁₆×C₃₇

—

C₄⋊Q₈×C₃₇

C₂²

yes

characteristic

C₄×Q₈×C₃₇

C₂²

yes

characteristic

C₄⋊C₈×C₃₇

C₂²

yes

characteristic

#103×C₃₇

—

C₂×Q₈×C₃₇

—

C₂×Q₈×C₃₇

D₄

yes

characteristic

C₂×Dic₇₄

—

Q₁₆×C₃₇

—

C₂×C₈×C₃₇

—

C₄⋊C₄×C₃₇

—

C₄⋊C₄×C₃₇

—

C₄⋊C₄×C₃₇

—

C₄⋊C₄×C₃₇

D₄

yes

characteristic

C₄²×C₃₇

—

C₄²×C₃₇

C₂³

yes

characteristic

C₄⋊Dic₃₇

—

C₄⋊Dic₃₇

—

C₂×#64436

—

C₂×#64436

—

C₂×#64436

D₄

yes

characteristic

C₂×#64436

D₄

yes

characteristic

Q₈×C₃₇

—

Q₈×C₃₇

—

Q₈×C₃₇

—

Q₈×C₃₇

SD₁₆

yes

—

C₂×C₄×C₃₇

—

C₂×C₄×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

—

Dic₇₄

—

Dic₇₄

—

C₈×C₃₇

—

—

—

—

—

C₂²×C₃₇

#121

yes

characteristic

C₄×C₃₇

—

C₄×C₃₇

—

C₄×C₃₇

—

C₄×C₃₇

—

C₄×C₃₇

C₂×SD₁₆

yes

C₄×C₃₇

yes

C₄×C₃₇

yes

C₄×C₃₇

yes

Dic₃₇

—

—

C₂×C₃₇

yes

C₂×C₃₇

yes

C₂×C₃₇

yes

—

—

D₃₇

yes

characteristic

C₈×Q₈

—

#334

—

#332

—

#332

—

C₃₇

#1680

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₃₇

yes

characteristic

C₄×Q₈

C₂×D₃₇

yes

characteristic

#106

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

C₂×D₃₇

yes

—

—

—

—

C₄×C₈

—

C₄×C₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂²⋊D₃₇

yes

characteristic

Q₁₆

—

C₂×C₈

—

100

C₂×C₈

—

101

C₂×C₈

—

102

C₂×C₈

—

103

C₂×C₈

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

106

C₄⋊C₄

—

107

C₄⋊C₄

—

108

C₄⋊C₄

—

109

C₄⋊C₄

C₂²⋊D₃₇

yes

characteristic

110

C₄²

—

111

C₄²

C₂²×D₃₇

yes

characteristic

112

Q₈

—

113

Q₈

—

114

Q₈

—

115

Q₈

—

148

—

116

Q₈

—

148

—

117

Q₈

#10596940

yes

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

148

—

122

C₂×C₄

D₄×D₃₇

yes

characteristic

123

C₂×C₄

#10596964

yes

characteristic

124

C₂×C₄

C₂×C₂²⋊D₃₇

yes

characteristic

125

C₈

—

126

C₈

—

127

C₈

—

128

C₈

—

148

—

129

C₈

—

148

—

130

C₂²

#11924508

yes

characteristic

131

C₄

—

132

C₄

—

133

C₄

—

134

C₄

—

135

C₄

—

296

—

136

C₄

C₂×#10596940

yes

characteristic

137

C₄

#11924556

yes

characteristic

138

C₄

#11924559

yes

characteristic

139

C₄

#11924568

yes

characteristic

140

C₂

#14557757

yes

characteristic

141

C₂

#14557760

yes

characteristic

142

C₂

#14557806

yes

characteristic

	subgroup	G/H
center	C₂²	#11924508
commutator, cosocle	C₄²×C₃₇	C₂³
Frattini	C₄²	C₂²×D₃₇
Fitting	C₃₇×#462	C₂
socle, prime radical	C₂²×C₃₇	#121
hypercenter	#462	D₃₇
hypocenter	C₃₇	#1680
LCS	G ▷ C₄²×C₃₇ ▷ C₂²×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #462

subgroup

G/H

center

C₂²

#11924508

commutator, cosocle

C₄²×C₃₇

C₂³

Frattini

C₄²

C₂²×D₃₇

Fitting

C₃₇×#462

C₂

C₂²×C₃₇

D₃₇

C₃₇

G ▷ C₄²×C₃₇ ▷ C₂²×C₃₇ ▷ C₃₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #462

N	H ≅ G/N	type
79. C₃₇	#1680	semidirect

H ≅ G/N

type

79. C₃₇

#1680

semidirect

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

subgroup	index
#14557885	2
#14557818	2
Q₈×#64436	2
#14557283	2
#14557279	2
#14557278	2
C₃₇×#462	2
#1680	37

subgroup

index

C₃₇×#462

Subgroups — #17994028

Conjugacy classes of subgroups

Complements

Maximal subgroups