Group #17994187 — subgroups

Subgroups — #17994187

Order 4736.

224 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 224 conjugacy classes of subgroups — too many to draw. Only its 65 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, complemented
2	#14557909	C₂	1	1	yes	maximal, characteristic
3	#14557907	C₂	1	1	yes	maximal, characteristic, complemented
4	#14557905	C₂	1	1	yes	maximal, characteristic, complemented
5	#14557880	C₂	1	1	yes	maximal, characteristic, complemented
6	C₈×#64436	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₇×#486	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#10596940	—	2	2	no	—
9	C₂×#10596941	—	2	2	no	—
10	C₂×#10596939	—	2	2	no	—
11	#11924536	C₂²	1	1	yes	characteristic
12	#11924533	C₂²	1	1	yes	characteristic
13	#11924469	—	2	2	no	—
14	#11924458	—	2	2	no	—
15	C₄×#64436	C₂²	1	1	yes	characteristic
16	C₄×#64436	C₂²	1	1	yes	characteristic
17	C₄⋊Q₈×C₃₇	C₂²	1	1	yes	characteristic
18	#127×C₃₇	C₂²	1	1	yes	characteristic
19	#102×C₃₇	—	2	2	no	—
20	C₄×C₈×C₃₇	C₂²	1	1	yes	characteristic
21	C₂×Q₈×C₃₇	—	2	1	no	—
22	C₂×D₄×C₃₇	—	2	1	no	—
23	C₂×D₄×C₃₇	—	2	1	no	—
24	C₂×D₄×C₃₇	—	2	1	no	—
25	C₂×D₁₄₈	—	2	1	no	—
26	C₂×D₁₄₈	—	2	1	no	—
27	C₂×D₁₄₈	—	2	1	no	—
28	C₂×Dic₇₄	—	2	1	no	—
29	C₂×C₈×C₃₇	D₄	1	2	yes	—
30	C₄⋊C₄×C₃₇	—	2	1	no	—
31	C₄⋊C₄×C₃₇	—	2	1	no	—
32	C₄²×C₃₇	C₂³	1	1	yes	characteristic
33	#10596941	—	4	4	no	—
34	#10596940	—	4	4	no	—
35	#10596939	—	4	4	no	—
36	C₄⋊Dic₃₇	—	2	1	no	—
37	C₄⋊Dic₃₇	—	2	1	no	—
38	C₂×#64436	D₄	1	2	yes	—
39	C₂×#64436	D₄	1	2	yes	—
40	C₂³×C₃₇	—	4	1	no	—
41	C₂²×D₃₇	—	4	1	no	—
42	Q₈×C₃₇	—	4	2	no	—
43	D₄×C₃₇	—	4	2	no	—
44	D₄×C₃₇	—	4	2	no	—
45	D₄×C₃₇	—	4	2	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	C₂×D₄	1	1	yes	characteristic
50	C₂×Dic₃₇	—	4	1	no	—
51	D₁₄₈	—	4	2	no	—
52	D₁₄₈	—	4	2	no	—
53	D₁₄₈	—	4	2	no	—
54	Dic₇₄	—	4	2	no	—
55	C₈×C₃₇	—	2	2	no	—
56	#64436	D₈	1	4	yes	complemented
57	#64436	SD₁₆	1	4	yes	complemented
58	C₂²×C₃₇	—	8	1	no	—
59	C₂²×C₃₇	—	8	1	no	—
60	C₂²×C₃₇	—	8	1	no	—
61	C₂²×C₃₇	#127	1	1	yes	characteristic
62	C₂×D₃₇	—	8	1	no	—
63	C₂×D₃₇	—	8	1	no	—
64	C₂×D₃₇	—	8	1	no	—
65	C₄×C₃₇	—	8	1	no	—
66	C₄×C₃₇	C₂×D₈	1	2	yes	—
67	C₄×C₃₇	C₂×SD₁₆	1	2	yes	—
68	C₄×C₃₇	#135	1	2	yes	—
69	Dic₃₇	—	8	1	no	—
70	#1706	—	37	1	no	maximal, Sylow
71	C₂×C₃₇	—	16	1	no	—
72	C₂×C₃₇	#492	1	1	yes	characteristic
73	C₂×C₃₇	#493	1	1	yes	characteristic
74	C₂×C₃₇	#495	1	1	yes	characteristic
75	D₃₇	—	16	1	no	—
76	#500	—	37	1	no	—
77	#493	—	37	1	no	—
78	#492	—	37	1	no	—
79	#492	—	37	1	no	—
80	#486	—	37	1	no	—
81	#486	D₃₇	1	1	yes	characteristic, complemented
82	C₈²	—	37	1	no	—
83	C₃₇	#1706	1	1	yes	characteristic, complemented, Sylow
84	C₂×SD₁₆	—	74	2	no	—
85	C₂×SD₁₆	—	74	2	no	—
86	C₂×D₈	—	74	2	no	—
87	C₄⋊Q₈	—	37	1	no	—
88	C₄⋊Q₈	C₂×D₃₇	1	1	yes	characteristic
89	#127	—	37	1	no	—
90	#127	C₂×D₃₇	1	1	yes	characteristic
91	#107	—	74	2	no	—
92	#102	—	2	2	no	—
93	#102	—	74	2	no	—
94	C₄×C₈	—	37	1	no	—
95	C₄×C₈	—	37	1	no	—
96	C₄×C₈	C₂×D₃₇	1	1	yes	characteristic
97	C₂×Q₈	—	2	1	no	—
98	C₂×Q₈	—	74	1	no	—
99	C₂×D₄	—	2	1	no	—
100	C₂×D₄	—	2	1	no	—
101	C₂×D₄	—	2	1	no	—
102	C₂×D₄	—	74	1	no	—
103	C₂×D₄	—	74	1	no	—
104	C₂×D₄	—	74	1	no	—
105	SD₁₆	—	148	4	no	—
106	SD₁₆	—	148	4	no	—
107	D₈	—	148	4	no	—
108	C₂×C₈	—	37	2	no	—
109	C₂×C₈	—	37	2	no	—
110	C₂×C₈	C₂²⋊D₃₇	1	2	yes	—
111	C₄⋊C₄	—	2	1	no	—
112	C₄⋊C₄	—	2	1	no	—
113	C₄⋊C₄	—	74	1	no	—
114	C₄⋊C₄	—	74	1	no	—
115	C₄²	C₂²×D₃₇	1	1	yes	characteristic
116	C₂³	—	4	1	no	—
117	C₂³	—	148	1	no	—
118	Q₈	—	4	2	no	—
119	Q₈	—	148	2	no	—
120	D₄	—	4	2	no	—
121	D₄	—	4	2	no	—
122	D₄	—	4	2	no	—
123	D₄	—	148	2	no	—
124	D₄	—	148	2	no	—
125	D₄	—	148	2	no	—
126	C₂×C₄	—	4	1	no	—
127	C₂×C₄	—	148	1	no	—
128	C₂×C₄	D₄×D₃₇	1	1	yes	characteristic
129	C₂×C₄	D₄×D₃₇	1	1	yes	characteristic
130	C₂×C₄	C₂×C₂²⋊D₃₇	1	1	yes	characteristic
131	C₈	—	2	2	no	—
132	C₈	—	37	4	no	—
133	C₈	—	37	4	no	—
134	C₂²	—	8	1	no	—
135	C₂²	—	8	1	no	—
136	C₂²	—	8	1	no	—
137	C₂²	—	296	1	no	—
138	C₂²	—	296	1	no	—
139	C₂²	—	296	1	no	—
140	C₂²	#11924532	1	1	yes	characteristic
141	C₄	—	8	1	no	—
142	C₄	—	296	1	no	—
143	C₄	D₈×D₃₇	1	2	yes	—
144	C₄	SD₁₆×D₃₇	1	2	yes	—
145	C₄	#11924556	1	2	yes	—
146	C₂	—	16	1	no	—
147	C₂	—	592	1	no	—
148	C₂	#14557904	1	1	yes	characteristic
149	C₂	#14557908	1	1	yes	characteristic
150	C₂	#14557915	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14557936

C₂

yes

maximal, characteristic, complemented

#14557909

C₂

yes

maximal, characteristic

#14557907

C₂

yes

maximal, characteristic, complemented

#14557905

C₂

yes

maximal, characteristic, complemented

#14557880

C₂

yes

maximal, characteristic, complemented

C₈×#64436

C₂

yes

maximal, characteristic, complemented

C₃₇×#486

C₂

yes

maximal, characteristic, complemented

—

—

—

C₂²

yes

characteristic

#11924533

C₂²

yes

—

—

C₂²

yes

characteristic

C₄×#64436

C₂²

yes

characteristic

C₄⋊Q₈×C₃₇

C₂²

yes

characteristic

#127×C₃₇

C₂²

yes

characteristic

#102×C₃₇

—

C₄×C₈×C₃₇

C₂²

yes

characteristic

C₂×Q₈×C₃₇

—

C₂×D₄×C₃₇

—

C₂×D₄×C₃₇

—

C₂×D₄×C₃₇

—

C₂×D₁₄₈

—

C₂×D₁₄₈

—

C₂×D₁₄₈

—

C₂×Dic₇₄

—

C₂×C₈×C₃₇

D₄

yes

—

C₄⋊C₄×C₃₇

—

C₄⋊C₄×C₃₇

—

C₄²×C₃₇

C₂³

yes

—

—

—

C₄⋊Dic₃₇

—

C₄⋊Dic₃₇

—

C₂×#64436

D₄

yes

—

C₂×#64436

D₄

yes

—

C₂³×C₃₇

—

C₂²×D₃₇

—

Q₈×C₃₇

—

D₄×C₃₇

—

D₄×C₃₇

—

D₄×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

C₂×D₄

yes

characteristic

C₂×C₄×C₃₇

C₂×D₄

yes

characteristic

C₂×C₄×C₃₇

C₂×D₄

yes

characteristic

C₂×Dic₃₇

—

D₁₄₈

—

D₁₄₈

—

D₁₄₈

—

Dic₇₄

—

C₈×C₃₇

—

#64436

D₈

yes

complemented

#64436

SD₁₆

yes

complemented

C₂²×C₃₇

—

C₂²×C₃₇

—

C₂²×C₃₇

—

C₂²×C₃₇

#127

yes

characteristic

C₂×D₃₇

—

C₂×D₃₇

—

C₂×D₃₇

—

C₄×C₃₇

—

C₄×C₃₇

C₂×D₈

yes

—

C₄×C₃₇

C₂×SD₁₆

yes

—

C₄×C₃₇

#135

yes

—

Dic₃₇

—

#1706

—

maximal, Sylow

C₂×C₃₇

—

C₂×C₃₇

yes

C₂×C₃₇

yes

C₂×C₃₇

yes

D₃₇

—

—

—

—

—

—

D₃₇

yes

characteristic, complemented

C₈²

—

C₃₇

#1706

yes

characteristic, complemented, Sylow

C₂×SD₁₆

—

C₂×SD₁₆

—

C₂×D₈

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₃₇

yes

characteristic

#127

—

#127

C₂×D₃₇

yes

—

—

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×D₃₇

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

C₂×D₄

—

100

C₂×D₄

—

101

C₂×D₄

—

102

C₂×D₄

—

103

C₂×D₄

—

104

C₂×D₄

—

105

SD₁₆

—

148

—

106

SD₁₆

—

148

—

107

D₈

—

148

—

108

C₂×C₈

—

109

C₂×C₈

—

110

C₂×C₈

C₂²⋊D₃₇

yes

—

111

C₄⋊C₄

—

112

C₄⋊C₄

—

113

C₄⋊C₄

—

114

C₄⋊C₄

—

115

C₄²

C₂²×D₃₇

yes

characteristic

116

C₂³

—

117

C₂³

—

148

—

118

Q₈

—

119

Q₈

—

148

—

120

D₄

—

121

D₄

—

122

D₄

—

123

D₄

—

148

—

124

D₄

—

148

—

125

D₄

—

148

—

126

C₂×C₄

—

127

C₂×C₄

—

148

—

128

C₂×C₄

D₄×D₃₇

yes

characteristic

129

C₂×C₄

D₄×D₃₇

yes

characteristic

130

C₂×C₄

C₂×C₂²⋊D₃₇

yes

characteristic

131

C₈

—

132

C₈

—

133

C₈

—

134

C₂²

—

135

C₂²

—

136

C₂²

—

137

C₂²

—

296

—

138

C₂²

—

296

—

139

C₂²

—

296

—

140

C₂²

#11924532

yes

characteristic

141

C₄

—

142

C₄

—

296

—

143

C₄

D₈×D₃₇

yes

—

144

C₄

SD₁₆×D₃₇

yes

—

145

C₄

#11924556

yes

—

146

C₂

—

147

C₂

—

592

—

148

C₂

#14557904

yes

characteristic

149

C₂

#14557908

yes

characteristic

150

C₂

#14557915

yes

characteristic

	subgroup	G/H
center	C₂²	#11924532
commutator, cosocle	C₄²×C₃₇	C₂³
Frattini	C₄²	C₂²×D₃₇
Fitting	C₃₇×#486	C₂
socle	C₂²×C₃₇	#127
prime radical	#14557909	C₂
hypercenter	#486	D₃₇
hypocenter	C₃₇	#1706
LCS	G ▷ C₄²×C₃₇ ▷ C₂²×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #486

subgroup

G/H

center

C₂²

#11924532

commutator, cosocle

C₄²×C₃₇

C₂³

Frattini

C₄²

C₂²×D₃₇

Fitting

C₃₇×#486

C₂

C₂²×C₃₇

C₂

D₃₇

C₃₇

G ▷ C₄²×C₃₇ ▷ C₂²×C₃₇ ▷ C₃₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #486

N	H ≅ G/N	type
1. #14557936	C₂	semidirect
3. #14557907	C₂	semidirect
4. #14557905	C₂	semidirect
5. #14557880	C₂	semidirect
6. C₈×#64436	C₂	semidirect
7. C₃₇×#486	C₂	semidirect
56. #64436	D₈	semidirect
57. #64436	SD₁₆	semidirect
81. #486	D₃₇	semidirect
83. C₃₇	#1706	semidirect

H ≅ G/N

type

1. #14557936

C₂

semidirect

3. #14557907

C₂

semidirect

4. #14557905

C₂

semidirect

5. #14557880

C₂

semidirect

6. C₈×#64436

C₂

semidirect

7. C₃₇×#486

C₂

semidirect

56. #64436

D₈

semidirect

57. #64436

SD₁₆

semidirect

81. #486

D₃₇

semidirect

83. C₃₇

#1706

semidirect

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

subgroup	index
#14557936	2
#14557909	2
#14557907	2
#14557905	2
#14557880	2
C₈×#64436	2
C₃₇×#486	2
#1706	37

subgroup

index

C₃₇×#486

Subgroups — #17994187

Conjugacy classes of subgroups

Complements

Maximal subgroups