Group #18010701 — subgroups

Subgroups — #18010701

Order 4736.

240 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 78 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	C₂×#11924604	C₂	1	1	yes	maximal, characteristic
2	C₂×#11924590	C₂	1	1	yes	maximal, characteristic
3	#14558484	C₂	1	2	yes	maximal
4	#14558406	C₂	1	2	yes	maximal
5	#107×D₃₇	C₂	1	1	yes	maximal, characteristic
6	C₂×D₃₇⋊C₈	C₂²	1	1	yes	characteristic
7	C₂×C₄⋊#3866	—	2	2	no	—
8	C₂×C₄⋊#3866	C₂²	1	2	yes	—
9	C₂×C₄×#3866	C₂²	1	1	yes	characteristic
10	#11924604	—	2	2	no	—
11	#11924590	—	2	2	no	—
12	(C₂×#3866)⋊C₄	—	2	2	no	—
13	C₂×C₈×D₃₇	C₂²	1	1	yes	characteristic
14	C₄⋊C₄×D₃₇	C₂²	1	2	yes	—
15	#11924470	C₄	1	1	yes	characteristic, complemented
16	#11924469	C₄	1	2	yes	complemented
17	#107×C₃₇	C₄	1	1	yes	characteristic, complemented
18	C₂²×#3866	—	2	2	no	—
19	C₂²×#3866	D₄	1	2	yes	—
20	C₂×C₄×D₃₇	—	2	2	no	—
21	C₂×C₄×D₃₇	C₂³	1	1	yes	characteristic
22	C₂×#64438	—	2	1	no	—
23	C₄⋊#3866	—	2	2	no	—
24	C₄⋊#3866	—	2	2	no	—
25	C₄⋊#3866	—	4	4	no	—
26	C₄×#3866	—	2	1	no	—
27	C₄×#3866	—	2	1	no	—
28	D₃₇⋊C₈	—	2	1	no	—
29	D₃₇⋊C₈	Q₈	1	2	yes	—
30	C₂×C₈×C₃₇	C₂×C₄	1	1	yes	characteristic
31	C₄⋊C₄×C₃₇	C₂×C₄	1	2	yes	—
32	C₄⋊Dic₃₇	C₂×C₄	1	2	yes	—
33	#10596936	—	2	2	no	—
34	C₂×#64436	C₂×C₄	1	1	yes	characteristic
35	C₈×D₃₇	D₄	1	2	yes	—
36	C₈×D₃₇	Q₈	1	2	yes	—
37	C₂²×D₃₇	C₂×D₄	1	1	yes	characteristic
38	C₂×#3866	—	2	2	no	—
39	C₂×#3866	—	2	2	no	—
40	C₂×#3866	—	2	2	no	—
41	C₂×#3866	—	4	2	no	—
42	C₂×#3866	—	4	2	no	—
43	C₂×#3866	—	4	2	no	—
44	C₂×C₄×C₃₇	—	2	2	no	—
45	C₂×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
46	C₂×Dic₃₇	—	2	2	no	—
47	C₂×Dic₃₇	#41	1	1	yes	characteristic
48	C₄×D₃₇	—	4	2	no	—
49	C₄×D₃₇	—	4	2	no	—
50	C₄×D₃₇	C₂×D₄	1	1	yes	characteristic
51	C₄×D₃₇	C₂×Q₈	1	1	yes	characteristic
52	C₄×D₃₇	C₂×Q₈	1	1	yes	characteristic
53	C₄×D₃₇	#41	1	1	yes	characteristic
54	#64438	—	2	2	no	—
55	C₈×C₃₇	C₄⋊C₄	1	2	yes	complemented
56	#64436	C₄⋊C₄	1	2	yes	complemented
57	C₂²×C₃₇	C₄×D₄	1	1	yes	characteristic
58	C₂×D₃₇	#121	1	1	yes	characteristic
59	C₂×D₃₇	C₄⋊Q₈	1	1	yes	characteristic
60	C₂×D₃₇	AGL(1,ℤ/8)	1	2	yes	—
61	C₂×D₃₇	#137	1	2	yes	—
62	#3866	—	4	2	no	—
63	#3866	—	8	2	no	—
64	C₄×C₃₇	—	4	2	no	—
65	C₄×C₃₇	C₂×C₄⋊C₄	1	1	yes	characteristic
66	C₄×C₃₇	C₄×Q₈	1	1	yes	characteristic
67	Dic₃₇	—	4	2	no	—
68	Dic₃₇	C₂²⋊Q₈	1	1	yes	characteristic
69	Dic₃₇	#125	1	1	yes	characteristic
70	#1943	—	37	1	no	maximal, Sylow
71	C₂×C₃₇	#389	1	1	yes	characteristic
72	C₂×C₃₇	#441	1	1	yes	characteristic
73	C₂×C₃₇	#442	1	1	yes	characteristic
74	D₃₇	#468	1	2	yes	—
75	D₃₇	#501	1	2	yes	—
76	C₂×#107	—	37	1	no	—
77	C₂×#106	—	37	1	no	—
78	C₂×#97	—	37	1	no	—
79	#389	—	37	2	no	—
80	#340	—	37	2	no	—
81	C₃₇	#1943	1	1	yes	characteristic, complemented, Sylow
82	C₂²×C₈	—	37	1	no	—
83	C₂²×C₈	—	37	1	no	—
84	C₂×C₄⋊C₄	—	37	2	no	—
85	C₂×C₄⋊C₄	—	37	2	no	—
86	C₂×C₄⋊C₄	—	74	2	no	—
87	C₂×C₄²	—	37	1	no	—
88	#107	—	37	1	no	—
89	#107	—	37	2	no	—
90	#107	#3866	1	1	yes	characteristic, complemented
91	#106	—	74	2	no	—
92	#97	—	74	2	no	—
93	#95	—	74	2	no	—
94	C₂²×C₄	—	37	1	no	—
95	C₂²×C₄	—	37	2	no	—
96	C₂²×C₄	—	74	2	no	—
97	C₂²×C₄	—	74	2	no	—
98	C₂×C₈	—	37	1	no	—
99	C₂×C₈	—	37	2	no	—
100	C₂×C₈	—	37	2	no	—
101	C₂×C₈	—	37	2	no	—
102	C₂×C₈	—	74	1	no	—
103	C₂×C₈	—	74	1	no	—
104	C₂×C₈	C₂×#3866	1	1	yes	characteristic
105	C₄⋊C₄	—	37	2	no	—
106	C₄⋊C₄	—	74	2	no	—
107	C₄⋊C₄	—	74	2	no	—
108	C₄⋊C₄	—	74	2	no	—
109	C₄⋊C₄	—	148	4	no	—
110	C₄⋊C₄	C₂×#3866	1	2	yes	—
111	C₄²	—	74	1	no	—
112	C₄²	—	74	1	no	—
113	C₂³	—	37	1	no	—
114	C₂×C₄	—	2	2	no	—
115	C₂×C₄	—	37	1	no	—
116	C₂×C₄	—	37	1	no	—
117	C₂×C₄	—	37	1	no	—
118	C₂×C₄	—	37	1	no	—
119	C₂×C₄	—	37	1	no	—
120	C₂×C₄	—	74	2	no	—
121	C₂×C₄	—	74	2	no	—
122	C₂×C₄	—	74	2	no	—
123	C₂×C₄	—	74	2	no	—
124	C₂×C₄	—	148	2	no	—
125	C₂×C₄	—	148	2	no	—
126	C₂×C₄	—	148	2	no	—
127	C₂×C₄	—	148	2	no	—
128	C₂×C₄	—	148	2	no	—
129	C₂×C₄	C₂²×#3866	1	1	yes	characteristic
130	C₈	—	37	2	no	—
131	C₈	—	74	2	no	—
132	C₈	C₄⋊#3866	1	2	yes	complemented
133	C₂²	—	37	1	no	—
134	C₂²	—	37	1	no	—
135	C₂²	—	37	2	no	—
136	C₂²	—	37	2	no	—
137	C₂²	D₄×#3866	1	1	yes	characteristic
138	C₄	—	4	2	no	—
139	C₄	—	37	1	no	—
140	C₄	—	37	1	no	—
141	C₄	—	296	2	no	—
142	C₄	—	148	2	no	—
143	C₄	—	148	2	no	—
144	C₄	C₂×C₄⋊#3866	1	1	yes	characteristic
145	C₄	Q₈×#3866	1	1	yes	characteristic
146	C₂	—	37	2	no	—
147	C₂	—	37	2	no	—
148	C₂	#14558484	1	1	yes	characteristic
149	C₂	#14558573	1	1	yes	characteristic
150	C₂	#14558574	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#11924604

C₂

yes

maximal, characteristic

C₂×#11924590

C₂

yes

maximal, characteristic

#14558484

C₂

yes

maximal

#14558406

C₂

yes

maximal

#107×D₃₇

C₂

yes

maximal, characteristic

C₂×D₃₇⋊C₈

C₂²

yes

characteristic

C₂×C₄⋊#3866

—

C₂×C₄⋊#3866

C₂²

yes

—

C₂×C₄×#3866

C₂²

yes

—

—

—

C₂×C₈×D₃₇

C₂²

yes

characteristic

C₄⋊C₄×D₃₇

C₂²

yes

—

#11924470

C₄

yes

characteristic, complemented

#11924469

C₄

yes

complemented

#107×C₃₇

C₄

yes

characteristic, complemented

C₂²×#3866

—

C₂²×#3866

D₄

yes

—

C₂×C₄×D₃₇

—

C₂×C₄×D₃₇

C₂³

yes

—

—

—

—

—

—

D₃₇⋊C₈

—

D₃₇⋊C₈

Q₈

yes

—

C₂×C₈×C₃₇

C₂×C₄

yes

characteristic

C₄⋊C₄×C₃₇

C₂×C₄

yes

—

C₄⋊Dic₃₇

C₂×C₄

yes

—

#10596936

—

C₂×#64436

C₂×C₄

yes

characteristic

C₈×D₃₇

D₄

yes

—

C₈×D₃₇

Q₈

yes

—

C₂²×D₃₇

C₂×D₄

yes

—

—

—

—

—

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

C₂²×C₄

yes

characteristic

C₂×Dic₃₇

—

C₂×Dic₃₇

#41

yes

characteristic

C₄×D₃₇

—

C₄×D₃₇

—

C₄×D₃₇

C₂×D₄

yes

characteristic

C₄×D₃₇

C₂×Q₈

yes

characteristic

C₄×D₃₇

C₂×Q₈

yes

C₄×D₃₇

yes

—

C₈×C₃₇

C₄⋊C₄

yes

complemented

#64436

C₄⋊C₄

yes

complemented

C₂²×C₃₇

C₄×D₄

yes

characteristic

C₂×D₃₇

#121

yes

characteristic

C₂×D₃₇

C₄⋊Q₈

yes

characteristic

C₂×D₃₇

AGL(1,ℤ/8)

yes

—

C₂×D₃₇

#137

yes

—

#3866

—

#3866

—

C₄×C₃₇

—

C₄×C₃₇

C₂×C₄⋊C₄

yes

characteristic

C₄×C₃₇

C₄×Q₈

yes

characteristic

Dic₃₇

—

Dic₃₇

C₂²⋊Q₈

yes

Dic₃₇

yes

—

C₂×C₃₇

yes

C₂×C₃₇

yes

C₂×C₃₇

yes

D₃₇

yes

—

D₃₇

#501

yes

—

C₂×#107

—

C₂×#106

—

C₂×#97

—

#389

—

#340

—

C₃₇

#1943

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂²×C₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

—

—

—

yes

characteristic, complemented

#106

—

#97

—

#95

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂×C₈

—

C₂×C₈

—

100

C₂×C₈

—

101

C₂×C₈

—

102

C₂×C₈

—

103

C₂×C₈

—

104

C₂×C₈

C₂×#3866

yes

characteristic

105

C₄⋊C₄

—

106

C₄⋊C₄

—

107

C₄⋊C₄

—

108

C₄⋊C₄

—

109

C₄⋊C₄

—

148

—

110

C₄⋊C₄

C₂×#3866

yes

—

111

C₄²

—

112

C₄²

—

113

C₂³

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

148

—

125

C₂×C₄

—

148

—

126

C₂×C₄

—

148

—

127

C₂×C₄

—

148

—

128

C₂×C₄

—

148

—

129

C₂×C₄

C₂²×#3866

yes

characteristic

130

C₈

—

131

C₈

—

132

C₈

C₄⋊#3866

yes

complemented

133

C₂²

—

134

C₂²

—

135

C₂²

—

136

C₂²

—

137

C₂²

D₄×#3866

yes

characteristic

138

C₄

—

139

C₄

—

140

C₄

—

141

C₄

—

296

—

142

C₄

—

148

—

143

C₄

—

148

—

144

C₄

C₂×C₄⋊#3866

yes

characteristic

145

C₄

Q₈×#3866

yes

characteristic

146

C₂

—

147

C₂

—

148

C₂

#14558484

yes

characteristic

149

C₂

#14558573

yes

characteristic

150

C₂

#14558574

yes

characteristic

	subgroup	G/H
center	C₂²	D₄×#3866
commutator	C₂×C₄×C₃₇	C₂²×C₄
Frattini	C₂×C₄	C₂²×#3866
Fitting	#107×C₃₇	C₄
socle	C₂²×C₃₇	C₄×D₄
prime radical	C₂²×D₃₇	C₂×D₄
cosocle	C₂×C₄×D₃₇	C₂³
hypercenter	#107	#3866
hypocenter	C₃₇	#1943
LCS	G ▷ C₂×C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #107

subgroup

G/H

center

C₂²

D₄×#3866

commutator

C₂×C₄×C₃₇

C₂²×C₄

Frattini

C₂×C₄

C₂²×#3866

Fitting

#107×C₃₇

C₄

socle

C₂²×C₃₇

C₄×D₄

prime radical

C₂²×D₃₇

C₂×D₄

cosocle

C₂×C₄×D₃₇

C₂³

C₃₇

G ▷ C₂×C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ #107

N	H ≅ G/N	type
15. #11924470	C₄	semidirect
16. #11924469	C₄	semidirect
17. #107×C₃₇	C₄	semidirect
55. C₈×C₃₇	C₄⋊C₄	semidirect
56. #64436	C₄⋊C₄	semidirect
81. C₃₇	#1943	semidirect
90. #107	#3866	semidirect
132. C₈	C₄⋊#3866	semidirect

H ≅ G/N

type

15. #11924470

C₄

semidirect

16. #11924469

C₄

semidirect

17. #107×C₃₇

C₄

semidirect

55. C₈×C₃₇

C₄⋊C₄

semidirect

56. #64436

C₄⋊C₄

semidirect

81. C₃₇

#1943

semidirect

90. #107

#3866

semidirect

132. C₈

C₄⋊#3866

semidirect

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

subgroup	index
C₂×#11924604	2
C₂×#11924590	2
#14558484	2
#14558406	2
#107×D₃₇	2
#1943	37

subgroup

index

#107×D₃₇

Subgroups — #18010701

Conjugacy classes of subgroups

Complements

Maximal subgroups