Group #18007274 — subgroups

Subgroups — #18007274

Order 4736.

336 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 336 conjugacy classes of subgroups — too many to draw. Only its 141 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	#14558262	C₂	1	1	yes	maximal, characteristic
2	#14557885	C₂	1	3	yes	maximal
3	#14557825	C₂	1	3	yes	maximal
4	#14557743	C₂	1	3	yes	maximal
5	Q₈×#64436	C₂	1	1	yes	maximal, characteristic
6	C₄×#10596942	C₂	1	3	yes	maximal
7	Q₈²×C₃₇	C₂	1	1	yes	maximal, characteristic
8	C₂×#10596942	C₂²	1	3	yes	—
9	#11924536	C₂²	1	3	yes	—
10	#11924527	—	2	3	no	—
11	Q₈×Dic₃₇	—	2	1	no	—
12	C₄×Dic₇₄	C₂²	1	3	yes	—
13	#11924469	C₂²	1	3	yes	—
14	C₄⋊#64436	C₂²	1	3	yes	—
15	#11924460	C₂²	1	1	yes	characteristic
16	#11924459	C₂²	1	6	yes	—
17	#11924459	C₂²	1	3	yes	—
18	C₄×#64436	C₂²	1	3	yes	—
19	C₄⋊Q₈×C₃₇	—	2	3	no	—
20	C₄⋊Q₈×C₃₇	C₂²	1	3	yes	—
21	C₄×Q₈×C₃₇	—	2	1	no	—
22	C₄×Q₈×C₃₇	C₂²	1	3	yes	—
23	C₄×Q₈×C₃₇	C₂²	1	1	yes	characteristic
24	C₂×Q₈×C₃₇	—	2	3	no	—
25	C₂×Q₈×C₃₇	D₄	1	1	yes	characteristic
26	C₂×Q₈×C₃₇	C₂³	1	1	yes	characteristic
27	C₂×Dic₇₄	C₂³	1	3	yes	—
28	C₄⋊C₄×C₃₇	—	2	3	no	—
29	C₄⋊C₄×C₃₇	—	2	3	no	—
30	C₄⋊C₄×C₃₇	D₄	1	3	yes	—
31	C₄⋊C₄×C₃₇	C₂³	1	3	yes	—
32	C₄²×C₃₇	—	2	3	no	—
33	C₄²×C₃₇	C₂³	1	3	yes	—
34	#10596942	—	2	6	no	—
35	C₄⋊Dic₃₇	—	2	3	no	—
36	C₄⋊Dic₃₇	—	2	3	no	—
37	C₄⋊Dic₃₇	C₂³	1	1	yes	characteristic
38	#10596936	—	2	3	no	—
39	C₄×Dic₃₇	—	2	3	no	—
40	C₂×#64436	C₂³	1	3	yes	—
41	C₂×#64436	C₂³	1	1	yes	characteristic
42	Q₈×C₃₇	—	2	1	no	—
43	Q₈×C₃₇	—	8	3	no	—
44	Q₈×C₃₇	Q₁₆	1	4	yes	complemented
45	Q₈×C₃₇	#41	1	2	yes	—
46	C₂×C₄×C₃₇	—	2	1	no	—
47	C₂×C₄×C₃₇	—	2	3	no	—
48	C₂×C₄×C₃₇	C₂×D₄	1	3	yes	—
49	C₂×C₄×C₃₇	C₂×D₄	1	3	yes	—
50	C₂×C₄×C₃₇	C₂⁴	1	1	yes	characteristic
51	C₂×Dic₃₇	—	2	1	no	—
52	C₂×Dic₃₇	—	2	3	no	—
53	Dic₇₄	—	2	3	no	—
54	Dic₇₄	—	2	3	no	—
55	#64436	—	2	3	no	—
56	#64436	—	2	1	no	—
57	C₂²×C₃₇	C₂²×D₄	1	1	yes	characteristic
58	C₄×C₃₇	—	2	2	no	—
59	C₄×C₃₇	—	2	3	no	—
60	C₄×C₃₇	—	4	3	no	—
61	C₄×C₃₇	C₂×Q₁₆	1	6	yes	—
62	C₄×C₃₇	C₂×#41	1	1	yes	characteristic
63	C₄×C₃₇	2₋¹⁺⁴	1	1	yes	characteristic
64	Dic₃₇	—	4	3	no	—
65	Dic₃₇	—	4	1	no	—
66	#3363	—	37	1	no	maximal, Sylow
67	C₂×C₃₇	#548	1	1	yes	characteristic
68	C₂×C₃₇	C₂²×Q₁₆	1	1	yes	characteristic
69	C₂×C₃₇	#577	1	1	yes	characteristic
70	Q₈²	D₃₇	1	1	yes	characteristic
71	#557	—	37	1	no	—
72	#487	—	37	3	no	—
73	#477	—	37	3	no	—
74	#462	—	37	3	no	—
75	C₈×Q₈	—	37	1	no	—
76	C₄×Q₁₆	—	37	3	no	—
77	C₃₇	#3363	1	1	yes	characteristic, complemented, Sylow
78	C₂×Q₁₆	—	37	3	no	—
79	C₄⋊Q₈	—	2	3	no	—
80	C₄⋊Q₈	—	37	3	no	—
81	C₄⋊Q₈	C₂×D₃₇	1	3	yes	—
82	#125	—	74	3	no	—
83	C₄×Q₈	—	2	1	no	—
84	C₄×Q₈	—	37	3	no	—
85	C₄×Q₈	—	74	1	no	—
86	C₄×Q₈	C₂×D₃₇	1	3	yes	—
87	C₄×Q₈	C₂×D₃₇	1	1	yes	characteristic
88	#107	—	37	3	no	—
89	C₄⋊C₈	—	37	3	no	—
90	#103	—	37	1	no	—
91	#103	—	37	3	no	—
92	#103	—	37	6	no	—
93	C₄×C₈	—	37	3	no	—
94	C₂×Q₈	—	2	3	no	—
95	C₂×Q₈	—	37	3	no	—
96	C₂×Q₈	C₂²⋊D₃₇	1	1	yes	characteristic
97	C₂×Q₈	C₂²×D₃₇	1	1	yes	characteristic
98	Q₁₆	—	74	6	no	—
99	C₂×C₈	—	37	1	no	—
100	C₂×C₈	—	37	3	no	—
101	C₄⋊C₄	—	2	3	no	—
102	C₄⋊C₄	—	2	3	no	—
103	C₄⋊C₄	—	37	1	no	—
104	C₄⋊C₄	—	74	3	no	—
105	C₄⋊C₄	—	74	3	no	—
106	C₄⋊C₄	—	74	3	no	—
107	C₄⋊C₄	C₂²⋊D₃₇	1	3	yes	—
108	C₄⋊C₄	C₂²×D₃₇	1	3	yes	—
109	C₄²	—	2	3	no	—
110	C₄²	—	74	3	no	—
111	C₄²	C₂²×D₃₇	1	3	yes	—
112	Q₈	—	2	1	no	—
113	Q₈	—	8	3	no	—
114	Q₈	—	74	3	no	—
115	Q₈	—	74	3	no	—
116	Q₈	#10596942	1	4	yes	complemented
117	Q₈	#10596966	1	2	yes	—
118	C₂×C₄	—	2	1	no	—
119	C₂×C₄	—	2	3	no	—
120	C₂×C₄	—	74	1	no	—
121	C₂×C₄	—	74	3	no	—
122	C₂×C₄	C₂×C₂²⋊D₃₇	1	3	yes	—
123	C₂×C₄	C₂×C₂²⋊D₃₇	1	3	yes	—
124	C₂×C₄	C₂³×D₃₇	1	1	yes	characteristic
125	C₈	—	74	3	no	—
126	C₈	—	74	1	no	—
127	C₂²	C₂²×C₂²⋊D₃₇	1	1	yes	characteristic
128	C₄	—	2	3	no	—
129	C₄	—	2	2	no	—
130	C₄	—	4	3	no	—
131	C₄	—	148	1	no	—
132	C₄	—	148	3	no	—
133	C₄	C₂×#10596942	1	6	yes	—
134	C₄	C₂×#10596966	1	1	yes	characteristic
135	C₄	#11924584	1	1	yes	characteristic
136	C₂	#14558194	1	1	yes	characteristic
137	C₂	C₂²×#10596942	1	1	yes	characteristic
138	C₂	#14558344	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14558262

C₂

yes

maximal, characteristic

#14557885

C₂

yes

maximal

#14557825

C₂

yes

maximal

#14557743

C₂

yes

maximal

Q₈×#64436

C₂

yes

maximal, characteristic

C₄×#10596942

C₂

yes

maximal

Q₈²×C₃₇

C₂

yes

maximal, characteristic

C₂×#10596942

C₂²

yes

—

#11924536

C₂²

yes

—

#11924527

—

Q₈×Dic₃₇

—

C₄×Dic₇₄

C₂²

yes

—

#11924469

C₂²

yes

—

C₄⋊#64436

C₂²

yes

—

#11924460

C₂²

yes

characteristic

#11924459

C₂²

yes

—

#11924459

C₂²

yes

—

C₄×#64436

C₂²

yes

—

C₄⋊Q₈×C₃₇

—

C₄⋊Q₈×C₃₇

C₂²

yes

—

C₄×Q₈×C₃₇

—

C₄×Q₈×C₃₇

C₂²

yes

—

C₄×Q₈×C₃₇

C₂²

yes

characteristic

C₂×Q₈×C₃₇

—

C₂×Q₈×C₃₇

D₄

yes

characteristic

C₂×Q₈×C₃₇

C₂³

yes

characteristic

C₂×Dic₇₄

C₂³

yes

—

C₄⋊C₄×C₃₇

—

C₄⋊C₄×C₃₇

—

C₄⋊C₄×C₃₇

D₄

yes

—

C₄⋊C₄×C₃₇

C₂³

yes

—

C₄²×C₃₇

—

C₄²×C₃₇

C₂³

yes

—

#10596942

—

C₄⋊Dic₃₇

—

C₄⋊Dic₃₇

—

C₄⋊Dic₃₇

C₂³

yes

characteristic

#10596936

—

C₄×Dic₃₇

—

C₂×#64436

C₂³

yes

—

C₂×#64436

C₂³

yes

characteristic

Q₈×C₃₇

—

Q₈×C₃₇

—

Q₈×C₃₇

Q₁₆

yes

complemented

Q₈×C₃₇

#41

yes

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

C₂×D₄

yes

—

C₂×C₄×C₃₇

C₂×D₄

yes

—

C₂×C₄×C₃₇

C₂⁴

yes

characteristic

C₂×Dic₃₇

—

C₂×Dic₃₇

—

Dic₇₄

—

Dic₇₄

—

#64436

—

#64436

—

C₂²×C₃₇

C₂²×D₄

yes

characteristic

C₄×C₃₇

—

C₄×C₃₇

—

C₄×C₃₇

—

C₄×C₃₇

C₂×Q₁₆

yes

—

C₄×C₃₇

C₂×#41

yes

characteristic

C₄×C₃₇

2₋¹⁺⁴

yes

characteristic

Dic₃₇

—

Dic₃₇

—

—

C₂×C₃₇

yes

C₂×C₃₇

C₂²×Q₁₆

yes

characteristic

C₂×C₃₇

#577

yes

characteristic

Q₈²

D₃₇

yes

—

—

—

—

C₈×Q₈

—

C₄×Q₁₆

—

C₃₇

#3363

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₃₇

yes

—

#125

—

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

C₂×D₃₇

yes

—

C₄×Q₈

C₂×D₃₇

yes

characteristic

#107

—

C₄⋊C₈

—

#103

—

#103

—

#103

—

C₄×C₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂²⋊D₃₇

yes

characteristic

C₂×Q₈

C₂²×D₃₇

yes

characteristic

Q₁₆

—

C₂×C₈

—

100

C₂×C₈

—

101

C₄⋊C₄

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

106

C₄⋊C₄

—

107

C₄⋊C₄

C₂²⋊D₃₇

yes

—

108

C₄⋊C₄

C₂²×D₃₇

yes

—

109

C₄²

—

110

C₄²

—

111

C₄²

C₂²×D₃₇

yes

—

112

Q₈

—

113

Q₈

—

114

Q₈

—

115

Q₈

—

116

Q₈

#10596942

yes

complemented

117

Q₈

#10596966

yes

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

C₂×C₂²⋊D₃₇

yes

—

123

C₂×C₄

C₂×C₂²⋊D₃₇

yes

—

124

C₂×C₄

C₂³×D₃₇

yes

characteristic

125

C₈

—

126

C₈

—

127

C₂²

C₂²×C₂²⋊D₃₇

yes

characteristic

128

C₄

—

129

C₄

—

130

C₄

—

131

C₄

—

148

—

132

C₄

—

148

—

133

C₄

C₂×#10596942

yes

—

134

C₄

C₂×#10596966

yes

characteristic

135

C₄

#11924584

yes

characteristic

136

C₂

#14558194

yes

characteristic

137

C₂

C₂²×#10596942

yes

characteristic

138

C₂

#14558344

yes

characteristic

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₃₇
commutator, cosocle	C₂×C₄×C₃₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₃₇
Fitting	Q₈²×C₃₇	C₂
socle, prime radical	C₂²×C₃₇	C₂²×D₄
hypercenter	Q₈²	D₃₇
hypocenter	C₃₇	#3363
LCS	G ▷ C₂×C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ Q₈²

subgroup

G/H

center

C₂²

C₂²×C₂²⋊D₃₇

commutator, cosocle

C₂×C₄×C₃₇

C₂⁴

Frattini

C₂×C₄

C₂³×D₃₇

Fitting

Q₈²×C₃₇

C₂

socle, prime radical

C₂²×C₃₇

C₂²×D₄

hypercenter

Q₈²

D₃₇

hypocenter

C₃₇

#3363

LCS

G ▷ C₂×C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇

UCS

C₁ ◁ C₂² ◁ C₄×Q₈ ◁ Q₈²

N	H ≅ G/N	type
44. Q₈×C₃₇	Q₁₆	semidirect
77. C₃₇	#3363	semidirect
116. Q₈	#10596942	semidirect

H ≅ G/N

type

44. Q₈×C₃₇

Q₁₆

semidirect

77. C₃₇

#3363

semidirect

116. Q₈

#10596942

semidirect

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

subgroup	index
#14558262	2
#14557885	2
#14557825	2
#14557743	2
Q₈×#64436	2
C₄×#10596942	2
Q₈²×C₃₇	2
#3363	37

subgroup

index

Q₈²×C₃₇

Subgroups — #18007274

Conjugacy classes of subgroups

Complements

Maximal subgroups