Group #18007263 — subgroups

Subgroups — #18007263

Order 4736.

392 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 392 conjugacy classes of subgroups — too many to draw. Only its 139 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	Q₈×Dic₇₄	C₂	1	1	yes	maximal, characteristic, complemented
2	#14557879	C₂	1	3	yes	maximal, complemented
3	#14557823	C₂	1	3	yes	maximal, complemented
4	#14557730	C₂	1	3	yes	maximal, complemented
5	Q₈×#64436	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄×#10596940	C₂	1	3	yes	maximal, complemented
7	C₃₇×#550	C₂	1	1	yes	maximal, characteristic
8	C₂×#10596940	C₂²	1	3	yes	—
9	#11924536	C₂²	1	3	yes	—
10	#11924535	—	2	3	no	—
11	Q₈×Dic₃₇	—	2	1	no	—
12	C₄×Dic₇₄	C₂²	1	3	yes	—
13	#11924467	C₂²	1	3	yes	—
14	C₄⋊#64436	C₂²	1	3	yes	—
15	#11924459	C₂²	1	1	yes	characteristic
16	#11924457	C₂²	1	6	yes	—
17	#11924457	C₂²	1	3	yes	—
18	C₄×#64436	C₂²	1	3	yes	—
19	C₂×#41×C₃₇	—	2	1	no	—
20	#127×C₃₇	C₂²	1	3	yes	—
21	#121×C₃₇	—	2	3	no	—
22	C₄×Q₈×C₃₇	C₂²	1	1	yes	characteristic
23	C₄×D₄×C₃₇	C₂²	1	3	yes	—
24	#41×C₃₇	—	4	4	no	—
25	C₂×Q₈×C₃₇	D₄	1	1	yes	characteristic
26	C₂×D₄×C₃₇	—	2	3	no	—
27	C₂×D₄×C₃₇	—	2	3	no	—
28	C₂×D₄×C₃₇	C₂³	1	3	yes	—
29	C₂²×C₄×C₃₇	—	2	3	no	—
30	C₂×Dic₇₄	—	2	3	no	—
31	C₂×Dic₇₄	C₂³	1	1	yes	characteristic
32	C₄⋊C₄×C₃₇	D₄	1	3	yes	—
33	C₄⋊C₄×C₃₇	C₂³	1	1	yes	characteristic
34	C₂²⋊C₄×C₃₇	—	2	3	no	—
35	C₄²×C₃₇	C₂³	1	3	yes	—
36	#10596940	—	2	6	no	—
37	C₄⋊Dic₃₇	—	2	3	no	—
38	C₄⋊Dic₃₇	C₂³	1	3	yes	—
39	#10596936	—	2	3	no	—
40	C₄×Dic₃₇	—	2	3	no	—
41	C₂×#64436	C₂³	1	3	yes	—
42	C₂×#64436	C₂³	1	1	yes	characteristic
43	C₂³×C₃₇	—	2	3	no	—
44	Q₈×C₃₇	SD₁₆	1	4	yes	complemented
45	D₄×C₃₇	—	2	3	no	—
46	D₄×C₃₇	—	2	3	no	—
47	D₄×C₃₇	—	4	6	no	—
48	D₄×C₃₇	—	8	3	no	—
49	C₂×C₄×C₃₇	—	2	1	no	—
50	C₂×C₄×C₃₇	—	4	6	no	—
51	C₂×C₄×C₃₇	C₂×D₄	1	3	yes	—
52	C₂×C₄×C₃₇	C₂×D₄	1	3	yes	—
53	C₂×C₄×C₃₇	C₂⁴	1	1	yes	characteristic
54	C₂×Dic₃₇	—	2	1	no	—
55	C₂×Dic₃₇	—	2	3	no	—
56	Dic₇₄	—	2	1	no	—
57	Dic₇₄	—	8	3	no	—
58	Dic₇₄	#41	1	2	yes	complemented
59	#64436	—	2	3	no	—
60	#64436	—	2	1	no	—
61	C₂²×C₃₇	—	4	3	no	—
62	C₂²×C₃₇	—	4	3	no	—
63	C₂²×C₃₇	—	4	3	no	—
64	C₂²×C₃₇	C₂²×D₄	1	1	yes	characteristic
65	C₄×C₃₇	—	2	3	no	—
66	C₄×C₃₇	—	4	1	no	—
67	C₄×C₃₇	C₂×SD₁₆	1	6	yes	—
68	C₄×C₃₇	C₂×#41	1	1	yes	characteristic
69	C₄×C₃₇	2₋¹⁺⁴	1	1	yes	characteristic
70	Dic₃₇	—	2	2	no	—
71	Dic₃₇	—	4	3	no	—
72	#3362	—	37	1	no	maximal, Sylow
73	C₂×C₃₇	—	8	3	no	—
74	C₂×C₃₇	#548	1	1	yes	characteristic
75	C₂×C₃₇	C₂²×SD₁₆	1	1	yes	characteristic
76	C₂×C₃₇	#576	1	1	yes	characteristic
77	Q₈²	—	37	1	no	—
78	#550	D₃₇	1	1	yes	characteristic
79	#486	—	37	3	no	—
80	#476	—	37	3	no	—
81	#460	—	37	3	no	—
82	C₈×Q₈	—	37	1	no	—
83	C₄×SD₁₆	—	37	3	no	—
84	C₃₇	#3362	1	1	yes	characteristic, complemented, Sylow
85	C₂×#41	—	2	1	no	—
86	C₂×SD₁₆	—	37	3	no	—
87	C₄⋊Q₈	—	37	3	no	—
88	C₄⋊Q₈	—	74	3	no	—
89	#127	C₂×D₃₇	1	3	yes	—
90	#121	—	2	3	no	—
91	C₄×Q₈	—	37	3	no	—
92	C₄×Q₈	—	74	1	no	—
93	C₄×Q₈	C₂×D₃₇	1	1	yes	characteristic
94	C₄×D₄	C₂×D₃₇	1	3	yes	—
95	#106	—	37	3	no	—
96	C₄⋊C₈	—	37	3	no	—
97	#103	—	37	1	no	—
98	#102	—	37	3	no	—
99	#102	—	37	6	no	—
100	C₄×C₈	—	37	3	no	—
101	#41	—	4	4	no	—
102	C₂×Q₈	—	37	1	no	—
103	C₂×Q₈	—	74	3	no	—
104	C₂×Q₈	C₂²⋊D₃₇	1	1	yes	characteristic
105	C₂×D₄	—	2	3	no	—
106	C₂×D₄	—	2	3	no	—
107	C₂×D₄	C₂²×D₃₇	1	3	yes	—
108	C₂²×C₄	—	2	3	no	—
109	SD₁₆	—	74	6	no	—
110	C₂×C₈	—	37	1	no	—
111	C₂×C₈	—	37	3	no	—
112	C₄⋊C₄	—	37	3	no	—
113	C₄⋊C₄	—	74	3	no	—
114	C₄⋊C₄	—	74	3	no	—
115	C₄⋊C₄	C₂²⋊D₃₇	1	3	yes	—
116	C₄⋊C₄	C₂²×D₃₇	1	1	yes	characteristic
117	C₂²⋊C₄	—	2	3	no	—
118	C₄²	—	74	3	no	—
119	C₄²	C₂²×D₃₇	1	3	yes	—
120	C₂³	—	2	3	no	—
121	Q₈	—	37	2	no	—
122	Q₈	—	296	3	no	—
123	Q₈	—	74	1	no	—
124	Q₈	#10596940	1	4	yes	complemented
125	D₄	—	2	3	no	—
126	D₄	—	2	3	no	—
127	D₄	—	4	6	no	—
128	D₄	—	8	3	no	—
129	C₂×C₄	—	2	1	no	—
130	C₂×C₄	—	4	6	no	—
131	C₂×C₄	—	74	1	no	—
132	C₂×C₄	—	74	3	no	—
133	C₂×C₄	C₂×C₂²⋊D₃₇	1	3	yes	—
134	C₂×C₄	C₂×C₂²⋊D₃₇	1	3	yes	—
135	C₂×C₄	C₂³×D₃₇	1	1	yes	characteristic
136	C₈	—	74	3	no	—
137	C₈	—	74	1	no	—
138	C₂²	—	4	3	no	—
139	C₂²	—	4	3	no	—
140	C₂²	—	4	3	no	—
141	C₂²	C₂²×C₂²⋊D₃₇	1	1	yes	characteristic
142	C₄	—	2	3	no	—
143	C₄	—	4	1	no	—
144	C₄	—	74	2	no	—
145	C₄	—	148	3	no	—
146	C₄	C₂×#10596940	1	6	yes	—
147	C₄	#41×D₃₇	1	1	yes	characteristic
148	C₄	#11924583	1	1	yes	characteristic
149	C₂	—	8	3	no	—
150	C₂	#14558190	1	1	yes	characteristic
151	C₂	C₂²×#10596940	1	1	yes	characteristic
152	C₂	#14558336	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

Q₈×Dic₇₄

C₂

yes

maximal, characteristic, complemented

#14557879

C₂

yes

maximal, complemented

#14557823

C₂

yes

maximal, complemented

#14557730

C₂

yes

maximal, complemented

Q₈×#64436

C₂

yes

maximal, characteristic, complemented

C₄×#10596940

C₂

yes

maximal, complemented

C₃₇×#550

C₂

yes

maximal, characteristic

C₂×#10596940

C₂²

yes

—

#11924536

C₂²

yes

—

#11924535

—

Q₈×Dic₃₇

—

C₄×Dic₇₄

C₂²

yes

—

#11924467

C₂²

yes

—

C₄⋊#64436

C₂²

yes

—

#11924459

C₂²

yes

characteristic

#11924457

C₂²

yes

—

#11924457

C₂²

yes

—

C₄×#64436

C₂²

yes

—

C₂×#41×C₃₇

—

#127×C₃₇

C₂²

yes

—

#121×C₃₇

—

C₄×Q₈×C₃₇

C₂²

yes

characteristic

C₄×D₄×C₃₇

C₂²

yes

—

#41×C₃₇

—

C₂×Q₈×C₃₇

D₄

yes

characteristic

C₂×D₄×C₃₇

—

C₂×D₄×C₃₇

—

C₂×D₄×C₃₇

C₂³

yes

—

C₂²×C₄×C₃₇

—

C₂×Dic₇₄

—

C₂×Dic₇₄

C₂³

yes

characteristic

C₄⋊C₄×C₃₇

D₄

yes

—

C₄⋊C₄×C₃₇

C₂³

yes

characteristic

C₂²⋊C₄×C₃₇

—

C₄²×C₃₇

C₂³

yes

—

#10596940

—

C₄⋊Dic₃₇

—

C₄⋊Dic₃₇

C₂³

yes

—

#10596936

—

C₄×Dic₃₇

—

C₂×#64436

C₂³

yes

—

C₂×#64436

C₂³

yes

characteristic

C₂³×C₃₇

—

Q₈×C₃₇

SD₁₆

yes

complemented

D₄×C₃₇

—

D₄×C₃₇

—

D₄×C₃₇

—

D₄×C₃₇

—

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

—

Dic₇₄

yes

—

—

C₂²×C₃₇

—

C₂²×C₃₇

—

C₂²×C₃₇

—

C₂²×C₃₇

C₂²×D₄

yes

characteristic

C₄×C₃₇

—

C₄×C₃₇

—

C₄×C₃₇

C₂×SD₁₆

yes

—

C₄×C₃₇

C₂×#41

yes

characteristic

C₄×C₃₇

2₋¹⁺⁴

yes

characteristic

Dic₃₇

—

Dic₃₇

—

#3362

—

maximal, Sylow

C₂×C₃₇

—

C₂×C₃₇

#548

yes

characteristic

C₂×C₃₇

C₂²×SD₁₆

yes

characteristic

C₂×C₃₇

#576

yes

characteristic

Q₈²

—

#550

D₃₇

yes

—

—

—

C₈×Q₈

—

C₄×SD₁₆

—

C₃₇

#3362

yes

characteristic, complemented, Sylow

C₂×#41

—

C₂×SD₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

—

#127

C₂×D₃₇

yes

—

#121

—

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

C₂×D₃₇

yes

characteristic

C₄×D₄

C₂×D₃₇

yes

—

#106

—

C₄⋊C₈

—

#103

—

#102

—

#102

—

100

C₄×C₈

—

101

#41

—

102

C₂×Q₈

—

103

C₂×Q₈

—

104

C₂×Q₈

C₂²⋊D₃₇

yes

characteristic

105

C₂×D₄

—

106

C₂×D₄

—

107

C₂×D₄

C₂²×D₃₇

yes

—

108

C₂²×C₄

—

109

SD₁₆

—

110

C₂×C₈

—

111

C₂×C₈

—

112

C₄⋊C₄

—

113

C₄⋊C₄

—

114

C₄⋊C₄

—

115

C₄⋊C₄

C₂²⋊D₃₇

yes

—

116

C₄⋊C₄

C₂²×D₃₇

yes

characteristic

117

C₂²⋊C₄

—

118

C₄²

—

119

C₄²

C₂²×D₃₇

yes

—

120

C₂³

—

121

Q₈

—

122

Q₈

—

296

—

123

Q₈

—

124

Q₈

#10596940

yes

complemented

125

D₄

—

126

D₄

—

127

D₄

—

128

D₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

C₂×C₂²⋊D₃₇

yes

—

134

C₂×C₄

C₂×C₂²⋊D₃₇

yes

—

135

C₂×C₄

C₂³×D₃₇

yes

characteristic

136

C₈

—

137

C₈

—

138

C₂²

—

139

C₂²

—

140

C₂²

—

141

C₂²

C₂²×C₂²⋊D₃₇

yes

characteristic

142

C₄

—

143

C₄

—

144

C₄

—

145

C₄

—

148

—

146

C₄

C₂×#10596940

yes

—

147

C₄

#41×D₃₇

yes

characteristic

148

C₄

#11924583

yes

characteristic

149

C₂

—

150

C₂

#14558190

yes

characteristic

151

C₂

C₂²×#10596940

yes

characteristic

152

C₂

#14558336

yes

characteristic

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₃₇
commutator, cosocle	C₂×C₄×C₃₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₃₇
Fitting, prime radical	C₃₇×#550	C₂
socle	C₂²×C₃₇	C₂²×D₄
hypercenter	#550	D₃₇
hypocenter	C₃₇	#3362
LCS	G ▷ C₂×C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ #550

subgroup

G/H

center

C₂²

C₂²×C₂²⋊D₃₇

commutator, cosocle

C₂×C₄×C₃₇

C₂⁴

Frattini

C₂×C₄

C₂³×D₃₇

Fitting, prime radical

C₃₇×#550

C₂

socle

C₂²×C₃₇

C₂²×D₄

D₃₇

C₃₇

G ▷ C₂×C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇

UCS

C₁ ◁ C₂² ◁ C₄×Q₈ ◁ #550

N	H ≅ G/N	type
1. Q₈×Dic₇₄	C₂	semidirect
2. #14557879	C₂	semidirect
3. #14557823	C₂	semidirect
4. #14557730	C₂	semidirect
5. Q₈×#64436	C₂	semidirect
6. C₄×#10596940	C₂	semidirect
44. Q₈×C₃₇	SD₁₆	semidirect
58. Dic₇₄	#41	semidirect
84. C₃₇	#3362	semidirect
124. Q₈	#10596940	semidirect

H ≅ G/N

type

1. Q₈×Dic₇₄

C₂

semidirect

2. #14557879

C₂

semidirect

3. #14557823

C₂

semidirect

4. #14557730

C₂

semidirect

5. Q₈×#64436

C₂

semidirect

6. C₄×#10596940

C₂

semidirect

44. Q₈×C₃₇

SD₁₆

semidirect

58. Dic₇₄

#41

semidirect

84. C₃₇

#3362

semidirect

124. Q₈

#10596940

semidirect

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

subgroup	index
Q₈×Dic₇₄	2
#14557879	2
#14557823	2
#14557730	2
Q₈×#64436	2
C₄×#10596940	2
C₃₇×#550	2
#3362	37

subgroup

index

Q₈×Dic₇₄

C₃₇×#550

Subgroups — #18007263

Conjugacy classes of subgroups

Complements

Maximal subgroups