Group #18010221 — subgroups

Subgroups — #18010221

Order 4736.

152 conjugacy classes of subgroups; only the 56 normal subgroups are shown.

normal subgroups only

This group has 152 conjugacy classes of subgroups — too many to draw. Only its 56 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	#14558584	C₂	1	1	yes	maximal, characteristic
2	Q₈×#64438	C₂	1	1	yes	maximal, characteristic
3	#14558395	C₂	1	1	yes	maximal, characteristic
4	Q₈⋊#64438	C₂	1	1	yes	maximal, characteristic
5	Q₈⋊#64438	C₂	1	1	yes	maximal, characteristic
6	#64438⋊C₈	C₂	1	1	yes	maximal, characteristic
7	Q₁₆×Dic₃₇	C₂	1	1	yes	maximal, characteristic
8	#11924603	—	2	1	no	—
9	C₄⋊#64438	—	2	1	no	—
10	C₄⋊#64438	C₂²	1	1	yes	characteristic
11	C₄⋊#64438	C₂²	1	1	yes	characteristic
12	#11924591	—	2	1	no	—
13	C₄×#64438	—	2	1	no	—
14	C₄×#64438	C₂²	1	1	yes	characteristic
15	C₄×#64438	C₂²	1	1	yes	characteristic
16	Q₈×Dic₃₇	C₂²	1	1	yes	characteristic
17	Q₈×Dic₃₇	C₂²	1	1	yes	characteristic
18	#11924470	C₄	1	1	yes	characteristic
19	#11924460	C₄	1	1	yes	characteristic
20	#11924460	C₄	1	1	yes	characteristic
21	C₈×Dic₃₇	C₂²	1	1	yes	characteristic
22	C₂×Q₁₆×C₃₇	C₄	1	1	yes	characteristic
23	C₂×Q₈×C₃₇	C₂×C₄	1	1	yes	characteristic
24	C₂×Q₈×C₃₇	C₂×C₄	1	1	yes	characteristic
25	C₂×#64438	—	2	1	no	—
26	C₂×#64438	—	2	1	no	—
27	C₂×#64438	—	2	1	no	—
28	C₂×#64438	D₄	1	1	yes	characteristic
29	C₂×#64438	D₄	1	1	yes	characteristic
30	Q₁₆×C₃₇	—	2	2	no	—
31	C₂×C₈×C₃₇	C₂×C₄	1	1	yes	characteristic
32	C₄⋊Dic₃₇	—	2	1	no	—
33	C₄⋊Dic₃₇	—	2	1	no	—
34	C₄⋊Dic₃₇	C₂×C₄	1	1	yes	characteristic
35	C₄⋊Dic₃₇	C₂×C₄	1	1	yes	characteristic
36	C₄×Dic₃₇	—	2	1	no	—
37	C₄×Dic₃₇	—	2	1	no	—
38	C₄×Dic₃₇	C₂³	1	1	yes	characteristic
39	C₂×#64436	C₂×C₄	1	1	yes	characteristic
40	Q₈×C₃₇	—	2	1	no	—
41	Q₈×C₃₇	—	2	1	no	—
42	Q₈×C₃₇	—	2	1	no	—
43	Q₈×C₃₇	M₁₆	1	2	yes	—
44	C₂×C₄×C₃₇	—	2	1	no	—
45	C₂×C₄×C₃₇	—	2	1	no	—
46	C₂×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
47	C₂×Dic₃₇	—	2	1	no	—
48	C₂×Dic₃₇	—	2	1	no	—
49	C₂×Dic₃₇	C₂×D₄	1	1	yes	characteristic
50	C₂×Dic₃₇	#41	1	1	yes	characteristic
51	#64438	—	2	2	no	—
52	#64438	—	2	1	no	—
53	#64438	—	2	1	no	—
54	#64438	—	4	1	no	—
55	#64438	—	4	1	no	—
56	C₈×C₃₇	—	2	1	no	—
57	#64436	—	2	1	no	—
58	C₂²×C₃₇	C₄×D₄	1	1	yes	characteristic
59	C₄×C₃₇	—	2	2	no	—
60	C₄×C₃₇	—	4	1	no	—
61	C₄×C₃₇	C₂×M₁₆	1	1	yes	characteristic
62	C₄×C₃₇	#131	1	1	yes	characteristic
63	Dic₃₇	—	2	1	no	—
64	Dic₃₇	—	4	1	no	—
65	Dic₃₇	—	4	1	no	—
66	Dic₃₇	#137	1	1	yes	characteristic
67	Dic₃₇	#137	1	1	yes	characteristic
68	#1587	—	37	1	no	maximal, Sylow
69	C₂×C₃₇	#435	1	1	yes	characteristic
70	C₂×C₃₇	#441	1	1	yes	characteristic
71	C₂×C₃₇	#443	1	1	yes	characteristic
72	#446	—	37	1	no	—
73	C₈×Q₈	—	37	1	no	—
74	C₄×Q₁₆	—	37	1	no	—
75	#334	—	37	1	no	—
76	Q₈⋊C₈	—	37	1	no	—
77	Q₈⋊C₈	—	37	1	no	—
78	#322	—	37	1	no	—
79	C₃₇	#1587	1	1	yes	characteristic, complemented, Sylow
80	C₂×Q₁₆	#3866	1	1	yes	characteristic
81	C₄×Q₈	—	37	1	no	—
82	C₄×Q₈	—	37	1	no	—
83	#107	—	37	1	no	—
84	C₄⋊C₈	—	37	1	no	—
85	C₄⋊C₈	—	37	1	no	—
86	C₄⋊C₈	—	74	1	no	—
87	C₄⋊C₈	—	74	1	no	—
88	#103	—	37	1	no	—
89	#103	—	37	1	no	—
90	#97	—	74	1	no	—
91	C₄×C₈	—	37	1	no	—
92	C₄×C₈	—	37	1	no	—
93	C₄×C₈	—	37	1	no	—
94	C₄×C₈	—	74	1	no	—
95	C₂×Q₈	C₂×#3866	1	1	yes	characteristic
96	C₂×Q₈	C₂×#3866	1	1	yes	characteristic
97	Q₁₆	—	2	2	no	—
98	C₂×C₈	—	37	1	no	—
99	C₂×C₈	—	37	1	no	—
100	C₂×C₈	—	37	1	no	—
101	C₂×C₈	—	74	1	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	1	no	—
106	C₄⋊C₄	—	37	1	no	—
107	C₄⋊C₄	—	74	1	no	—
108	C₄⋊C₄	—	74	1	no	—
109	C₄²	—	37	1	no	—
110	C₄²	—	74	1	no	—
111	C₄²	—	74	1	no	—
112	Q₈	—	2	1	no	—
113	Q₈	—	2	1	no	—
114	Q₈	—	2	1	no	—
115	Q₈	#10596957	1	2	yes	—
116	C₂×C₄	—	2	1	no	—
117	C₂×C₄	—	2	1	no	—
118	C₂×C₄	—	37	1	no	—
119	C₂×C₄	—	37	1	no	—
120	C₂×C₄	—	74	1	no	—
121	C₂×C₄	—	74	1	no	—
122	C₂×C₄	C₂²×#3866	1	1	yes	characteristic
123	C₈	—	2	1	no	—
124	C₈	—	74	1	no	—
125	C₈	—	74	1	no	—
126	C₈	—	74	2	no	—
127	C₈	—	74	1	no	—
128	C₈	—	148	1	no	—
129	C₈	—	148	1	no	—
130	C₂²	D₄×#3866	1	1	yes	characteristic
131	C₄	—	2	2	no	—
132	C₄	—	4	1	no	—
133	C₄	—	37	1	no	—
134	C₄	—	37	1	no	—
135	C₄	—	74	1	no	—
136	C₄	—	148	1	no	—
137	C₄	—	148	1	no	—
138	C₄	C₂×#10596957	1	1	yes	characteristic
139	C₄	#11924623	1	1	yes	characteristic
140	C₂	#14558564	1	1	yes	characteristic
141	C₂	#14558573	1	1	yes	characteristic
142	C₂	#14558577	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14558584

C₂

yes

maximal, characteristic

Q₈×#64438

C₂

yes

maximal, characteristic

#14558395

C₂

yes

maximal, characteristic

Q₈⋊#64438

C₂

yes

maximal, characteristic

Q₈⋊#64438

C₂

yes

maximal, characteristic

#64438⋊C₈

C₂

yes

maximal, characteristic

Q₁₆×Dic₃₇

C₂

yes

maximal, characteristic

#11924603

—

C₄⋊#64438

—

C₄⋊#64438

C₂²

yes

characteristic

C₄⋊#64438

C₂²

yes

—

—

C₂²

yes

characteristic

C₄×#64438

C₂²

yes

characteristic

Q₈×Dic₃₇

C₂²

yes

characteristic

Q₈×Dic₃₇

C₂²

yes

characteristic

#11924470

C₄

yes

characteristic

#11924460

C₄

yes

characteristic

#11924460

C₄

yes

characteristic

C₈×Dic₃₇

C₂²

yes

characteristic

C₂×Q₁₆×C₃₇

C₄

yes

characteristic

C₂×Q₈×C₃₇

C₂×C₄

yes

characteristic

C₂×Q₈×C₃₇

C₂×C₄

yes

—

—

—

D₄

yes

characteristic

C₂×#64438

D₄

yes

characteristic

Q₁₆×C₃₇

—

C₂×C₈×C₃₇

C₂×C₄

yes

characteristic

C₄⋊Dic₃₇

—

C₄⋊Dic₃₇

—

C₄⋊Dic₃₇

C₂×C₄

yes

characteristic

C₄⋊Dic₃₇

C₂×C₄

yes

characteristic

C₄×Dic₃₇

—

C₄×Dic₃₇

—

C₄×Dic₃₇

C₂³

yes

characteristic

C₂×#64436

C₂×C₄

yes

characteristic

Q₈×C₃₇

—

Q₈×C₃₇

—

Q₈×C₃₇

—

Q₈×C₃₇

M₁₆

yes

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

C₂²×C₄

yes

characteristic

C₂×Dic₃₇

—

C₂×Dic₃₇

—

C₂×Dic₃₇

C₂×D₄

yes

C₂×Dic₃₇

yes

—

—

—

—

—

C₈×C₃₇

—

#64436

—

C₂²×C₃₇

C₄×D₄

yes

characteristic

C₄×C₃₇

—

C₄×C₃₇

—

C₄×C₃₇

C₂×M₁₆

yes

characteristic

C₄×C₃₇

#131

yes

characteristic

Dic₃₇

—

Dic₃₇

—

Dic₃₇

—

Dic₃₇

yes

Dic₃₇

yes

—

C₂×C₃₇

yes

C₂×C₃₇

yes

C₂×C₃₇

yes

—

C₈×Q₈

—

C₄×Q₁₆

—

#334

—

Q₈⋊C₈

—

Q₈⋊C₈

—

#322

—

C₃₇

#1587

yes

characteristic, complemented, Sylow

C₂×Q₁₆

#3866

yes

characteristic

C₄×Q₈

—

C₄×Q₈

—

#107

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

#103

—

#103

—

#97

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₂×Q₈

yes

C₂×Q₈

yes

Q₁₆

—

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

C₄²

—

110

C₄²

—

111

C₄²

—

112

Q₈

—

113

Q₈

—

114

Q₈

—

115

Q₈

#10596957

yes

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

C₂²×#3866

yes

characteristic

123

C₈

—

124

C₈

—

125

C₈

—

126

C₈

—

127

C₈

—

128

C₈

—

148

—

129

C₈

—

148

—

130

C₂²

D₄×#3866

yes

characteristic

131

C₄

—

132

C₄

—

133

C₄

—

134

C₄

—

135

C₄

—

136

C₄

—

148

—

137

C₄

—

148

—

138

C₄

C₂×#10596957

yes

characteristic

139

C₄

#11924623

yes

characteristic

140

C₂

#14558564

yes

characteristic

141

C₂

#14558573

yes

characteristic

142

C₂

#14558577

yes

characteristic

	subgroup	G/H
center	C₂²	D₄×#3866
commutator	C₂×C₄×C₃₇	C₂²×C₄
Frattini	C₂×C₄	C₂²×#3866
Fitting	C₂×Q₁₆×C₃₇	C₄
socle, prime radical	C₂²×C₃₇	C₄×D₄
cosocle	C₄×Dic₃₇	C₂³
hypercenter	C₂×Q₁₆	#3866
hypocenter	C₃₇	#1587
LCS	G ▷ C₂×C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×Q₁₆

subgroup

G/H

center

C₂²

D₄×#3866

commutator

C₂×C₄×C₃₇

C₂²×C₄

Frattini

C₂×C₄

C₂²×#3866

Fitting

C₂×Q₁₆×C₃₇

C₄

socle, prime radical

C₂²×C₃₇

C₄×D₄

cosocle

C₄×Dic₃₇

C₂³

C₂×Q₁₆

C₃₇

G ▷ C₂×C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×Q₁₆

N	H ≅ G/N	type
79. C₃₇	#1587	semidirect

H ≅ G/N

type

79. C₃₇

#1587

semidirect

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

subgroup	index
#14558584	2
Q₈×#64438	2
#14558395	2
Q₈⋊#64438	2
#64438⋊C₈	2
Q₁₆×Dic₃₇	2
#1587	37

subgroup

index

Q₁₆×Dic₃₇

Subgroups — #18010221

Conjugacy classes of subgroups

Complements

Maximal subgroups