Group #18009447 — subgroups

Subgroups — #18009447

Order 4736.

164 conjugacy classes of subgroups; only the 46 normal subgroups are shown.

normal subgroups only

This group has 164 conjugacy classes of subgroups — too many to draw. Only its 46 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	#14558553	C₂	1	1	yes	maximal, characteristic
2	#14558553	C₂	1	1	yes	maximal, characteristic
3	C₄×#10596936	C₂	1	1	yes	maximal, characteristic
4	#11924603	—	2	1	no	—
5	#11924603	C₄	1	2	yes	complemented
6	C₂²⋊#64438	—	2	1	no	—
7	C₂²⋊#64438	—	2	1	no	—
8	#11924591	—	2	1	no	—
9	#11924591	C₄	1	2	yes	complemented
10	C₂×#10596936	—	2	1	no	—
11	C₂×C₄×Dic₃₇	C₄	1	1	yes	characteristic
12	C₂×C₄×Dic₃₇	C₂²	1	1	yes	characteristic
13	#11924443	—	2	1	no	—
14	C₂×C₄²×C₃₇	C₄	1	1	yes	characteristic
15	C₂²×C₄×C₃₇	—	2	1	no	—
16	C₂²×C₄×C₃₇	C₂×C₄	1	1	yes	characteristic
17	C₂²×Dic₃₇	—	2	1	no	—
18	C₂²×Dic₃₇	D₄	1	1	yes	characteristic
19	C₂²×Dic₃₇	D₄	1	1	yes	characteristic
20	C₂×#64438	—	2	2	no	—
21	C₂×#64438	—	2	2	no	—
22	C₄²×C₃₇	—	4	1	no	—
23	#10596936	—	2	4	no	—
24	C₄×Dic₃₇	—	4	1	no	—
25	C₄×Dic₃₇	C₂×C₄	1	1	yes	characteristic
26	C₄×Dic₃₇	C₂×C₄	1	1	yes	characteristic
27	C₄×Dic₃₇	D₄	1	1	yes	characteristic
28	C₄×Dic₃₇	Q₈	1	1	yes	characteristic
29	C₂³×C₃₇	C₂²⋊C₄	1	1	yes	characteristic
30	C₂×C₄×C₃₇	—	2	1	no	—
31	C₂×C₄×C₃₇	—	2	1	no	—
32	C₂×C₄×C₃₇	—	2	2	no	—
33	C₂×C₄×C₃₇	—	4	1	no	—
34	C₂×C₄×C₃₇	—	4	1	no	—
35	C₂×C₄×C₃₇	C₄²	1	1	yes	characteristic
36	C₂×C₄×C₃₇	C₄⋊C₄	1	1	yes	characteristic
37	C₂×Dic₃₇	—	2	2	no	—
38	C₂×Dic₃₇	—	2	2	no	—
39	C₂×Dic₃₇	—	4	1	no	—
40	C₂×Dic₃₇	—	4	1	no	—
41	C₂×Dic₃₇	—	4	1	no	—
42	C₂×Dic₃₇	C₂²⋊C₄	1	1	yes	characteristic
43	C₂×Dic₃₇	C₂²⋊C₄	1	1	yes	characteristic
44	C₂×Dic₃₇	C₄⋊C₄	1	1	yes	characteristic
45	C₂×Dic₃₇	C₄⋊C₄	1	1	yes	characteristic
46	#64438	—	4	2	no	—
47	#64438	—	4	2	no	—
48	C₂²×C₃₇	—	2	1	no	—
49	C₂²×C₃₇	—	2	1	no	—
50	C₂²×C₃₇	#95	1	1	yes	characteristic
51	C₂²×C₃₇	#100	1	1	yes	characteristic
52	C₂²×C₃₇	#101	1	1	yes	characteristic
53	C₄×C₃₇	—	2	1	no	—
54	C₄×C₃₇	—	2	1	no	—
55	C₄×C₃₇	—	4	2	no	—
56	Dic₃₇	—	2	1	no	—
57	Dic₃₇	—	2	1	no	—
58	Dic₃₇	—	2	1	no	—
59	Dic₃₇	—	4	2	no	—
60	Dic₃₇	#104	1	2	yes	—
61	#1285	—	37	1	no	maximal, Sylow
62	C₂×C₃₇	—	2	1	no	—
63	C₂×C₃₇	—	2	1	no	—
64	C₂×C₃₇	H(3,ℤ/4)	1	1	yes	characteristic
65	C₂×C₃₇	#339	1	1	yes	characteristic
66	C₂×C₃₇	#343	1	1	yes	characteristic
67	#432	—	37	1	no	—
68	#432	—	37	1	no	—
69	C₄×C₄⋊C₄	—	37	1	no	—
70	C₃₇	#1285	1	1	yes	characteristic, complemented, Sylow
71	C₂×C₄⋊C₄	—	74	1	no	—
72	C₂×C₄²	—	37	1	no	—
73	C₂×C₄²	—	37	1	no	—
74	C₂×C₄²	#3866	1	1	yes	characteristic
75	C₄⋊C₈	—	37	2	no	—
76	C₄⋊C₈	—	74	1	no	—
77	C₂²⋊C₈	—	74	1	no	—
78	C₂²⋊C₈	—	74	1	no	—
79	#97	—	37	2	no	—
80	#97	—	74	1	no	—
81	#95	—	74	1	no	—
82	C₂²×C₄	—	2	1	no	—
83	C₂²×C₄	—	37	1	no	—
84	C₂²×C₄	—	37	1	no	—
85	C₂²×C₄	—	74	1	no	—
86	C₂²×C₄	C₂×#3866	1	1	yes	characteristic
87	C₂×C₈	—	74	2	no	—
88	C₂×C₈	—	74	2	no	—
89	C₄⋊C₄	—	74	4	no	—
90	C₄²	—	4	1	no	—
91	C₄²	—	37	1	no	—
92	C₄²	—	37	1	no	—
93	C₄²	—	37	1	no	—
94	C₄²	—	37	1	no	—
95	C₄²	—	148	1	no	—
96	C₂³	C₂²⋊#3866	1	1	yes	characteristic
97	C₂×C₄	—	2	1	no	—
98	C₂×C₄	—	2	1	no	—
99	C₂×C₄	—	2	2	no	—
100	C₂×C₄	—	4	1	no	—
101	C₂×C₄	—	4	1	no	—
102	C₂×C₄	—	37	1	no	—
103	C₂×C₄	—	37	1	no	—
104	C₂×C₄	—	37	1	no	—
105	C₂×C₄	—	37	1	no	—
106	C₂×C₄	—	74	2	no	—
107	C₂×C₄	—	74	2	no	—
108	C₂×C₄	—	148	1	no	—
109	C₂×C₄	—	148	1	no	—
110	C₂×C₄	—	148	1	no	—
111	C₂×C₄	C₄×#3866	1	1	yes	characteristic
112	C₂×C₄	C₄⋊#3866	1	1	yes	characteristic
113	C₈	—	148	2	no	—
114	C₈	—	148	2	no	—
115	C₂²	—	2	1	no	—
116	C₂²	—	2	1	no	—
117	C₂²	(C₂×#3866)⋊C₄	1	1	yes	characteristic
118	C₂²	#11924596	1	1	yes	characteristic
119	C₂²	#11924597	1	1	yes	characteristic
120	C₄	—	2	1	no	—
121	C₄	—	2	1	no	—
122	C₄	—	4	2	no	—
123	C₄	—	37	2	no	—
124	C₄	—	74	1	no	—
125	C₄	—	74	1	no	—
126	C₄	—	74	1	no	—
127	C₄	—	148	2	no	—
128	C₂	—	2	1	no	—
129	C₂	—	2	1	no	—
130	C₂	#14558400	1	1	yes	characteristic
131	C₂	#14558404	1	1	yes	characteristic
132	C₂	#14558413	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14558553

C₂

yes

maximal, characteristic

#14558553

C₂

yes

maximal, characteristic

C₄×#10596936

C₂

yes

maximal, characteristic

#11924603

—

#11924603

C₄

yes

—

—

—

C₄

yes

complemented

C₂×#10596936

—

C₂×C₄×Dic₃₇

C₄

yes

characteristic

C₂×C₄×Dic₃₇

C₂²

yes

characteristic

#11924443

—

C₂×C₄²×C₃₇

C₄

yes

characteristic

C₂²×C₄×C₃₇

—

C₂²×C₄×C₃₇

C₂×C₄

yes

characteristic

C₂²×Dic₃₇

—

C₂²×Dic₃₇

D₄

yes

characteristic

C₂²×Dic₃₇

D₄

yes

characteristic

C₂×#64438

—

C₂×#64438

—

C₄²×C₃₇

—

#10596936

—

C₄×Dic₃₇

—

C₄×Dic₃₇

C₂×C₄

yes

characteristic

C₄×Dic₃₇

C₂×C₄

yes

characteristic

C₄×Dic₃₇

D₄

yes

characteristic

C₄×Dic₃₇

Q₈

yes

characteristic

C₂³×C₃₇

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

C₄²

yes

characteristic

C₂×C₄×C₃₇

C₄⋊C₄

yes

characteristic

C₂×Dic₃₇

—

C₂×Dic₃₇

—

C₂×Dic₃₇

—

C₂×Dic₃₇

—

C₂×Dic₃₇

—

C₂×Dic₃₇

C₂²⋊C₄

yes

characteristic

C₂×Dic₃₇

C₂²⋊C₄

yes

characteristic

C₂×Dic₃₇

C₄⋊C₄

yes

characteristic

C₂×Dic₃₇

C₄⋊C₄

yes

characteristic

#64438

—

#64438

—

C₂²×C₃₇

—

C₂²×C₃₇

—

C₂²×C₃₇

yes

C₂²×C₃₇

yes

C₂²×C₃₇

yes

C₄×C₃₇

—

C₄×C₃₇

—

C₄×C₃₇

—

Dic₃₇

—

Dic₃₇

—

Dic₃₇

—

Dic₃₇

—

Dic₃₇

#104

yes

—

#1285

—

maximal, Sylow

C₂×C₃₇

—

C₂×C₃₇

—

C₂×C₃₇

yes

C₂×C₃₇

yes

C₂×C₃₇

yes

—

—

C₄×C₄⋊C₄

—

C₃₇

#1285

yes

characteristic, complemented, Sylow

C₂×C₄⋊C₄

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

#3866

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₂²⋊C₈

—

C₂²⋊C₈

—

#97

—

#97

—

#95

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×#3866

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

148

—

C₂³

C₂²⋊#3866

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

148

—

109

C₂×C₄

—

148

—

110

C₂×C₄

—

148

—

111

C₂×C₄

C₄×#3866

yes

characteristic

112

C₂×C₄

C₄⋊#3866

yes

characteristic

113

C₈

—

148

—

114

C₈

—

148

—

115

C₂²

—

116

C₂²

—

117

C₂²

(C₂×#3866)⋊C₄

yes

characteristic

118

C₂²

#11924596

yes

characteristic

119

C₂²

#11924597

yes

characteristic

120

C₄

—

121

C₄

—

122

C₄

—

123

C₄

—

124

C₄

—

125

C₄

—

126

C₄

—

127

C₄

—

148

—

128

C₂

—

129

C₂

—

130

C₂

#14558400

yes

characteristic

131

C₂

#14558404

yes

characteristic

132

C₂

#14558413

yes

characteristic

	subgroup	G/H
center	C₂²	(C₂×#3866)⋊C₄
commutator	C₂×C₄×C₃₇	C₄²
Frattini	C₂²×C₄	C₂×#3866
Fitting	C₂×C₄²×C₃₇	C₄
socle	C₂²×C₃₇	#95
prime radical	C₂³×C₃₇	C₂²⋊C₄
cosocle	C₂×C₄×Dic₃₇	C₂²
hypercenter	C₂×C₄²	#3866
hypocenter	C₃₇	#1285
LCS	G ▷ C₂×C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄²

subgroup

G/H

center

C₂²

(C₂×#3866)⋊C₄

commutator

C₂×C₄×C₃₇

C₄²

Frattini

C₂²×C₄

C₂×#3866

Fitting

C₂×C₄²×C₃₇

C₄

socle

C₂²×C₃₇

#95

prime radical

C₂³×C₃₇

C₂²⋊C₄

cosocle

C₂×C₄×Dic₃₇

C₂²

C₂×C₄²

C₃₇

G ▷ C₂×C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄²

N	H ≅ G/N	type
5. #11924603	C₄	semidirect
9. #11924591	C₄	semidirect
70. C₃₇	#1285	semidirect

H ≅ G/N

type

5. #11924603

C₄

semidirect

9. #11924591

C₄

semidirect

70. C₃₇

#1285

semidirect

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

subgroup	index
#14558553	2
C₄×#10596936	2
#1285	37

subgroup

index

#14558553

C₄×#10596936

#1285

Subgroups — #18009447

Conjugacy classes of subgroups

Complements

Maximal subgroups