Group #18010164 — subgroups

Subgroups — #18010164

Order 4736.

168 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 168 conjugacy classes of subgroups — too many to draw. Only its 70 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	#14558537	C₂	1	1	yes	maximal, characteristic
2	D₃₇⋊(C₄⋊C₈)	C₂	1	1	yes	maximal, characteristic
3	#14558395	C₂	1	2	yes	maximal, complemented
4	#14558394	C₂	1	2	yes	maximal, complemented
5	#14557585	C₂	1	1	yes	maximal, characteristic
6	C₂×#10596953	—	2	1	no	—
7	C₂×D₃₇⋊C₈	—	2	1	no	—
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	C₄⋊#64438	C₂²	1	2	yes	—
13	C₄²×D₃₇	C₂²	1	1	yes	characteristic
14	C₄⋊#64436	C₄	1	1	yes	characteristic
15	#11924448	C₄	1	2	yes	—
16	C₈×Dic₃₇	C₂²	1	2	yes	—
17	C₄⋊C₈×C₃₇	C₄	1	1	yes	characteristic
18	C₂×C₄×D₃₇	C₂×C₄	1	1	yes	characteristic
19	C₂×C₄×D₃₇	D₄	1	1	yes	characteristic
20	C₂×C₄×D₃₇	Q₈	1	1	yes	characteristic
21	C₂×#64438	—	2	2	no	—
22	C₂×#64438	—	2	1	no	—
23	C₂×#64438	—	2	1	no	—
24	#10596953	—	4	2	no	—
25	D₃₇⋊C₈	—	4	2	no	—
26	C₂×C₈×C₃₇	C₂×C₄	1	2	yes	—
27	C₄²×C₃₇	C₂×C₄	1	1	yes	characteristic
28	C₄×Dic₃₇	D₄	1	1	yes	characteristic
29	C₄×Dic₃₇	Q₈	1	1	yes	characteristic
30	C₄×Dic₃₇	C₂³	1	1	yes	characteristic
31	C₂×#64436	C₂×C₄	1	2	yes	—
32	C₂²×D₃₇	C₄⋊C₄	1	1	yes	characteristic
33	C₂×C₄×C₃₇	C₄⋊C₄	1	1	yes	characteristic
34	C₂×C₄×C₃₇	C₄⋊C₄	1	1	yes	characteristic
35	C₂×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
36	C₂×Dic₃₇	C₄⋊C₄	1	1	yes	characteristic
37	C₂×Dic₃₇	C₂×D₄	1	1	yes	characteristic
38	C₂×Dic₃₇	C₂×Q₈	1	1	yes	characteristic
39	C₄×D₃₇	—	2	1	no	—
40	C₄×D₃₇	—	2	1	no	—
41	C₄×D₃₇	—	4	1	no	—
42	C₄×D₃₇	SD₁₆	1	2	yes	—
43	C₄×D₃₇	SD₁₆	1	2	yes	—
44	#64438	—	4	1	no	—
45	#64438	—	4	1	no	—
46	#64438	—	4	2	no	—
47	C₈×C₃₇	—	2	2	no	—
48	#64436	—	2	2	no	—
49	C₂²×C₃₇	C₂×C₄⋊C₄	1	1	yes	characteristic
50	C₂×D₃₇	—	2	1	no	—
51	C₂×D₃₇	—	2	1	no	—
52	C₂×D₃₇	#106	1	2	yes	—
53	C₄×C₃₇	—	2	1	no	—
54	C₄×C₃₇	#106	1	2	yes	—
55	C₄×C₃₇	#131	1	1	yes	characteristic
56	C₄×C₃₇	#131	1	1	yes	characteristic
57	Dic₃₇	—	2	1	no	—
58	Dic₃₇	—	2	1	no	—
59	Dic₃₇	C₂×SD₁₆	1	1	yes	characteristic
60	Dic₃₇	C₂×SD₁₆	1	1	yes	characteristic
61	#1570	—	37	1	no	maximal, Sylow
62	C₂×C₃₇	#424	1	1	yes	characteristic
63	C₂×C₃₇	C₂×#106	1	1	yes	characteristic
64	C₂×C₃₇	#430	1	1	yes	characteristic
65	D₃₇	—	2	2	no	—
66	#431	—	37	1	no	—
67	#423	—	37	1	no	—
68	C₂×C₄⋊C₈	—	37	1	no	—
69	#334	—	37	2	no	—
70	#334	—	37	2	no	—
71	C₃₇	#1570	1	1	yes	characteristic, complemented, Sylow
72	C₂×M₁₆	—	74	1	no	—
73	C₂²×C₈	—	74	1	no	—
74	C₂×C₄²	—	37	1	no	—
75	C₄⋊C₈	—	37	1	no	—
76	C₄⋊C₈	—	37	1	no	—
77	C₄⋊C₈	—	37	2	no	—
78	C₄⋊C₈	—	37	1	no	—
79	C₄⋊C₈	—	74	1	no	—
80	C₄⋊C₈	—	74	1	no	—
81	C₄⋊C₈	#3866	1	1	yes	characteristic
82	C₂²⋊C₈	—	37	2	no	—
83	C₄×C₈	—	37	2	no	—
84	C₂²×C₄	—	37	1	no	—
85	C₂²×C₄	—	37	1	no	—
86	C₂²×C₄	—	37	1	no	—
87	M₁₆	—	148	2	no	—
88	C₂×C₈	—	37	2	no	—
89	C₂×C₈	—	74	1	no	—
90	C₂×C₈	—	74	1	no	—
91	C₂×C₈	—	74	2	no	—
92	C₂×C₈	—	148	2	no	—
93	C₂×C₈	C₂×#3866	1	2	yes	—
94	C₄²	—	37	1	no	—
95	C₄²	—	37	1	no	—
96	C₄²	—	37	1	no	—
97	C₄²	C₂×#3866	1	1	yes	characteristic
98	C₂³	—	37	1	no	—
99	C₂×C₄	—	37	1	no	—
100	C₂×C₄	—	37	2	no	—
101	C₂×C₄	—	37	2	no	—
102	C₂×C₄	—	37	1	no	—
103	C₂×C₄	—	37	1	no	—
104	C₂×C₄	—	74	1	no	—
105	C₂×C₄	—	74	1	no	—
106	C₂×C₄	—	148	1	no	—
107	C₂×C₄	C₄⋊#3866	1	1	yes	characteristic
108	C₂×C₄	C₄⋊#3866	1	1	yes	characteristic
109	C₂×C₄	C₂²×#3866	1	1	yes	characteristic
110	C₈	—	2	2	no	—
111	C₈	—	74	2	no	—
112	C₈	—	148	1	no	—
113	C₈	—	148	1	no	—
114	C₈	—	148	2	no	—
115	C₂²	—	37	2	no	—
116	C₂²	—	74	1	no	—
117	C₂²	—	74	1	no	—
118	C₂²	C₂×C₄⋊#3866	1	1	yes	characteristic
119	C₄	—	2	1	no	—
120	C₄	—	37	1	no	—
121	C₄	—	37	1	no	—
122	C₄	—	74	1	no	—
123	C₄	—	74	1	no	—
124	C₄	#11924604	1	2	yes	—
125	C₄	#11924623	1	1	yes	characteristic
126	C₄	#11924624	1	1	yes	characteristic
127	C₂	—	74	2	no	—
128	C₂	#14558540	1	1	yes	characteristic
129	C₂	C₂×#11924604	1	1	yes	characteristic
130	C₂	#14558548	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14558537

C₂

yes

maximal, characteristic

D₃₇⋊(C₄⋊C₈)

C₂

yes

maximal, characteristic

#14558395

C₂

yes

maximal, complemented

#14558394

C₂

yes

maximal, complemented

#14557585

C₂

yes

maximal, characteristic

C₂×#10596953

—

C₂×D₃₇⋊C₈

—

#11924603

—

C₄⋊#64438

—

C₄⋊#64438

C₂²

yes

characteristic

C₄⋊#64438

C₂²

yes

characteristic

C₄⋊#64438

C₂²

yes

—

C₄²×D₃₇

C₂²

yes

characteristic

C₄⋊#64436

C₄

yes

characteristic

#11924448

C₄

yes

—

C₈×Dic₃₇

C₂²

yes

—

C₄⋊C₈×C₃₇

C₄

yes

characteristic

C₂×C₄×D₃₇

C₂×C₄

yes

characteristic

C₂×C₄×D₃₇

D₄

yes

characteristic

C₂×C₄×D₃₇

Q₈

yes

—

—

—

—

D₃₇⋊C₈

—

C₂×C₈×C₃₇

C₂×C₄

yes

—

C₄²×C₃₇

C₂×C₄

yes

characteristic

C₄×Dic₃₇

D₄

yes

characteristic

C₄×Dic₃₇

Q₈

yes

characteristic

C₄×Dic₃₇

C₂³

yes

characteristic

C₂×#64436

C₂×C₄

yes

—

C₂²×D₃₇

C₄⋊C₄

yes

characteristic

C₂×C₄×C₃₇

C₄⋊C₄

yes

characteristic

C₂×C₄×C₃₇

C₄⋊C₄

yes

characteristic

C₂×C₄×C₃₇

C₂²×C₄

yes

characteristic

C₂×Dic₃₇

C₄⋊C₄

yes

characteristic

C₂×Dic₃₇

C₂×D₄

yes

characteristic

C₂×Dic₃₇

C₂×Q₈

yes

characteristic

C₄×D₃₇

—

C₄×D₃₇

—

C₄×D₃₇

—

C₄×D₃₇

SD₁₆

yes

—

C₄×D₃₇

SD₁₆

yes

—

#64438

—

#64438

—

#64438

—

C₈×C₃₇

—

#64436

—

C₂²×C₃₇

C₂×C₄⋊C₄

yes

characteristic

C₂×D₃₇

—

C₂×D₃₇

—

C₂×D₃₇

#106

yes

—

C₄×C₃₇

—

C₄×C₃₇

#106

yes

—

C₄×C₃₇

yes

C₄×C₃₇

yes

Dic₃₇

—

Dic₃₇

—

Dic₃₇

C₂×SD₁₆

yes

characteristic

Dic₃₇

C₂×SD₁₆

yes

—

C₂×C₃₇

yes

C₂×C₃₇

yes

C₂×C₃₇

yes

D₃₇

—

#431

—

#423

—

C₂×C₄⋊C₈

—

#334

—

#334

—

C₃₇

#1570

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

C₂²×C₈

—

C₂×C₄²

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

#3866

yes

characteristic

C₂²⋊C₈

—

C₄×C₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

M₁₆

—

148

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

148

—

C₂×C₈

C₂×#3866

yes

—

C₄²

—

C₄²

—

C₄²

—

C₄²

C₂×#3866

yes

characteristic

C₂³

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

148

—

107

C₂×C₄

C₄⋊#3866

yes

characteristic

108

C₂×C₄

C₄⋊#3866

yes

characteristic

109

C₂×C₄

C₂²×#3866

yes

characteristic

110

C₈

—

111

C₈

—

112

C₈

—

148

—

113

C₈

—

148

—

114

C₈

—

148

—

115

C₂²

—

116

C₂²

—

117

C₂²

—

118

C₂²

C₂×C₄⋊#3866

yes

characteristic

119

C₄

—

120

C₄

—

121

C₄

—

122

C₄

—

123

C₄

—

124

C₄

#11924604

yes

—

125

C₄

#11924623

yes

characteristic

126

C₄

#11924624

yes

characteristic

127

C₂

—

128

C₂

#14558540

yes

characteristic

129

C₂

C₂×#11924604

yes

characteristic

130

C₂

#14558548

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×C₄⋊#3866
commutator	C₂×C₄×C₃₇	C₂²×C₄
Frattini	C₂×C₄	C₂²×#3866
Fitting	C₄⋊C₈×C₃₇	C₄
socle	C₂²×C₃₇	C₂×C₄⋊C₄
prime radical	C₂²×D₃₇	C₄⋊C₄
cosocle	C₄×Dic₃₇	C₂³
hypercenter	C₄⋊C₈	#3866
hypocenter	C₃₇	#1570
LCS	G ▷ C₂×C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊C₈

subgroup

G/H

center

C₂²

C₂×C₄⋊#3866

commutator

C₂×C₄×C₃₇

C₂²×C₄

Frattini

C₂×C₄

C₂²×#3866

Fitting

C₄⋊C₈×C₃₇

C₄

socle

C₂²×C₃₇

C₂×C₄⋊C₄

prime radical

C₂²×D₃₇

C₄⋊C₄

cosocle

C₄×Dic₃₇

C₂³

C₄⋊C₈

C₃₇

G ▷ C₂×C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊C₈

N	H ≅ G/N	type
3. #14558395	C₂	semidirect
4. #14558394	C₂	semidirect
71. C₃₇	#1570	semidirect

H ≅ G/N

type

3. #14558395

C₂

semidirect

4. #14558394

C₂

semidirect

71. C₃₇

#1570

semidirect

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

subgroup	index
#14558537	2
D₃₇⋊(C₄⋊C₈)	2
#14558395	2
#14558394	2
#14557585	2
#1570	37

subgroup

index

D₃₇⋊(C₄⋊C₈)

Subgroups — #18010164

Conjugacy classes of subgroups

Complements

Maximal subgroups