Group #18009802 — subgroups

Subgroups — #18009802

Order 4736.

212 conjugacy classes of subgroups; only the 104 normal subgroups are shown.

normal subgroups only

This group has 212 conjugacy classes of subgroups — too many to draw. Only its 104 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	#14558552	C₂	1	1	yes	maximal, characteristic
2	C₄×D₃₇⋊C₈	C₂	1	1	yes	maximal, characteristic
3	#64438⋊C₈	C₂	1	2	yes	maximal, complemented
4	C₈×#64438	C₂	1	2	yes	maximal, complemented
5	#14557585	C₂	1	1	yes	maximal, characteristic
6	C₂×D₃₇⋊C₈	C₄	1	2	yes	—
7	C₄⋊#64438	C₄	1	2	yes	—
8	#11924593	C₄	1	2	yes	—
9	C₄×#64438	C₄	1	2	yes	—
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	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄×D₃₇	C₂×C₄	1	1	yes	characteristic
21	C₂×#64438	C₂×C₄	1	4	yes	—
22	C₂×#64438	C₂×C₄	1	2	yes	—
23	C₂×#64438	C₂×C₄	1	2	yes	—
24	D₃₇⋊C₈	C₈	1	8	yes	complemented
25	C₂×C₈×C₃₇	C₂×C₄	1	2	yes	—
26	C₄²×C₃₇	C₂×C₄	1	1	yes	characteristic
27	C₄×Dic₃₇	C₂×C₄	1	1	yes	characteristic
28	C₄×Dic₃₇	C₂×C₄	1	1	yes	characteristic
29	C₄×Dic₃₇	C₂³	1	1	yes	characteristic
30	C₂×#64436	C₂×C₄	1	2	yes	—
31	C₂²×D₃₇	C₄²	1	1	yes	characteristic
32	C₂×C₄×C₃₇	C₄²	1	1	yes	characteristic
33	C₂×C₄×C₃₇	C₄²	1	1	yes	characteristic
34	C₂×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
35	C₂×Dic₃₇	C₄²	1	1	yes	characteristic
36	C₂×Dic₃₇	C₂²×C₄	1	1	yes	characteristic
37	C₂×Dic₃₇	C₂²×C₄	1	1	yes	characteristic
38	C₄×D₃₇	—	2	1	no	—
39	C₄×D₃₇	—	2	1	no	—
40	C₄×D₃₇	—	2	2	no	—
41	C₄×D₃₇	C₂×C₈	1	4	yes	—
42	#64438	—	2	4	no	—
43	#64438	C₂×C₈	1	4	yes	—
44	#64438	C₂×C₈	1	4	yes	—
45	C₈×C₃₇	—	2	2	no	—
46	#64436	—	2	2	no	—
47	C₂²×C₃₇	C₂×C₄²	1	1	yes	characteristic
48	C₂×D₃₇	—	2	1	no	—
49	C₂×D₃₇	—	2	1	no	—
50	C₂×D₃₇	C₄×C₈	1	2	yes	—
51	C₄×C₃₇	—	2	1	no	—
52	C₄×C₃₇	C₄×C₈	1	2	yes	—
53	C₄×C₃₇	#131	1	1	yes	characteristic
54	C₄×C₃₇	#131	1	1	yes	characteristic
55	Dic₃₇	—	2	1	no	—
56	Dic₃₇	C₂²×C₈	1	2	yes	—
57	Dic₃₇	#131	1	2	yes	—
58	#1450	—	37	1	no	maximal, Sylow
59	C₂×C₃₇	C₂×C₄×C₈	1	1	yes	characteristic
60	C₂×C₃₇	#405	1	1	yes	characteristic
61	C₂×C₃₇	#405	1	1	yes	characteristic
62	D₃₇	—	2	2	no	—
63	#431	—	37	1	no	—
64	#431	—	37	1	no	—
65	C₂×C₄×C₈	—	37	1	no	—
66	#322	—	37	2	no	—
67	C₈²	—	37	2	no	—
68	C₃₇	#1450	1	1	yes	characteristic, complemented, Sylow
69	C₂²×C₈	—	37	2	no	—
70	C₂×C₄²	—	37	1	no	—
71	C₄⋊C₈	—	37	2	no	—
72	C₄⋊C₈	—	37	1	no	—
73	C₄⋊C₈	#3866	1	1	yes	characteristic
74	C₂²⋊C₈	—	37	2	no	—
75	C₂²⋊C₈	—	37	2	no	—
76	C₄×C₈	—	37	1	no	—
77	C₄×C₈	—	37	2	no	—
78	C₄×C₈	—	37	1	no	—
79	C₄×C₈	—	37	2	no	—
80	C₄×C₈	—	37	2	no	—
81	C₂²×C₄	—	37	1	no	—
82	C₂²×C₄	—	37	1	no	—
83	C₂²×C₄	—	37	1	no	—
84	C₂×C₈	—	37	8	no	—
85	C₂×C₈	—	37	2	no	—
86	C₂×C₈	—	37	2	no	—
87	C₂×C₈	—	37	4	no	—
88	C₂×C₈	—	37	2	no	—
89	C₂×C₈	C₂×#3866	1	2	yes	—
90	C₄²	—	37	1	no	—
91	C₄²	—	37	1	no	—
92	C₄²	—	37	1	no	—
93	C₄²	C₂×#3866	1	1	yes	characteristic
94	C₂³	—	37	1	no	—
95	C₂×C₄	—	37	1	no	—
96	C₂×C₄	—	37	1	no	—
97	C₂×C₄	—	37	4	no	—
98	C₂×C₄	—	37	1	no	—
99	C₂×C₄	—	74	2	no	—
100	C₂×C₄	—	74	1	no	—
101	C₂×C₄	—	74	1	no	—
102	C₂×C₄	C₄×#3866	1	1	yes	characteristic
103	C₂×C₄	C₄×#3866	1	1	yes	characteristic
104	C₂×C₄	C₂²×#3866	1	1	yes	characteristic
105	C₈	—	2	2	no	—
106	C₈	—	37	4	no	—
107	C₈	—	37	4	no	—
108	C₈	—	74	4	no	—
109	C₈	—	74	2	no	—
110	C₂²	—	37	2	no	—
111	C₂²	—	74	1	no	—
112	C₂²	—	74	1	no	—
113	C₂²	C₂×C₄×#3866	1	1	yes	characteristic
114	C₄	—	2	1	no	—
115	C₄	—	37	2	no	—
116	C₄	—	37	2	no	—
117	C₄	—	74	1	no	—
118	C₄	C₈×#3866	1	2	yes	—
119	C₄	#11924623	1	1	yes	characteristic
120	C₄	#11924624	1	1	yes	characteristic
121	C₂	—	74	2	no	—
122	C₂	C₂×C₈×#3866	1	1	yes	characteristic
123	C₂	#14558498	1	1	yes	characteristic
124	C₂	#14558499	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14558552

C₂

yes

maximal, characteristic

C₄×D₃₇⋊C₈

C₂

yes

maximal, characteristic

#64438⋊C₈

C₂

yes

maximal, complemented

C₈×#64438

C₂

yes

maximal, complemented

#14557585

C₂

yes

maximal, characteristic

C₂×D₃₇⋊C₈

C₄

yes

—

C₄⋊#64438

C₄

yes

—

#11924593

C₄

yes

—

C₄×#64438

C₄

yes

—

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

C₂×C₄

yes

characteristic

C₂×C₄×D₃₇

C₂×C₄

yes

characteristic

C₂×#64438

C₂×C₄

yes

—

C₂×#64438

C₂×C₄

yes

—

C₂×#64438

C₂×C₄

yes

—

D₃₇⋊C₈

C₈

yes

complemented

C₂×C₈×C₃₇

C₂×C₄

yes

—

C₄²×C₃₇

C₂×C₄

yes

characteristic

C₄×Dic₃₇

C₂×C₄

yes

characteristic

C₄×Dic₃₇

C₂×C₄

yes

characteristic

C₄×Dic₃₇

C₂³

yes

characteristic

C₂×#64436

C₂×C₄

yes

—

C₂²×D₃₇

C₄²

yes

characteristic

C₂×C₄×C₃₇

C₄²

yes

characteristic

C₂×C₄×C₃₇

C₄²

yes

characteristic

C₂×C₄×C₃₇

C₂²×C₄

yes

characteristic

C₂×Dic₃₇

C₄²

yes

characteristic

C₂×Dic₃₇

C₂²×C₄

yes

characteristic

C₂×Dic₃₇

C₂²×C₄

yes

characteristic

C₄×D₃₇

—

C₄×D₃₇

—

C₄×D₃₇

—

C₄×D₃₇

C₂×C₈

yes

—

#64438

—

#64438

C₂×C₈

yes

—

#64438

C₂×C₈

yes

—

C₈×C₃₇

—

#64436

—

C₂²×C₃₇

C₂×C₄²

yes

characteristic

C₂×D₃₇

—

C₂×D₃₇

—

C₂×D₃₇

C₄×C₈

yes

—

C₄×C₃₇

—

C₄×C₃₇

C₄×C₈

yes

—

C₄×C₃₇

yes

C₄×C₃₇

yes

Dic₃₇

—

Dic₃₇

C₂²×C₈

yes

—

Dic₃₇

#131

yes

—

#1450

—

maximal, Sylow

C₂×C₃₇

C₂×C₄×C₈

yes

C₂×C₃₇

yes

C₂×C₃₇

yes

D₃₇

—

#431

—

#431

—

C₂×C₄×C₈

—

#322

—

C₈²

—

C₃₇

#1450

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂×C₄²

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

#3866

yes

characteristic

C₂²⋊C₈

—

C₂²⋊C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂×#3866

yes

—

C₄²

—

C₄²

—

C₄²

—

C₄²

C₂×#3866

yes

characteristic

C₂³

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

C₄×#3866

yes

characteristic

103

C₂×C₄

C₄×#3866

yes

characteristic

104

C₂×C₄

C₂²×#3866

yes

characteristic

105

C₈

—

106

C₈

—

107

C₈

—

108

C₈

—

109

C₈

—

110

C₂²

—

111

C₂²

—

112

C₂²

—

113

C₂²

C₂×C₄×#3866

yes

characteristic

114

C₄

—

115

C₄

—

116

C₄

—

117

C₄

—

118

C₄

C₈×#3866

yes

—

119

C₄

#11924623

yes

characteristic

120

C₄

#11924624

yes

characteristic

121

C₂

—

122

C₂

C₂×C₈×#3866

yes

characteristic

123

C₂

#14558498

yes

characteristic

124

C₂

#14558499

yes

characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×#3866
commutator	C₂×C₃₇	C₂×C₄×C₈
Fitting	C₄⋊C₈×C₃₇	C₄
socle	C₂²×C₃₇	C₂×C₄²
prime radical	C₂²×D₃₇	C₄²
cosocle	C₄×Dic₃₇	C₂³
hypercenter	C₄⋊C₈	#3866
hypocenter	C₃₇	#1450
LCS	G ▷ C₂×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄⋊C₈

subgroup

G/H

center, Frattini

C₂×C₄

C₂²×#3866

commutator

C₂×C₃₇

C₂×C₄×C₈

Fitting

C₄⋊C₈×C₃₇

C₄

socle

C₂²×C₃₇

C₂×C₄²

prime radical

C₂²×D₃₇

C₄²

cosocle

C₄×Dic₃₇

C₂³

C₄⋊C₈

C₃₇

G ▷ C₂×C₃₇ ▷ C₃₇

C₁ ◁ C₂×C₄ ◁ C₄⋊C₈

N	H ≅ G/N	type
3. #64438⋊C₈	C₂	semidirect
4. C₈×#64438	C₂	semidirect
24. D₃₇⋊C₈	C₈	semidirect
68. C₃₇	#1450	semidirect

H ≅ G/N

type

3. #64438⋊C₈

C₂

semidirect

4. C₈×#64438

C₂

semidirect

24. D₃₇⋊C₈

C₈

semidirect

68. C₃₇

#1450

semidirect

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

subgroup	index
#14558552	2
C₄×D₃₇⋊C₈	2
#64438⋊C₈	2
C₈×#64438	2
#14557585	2
#1450	37

subgroup

index

C₄×D₃₇⋊C₈

Subgroups — #18009802

Conjugacy classes of subgroups

Complements

Maximal subgroups