Group #18011168 — subgroups

Subgroups — #18011168

Order 4736.

556 conjugacy classes of subgroups; only the 282 normal subgroups are shown.

normal subgroups only

This group has 556 conjugacy classes of subgroups — too many to draw. Only its 282 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	C₂×#11924624	C₂	1	2	yes	maximal
2	#14558498	C₂	1	6	yes	maximal, complemented
3	C₄×#10596953	C₂	1	3	yes	maximal, complemented
4	#14558492	C₂	1	3	yes	maximal, complemented
5	C₄×#10596966	C₂	1	1	yes	maximal, characteristic
6	#11924624	C₄	1	16	yes	complemented
7	C₂×#10596953	C₂²	1	6	yes	—
8	C₂×D₃₇⋊C₈	C₂²	1	6	yes	—
9	#11924591	C₂²	1	1	yes	characteristic
10	#11924591	C₂²	1	3	yes	—
11	#11924591	C₂²	1	6	yes	—
12	C₄×#64438	C₂²	1	6	yes	—
13	C₂×#10596966	C₂²	1	1	yes	characteristic
14	Q₈×Dic₃₇	C₄	1	1	yes	characteristic
15	C₄×D₁₄₈	C₄	1	3	yes	—
16	#11924498	C₄	1	3	yes	—
17	#11924494	C₂²	1	3	yes	—
18	C₄²×D₃₇	C₂²	1	3	yes	—
19	C₄×Q₈×C₃₇	C₄	1	1	yes	characteristic
20	C₂×Q₈×C₃₇	C₂×C₄	1	1	yes	characteristic
21	#10596966	C₂×C₄	1	8	yes	—
22	C₂×D₁₄₈	C₂×C₄	1	3	yes	—
23	C₂×C₄×D₃₇	C₂×C₄	1	3	yes	—
24	C₂×C₄×D₃₇	C₂×C₄	1	3	yes	—
25	C₂×C₄×D₃₇	C₂³	1	3	yes	—
26	C₂×#64438	C₂³	1	6	yes	—
27	C₂×#64438	C₂³	1	2	yes	—
28	#10596953	C₂×C₄	1	24	yes	complemented
29	D₃₇⋊C₈	C₂×C₄	1	24	yes	complemented
30	C₄⋊C₄×C₃₇	C₂×C₄	1	3	yes	—
31	C₄²×C₃₇	C₂×C₄	1	3	yes	—
32	#10596938	C₂×C₄	1	6	yes	—
33	C₄⋊Dic₃₇	C₂×C₄	1	3	yes	—
34	C₄×Dic₃₇	C₂×C₄	1	3	yes	—
35	C₄×Dic₃₇	C₂³	1	1	yes	characteristic
36	C₄×Dic₃₇	C₂³	1	3	yes	—
37	C₂²×D₃₇	C₂²×C₄	1	3	yes	—
38	Q₈×C₃₇	C₄²	1	4	yes	—
39	C₂×C₄×C₃₇	C₂²×C₄	1	3	yes	—
40	C₂×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₄×C₃₇	C₂²×C₄	1	3	yes	—
42	C₂×Dic₃₇	C₂²×C₄	1	1	yes	characteristic
43	C₂×Dic₃₇	C₂²×C₄	1	3	yes	—
44	C₂×Dic₃₇	C₂⁴	1	1	yes	characteristic
45	D₁₄₈	C₄²	1	12	yes	—
46	C₄×D₃₇	—	4	3	no	—
47	C₄×D₃₇	—	4	3	no	—
48	C₄×D₃₇	C₂²×C₄	1	12	yes	—
49	#64438	C₂²×C₄	1	4	yes	—
50	#64438	C₂²×C₄	1	12	yes	—
51	C₂²×C₃₇	C₂³×C₄	1	1	yes	characteristic
52	C₂×D₃₇	—	2	3	no	—
53	C₂×D₃₇	—	2	3	no	—
54	C₂×D₃₇	C₂×C₄²	1	6	yes	—
55	C₄×C₃₇	—	2	1	no	—
56	C₄×C₃₇	—	2	3	no	—
57	C₄×C₃₇	C₂×C₄²	1	6	yes	—
58	Dic₃₇	—	2	1	no	—
59	Dic₃₇	—	2	3	no	—
60	Dic₃₇	C₂³×C₄	1	2	yes	—
61	#2875	—	37	1	no	maximal, Sylow
62	C₂×C₃₇	C₂²×C₄²	1	1	yes	characteristic
63	C₂×C₃₇	#568	1	1	yes	characteristic
64	C₂×C₃₇	#568	1	1	yes	characteristic
65	D₃₇	—	2	6	no	—
66	C₂×#131	—	37	2	no	—
67	C₄×#41	—	37	1	no	—
68	#405	—	37	6	no	—
69	C₄×M₁₆	—	37	3	no	—
70	C₂×#97	—	37	3	no	—
71	C₃₇	#2875	1	1	yes	characteristic, complemented, Sylow
72	C₂×#41	—	37	1	no	—
73	#131	—	37	16	no	—
74	C₂×M₁₆	—	37	6	no	—
75	C₂²×C₈	—	37	6	no	—
76	C₄×Q₈	—	37	1	no	—
77	C₄×Q₈	#3866	1	1	yes	characteristic
78	C₄×D₄	—	37	3	no	—
79	C₄×D₄	—	37	3	no	—
80	#117	—	37	3	no	—
81	C₂×C₄²	—	37	3	no	—
82	#97	—	37	1	no	—
83	#97	—	37	3	no	—
84	#97	—	37	6	no	—
85	C₄×C₈	—	37	6	no	—
86	#41	—	37	8	no	—
87	C₂×Q₈	C₂×#3866	1	1	yes	characteristic
88	C₂×D₄	—	37	3	no	—
89	C₂²×C₄	—	37	3	no	—
90	C₂²×C₄	—	37	3	no	—
91	C₂²×C₄	—	37	3	no	—
92	M₁₆	—	37	24	no	—
93	C₂×C₈	—	37	24	no	—
94	C₂×C₈	—	37	6	no	—
95	C₂×C₈	—	37	2	no	—
96	C₄⋊C₄	—	37	3	no	—
97	C₄⋊C₄	C₂×#3866	1	3	yes	—
98	C₂²⋊C₄	—	37	6	no	—
99	C₄²	—	37	3	no	—
100	C₄²	—	37	3	no	—
101	C₄²	—	37	1	no	—
102	C₄²	C₂×#3866	1	3	yes	—
103	C₂³	—	37	3	no	—
104	Q₈	C₄×#3866	1	4	yes	—
105	D₄	—	37	12	no	—
106	C₂×C₄	—	37	1	no	—
107	C₂×C₄	—	37	12	no	—
108	C₂×C₄	—	37	1	no	—
109	C₂×C₄	—	37	3	no	—
110	C₂×C₄	—	148	3	no	—
111	C₂×C₄	—	148	3	no	—
112	C₂×C₄	C₂²×#3866	1	1	yes	characteristic
113	C₂×C₄	C₂²×#3866	1	3	yes	—
114	C₂×C₄	C₂²×#3866	1	3	yes	—
115	C₈	—	37	4	no	—
116	C₈	—	37	12	no	—
117	C₂²	—	37	6	no	—
118	C₂²	—	74	3	no	—
119	C₂²	—	74	3	no	—
120	C₂²	C₂³×#3866	1	1	yes	characteristic
121	C₄	—	2	3	no	—
122	C₄	—	2	1	no	—
123	C₄	—	37	2	no	—
124	C₄	—	74	1	no	—
125	C₄	—	74	3	no	—
126	C₄	C₂×C₄×#3866	1	6	yes	—
127	C₂	—	74	6	no	—
128	C₂	C₂²×C₄×#3866	1	1	yes	characteristic
129	C₂	#14558612	1	1	yes	characteristic
130	C₂	#14558613	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#11924624

C₂

yes

maximal

#14558498

C₂

yes

maximal, complemented

C₄×#10596953

C₂

yes

maximal, complemented

#14558492

C₂

yes

maximal, complemented

C₄×#10596966

C₂

yes

maximal, characteristic

#11924624

C₄

yes

complemented

C₂×#10596953

C₂²

yes

—

C₂×D₃₇⋊C₈

C₂²

yes

—

#11924591

C₂²

yes

characteristic

#11924591

C₂²

yes

—

#11924591

C₂²

yes

—

C₄×#64438

C₂²

yes

—

C₂×#10596966

C₂²

yes

characteristic

Q₈×Dic₃₇

C₄

yes

characteristic

C₄×D₁₄₈

C₄

yes

—

#11924498

C₄

yes

—

#11924494

C₂²

yes

—

C₄²×D₃₇

C₂²

yes

—

C₄×Q₈×C₃₇

C₄

yes

characteristic

C₂×Q₈×C₃₇

C₂×C₄

yes

characteristic

#10596966

C₂×C₄

yes

—

C₂×D₁₄₈

C₂×C₄

yes

—

C₂×C₄×D₃₇

C₂×C₄

yes

—

C₂×C₄×D₃₇

C₂×C₄

yes

—

C₂×C₄×D₃₇

C₂³

yes

—

C₂×#64438

C₂³

yes

—

C₂×#64438

C₂³

yes

—

#10596953

C₂×C₄

yes

complemented

D₃₇⋊C₈

C₂×C₄

yes

complemented

C₄⋊C₄×C₃₇

C₂×C₄

yes

—

C₄²×C₃₇

C₂×C₄

yes

—

#10596938

C₂×C₄

yes

—

C₄⋊Dic₃₇

C₂×C₄

yes

—

C₄×Dic₃₇

C₂×C₄

yes

—

C₄×Dic₃₇

C₂³

yes

characteristic

C₄×Dic₃₇

C₂³

yes

—

C₂²×D₃₇

C₂²×C₄

yes

—

Q₈×C₃₇

C₄²

yes

—

C₂×C₄×C₃₇

C₂²×C₄

yes

—

C₂×C₄×C₃₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₃₇

C₂²×C₄

yes

—

C₂×Dic₃₇

C₂²×C₄

yes

characteristic

C₂×Dic₃₇

C₂²×C₄

yes

—

C₂×Dic₃₇

C₂⁴

yes

characteristic

D₁₄₈

C₄²

yes

—

C₄×D₃₇

—

C₄×D₃₇

—

C₄×D₃₇

C₂²×C₄

yes

—

#64438

C₂²×C₄

yes

—

#64438

C₂²×C₄

yes

—

C₂²×C₃₇

C₂³×C₄

yes

characteristic

C₂×D₃₇

—

C₂×D₃₇

—

C₂×D₃₇

C₂×C₄²

yes

—

C₄×C₃₇

—

C₄×C₃₇

—

C₄×C₃₇

C₂×C₄²

yes

—

Dic₃₇

—

Dic₃₇

—

Dic₃₇

C₂³×C₄

yes

—

#2875

—

maximal, Sylow

C₂×C₃₇

C₂²×C₄²

yes

C₂×C₃₇

yes

C₂×C₃₇

yes

D₃₇

—

C₂×#131

—

C₄×#41

—

#405

—

C₄×M₁₆

—

C₂×#97

—

C₃₇

#2875

yes

characteristic, complemented, Sylow

C₂×#41

—

#131

—

C₂×M₁₆

—

C₂²×C₈

—

C₄×Q₈

—

C₄×Q₈

#3866

yes

characteristic

C₄×D₄

—

C₄×D₄

—

#117

—

C₂×C₄²

—

#97

—

#97

—

#97

—

C₄×C₈

—

#41

—

C₂×Q₈

C₂×#3866

yes

characteristic

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

C₂×#3866

yes

—

C₂²⋊C₄

—

C₄²

—

100

C₄²

—

101

C₄²

—

102

C₄²

C₂×#3866

yes

—

103

C₂³

—

104

Q₈

C₄×#3866

yes

—

105

D₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

148

—

111

C₂×C₄

—

148

—

112

C₂×C₄

C₂²×#3866

yes

characteristic

113

C₂×C₄

C₂²×#3866

yes

—

114

C₂×C₄

C₂²×#3866

yes

—

115

C₈

—

116

C₈

—

117

C₂²

—

118

C₂²

—

119

C₂²

—

120

C₂²

C₂³×#3866

yes

characteristic

121

C₄

—

122

C₄

—

123

C₄

—

124

C₄

—

125

C₄

—

126

C₄

C₂×C₄×#3866

yes

—

127

C₂

—

128

C₂

C₂²×C₄×#3866

yes

characteristic

129

C₂

#14558612

yes

characteristic

130

C₂

#14558613

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×#3866
commutator	C₂×C₃₇	C₂²×C₄²
Fitting	C₄×Q₈×C₃₇	C₄
socle	C₂²×C₃₇	C₂³×C₄
prime radical	C₂×#10596966	C₂²
cosocle	C₂×Dic₃₇	C₂⁴
hypercenter	C₄×Q₈	#3866
hypocenter	C₃₇	#2875
LCS	G ▷ C₂×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂² ◁ C₄×Q₈

subgroup

G/H

center, Frattini

C₂²

C₂³×#3866

commutator

C₂×C₃₇

C₂²×C₄²

Fitting

C₄×Q₈×C₃₇

C₄

socle

C₂²×C₃₇

C₂³×C₄

prime radical

C₂×#10596966

C₂²

cosocle

C₂×Dic₃₇

C₂⁴

C₄×Q₈

C₃₇

G ▷ C₂×C₃₇ ▷ C₃₇

C₁ ◁ C₂² ◁ C₄×Q₈

N	H ≅ G/N	type
2. #14558498	C₂	semidirect
3. C₄×#10596953	C₂	semidirect
4. #14558492	C₂	semidirect
6. #11924624	C₄	semidirect
28. #10596953	C₂×C₄	semidirect
29. D₃₇⋊C₈	C₂×C₄	semidirect
71. C₃₇	#2875	semidirect

H ≅ G/N

type

2. #14558498

C₂

semidirect

3. C₄×#10596953

C₂

semidirect

4. #14558492

C₂

semidirect

6. #11924624

C₄

semidirect

28. #10596953

C₂×C₄

semidirect

29. D₃₇⋊C₈

C₂×C₄

semidirect

71. C₃₇

#2875

semidirect

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

subgroup	index
C₂×#11924624	2
#14558498	2
C₄×#10596953	2
#14558492	2
C₄×#10596966	2
#2875	37

subgroup

index

Subgroups — #18011168

Conjugacy classes of subgroups

Complements

Maximal subgroups