Group #18011530 — subgroups

Subgroups — #18011530

Order 4736.

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

normal subgroups only

This group has 394 conjugacy classes of subgroups — too many to draw. Only its 168 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	Q₈×#64438	C₂	1	2	yes	maximal, complemented
2	#14558567	C₂	1	6	yes	maximal, complemented
3	#14558552	C₂	1	3	yes	maximal, complemented
4	#14558537	C₂	1	3	yes	maximal, complemented
5	C₄×#10596966	C₂	1	1	yes	maximal, characteristic
6	C₂×#10596953	C₂²	1	6	yes	—
7	C₄⋊#64438	C₂²	1	3	yes	—
8	C₄⋊#64438	C₂²	1	6	yes	—
9	C₄⋊#64438	C₂²	1	1	yes	characteristic
10	#11924593	C₂²	1	6	yes	—
11	C₄×#64438	C₂²	1	6	yes	—
12	C₂×#10596966	C₄	1	1	yes	characteristic
13	Q₈×Dic₃₇	C₂²	1	1	yes	characteristic
14	C₄×D₁₄₈	C₄	1	3	yes	—
15	#11924498	C₂²	1	3	yes	—
16	#11924494	C₄	1	3	yes	—
17	C₄²×D₃₇	C₂²	1	3	yes	—
18	C₄×Q₈×C₃₇	C₄	1	1	yes	characteristic
19	C₂×Q₈×C₃₇	C₂×C₄	1	1	yes	characteristic
20	#10596966	—	2	4	no	—
21	C₂×D₁₄₈	C₂×C₄	1	3	yes	—
22	C₂×C₄×D₃₇	C₂×C₄	1	3	yes	—
23	C₂×C₄×D₃₇	C₂×C₄	1	3	yes	—
24	C₂×C₄×D₃₇	C₂³	1	3	yes	—
25	C₂×#64438	C₂³	1	2	yes	—
26	C₂×#64438	C₂³	1	6	yes	—
27	#10596953	—	4	6	no	—
28	C₄⋊C₄×C₃₇	C₂×C₄	1	3	yes	—
29	C₄²×C₃₇	C₂×C₄	1	3	yes	—
30	#10596938	C₂×C₄	1	6	yes	—
31	C₄⋊Dic₃₇	C₂×C₄	1	3	yes	—
32	C₄×Dic₃₇	C₂×C₄	1	3	yes	—
33	C₄×Dic₃₇	C₂³	1	1	yes	characteristic
34	C₄×Dic₃₇	C₂³	1	3	yes	—
35	C₂²×D₃₇	C₂²×C₄	1	3	yes	—
36	Q₈×C₃₇	M₁₆	1	4	yes	complemented
37	C₂×C₄×C₃₇	C₂²×C₄	1	3	yes	—
38	C₂×C₄×C₃₇	C₂²×C₄	1	3	yes	—
39	C₂×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
40	C₂×Dic₃₇	C₂²×C₄	1	1	yes	characteristic
41	C₂×Dic₃₇	C₂²×C₄	1	3	yes	—
42	C₂×Dic₃₇	C₂⁴	1	1	yes	characteristic
43	D₁₄₈	—	2	6	no	—
44	C₄×D₃₇	—	2	6	no	—
45	C₄×D₃₇	—	2	6	no	—
46	C₄×D₃₇	—	2	3	no	—
47	C₄×D₃₇	—	2	3	no	—
48	#64438	—	2	6	no	—
49	#64438	—	2	2	no	—
50	C₂²×C₃₇	C₂³×C₄	1	1	yes	characteristic
51	C₂×D₃₇	—	2	3	no	—
52	C₂×D₃₇	—	2	3	no	—
53	C₂×D₃₇	—	2	3	no	—
54	C₄×C₃₇	—	2	3	no	—
55	C₄×C₃₇	C₂×M₁₆	1	6	yes	—
56	C₄×C₃₇	#131	1	2	yes	—
57	Dic₃₇	—	2	1	no	—
58	Dic₃₇	—	2	3	no	—
59	Dic₃₇	H(5,𝔽₂)	1	1	yes	characteristic
60	Dic₃₇	2₋¹⁺⁴	1	1	yes	characteristic
61	#2991	—	37	1	no	maximal, Sylow
62	C₂×C₃₇	#520	1	1	yes	characteristic
63	C₂×C₃₇	C₂²×M₁₆	1	1	yes	characteristic
64	C₂×C₃₇	C₂×#131	1	1	yes	characteristic
65	D₃₇	—	4	3	no	—
66	C₄×#41	—	37	1	no	—
67	C₈×Q₈	—	37	2	no	—
68	#436	—	37	6	no	—
69	#431	—	37	3	no	—
70	#423	—	37	3	no	—
71	C₃₇	#2991	1	1	yes	characteristic, complemented, Sylow
72	C₂×#41	—	37	1	no	—
73	C₂×M₁₆	—	37	6	no	—
74	C₄×Q₈	—	37	1	no	—
75	C₄×Q₈	#3866	1	1	yes	characteristic
76	C₄×D₄	—	37	3	no	—
77	C₄×D₄	—	37	3	no	—
78	#117	—	37	3	no	—
79	C₂×C₄²	—	37	3	no	—
80	C₄⋊C₈	—	37	3	no	—
81	C₄⋊C₈	—	37	6	no	—
82	C₄⋊C₈	—	37	1	no	—
83	C₂²⋊C₈	—	37	6	no	—
84	C₄×C₈	—	37	6	no	—
85	#41	—	74	4	no	—
86	C₂×Q₈	C₂×#3866	1	1	yes	characteristic
87	C₂×D₄	—	37	3	no	—
88	C₂²×C₄	—	37	3	no	—
89	C₂²×C₄	—	37	3	no	—
90	C₂²×C₄	—	37	3	no	—
91	M₁₆	—	148	6	no	—
92	C₂×C₈	—	37	2	no	—
93	C₂×C₈	—	37	6	no	—
94	C₄⋊C₄	—	37	3	no	—
95	C₄⋊C₄	C₂×#3866	1	3	yes	—
96	C₂²⋊C₄	—	37	6	no	—
97	C₄²	—	37	3	no	—
98	C₄²	—	37	3	no	—
99	C₄²	—	37	1	no	—
100	C₄²	C₂×#3866	1	3	yes	—
101	C₂³	—	37	3	no	—
102	Q₈	#10596953	1	4	yes	complemented
103	D₄	—	74	6	no	—
104	C₂×C₄	—	37	1	no	—
105	C₂×C₄	—	37	1	no	—
106	C₂×C₄	—	37	3	no	—
107	C₂×C₄	—	74	3	no	—
108	C₂×C₄	—	74	3	no	—
109	C₂×C₄	—	74	6	no	—
110	C₂×C₄	—	74	6	no	—
111	C₂×C₄	C₂²×#3866	1	3	yes	—
112	C₂×C₄	C₂²×#3866	1	3	yes	—
113	C₂×C₄	C₂²×#3866	1	1	yes	characteristic
114	C₈	—	74	6	no	—
115	C₈	—	74	2	no	—
116	C₂²	—	74	3	no	—
117	C₂²	—	74	3	no	—
118	C₂²	—	74	3	no	—
119	C₂²	C₂³×#3866	1	1	yes	characteristic
120	C₄	—	2	3	no	—
121	C₄	—	37	1	no	—
122	C₄	—	37	1	no	—
123	C₄	—	74	1	no	—
124	C₄	—	74	3	no	—
125	C₄	C₂×#10596953	1	6	yes	—
126	C₄	#11924624	1	2	yes	—
127	C₂	—	148	3	no	—
128	C₂	#14558602	1	1	yes	characteristic
129	C₂	C₂²×#10596953	1	1	yes	characteristic
130	C₂	C₂×#11924624	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

Q₈×#64438

C₂

yes

maximal, complemented

#14558567

C₂

yes

maximal, complemented

#14558552

C₂

yes

maximal, complemented

#14558537

C₂

yes

maximal, complemented

C₄×#10596966

C₂

yes

maximal, characteristic

C₂×#10596953

C₂²

yes

—

C₄⋊#64438

C₂²

yes

—

C₄⋊#64438

C₂²

yes

—

C₄⋊#64438

C₂²

yes

characteristic

#11924593

C₂²

yes

—

C₄×#64438

C₂²

yes

—

C₂×#10596966

C₄

yes

characteristic

Q₈×Dic₃₇

C₂²

yes

characteristic

C₄×D₁₄₈

C₄

yes

—

#11924498

C₂²

yes

—

#11924494

C₄

yes

—

C₄²×D₃₇

C₂²

yes

—

C₄×Q₈×C₃₇

C₄

yes

characteristic

C₂×Q₈×C₃₇

C₂×C₄

yes

characteristic

#10596966

—

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

—

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

M₁₆

yes

complemented

C₂×C₄×C₃₇

C₂²×C₄

yes

—

C₂×C₄×C₃₇

C₂²×C₄

yes

—

C₂×C₄×C₃₇

C₂²×C₄

yes

characteristic

C₂×Dic₃₇

C₂²×C₄

yes

characteristic

C₂×Dic₃₇

C₂²×C₄

yes

—

C₂×Dic₃₇

C₂⁴

yes

characteristic

D₁₄₈

—

C₄×D₃₇

—

C₄×D₃₇

—

C₄×D₃₇

—

C₄×D₃₇

—

#64438

—

#64438

—

C₂²×C₃₇

C₂³×C₄

yes

characteristic

C₂×D₃₇

—

C₂×D₃₇

—

C₂×D₃₇

—

C₄×C₃₇

—

C₄×C₃₇

C₂×M₁₆

yes

—

C₄×C₃₇

#131

yes

—

Dic₃₇

—

Dic₃₇

—

Dic₃₇

H(5,𝔽₂)

yes

characteristic

Dic₃₇

2₋¹⁺⁴

yes

—

C₂×C₃₇

yes

C₂×C₃₇

C₂²×M₁₆

yes

characteristic

C₂×C₃₇

C₂×#131

yes

characteristic

D₃₇

—

C₄×#41

—

C₈×Q₈

—

#436

—

#431

—

#423

—

C₃₇

#2991

yes

characteristic, complemented, Sylow

C₂×#41

—

C₂×M₁₆

—

C₄×Q₈

—

C₄×Q₈

#3866

yes

characteristic

C₄×D₄

—

C₄×D₄

—

#117

—

C₂×C₄²

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₂²⋊C₈

—

C₄×C₈

—

#41

—

C₂×Q₈

C₂×#3866

yes

characteristic

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

M₁₆

—

148

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

C₂×#3866

yes

—

C₂²⋊C₄

—

C₄²

—

C₄²

—

C₄²

—

100

C₄²

C₂×#3866

yes

—

101

C₂³

—

102

Q₈

#10596953

yes

complemented

103

D₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

C₂²×#3866

yes

—

112

C₂×C₄

C₂²×#3866

yes

—

113

C₂×C₄

C₂²×#3866

yes

characteristic

114

C₈

—

115

C₈

—

116

C₂²

—

117

C₂²

—

118

C₂²

—

119

C₂²

C₂³×#3866

yes

characteristic

120

C₄

—

121

C₄

—

122

C₄

—

123

C₄

—

124

C₄

—

125

C₄

C₂×#10596953

yes

—

126

C₄

#11924624

yes

—

127

C₂

—

148

—

128

C₂

#14558602

yes

characteristic

129

C₂

C₂²×#10596953

yes

characteristic

130

C₂

C₂×#11924624

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×#3866
commutator, socle	C₂²×C₃₇	C₂³×C₄
Fitting	C₄×Q₈×C₃₇	C₄
prime radical	C₂×#10596966	C₄
cosocle	C₂×Dic₃₇	C₂⁴
hypercenter	C₄×Q₈	#3866
hypocenter	C₃₇	#2991
LCS	G ▷ C₂²×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂² ◁ C₄×Q₈

subgroup

G/H

center, Frattini

C₂²

C₂³×#3866

commutator, socle

C₂²×C₃₇

C₂³×C₄

Fitting

C₄×Q₈×C₃₇

C₄

prime radical

C₂×#10596966

C₄

cosocle

C₂×Dic₃₇

C₂⁴

C₄×Q₈

C₃₇

G ▷ C₂²×C₃₇ ▷ C₃₇

C₁ ◁ C₂² ◁ C₄×Q₈

N	H ≅ G/N	type
1. Q₈×#64438	C₂	semidirect
2. #14558567	C₂	semidirect
3. #14558552	C₂	semidirect
4. #14558537	C₂	semidirect
36. Q₈×C₃₇	M₁₆	semidirect
71. C₃₇	#2991	semidirect
102. Q₈	#10596953	semidirect

H ≅ G/N

type

1. Q₈×#64438

C₂

semidirect

2. #14558567

C₂

semidirect

3. #14558552

C₂

semidirect

4. #14558537

C₂

semidirect

36. Q₈×C₃₇

M₁₆

semidirect

71. C₃₇

#2991

semidirect

102. Q₈

#10596953

semidirect

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

subgroup	index
Q₈×#64438	2
#14558567	2
#14558552	2
#14558537	2
C₄×#10596966	2
#2991	37

subgroup

index

Subgroups — #18011530

Conjugacy classes of subgroups

Complements

Maximal subgroups