Group #18005342 — subgroups

Subgroups — #18005342

Order 4736.

358 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 358 conjugacy classes of subgroups — too many to draw. Only its 125 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	#14558088	C₂	1	2	yes	maximal
2	#14557913	C₂	1	1	yes	maximal, characteristic, complemented
3	#14557885	C₂	1	2	yes	maximal, complemented
4	#14557792	C₂	1	4	yes	maximal
5	#14557639	C₂	1	4	yes	maximal, complemented
6	C₄×#10596929	C₂	1	1	yes	maximal, characteristic
7	C₃₇×#498	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×Dic₁₄₈	C₂²	1	4	yes	—
9	C₂×#10596929	C₂²	1	2	yes	—
10	#11924536	C₂²	1	2	yes	—
11	#11924527	—	2	2	no	—
12	#11924516	C₂²	1	4	yes	—
13	#11924512	—	2	2	no	—
14	Q₈×Dic₃₇	—	2	2	no	—
15	#11924501	C₂²	1	4	yes	—
16	#11924494	—	2	2	no	—
17	C₄²×D₃₇	C₂²	1	1	yes	characteristic
18	#11924459	C₂²	1	8	yes	—
19	#11924446	C₂²	1	2	yes	—
20	C₄×#64436	C₂²	1	1	yes	characteristic
21	C₄⋊Q₈×C₃₇	C₂²	1	2	yes	—
22	#106×C₃₇	C₂²	1	4	yes	—
23	C₄×C₈×C₃₇	C₂²	1	1	yes	characteristic
24	C₂×Q₈×C₃₇	—	2	2	no	—
25	C₂×C₄×D₃₇	D₄	1	2	yes	—
26	C₂×C₄×D₃₇	C₂³	1	1	yes	characteristic
27	C₂×Dic₇₄	C₂³	1	4	yes	—
28	C₂×C₈×C₃₇	C₂³	1	2	yes	—
29	C₄⋊C₄×C₃₇	—	2	2	no	—
30	C₄⋊C₄×C₃₇	C₂³	1	4	yes	—
31	C₄²×C₃₇	C₂³	1	1	yes	characteristic
32	#10596938	—	2	4	no	—
33	C₄⋊Dic₃₇	—	2	2	no	—
34	C₄⋊Dic₃₇	—	2	2	no	—
35	C₄⋊Dic₃₇	—	2	4	no	—
36	#10596936	—	2	4	no	—
37	C₄×Dic₃₇	—	2	4	no	—
38	C₄×Dic₃₇	D₄	1	2	yes	—
39	C₄×Dic₃₇	C₂³	1	1	yes	characteristic
40	C₂×#64436	C₂³	1	2	yes	—
41	Dic₁₄₈	—	2	8	no	—
42	#10596929	—	2	4	no	—
43	C₂²×D₃₇	C₂×D₄	1	1	yes	characteristic
44	Q₈×C₃₇	—	4	4	no	—
45	C₂×C₄×C₃₇	—	2	4	no	—
46	C₂×C₄×C₃₇	C₂×D₄	1	2	yes	—
47	C₂×C₄×C₃₇	C₂⁴	1	1	yes	characteristic
48	C₂×Dic₃₇	—	2	4	no	—
49	C₂×Dic₃₇	C₂×D₄	1	2	yes	—
50	C₂×Dic₃₇	C₂×D₄	1	1	yes	characteristic
51	C₄×D₃₇	—	2	4	no	—
52	C₄×D₃₇	—	2	2	no	—
53	Dic₇₄	—	2	8	no	—
54	C₈×C₃₇	#41	1	4	yes	—
55	#64436	—	2	2	no	—
56	C₂²×C₃₇	C₂²×D₄	1	1	yes	characteristic
57	C₂×D₃₇	—	2	1	no	—
58	C₂×D₃₇	—	2	2	no	—
59	C₄×C₃₇	—	4	4	no	—
60	C₄×C₃₇	#137	1	4	yes	—
61	C₄×C₃₇	C₂×#41	1	2	yes	—
62	Dic₃₇	—	2	1	no	—
63	Dic₃₇	—	2	2	no	—
64	Dic₃₇	—	4	4	no	—
65	#3185	—	37	1	no	maximal, Sylow
66	C₂×C₃₇	#532	1	1	yes	characteristic
67	C₂×C₃₇	C₂×#137	1	2	yes	—
68	D₃₇	—	4	1	no	—
69	#533	—	37	2	no	—
70	#498	D₃₇	1	1	yes	characteristic, complemented
71	#494	—	37	1	no	—
72	#487	—	37	2	no	—
73	#470	—	37	4	no	—
74	#441	—	37	4	no	—
75	C₄×M₁₆	—	37	1	no	—
76	C₃₇	#3185	1	1	yes	characteristic, complemented, Sylow
77	C₂×Q₁₆	—	37	4	no	—
78	C₂×M₁₆	—	37	2	no	—
79	C₄⋊Q₈	—	37	2	no	—
80	C₄⋊Q₈	C₂×D₃₇	1	2	yes	—
81	#125	—	74	2	no	—
82	C₂²⋊Q₈	—	37	4	no	—
83	C₂²⋊Q₈	—	74	2	no	—
84	C₄×Q₈	—	37	4	no	—
85	C₄×Q₈	—	74	2	no	—
86	#117	—	74	2	no	—
87	C₂×C₄²	—	37	1	no	—
88	#106	C₂×D₃₇	1	4	yes	—
89	#103	—	37	8	no	—
90	#97	—	37	2	no	—
91	C₄×C₈	—	37	1	no	—
92	C₄×C₈	C₂×D₃₇	1	1	yes	characteristic
93	C₂×Q₈	—	2	2	no	—
94	C₂×Q₈	—	37	4	no	—
95	C₂²×C₄	—	37	2	no	—
96	C₂²×C₄	—	37	1	no	—
97	Q₁₆	—	74	8	no	—
98	M₁₆	—	74	4	no	—
99	C₂×C₈	—	37	2	no	—
100	C₂×C₈	C₂²×D₃₇	1	2	yes	—
101	C₄⋊C₄	—	2	2	no	—
102	C₄⋊C₄	—	74	4	no	—
103	C₄⋊C₄	—	74	4	no	—
104	C₄⋊C₄	—	74	2	no	—
105	C₄⋊C₄	—	74	2	no	—
106	C₄⋊C₄	C₂²×D₃₇	1	4	yes	—
107	C₂²⋊C₄	—	74	4	no	—
108	C₄²	—	37	2	no	—
109	C₄²	—	37	1	no	—
110	C₄²	—	74	4	no	—
111	C₄²	C₂²×D₃₇	1	1	yes	characteristic
112	C₂³	—	37	1	no	—
113	Q₈	—	4	4	no	—
114	Q₈	—	74	8	no	—
115	C₂×C₄	—	2	4	no	—
116	C₂×C₄	—	37	2	no	—
117	C₂×C₄	—	37	1	no	—
118	C₂×C₄	—	74	4	no	—
119	C₂×C₄	—	74	2	no	—
120	C₂×C₄	—	74	4	no	—
121	C₂×C₄	D₄×D₃₇	1	2	yes	—
122	C₂×C₄	C₂³×D₃₇	1	1	yes	characteristic
123	C₈	—	74	2	no	—
124	C₈	#10596966	1	4	yes	—
125	C₂²	—	74	2	no	—
126	C₂²	—	74	1	no	—
127	C₂²	C₂×D₄×D₃₇	1	1	yes	characteristic
128	C₄	—	4	4	no	—
129	C₄	—	74	2	no	—
130	C₄	—	74	1	no	—
131	C₄	—	148	4	no	—
132	C₄	#11924565	1	4	yes	—
133	C₄	C₂×#10596966	1	2	yes	—
134	C₂	—	148	1	no	—
135	C₂	#14558084	1	1	yes	characteristic
136	C₂	C₂×#11924565	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#14558088

C₂

yes

maximal

#14557913

C₂

yes

maximal, characteristic, complemented

#14557885

C₂

yes

maximal, complemented

#14557792

C₂

yes

maximal

#14557639

C₂

yes

maximal, complemented

C₄×#10596929

C₂

yes

maximal, characteristic

C₃₇×#498

C₂

yes

maximal, characteristic, complemented

C₂×Dic₁₄₈

C₂²

yes

—

C₂×#10596929

C₂²

yes

—

#11924536

C₂²

yes

—

#11924527

—

#11924516

C₂²

yes

—

#11924512

—

Q₈×Dic₃₇

—

#11924501

C₂²

yes

—

#11924494

—

C₄²×D₃₇

C₂²

yes

characteristic

#11924459

C₂²

yes

—

#11924446

C₂²

yes

—

C₄×#64436

C₂²

yes

characteristic

C₄⋊Q₈×C₃₇

C₂²

yes

—

#106×C₃₇

C₂²

yes

—

C₄×C₈×C₃₇

C₂²

yes

characteristic

C₂×Q₈×C₃₇

—

C₂×C₄×D₃₇

D₄

yes

—

C₂×C₄×D₃₇

C₂³

yes

characteristic

C₂×Dic₇₄

C₂³

yes

—

C₂×C₈×C₃₇

C₂³

yes

—

C₄⋊C₄×C₃₇

—

C₄⋊C₄×C₃₇

C₂³

yes

—

C₄²×C₃₇

C₂³

yes

characteristic

#10596938

—

C₄⋊Dic₃₇

—

C₄⋊Dic₃₇

—

C₄⋊Dic₃₇

—

#10596936

—

C₄×Dic₃₇

—

C₄×Dic₃₇

D₄

yes

—

C₄×Dic₃₇

C₂³

yes

characteristic

C₂×#64436

C₂³

yes

—

Dic₁₄₈

—

#10596929

—

C₂²×D₃₇

C₂×D₄

yes

characteristic

Q₈×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

C₂×D₄

yes

—

C₂×C₄×C₃₇

C₂⁴

yes

characteristic

C₂×Dic₃₇

—

C₂×Dic₃₇

C₂×D₄

yes

—

C₂×Dic₃₇

C₂×D₄

yes

characteristic

C₄×D₃₇

—

C₄×D₃₇

—

Dic₇₄

—

C₈×C₃₇

#41

yes

—

#64436

—

C₂²×C₃₇

C₂²×D₄

yes

characteristic

C₂×D₃₇

—

C₂×D₃₇

—

C₄×C₃₇

—

C₄×C₃₇

#137

yes

—

C₄×C₃₇

C₂×#41

yes

—

Dic₃₇

—

Dic₃₇

—

Dic₃₇

—

—

C₂×C₃₇

yes

C₂×C₃₇

yes

—

D₃₇

—

#533

—

#498

D₃₇

yes

characteristic, complemented

—

—

—

—

C₄×M₁₆

—

C₃₇

#3185

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

C₂×M₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₃₇

yes

—

#125

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

C₄×Q₈

—

C₄×Q₈

—

#117

—

C₂×C₄²

—

#106

C₂×D₃₇

yes

—

#103

—

#97

—

C₄×C₈

—

C₄×C₈

C₂×D₃₇

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

Q₁₆

—

M₁₆

—

C₂×C₈

—

100

C₂×C₈

C₂²×D₃₇

yes

—

101

C₄⋊C₄

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

106

C₄⋊C₄

C₂²×D₃₇

yes

—

107

C₂²⋊C₄

—

108

C₄²

—

109

C₄²

—

110

C₄²

—

111

C₄²

C₂²×D₃₇

yes

characteristic

112

C₂³

—

113

Q₈

—

114

Q₈

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

D₄×D₃₇

yes

—

122

C₂×C₄

C₂³×D₃₇

yes

characteristic

123

C₈

—

124

C₈

#10596966

yes

—

125

C₂²

—

126

C₂²

—

127

C₂²

C₂×D₄×D₃₇

yes

characteristic

128

C₄

—

129

C₄

—

130

C₄

—

131

C₄

—

148

—

132

C₄

#11924565

yes

—

133

C₄

C₂×#10596966

yes

—

134

C₂

—

148

—

135

C₂

#14558084

yes

characteristic

136

C₂

C₂×#11924565

yes

—

	subgroup	G/H
center	C₂²	C₂×D₄×D₃₇
commutator, cosocle	C₂×C₄×C₃₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₃₇
Fitting	C₃₇×#498	C₂
socle	C₂²×C₃₇	C₂²×D₄
prime radical	C₂²×D₃₇	C₂×D₄
hypercenter	#498	D₃₇
hypocenter	C₃₇	#3185
LCS	G ▷ C₂×C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #498

subgroup

G/H

center

C₂²

C₂×D₄×D₃₇

commutator, cosocle

C₂×C₄×C₃₇

C₂⁴

Frattini

C₂×C₄

C₂³×D₃₇

Fitting

C₃₇×#498

C₂

socle

C₂²×C₃₇

C₂²×D₄

prime radical

C₂²×D₃₇

C₂×D₄

D₃₇

C₃₇

G ▷ C₂×C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #498

N	H ≅ G/N	type
2. #14557913	C₂	semidirect
3. #14557885	C₂	semidirect
5. #14557639	C₂	semidirect
7. C₃₇×#498	C₂	semidirect
70. #498	D₃₇	semidirect
76. C₃₇	#3185	semidirect

H ≅ G/N

type

2. #14557913

C₂

semidirect

3. #14557885

C₂

semidirect

5. #14557639

C₂

semidirect

7. C₃₇×#498

C₂

semidirect

70. #498

D₃₇

semidirect

76. C₃₇

#3185

semidirect

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

subgroup	index
#14558088	2
#14557913	2
#14557885	2
#14557792	2
#14557639	2
C₄×#10596929	2
C₃₇×#498	2
#3185	37

subgroup

index

C₃₇×#498

Subgroups — #18005342

Conjugacy classes of subgroups

Complements

Maximal subgroups