Group #18004977 — subgroups

Subgroups — #18004977

Order 4736.

380 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 380 conjugacy classes of subgroups — too many to draw. Only its 141 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	#14558090	C₂	1	1	yes	maximal, characteristic
2	C₂×#11924536	C₂	1	1	yes	maximal, characteristic
3	#14557902	C₂	1	4	yes	maximal, complemented
4	#14557884	C₂	1	4	yes	maximal, complemented
5	#14557543	C₂	1	4	yes	maximal
6	C₄×M₁₆×C₃₇	C₂	1	1	yes	maximal, characteristic
7	C₂²×Dic₇₄	C₂²	1	2	yes	—
8	#11924536	—	2	2	no	—
9	#11924536	C₂²	1	2	yes	—
10	#11924536	C₂²	1	2	yes	—
11	#11924536	C₂²	1	2	yes	—
12	#11924528	C₂²	1	2	yes	—
13	#11924515	—	2	2	no	—
14	C₄×Dic₇₄	—	2	2	no	—
15	#11924495	C₂²	1	2	yes	—
16	C₂×C₄⋊Dic₃₇	—	2	2	no	—
17	#11924461	C₂²	1	16	yes	—
18	C₂×M₁₆×C₃₇	C₂²	1	2	yes	—
19	C₂×C₄²×C₃₇	C₂²	1	1	yes	characteristic
20	#97×C₃₇	C₂²	1	2	yes	—
21	C₄×C₈×C₃₇	C₂²	1	2	yes	—
22	C₂²×C₄×C₃₇	D₄	1	2	yes	—
23	C₂²×C₄×C₃₇	C₂³	1	1	yes	characteristic
24	C₂²×Dic₃₇	—	2	2	no	—
25	C₂×Dic₇₄	—	2	8	no	—
26	C₂×Dic₇₄	—	2	2	no	—
27	C₂×Dic₇₄	—	2	2	no	—
28	C₂×Dic₇₄	C₂³	1	4	yes	—
29	M₁₆×C₃₇	—	2	4	no	—
30	C₂×C₈×C₃₇	C₂³	1	4	yes	—
31	C₄²×C₃₇	D₄	1	2	yes	—
32	C₄²×C₃₇	C₂³	1	1	yes	characteristic
33	C₄²×C₃₇	C₂³	1	1	yes	characteristic
34	C₂²⋊Dic₃₇	—	2	2	no	—
35	C₄⋊Dic₃₇	—	2	4	no	—
36	C₄⋊Dic₃₇	—	2	4	no	—
37	C₄⋊Dic₃₇	—	2	2	no	—
38	C₄⋊Dic₃₇	C₂³	1	4	yes	—
39	#10596936	—	2	4	no	—
40	C₄×Dic₃₇	—	2	2	no	—
41	C₂³×C₃₇	C₂×D₄	1	1	yes	characteristic
42	C₂×C₄×C₃₇	—	2	4	no	—
43	C₂×C₄×C₃₇	C₂×D₄	1	2	yes	—
44	C₂×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
45	C₂×C₄×C₃₇	C₂×D₄	1	2	yes	—
46	C₂×C₄×C₃₇	#41	1	4	yes	—
47	C₂×C₄×C₃₇	C₂⁴	1	1	yes	characteristic
48	C₂×Dic₃₇	—	2	4	no	—
49	C₂×Dic₃₇	—	2	4	no	—
50	C₂×Dic₃₇	—	4	4	no	—
51	Dic₇₄	—	2	8	no	—
52	Dic₇₄	—	4	4	no	—
53	Dic₇₄	—	4	4	no	—
54	C₈×C₃₇	—	2	4	no	—
55	C₂²×C₃₇	—	2	2	no	—
56	C₂²×C₃₇	#124	1	2	yes	—
57	C₂²×C₃₇	C₂²×D₄	1	1	yes	characteristic
58	C₄×C₃₇	—	2	2	no	—
59	C₄×C₃₇	#124	1	2	yes	—
60	C₄×C₃₇	#137	1	4	yes	—
61	C₄×C₃₇	C₂×#41	1	2	yes	—
62	Dic₃₇	—	4	4	no	—
63	Dic₃₇	—	4	4	no	—
64	#3139	—	37	1	no	maximal, Sylow
65	C₂×C₃₇	—	2	2	no	—
66	C₂×C₃₇	C₂×#124	1	1	yes	characteristic
67	C₂×C₃₇	C₂×#137	1	2	yes	—
68	#533	—	37	1	no	—
69	C₂×C₄⋊Q₈	—	37	1	no	—
70	#491	—	37	4	no	—
71	#487	—	37	4	no	—
72	#419	—	37	4	no	—
73	C₄×M₁₆	D₃₇	1	1	yes	characteristic
74	C₃₇	#3139	1	1	yes	characteristic, complemented, Sylow
75	C₂²×Q₈	—	37	2	no	—
76	C₂×M₁₆	C₂×D₃₇	1	2	yes	—
77	C₄⋊Q₈	—	37	2	no	—
78	C₄⋊Q₈	—	37	2	no	—
79	C₄⋊Q₈	—	37	2	no	—
80	C₄⋊Q₈	—	74	2	no	—
81	#125	—	37	2	no	—
82	C₂²⋊Q₈	—	74	2	no	—
83	C₄×Q₈	—	74	2	no	—
84	#117	—	37	2	no	—
85	C₂×C₄⋊C₄	—	74	2	no	—
86	C₂×C₄²	C₂×D₃₇	1	1	yes	characteristic
87	#103	—	37	16	no	—
88	#97	C₂×D₃₇	1	2	yes	—
89	C₄×C₈	C₂×D₃₇	1	2	yes	—
90	C₂×Q₈	—	37	4	no	—
91	C₂×Q₈	—	74	2	no	—
92	C₂×Q₈	—	74	2	no	—
93	C₂×Q₈	—	74	8	no	—
94	C₂²×C₄	—	74	2	no	—
95	C₂²×C₄	D₁₄₈	1	2	yes	—
96	C₂²×C₄	C₂²×D₃₇	1	1	yes	characteristic
97	M₁₆	—	2	4	no	—
98	C₂×C₈	C₂²×D₃₇	1	4	yes	—
99	C₄⋊C₄	—	37	4	no	—
100	C₄⋊C₄	—	74	4	no	—
101	C₄⋊C₄	—	74	4	no	—
102	C₄⋊C₄	—	74	4	no	—
103	C₄⋊C₄	—	74	2	no	—
104	C₂²⋊C₄	—	74	2	no	—
105	C₄²	—	74	2	no	—
106	C₄²	D₁₄₈	1	2	yes	—
107	C₄²	C₂²×D₃₇	1	1	yes	characteristic
108	C₄²	C₂²×D₃₇	1	1	yes	characteristic
109	C₂³	C₂×D₁₄₈	1	1	yes	characteristic
110	Q₈	—	74	8	no	—
111	Q₈	—	148	4	no	—
112	Q₈	—	148	4	no	—
113	C₂×C₄	—	2	4	no	—
114	C₂×C₄	—	74	4	no	—
115	C₂×C₄	—	74	4	no	—
116	C₂×C₄	—	148	4	no	—
117	C₂×C₄	C₂×D₁₄₈	1	2	yes	—
118	C₂×C₄	C₂×D₁₄₈	1	1	yes	characteristic
119	C₂×C₄	C₂×D₁₄₈	1	2	yes	—
120	C₂×C₄	#10596962	1	4	yes	—
121	C₂×C₄	C₂³×D₃₇	1	1	yes	characteristic
122	C₈	—	2	4	no	—
123	C₂²	—	2	2	no	—
124	C₂²	#11924524	1	2	yes	—
125	C₂²	C₂²×D₁₄₈	1	1	yes	characteristic
126	C₄	—	2	2	no	—
127	C₄	—	148	4	no	—
128	C₄	—	148	4	no	—
129	C₄	#11924524	1	2	yes	—
130	C₄	#11924569	1	4	yes	—
131	C₄	C₂×#10596962	1	2	yes	—
132	C₂	—	2	2	no	—
133	C₂	C₂×#11924524	1	1	yes	characteristic
134	C₂	C₂×#11924569	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#14558090

C₂

yes

maximal, characteristic

C₂×#11924536

C₂

yes

maximal, characteristic

#14557902

C₂

yes

maximal, complemented

#14557884

C₂

yes

maximal, complemented

#14557543

C₂

yes

maximal

C₄×M₁₆×C₃₇

C₂

yes

maximal, characteristic

C₂²×Dic₇₄

C₂²

yes

—

#11924536

—

#11924536

C₂²

yes

—

#11924536

C₂²

yes

—

#11924536

C₂²

yes

—

#11924528

C₂²

yes

—

#11924515

—

C₄×Dic₇₄

—

#11924495

C₂²

yes

—

C₂×C₄⋊Dic₃₇

—

#11924461

C₂²

yes

—

C₂×M₁₆×C₃₇

C₂²

yes

—

C₂×C₄²×C₃₇

C₂²

yes

characteristic

#97×C₃₇

C₂²

yes

—

C₄×C₈×C₃₇

C₂²

yes

—

C₂²×C₄×C₃₇

D₄

yes

—

C₂²×C₄×C₃₇

C₂³

yes

characteristic

C₂²×Dic₃₇

—

C₂×Dic₇₄

—

C₂×Dic₇₄

—

C₂×Dic₇₄

—

C₂×Dic₇₄

C₂³

yes

—

M₁₆×C₃₇

—

C₂×C₈×C₃₇

C₂³

yes

—

C₄²×C₃₇

D₄

yes

—

C₄²×C₃₇

C₂³

yes

characteristic

C₄²×C₃₇

C₂³

yes

characteristic

C₂²⋊Dic₃₇

—

C₄⋊Dic₃₇

—

C₄⋊Dic₃₇

—

C₄⋊Dic₃₇

—

C₄⋊Dic₃₇

C₂³

yes

—

#10596936

—

C₄×Dic₃₇

—

C₂³×C₃₇

C₂×D₄

yes

characteristic

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

C₂×D₄

yes

—

C₂×C₄×C₃₇

C₂×D₄

yes

characteristic

C₂×C₄×C₃₇

C₂×D₄

yes

—

C₂×C₄×C₃₇

#41

yes

—

C₂×C₄×C₃₇

C₂⁴

yes

characteristic

C₂×Dic₃₇

—

C₂×Dic₃₇

—

C₂×Dic₃₇

—

Dic₇₄

—

Dic₇₄

—

Dic₇₄

—

C₈×C₃₇

—

C₂²×C₃₇

—

C₂²×C₃₇

#124

yes

—

C₂²×C₃₇

C₂²×D₄

yes

characteristic

C₄×C₃₇

—

C₄×C₃₇

#124

yes

—

C₄×C₃₇

#137

yes

—

C₄×C₃₇

C₂×#41

yes

—

Dic₃₇

—

Dic₃₇

—

#3139

—

maximal, Sylow

C₂×C₃₇

—

C₂×C₃₇

C₂×#124

yes

characteristic

C₂×C₃₇

C₂×#137

yes

—

#533

—

C₂×C₄⋊Q₈

—

#491

—

#487

—

#419

—

C₄×M₁₆

D₃₇

yes

characteristic

C₃₇

#3139

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₂×M₁₆

C₂×D₃₇

yes

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

#125

—

C₂²⋊Q₈

—

C₄×Q₈

—

#117

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂×D₃₇

yes

characteristic

#103

—

#97

C₂×D₃₇

yes

—

C₄×C₈

C₂×D₃₇

yes

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

D₁₄₈

yes

—

C₂²×C₄

C₂²×D₃₇

yes

characteristic

M₁₆

—

C₂×C₈

C₂²×D₃₇

yes

—

C₄⋊C₄

—

100

C₄⋊C₄

—

101

C₄⋊C₄

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

104

C₂²⋊C₄

—

105

C₄²

—

106

C₄²

D₁₄₈

yes

—

107

C₄²

C₂²×D₃₇

yes

characteristic

108

C₄²

C₂²×D₃₇

yes

characteristic

109

C₂³

C₂×D₁₄₈

yes

characteristic

110

Q₈

—

111

Q₈

—

148

—

112

Q₈

—

148

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

148

—

117

C₂×C₄

C₂×D₁₄₈

yes

—

118

C₂×C₄

C₂×D₁₄₈

yes

characteristic

119

C₂×C₄

C₂×D₁₄₈

yes

—

120

C₂×C₄

#10596962

yes

—

121

C₂×C₄

C₂³×D₃₇

yes

characteristic

122

C₈

—

123

C₂²

—

124

C₂²

#11924524

yes

—

125

C₂²

C₂²×D₁₄₈

yes

characteristic

126

C₄

—

127

C₄

—

148

—

128

C₄

—

148

—

129

C₄

#11924524

yes

—

130

C₄

#11924569

yes

—

131

C₄

C₂×#10596962

yes

—

132

C₂

—

133

C₂

C₂×#11924524

yes

characteristic

134

C₂

C₂×#11924569

yes

—

	subgroup	G/H
center	C₂²	C₂²×D₁₄₈
commutator, cosocle	C₂×C₄×C₃₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₃₇
Fitting	C₄×M₁₆×C₃₇	C₂
socle	C₂²×C₃₇	C₂²×D₄
prime radical	C₂³×C₃₇	C₂×D₄
hypercenter	C₄×M₁₆	D₃₇
hypocenter	C₃₇	#3139
LCS	G ▷ C₂×C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄×M₁₆

subgroup

G/H

center

C₂²

C₂²×D₁₄₈

commutator, cosocle

C₂×C₄×C₃₇

C₂⁴

Frattini

C₂×C₄

C₂³×D₃₇

Fitting

C₄×M₁₆×C₃₇

C₂

socle

C₂²×C₃₇

C₂²×D₄

prime radical

C₂³×C₃₇

C₂×D₄

hypercenter

C₄×M₁₆

D₃₇

hypocenter

C₃₇

#3139

LCS

G ▷ C₂×C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄×M₁₆

N	H ≅ G/N	type
3. #14557902	C₂	semidirect
4. #14557884	C₂	semidirect
74. C₃₇	#3139	semidirect

H ≅ G/N

type

3. #14557902

C₂

semidirect

4. #14557884

C₂

semidirect

74. C₃₇

#3139

semidirect

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

subgroup	index
#14558090	2
C₂×#11924536	2
#14557902	2
#14557884	2
#14557543	2
C₄×M₁₆×C₃₇	2
#3139	37

subgroup

index

C₄×M₁₆×C₃₇

Subgroups — #18004977

Conjugacy classes of subgroups

Complements

Maximal subgroups