Group #17992003 — subgroups

Subgroups — C₃₇×#3185 #17992003

Order 4736.

358 conjugacy classes of subgroups; only the 184 normal subgroups are shown.

normal subgroups only

This group has 358 conjugacy classes of subgroups — too many to draw. Only its 184 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	C₃₇×#533	C₂	1	2	yes	maximal
2	C₃₇×#498	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₇×#494	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₇×#487	C₂	1	2	yes	maximal, complemented
5	C₃₇×#470	C₂	1	4	yes	maximal
6	C₃₇×#441	C₂	1	4	yes	maximal, complemented
7	C₄×M₁₆×C₃₇	C₂	1	1	yes	maximal, characteristic
8	C₂×Q₁₆×C₃₇	C₂²	1	4	yes	—
9	C₂×M₁₆×C₃₇	C₂²	1	2	yes	—
10	C₄⋊Q₈×C₃₇	C₂²	1	2	yes	—
11	C₄⋊Q₈×C₃₇	C₂²	1	2	yes	—
12	#125×C₃₇	—	2	2	no	—
13	C₂²⋊Q₈×C₃₇	—	2	2	no	—
14	C₂²⋊Q₈×C₃₇	C₂²	1	4	yes	—
15	C₄×Q₈×C₃₇	—	2	2	no	—
16	C₄×Q₈×C₃₇	C₂²	1	4	yes	—
17	#117×C₃₇	—	2	2	no	—
18	C₂×C₄²×C₃₇	C₂²	1	1	yes	characteristic
19	#106×C₃₇	C₂²	1	4	yes	—
20	#103×C₃₇	C₂²	1	8	yes	—
21	#97×C₃₇	C₂²	1	2	yes	—
22	C₄×C₈×C₃₇	C₂²	1	1	yes	characteristic
23	C₄×C₈×C₃₇	C₂²	1	1	yes	characteristic
24	C₂×Q₈×C₃₇	—	2	2	no	—
25	C₂×Q₈×C₃₇	C₂³	1	4	yes	—
26	C₂²×C₄×C₃₇	D₄	1	2	yes	—
27	C₂²×C₄×C₃₇	C₂³	1	1	yes	characteristic
28	Q₁₆×C₃₇	—	2	8	no	—
29	M₁₆×C₃₇	—	2	4	no	—
30	C₂×C₈×C₃₇	C₂³	1	2	yes	—
31	C₂×C₈×C₃₇	C₂³	1	2	yes	—
32	C₄⋊C₄×C₃₇	—	2	2	no	—
33	C₄⋊C₄×C₃₇	—	2	2	no	—
34	C₄⋊C₄×C₃₇	—	2	2	no	—
35	C₄⋊C₄×C₃₇	—	2	4	no	—
36	C₄⋊C₄×C₃₇	—	2	4	no	—
37	C₄⋊C₄×C₃₇	C₂³	1	4	yes	—
38	C₂²⋊C₄×C₃₇	—	2	4	no	—
39	C₄²×C₃₇	—	2	4	no	—
40	C₄²×C₃₇	D₄	1	2	yes	—
41	C₄²×C₃₇	C₂³	1	1	yes	characteristic
42	C₄²×C₃₇	C₂³	1	1	yes	characteristic
43	C₂³×C₃₇	C₂×D₄	1	1	yes	characteristic
44	Q₈×C₃₇	—	2	8	no	—
45	Q₈×C₃₇	—	4	4	no	—
46	C₂×C₄×C₃₇	—	2	4	no	—
47	C₂×C₄×C₃₇	—	2	2	no	—
48	C₂×C₄×C₃₇	—	2	4	no	—
49	C₂×C₄×C₃₇	—	2	4	no	—
50	C₂×C₄×C₃₇	C₂×D₄	1	2	yes	—
51	C₂×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
52	C₂×C₄×C₃₇	C₂×D₄	1	2	yes	—
53	C₂×C₄×C₃₇	C₂⁴	1	1	yes	characteristic
54	C₈×C₃₇	—	2	2	no	—
55	C₈×C₃₇	#41	1	4	yes	—
56	C₂²×C₃₇	—	2	1	no	—
57	C₂²×C₃₇	—	2	2	no	—
58	C₂²×C₃₇	C₂²×D₄	1	1	yes	characteristic
59	C₄×C₃₇	—	2	1	no	—
60	C₄×C₃₇	—	2	2	no	—
61	C₄×C₃₇	—	4	4	no	—
62	C₄×C₃₇	—	4	4	no	—
63	C₄×C₃₇	#137	1	4	yes	—
64	C₄×C₃₇	C₂×#41	1	2	yes	—
65	#3185	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
66	C₂×C₃₇	—	4	1	no	—
67	C₂×C₃₇	#532	1	1	yes	characteristic
68	C₂×C₃₇	C₂×#137	1	2	yes	—
69	#533	C₂×C₃₇	1	2	yes	—
70	#498	C₂×C₃₇	1	1	yes	characteristic, complemented
71	#494	C₂×C₃₇	1	1	yes	characteristic, complemented
72	#487	C₂×C₃₇	1	2	yes	complemented
73	#470	C₂×C₃₇	1	4	yes	—
74	#441	C₂×C₃₇	1	4	yes	complemented
75	C₄×M₁₆	C₂×C₃₇	1	1	yes	characteristic
76	C₃₇	#3185	1	1	yes	characteristic, direct product factor, Sylow
77	C₂×Q₁₆	C₂²×C₃₇	1	4	yes	—
78	C₂×M₁₆	C₂²×C₃₇	1	2	yes	—
79	C₄⋊Q₈	C₂²×C₃₇	1	2	yes	—
80	C₄⋊Q₈	C₂²×C₃₇	1	2	yes	—
81	#125	—	2	2	no	—
82	C₂²⋊Q₈	—	2	2	no	—
83	C₂²⋊Q₈	C₂²×C₃₇	1	4	yes	—
84	C₄×Q₈	—	2	2	no	—
85	C₄×Q₈	C₂²×C₃₇	1	4	yes	—
86	#117	—	2	2	no	—
87	C₂×C₄²	C₂²×C₃₇	1	1	yes	characteristic
88	#106	C₂²×C₃₇	1	4	yes	—
89	#103	C₂²×C₃₇	1	8	yes	—
90	#97	C₂²×C₃₇	1	2	yes	—
91	C₄×C₈	C₂²×C₃₇	1	1	yes	characteristic
92	C₄×C₈	C₂²×C₃₇	1	1	yes	characteristic
93	C₂×Q₈	—	2	2	no	—
94	C₂×Q₈	C₂³×C₃₇	1	4	yes	—
95	C₂²×C₄	D₄×C₃₇	1	2	yes	—
96	C₂²×C₄	C₂³×C₃₇	1	1	yes	characteristic
97	Q₁₆	—	2	8	no	—
98	M₁₆	—	2	4	no	—
99	C₂×C₈	C₂³×C₃₇	1	2	yes	—
100	C₂×C₈	C₂³×C₃₇	1	2	yes	—
101	C₄⋊C₄	—	2	2	no	—
102	C₄⋊C₄	—	2	2	no	—
103	C₄⋊C₄	—	2	2	no	—
104	C₄⋊C₄	—	2	4	no	—
105	C₄⋊C₄	—	2	4	no	—
106	C₄⋊C₄	C₂³×C₃₇	1	4	yes	—
107	C₂²⋊C₄	—	2	4	no	—
108	C₄²	—	2	4	no	—
109	C₄²	D₄×C₃₇	1	2	yes	—
110	C₄²	C₂³×C₃₇	1	1	yes	characteristic
111	C₄²	C₂³×C₃₇	1	1	yes	characteristic
112	C₂³	C₂×D₄×C₃₇	1	1	yes	characteristic
113	Q₈	—	2	8	no	—
114	Q₈	—	4	4	no	—
115	C₂×C₄	—	2	4	no	—
116	C₂×C₄	—	2	2	no	—
117	C₂×C₄	—	2	4	no	—
118	C₂×C₄	—	2	4	no	—
119	C₂×C₄	C₂×D₄×C₃₇	1	2	yes	—
120	C₂×C₄	C₂×D₄×C₃₇	1	1	yes	characteristic
121	C₂×C₄	C₂×D₄×C₃₇	1	2	yes	—
122	C₂×C₄	C₂⁴×C₃₇	1	1	yes	characteristic
123	C₈	—	2	2	no	—
124	C₈	#41×C₃₇	1	4	yes	—
125	C₂²	—	2	1	no	—
126	C₂²	—	2	2	no	—
127	C₂²	C₂²×D₄×C₃₇	1	1	yes	characteristic
128	C₄	—	2	1	no	—
129	C₄	—	2	2	no	—
130	C₄	—	4	4	no	—
131	C₄	—	4	4	no	—
132	C₄	#137×C₃₇	1	4	yes	—
133	C₄	C₂×#41×C₃₇	1	2	yes	—
134	C₂	—	4	1	no	—
135	C₂	C₃₇×#532	1	1	yes	characteristic
136	C₂	C₂×#137×C₃₇	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃₇×#533

C₂

yes

maximal

C₃₇×#498

C₂

yes

maximal, characteristic, complemented

C₃₇×#494

C₂

yes

maximal, characteristic, complemented

C₃₇×#487

C₂

yes

maximal, complemented

C₃₇×#470

C₂

yes

maximal

C₃₇×#441

C₂

yes

maximal, complemented

C₄×M₁₆×C₃₇

C₂

yes

maximal, characteristic

C₂×Q₁₆×C₃₇

C₂²

yes

—

C₂×M₁₆×C₃₇

C₂²

yes

—

C₄⋊Q₈×C₃₇

C₂²

yes

—

C₄⋊Q₈×C₃₇

C₂²

yes

—

#125×C₃₇

—

C₂²⋊Q₈×C₃₇

—

C₂²⋊Q₈×C₃₇

C₂²

yes

—

C₄×Q₈×C₃₇

—

C₄×Q₈×C₃₇

C₂²

yes

—

#117×C₃₇

—

C₂×C₄²×C₃₇

C₂²

yes

characteristic

#106×C₃₇

C₂²

yes

—

#103×C₃₇

C₂²

yes

—

#97×C₃₇

C₂²

yes

—

C₄×C₈×C₃₇

C₂²

yes

characteristic

C₄×C₈×C₃₇

C₂²

yes

characteristic

C₂×Q₈×C₃₇

—

C₂×Q₈×C₃₇

C₂³

yes

—

C₂²×C₄×C₃₇

D₄

yes

—

C₂²×C₄×C₃₇

C₂³

yes

characteristic

Q₁₆×C₃₇

—

M₁₆×C₃₇

—

C₂×C₈×C₃₇

C₂³

yes

—

C₂×C₈×C₃₇

C₂³

yes

—

C₄⋊C₄×C₃₇

—

C₄⋊C₄×C₃₇

—

C₄⋊C₄×C₃₇

—

C₄⋊C₄×C₃₇

—

C₄⋊C₄×C₃₇

—

C₄⋊C₄×C₃₇

C₂³

yes

—

C₂²⋊C₄×C₃₇

—

C₄²×C₃₇

—

C₄²×C₃₇

D₄

yes

—

C₄²×C₃₇

C₂³

yes

characteristic

C₄²×C₃₇

C₂³

yes

characteristic

C₂³×C₃₇

C₂×D₄

yes

characteristic

Q₈×C₃₇

—

Q₈×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

—

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

C₂⁴

yes

characteristic

C₈×C₃₇

—

C₈×C₃₇

#41

yes

—

C₂²×C₃₇

—

C₂²×C₃₇

—

C₂²×C₃₇

C₂²×D₄

yes

characteristic

C₄×C₃₇

—

C₄×C₃₇

—

C₄×C₃₇

—

C₄×C₃₇

—

C₄×C₃₇

#137

yes

—

C₄×C₃₇

C₂×#41

yes

—

#3185

C₃₇

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₃₇

—

C₂×C₃₇

#532

yes

characteristic

C₂×C₃₇

C₂×#137

yes

—

#533

C₂×C₃₇

yes

—

#498

C₂×C₃₇

yes

characteristic, complemented

#494

C₂×C₃₇

yes

characteristic, complemented

#487

C₂×C₃₇

yes

complemented

#470

C₂×C₃₇

yes

—

#441

C₂×C₃₇

yes

complemented

C₄×M₁₆

C₂×C₃₇

yes

characteristic

C₃₇

#3185

yes

characteristic, direct product factor, Sylow

C₂×Q₁₆

C₂²×C₃₇

yes

—

C₂×M₁₆

C₂²×C₃₇

yes

—

C₄⋊Q₈

C₂²×C₃₇

yes

—

C₄⋊Q₈

C₂²×C₃₇

yes

—

#125

—

C₂²⋊Q₈

—

C₂²⋊Q₈

C₂²×C₃₇

yes

—

C₄×Q₈

—

C₄×Q₈

C₂²×C₃₇

yes

—

#117

—

C₂×C₄²

C₂²×C₃₇

yes

characteristic

#106

C₂²×C₃₇

yes

—

#103

C₂²×C₃₇

yes

—

#97

C₂²×C₃₇

yes

—

C₄×C₈

C₂²×C₃₇

yes

characteristic

C₄×C₈

C₂²×C₃₇

yes

characteristic

C₂×Q₈

—

C₂×Q₈

C₂³×C₃₇

yes

—

C₂²×C₄

D₄×C₃₇

yes

—

C₂²×C₄

C₂³×C₃₇

yes

characteristic

Q₁₆

—

M₁₆

—

C₂×C₈

C₂³×C₃₇

yes

—

100

C₂×C₈

C₂³×C₃₇

yes

—

101

C₄⋊C₄

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

106

C₄⋊C₄

C₂³×C₃₇

yes

—

107

C₂²⋊C₄

—

108

C₄²

—

109

C₄²

D₄×C₃₇

yes

—

110

C₄²

C₂³×C₃₇

yes

characteristic

111

C₄²

C₂³×C₃₇

yes

characteristic

112

C₂³

C₂×D₄×C₃₇

yes

characteristic

113

Q₈

—

114

Q₈

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

C₂×D₄×C₃₇

yes

—

120

C₂×C₄

C₂×D₄×C₃₇

yes

characteristic

121

C₂×C₄

C₂×D₄×C₃₇

yes

—

122

C₂×C₄

C₂⁴×C₃₇

yes

characteristic

123

C₈

—

124

C₈

#41×C₃₇

yes

—

125

C₂²

—

126

C₂²

—

127

C₂²

C₂²×D₄×C₃₇

yes

characteristic

128

C₄

—

129

C₄

—

130

C₄

—

131

C₄

—

132

C₄

#137×C₃₇

yes

—

133

C₄

C₂×#41×C₃₇

yes

—

134

C₂

—

135

C₂

C₃₇×#532

yes

characteristic

136

C₂

C₂×#137×C₃₇

yes

—

	subgroup	G/H
center, socle	C₂²×C₃₇	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₃₇
prime radical	C₂³×C₃₇	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₇ ◁ C₂×C₄²×C₃₇ ◁ G

subgroup

G/H

center, socle

C₂²×C₃₇

C₂²×D₄

commutator, Frattini, cosocle

C₂×C₄

C₂⁴×C₃₇

prime radical

C₂³×C₃₇

C₂×D₄

LCS

G ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₃₇ ◁ C₂×C₄²×C₃₇ ◁ G

N	H ≅ G/N	type
2. C₃₇×#498	C₂	semidirect
3. C₃₇×#494	C₂	semidirect
4. C₃₇×#487	C₂	semidirect
6. C₃₇×#441	C₂	semidirect
65. #3185	C₃₇	direct (N × H)
70. #498	C₂×C₃₇	semidirect
71. #494	C₂×C₃₇	semidirect
72. #487	C₂×C₃₇	semidirect
74. #441	C₂×C₃₇	semidirect
76. C₃₇	#3185	direct (N × H)

H ≅ G/N

type

2. C₃₇×#498

C₂

semidirect

3. C₃₇×#494

C₂

semidirect

4. C₃₇×#487

C₂

semidirect

6. C₃₇×#441

C₂

semidirect

65. #3185

C₃₇

direct (N × H)

70. #498

C₂×C₃₇

semidirect

71. #494

C₂×C₃₇

semidirect

72. #487

C₂×C₃₇

semidirect

74. #441

C₂×C₃₇

semidirect

76. C₃₇

#3185

direct (N × H)

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

subgroup	index
C₃₇×#533	2
C₃₇×#498	2
C₃₇×#494	2
C₃₇×#487	2
C₃₇×#470	2
C₃₇×#441	2
C₄×M₁₆×C₃₇	2
#3185	37

subgroup

index

C₃₇×#533

C₃₇×#498

C₃₇×#494

C₃₇×#487

C₃₇×#470

C₃₇×#441

C₄×M₁₆×C₃₇

#3185

Subgroups — C₃₇×#3185 #17992003

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C37×#3185 #17992003

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₇×#3185 #17992003