Group #18003998 — subgroups

Subgroups — #18003998

Order 4736.

772 conjugacy classes of subgroups; only the 161 normal subgroups are shown.

normal subgroups only

This group has 772 conjugacy classes of subgroups — too many to draw. Only its 161 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	#14558097	C₂	1	3	yes	maximal, complemented
2	#14557720	C₂	1	8	yes	maximal, complemented
3	#14557507	C₂	1	3	yes	maximal, complemented
4	C₃₇×#585	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×D₁₄₈	C₂²	1	1	yes	characteristic, complemented
6	#11924532	—	2	3	no	—
7	#11924525	—	2	3	no	—
8	#11924510	—	2	6	no	—
9	#11924504	C₂²	1	12	yes	complemented
10	#11924495	C₂²	1	3	yes	complemented
11	#11924451	C₂²	1	12	yes	complemented
12	2₋¹⁺⁴×C₃₇	—	2	3	no	—
13	H(5,𝔽₂)×C₃₇	—	2	3	no	—
14	H(5,𝔽₂)×C₃₇	C₂²	1	4	yes	—
15	C₂×#41×C₃₇	—	2	6	no	—
16	C₂×#41×C₃₇	C₂²	1	3	yes	—
17	C₂³×D₃₇	C₂³	1	2	yes	—
18	#41×C₃₇	—	2	4	no	—
19	#41×C₃₇	—	2	12	no	—
20	#41×C₃₇	—	2	6	no	—
21	#41×C₃₇	—	2	12	no	—
22	#41×C₃₇	—	2	6	no	—
23	C₂×Q₈×C₃₇	—	2	6	no	—
24	C₂×Q₈×C₃₇	D₄	1	3	yes	complemented
25	C₂×D₄×C₃₇	—	2	6	no	—
26	C₂×D₄×C₃₇	—	2	12	no	—
27	C₂×D₄×C₃₇	D₄	1	3	yes	complemented
28	C₂×D₄×C₃₇	C₂³	1	6	yes	—
29	C₂²×C₄×C₃₇	—	2	4	no	—
30	C₂²×C₄×C₃₇	D₄	1	6	yes	complemented
31	C₂²×C₄×C₃₇	C₂³	1	1	yes	characteristic
32	C₂×C₂²⋊D₃₇	—	2	12	no	—
33	C₂×D₁₄₈	—	2	3	no	—
34	C₂×D₁₄₈	—	2	3	no	—
35	C₂²⋊Dic₃₇	C₂³	1	6	yes	—
36	#10596938	—	2	12	no	—
37	C₄⋊Dic₃₇	—	2	3	no	—
38	C₄×Dic₃₇	—	2	3	no	—
39	C₂³×C₃₇	—	2	4	no	—
40	C₂³×C₃₇	C₂×D₄	1	6	yes	—
41	C₂³×C₃₇	C₂⁴	1	1	yes	characteristic
42	C₂²×D₃₇	—	2	6	no	—
43	C₂²×D₃₇	—	8	2	no	—
44	Q₈×C₃₇	—	2	4	no	—
45	Q₈×C₃₇	—	2	6	no	—
46	D₄×C₃₇	—	2	12	no	—
47	D₄×C₃₇	—	2	12	no	—
48	D₄×C₃₇	—	2	6	no	—
49	C₂×C₄×C₃₇	—	2	12	no	—
50	C₂×C₄×C₃₇	—	2	6	no	—
51	C₂×C₄×C₃₇	—	2	6	no	—
52	C₂×C₄×C₃₇	C₂×D₄	1	6	yes	—
53	C₂×C₄×C₃₇	C₂×D₄	1	3	yes	—
54	C₂×C₄×C₃₇	C₂×D₄	1	3	yes	—
55	C₂²⋊D₃₇	—	8	12	no	—
56	C₂×Dic₃₇	—	2	6	no	—
57	D₁₄₈	—	4	1	no	—
58	D₁₄₈	—	4	3	no	—
59	C₂²×C₃₇	—	2	6	no	—
60	C₂²×C₃₇	—	4	1	no	—
61	C₂²×C₃₇	—	4	6	no	—
62	C₂²×C₃₇	—	8	4	no	—
63	C₂²×C₃₇	C₂²×D₄	1	3	yes	—
64	C₂×D₃₇	—	4	12	no	—
65	C₂×D₃₇	—	4	2	no	—
66	C₄×C₃₇	—	2	6	no	—
67	C₄×C₃₇	#120	1	1	yes	characteristic
68	C₄×C₃₇	#120	1	3	yes	—
69	Dic₃₇	—	4	6	no	—
70	#3027	—	37	1	no	maximal, Sylow
71	C₂×C₃₇	—	2	3	no	—
72	C₂×C₃₇	—	4	6	no	—
73	C₂×C₃₇	C₂×#120	1	1	yes	characteristic
74	D₃₇	—	8	2	no	—
75	#585	D₃₇	1	1	yes	characteristic, complemented
76	#535	—	37	3	no	—
77	B(4,𝔽₂)	—	37	8	no	—
78	#410	—	37	3	no	—
79	C₃₇	#3027	1	1	yes	characteristic, complemented, Sylow
80	2₋¹⁺⁴	—	2	3	no	—
81	H(5,𝔽₂)	—	2	3	no	—
82	H(5,𝔽₂)	C₂×D₃₇	1	4	yes	—
83	C₂×#41	—	2	6	no	—
84	C₂×#41	C₂×D₃₇	1	3	yes	—
85	C₂²×D₄	—	37	1	no	—
86	#127	—	74	3	no	—
87	#124	—	74	3	no	—
88	#121	—	74	6	no	—
89	#120	—	37	12	no	—
90	#117	—	37	3	no	—
91	#99	—	37	12	no	—
92	C₂⁴	—	37	2	no	—
93	#41	—	2	4	no	—
94	#41	—	2	12	no	—
95	#41	—	2	6	no	—
96	#41	—	2	12	no	—
97	#41	—	2	6	no	—
98	C₂×Q₈	—	2	6	no	—
99	C₂×Q₈	C₂²⋊D₃₇	1	3	yes	complemented
100	C₂×D₄	—	2	6	no	—
101	C₂×D₄	—	2	12	no	—
102	C₂×D₄	—	74	12	no	—
103	C₂×D₄	—	74	3	no	—
104	C₂×D₄	—	74	3	no	—
105	C₂×D₄	C₂²⋊D₃₇	1	3	yes	complemented
106	C₂×D₄	C₂²×D₃₇	1	6	yes	—
107	C₂²×C₄	—	2	4	no	—
108	C₂²×C₄	C₂²⋊D₃₇	1	6	yes	complemented
109	C₂²×C₄	C₂²×D₃₇	1	1	yes	characteristic
110	C₄⋊C₄	—	74	3	no	—
111	C₂²⋊C₄	—	37	6	no	—
112	C₂²⋊C₄	—	74	12	no	—
113	C₄²	—	74	3	no	—
114	C₂³	—	2	4	no	—
115	C₂³	—	296	2	no	—
116	C₂³	—	74	6	no	—
117	C₂³	C₂×C₂²⋊D₃₇	1	6	yes	—
118	C₂³	C₂³×D₃₇	1	1	yes	characteristic
119	Q₈	—	2	4	no	—
120	Q₈	—	2	6	no	—
121	D₄	—	2	12	no	—
122	D₄	—	2	12	no	—
123	D₄	—	2	6	no	—
124	D₄	—	296	12	no	—
125	D₄	—	148	1	no	—
126	D₄	—	148	3	no	—
127	C₂×C₄	—	2	12	no	—
128	C₂×C₄	—	2	6	no	—
129	C₂×C₄	—	2	6	no	—
130	C₂×C₄	—	74	6	no	—
131	C₂×C₄	C₂×C₂²⋊D₃₇	1	6	yes	—
132	C₂×C₄	C₂×C₂²⋊D₃₇	1	3	yes	—
133	C₂×C₄	C₂×C₂²⋊D₃₇	1	3	yes	—
134	C₂²	—	2	6	no	—
135	C₂²	—	4	1	no	—
136	C₂²	—	4	6	no	—
137	C₂²	—	8	4	no	—
138	C₂²	—	148	12	no	—
139	C₂²	—	148	2	no	—
140	C₂²	C₂²×C₂²⋊D₃₇	1	3	yes	—
141	C₄	—	2	6	no	—
142	C₄	—	148	6	no	—
143	C₄	#11924505	1	1	yes	characteristic
144	C₄	#11924505	1	3	yes	—
145	C₂	—	2	3	no	—
146	C₂	—	4	6	no	—
147	C₂	—	296	2	no	—
148	C₂	C₂×#11924505	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14558097

C₂

yes

maximal, complemented

#14557720

C₂

yes

maximal, complemented

#14557507

C₂

yes

maximal, complemented

C₃₇×#585

C₂

yes

maximal, characteristic, complemented

C₂²×D₁₄₈

C₂²

yes

characteristic, complemented

—

—

—

C₂²

yes

complemented

#11924495

C₂²

yes

complemented

#11924451

C₂²

yes

complemented

2₋¹⁺⁴×C₃₇

—

H(5,𝔽₂)×C₃₇

—

H(5,𝔽₂)×C₃₇

C₂²

yes

—

C₂×#41×C₃₇

—

C₂×#41×C₃₇

C₂²

yes

—

C₂³×D₃₇

C₂³

yes

—

#41×C₃₇

—

#41×C₃₇

—

#41×C₃₇

—

#41×C₃₇

—

#41×C₃₇

—

C₂×Q₈×C₃₇

—

C₂×Q₈×C₃₇

D₄

yes

complemented

C₂×D₄×C₃₇

—

C₂×D₄×C₃₇

—

C₂×D₄×C₃₇

D₄

yes

complemented

C₂×D₄×C₃₇

C₂³

yes

—

C₂²×C₄×C₃₇

—

C₂²×C₄×C₃₇

D₄

yes

complemented

C₂²×C₄×C₃₇

C₂³

yes

characteristic

C₂×C₂²⋊D₃₇

—

C₂×D₁₄₈

—

C₂×D₁₄₈

—

C₂²⋊Dic₃₇

C₂³

yes

—

#10596938

—

C₄⋊Dic₃₇

—

C₄×Dic₃₇

—

C₂³×C₃₇

—

C₂³×C₃₇

C₂×D₄

yes

—

C₂³×C₃₇

C₂⁴

yes

characteristic

C₂²×D₃₇

—

C₂²×D₃₇

—

Q₈×C₃₇

—

Q₈×C₃₇

—

D₄×C₃₇

—

D₄×C₃₇

—

D₄×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

C₂×D₄

yes

—

C₂×C₄×C₃₇

C₂×D₄

yes

—

C₂×C₄×C₃₇

C₂×D₄

yes

—

C₂²⋊D₃₇

—

C₂×Dic₃₇

—

D₁₄₈

—

D₁₄₈

—

C₂²×C₃₇

—

C₂²×C₃₇

—

C₂²×C₃₇

—

C₂²×C₃₇

—

C₂²×C₃₇

C₂²×D₄

yes

—

C₂×D₃₇

—

C₂×D₃₇

—

C₄×C₃₇

—

C₄×C₃₇

#120

yes

characteristic

C₄×C₃₇

#120

yes

—

Dic₃₇

—

#3027

—

maximal, Sylow

C₂×C₃₇

—

C₂×C₃₇

—

C₂×C₃₇

C₂×#120

yes

characteristic

D₃₇

—

#585

D₃₇

yes

characteristic, complemented

#535

—

B(4,𝔽₂)

—

#410

—

C₃₇

#3027

yes

characteristic, complemented, Sylow

2₋¹⁺⁴

—

H(5,𝔽₂)

—

H(5,𝔽₂)

C₂×D₃₇

yes

—

C₂×#41

—

C₂×#41

C₂×D₃₇

yes

—

C₂²×D₄

—

—

—

—

—

—

—

C₂⁴

—

—

—

—

—

—

C₂×Q₈

—

C₂×Q₈

C₂²⋊D₃₇

yes

complemented

100

C₂×D₄

—

101

C₂×D₄

—

102

C₂×D₄

—

103

C₂×D₄

—

104

C₂×D₄

—

105

C₂×D₄

C₂²⋊D₃₇

yes

complemented

106

C₂×D₄

C₂²×D₃₇

yes

—

107

C₂²×C₄

—

108

C₂²×C₄

C₂²⋊D₃₇

yes

complemented

109

C₂²×C₄

C₂²×D₃₇

yes

characteristic

110

C₄⋊C₄

—

111

C₂²⋊C₄

—

112

C₂²⋊C₄

—

113

C₄²

—

114

C₂³

—

115

C₂³

—

296

—

116

C₂³

—

117

C₂³

C₂×C₂²⋊D₃₇

yes

—

118

C₂³

C₂³×D₃₇

yes

characteristic

119

Q₈

—

120

Q₈

—

121

D₄

—

122

D₄

—

123

D₄

—

124

D₄

—

296

—

125

D₄

—

148

—

126

D₄

—

148

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

C₂×C₂²⋊D₃₇

yes

—

132

C₂×C₄

C₂×C₂²⋊D₃₇

yes

—

133

C₂×C₄

C₂×C₂²⋊D₃₇

yes

—

134

C₂²

—

135

C₂²

—

136

C₂²

—

137

C₂²

—

138

C₂²

—

148

—

139

C₂²

—

148

—

140

C₂²

C₂²×C₂²⋊D₃₇

yes

—

141

C₄

—

142

C₄

—

148

—

143

C₄

#11924505

yes

characteristic

144

C₄

#11924505

yes

—

145

C₂

—

146

C₂

—

147

C₂

—

296

—

148

C₂

C₂×#11924505

yes

characteristic

	subgroup	G/H
center	C₂	C₂×#11924505
commutator, cosocle	C₂³×C₃₇	C₂⁴
Frattini	C₂³	C₂³×D₃₇
Fitting	C₃₇×#585	C₂
socle	C₂×C₃₇	C₂×#120
hypercenter	#585	D₃₇
hypocenter	C₃₇	#3027
LCS	G ▷ C₂³×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #585

subgroup

G/H

center

C₂

C₂×#11924505

commutator, cosocle

C₂³×C₃₇

C₂⁴

Frattini

C₂³

C₂³×D₃₇

Fitting

C₃₇×#585

C₂

C₂×C₃₇

D₃₇

C₃₇

G ▷ C₂³×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇

UCS

C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #585

N	H ≅ G/N	type
1. #14558097	C₂	semidirect
2. #14557720	C₂	semidirect
3. #14557507	C₂	semidirect
4. C₃₇×#585	C₂	semidirect
5. C₂²×D₁₄₈	C₂²	semidirect
9. #11924504	C₂²	semidirect
10. #11924495	C₂²	semidirect
11. #11924451	C₂²	semidirect
24. C₂×Q₈×C₃₇	D₄	semidirect
27. C₂×D₄×C₃₇	D₄	semidirect
30. C₂²×C₄×C₃₇	D₄	semidirect
75. #585	D₃₇	semidirect
79. C₃₇	#3027	semidirect
99. C₂×Q₈	C₂²⋊D₃₇	semidirect
105. C₂×D₄	C₂²⋊D₃₇	semidirect
108. C₂²×C₄	C₂²⋊D₃₇	semidirect

H ≅ G/N

type

1. #14558097

C₂

semidirect

2. #14557720

C₂

semidirect

3. #14557507

C₂

semidirect

4. C₃₇×#585

C₂

semidirect

5. C₂²×D₁₄₈

C₂²

semidirect

9. #11924504

C₂²

semidirect

10. #11924495

C₂²

semidirect

11. #11924451

C₂²

semidirect

24. C₂×Q₈×C₃₇

D₄

semidirect

27. C₂×D₄×C₃₇

D₄

semidirect

30. C₂²×C₄×C₃₇

D₄

semidirect

75. #585

D₃₇

semidirect

79. C₃₇

#3027

semidirect

99. C₂×Q₈

C₂²⋊D₃₇

semidirect

105. C₂×D₄

C₂²⋊D₃₇

semidirect

108. C₂²×C₄

C₂²⋊D₃₇

semidirect

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

subgroup	index
#14558097	2
#14557720	2
#14557507	2
C₃₇×#585	2
#3027	37

subgroup

index

C₃₇×#585

Subgroups — #18003998

Conjugacy classes of subgroups

Complements

Maximal subgroups