Group #18010729 — subgroups

Subgroups — #18010729

Order 4736.

360 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 78 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	#14558537	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#11924601	C₂	1	4	yes	maximal, complemented
3	C₄²×#3866	C₂	1	1	yes	maximal, characteristic, complemented
4	#14558084	C₂	1	1	yes	maximal, characteristic
5	C₂×#10596953	C₂²	1	2	yes	—
6	C₂×C₄×#3866	—	2	2	no	—
7	C₂×C₄×#3866	C₂²	1	2	yes	—
8	C₄⋊#64438	—	2	2	no	—
9	#11924601	—	2	8	no	—
10	C₂×#10596966	C₂²	1	2	yes	—
11	#11924533	C₄	1	1	yes	characteristic, complemented
12	#11924525	C₄	1	2	yes	complemented
13	#11924511	—	2	2	no	—
14	#11924498	—	2	2	no	—
15	C₄²×D₃₇	C₂²	1	1	yes	characteristic
16	C₄⋊Q₈×C₃₇	C₄	1	1	yes	characteristic, complemented
17	C₂²×#3866	—	2	2	no	—
18	C₂×Q₈×C₃₇	C₂×C₄	1	2	yes	—
19	#10596966	—	2	4	no	—
20	#10596966	—	2	4	no	—
21	C₂×D₁₄₈	—	2	2	no	—
22	C₂×D₁₄₈	—	2	2	no	—
23	C₂×D₁₄₈	C₂×C₄	1	2	yes	—
24	C₂×C₄×D₃₇	—	2	2	no	—
25	C₂×C₄×D₃₇	D₄	1	2	yes	—
26	C₂×C₄×D₃₇	C₂³	1	1	yes	characteristic
27	C₂×#64438	—	2	2	no	—
28	C₄×#3866	—	2	8	no	—
29	C₄×#3866	—	2	2	no	—
30	C₄×#3866	D₄	1	4	yes	complemented
31	#10596953	—	2	4	no	—
32	C₄⋊C₄×C₃₇	—	2	2	no	—
33	C₄²×C₃₇	C₂×C₄	1	1	yes	characteristic
34	#10596938	—	2	4	no	—
35	C₄×Dic₃₇	C₂×C₄	1	1	yes	characteristic
36	C₄×Dic₃₇	D₄	1	2	yes	—
37	C₂²×D₃₇	—	2	2	no	—
38	C₂²×D₃₇	C₂×D₄	1	1	yes	characteristic
39	C₂×#3866	—	2	8	no	—
40	C₂×#3866	—	2	4	no	—
41	Q₈×C₃₇	—	2	4	no	—
42	C₂×C₄×C₃₇	—	2	2	no	—
43	C₂×C₄×C₃₇	C₂²⋊C₄	1	2	yes	—
44	C₂×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
45	C₂×Dic₃₇	C₂²⋊C₄	1	2	yes	—
46	C₂×Dic₃₇	C₂×D₄	1	1	yes	characteristic
47	D₁₄₈	—	2	4	no	—
48	D₁₄₈	—	4	4	no	—
49	D₁₄₈	—	4	4	no	—
50	C₄×D₃₇	—	2	4	no	—
51	C₄×D₃₇	—	4	2	no	—
52	C₄×D₃₇	—	4	2	no	—
53	C₄×D₃₇	C₂×D₄	1	2	yes	—
54	C₄×D₃₇	#41	1	2	yes	—
55	#64438	—	4	2	no	—
56	C₂²×C₃₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
57	C₂×D₃₇	—	2	2	no	—
58	C₂×D₃₇	—	4	2	no	—
59	C₂×D₃₇	—	4	4	no	—
60	C₂×D₃₇	#121	1	2	yes	—
61	#3866	—	2	8	no	—
62	C₄×C₃₇	—	4	2	no	—
63	C₄×C₃₇	#104	1	4	yes	complemented
64	C₄×C₃₇	C₄×D₄	1	2	yes	—
65	Dic₃₇	—	2	2	no	—
66	Dic₃₇	#124	1	1	yes	characteristic
67	Dic₃₇	#127	1	1	yes	characteristic
68	#1962	—	37	1	no	maximal, Sylow
69	C₂×C₃₇	#390	1	1	yes	characteristic
70	C₂×C₃₇	C₂×#104	1	2	yes	—
71	D₃₇	—	2	2	no	—
72	D₃₇	—	8	2	no	—
73	#532	—	37	1	no	—
74	#423	—	37	1	no	—
75	C₂×#104	—	37	4	no	—
76	C₄³	—	37	1	no	—
77	C₃₇	#1962	1	1	yes	characteristic, complemented, Sylow
78	C₂×#41	—	37	2	no	—
79	C₂×M₁₆	—	37	2	no	—
80	C₄⋊Q₈	#3866	1	1	yes	characteristic, complemented
81	#127	—	37	1	no	—
82	#124	—	37	2	no	—
83	#121	—	74	2	no	—
84	C₄×D₄	—	74	2	no	—
85	C₂×C₄²	—	37	1	no	—
86	C₂×C₄²	—	37	2	no	—
87	C₂×C₄²	—	74	2	no	—
88	C₄⋊C₈	—	74	2	no	—
89	#104	—	74	8	no	—
90	#41	—	74	4	no	—
91	#41	—	74	4	no	—
92	C₂×Q₈	C₂×#3866	1	2	yes	—
93	C₂×D₄	—	37	2	no	—
94	C₂×D₄	—	74	2	no	—
95	C₂×D₄	—	74	2	no	—
96	C₂²×C₄	—	37	1	no	—
97	C₂²×C₄	—	37	2	no	—
98	C₂²×C₄	—	74	2	no	—
99	C₂²×C₄	—	74	2	no	—
100	M₁₆	—	74	4	no	—
101	C₂×C₈	—	74	2	no	—
102	C₄⋊C₄	—	2	2	no	—
103	C₂²⋊C₄	—	74	4	no	—
104	C₄²	—	37	2	no	—
105	C₄²	—	37	1	no	—
106	C₄²	—	37	4	no	—
107	C₄²	—	74	2	no	—
108	C₄²	—	74	8	no	—
109	C₄²	C₂×#3866	1	1	yes	characteristic
110	C₂³	—	37	1	no	—
111	C₂³	—	74	2	no	—
112	Q₈	—	2	4	no	—
113	D₄	—	74	4	no	—
114	D₄	—	148	4	no	—
115	D₄	—	148	4	no	—
116	C₂×C₄	—	2	2	no	—
117	C₂×C₄	—	37	1	no	—
118	C₂×C₄	—	37	2	no	—
119	C₂×C₄	—	37	2	no	—
120	C₂×C₄	—	37	2	no	—
121	C₂×C₄	—	74	4	no	—
122	C₂×C₄	—	74	8	no	—
123	C₂×C₄	—	74	4	no	—
124	C₂×C₄	—	148	2	no	—
125	C₂×C₄	—	148	2	no	—
126	C₂×C₄	C₂²⋊#3866	1	2	yes	—
127	C₂×C₄	C₂²×#3866	1	1	yes	characteristic
128	C₈	—	148	2	no	—
129	C₂²	—	37	2	no	—
130	C₂²	—	74	2	no	—
131	C₂²	—	148	2	no	—
132	C₂²	—	148	4	no	—
133	C₂²	C₂×C₂²⋊#3866	1	1	yes	characteristic
134	C₄	—	4	2	no	—
135	C₄	—	37	1	no	—
136	C₄	—	37	1	no	—
137	C₄	—	74	8	no	—
138	C₄	—	74	2	no	—
139	C₄	#11924601	1	4	yes	complemented
140	C₄	D₄×#3866	1	2	yes	—
141	C₂	—	296	2	no	—
142	C₂	—	74	2	no	—
143	C₂	#14558485	1	1	yes	characteristic
144	C₂	C₂×#11924601	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#14558537

C₂

yes

maximal, characteristic, complemented

C₂×#11924601

C₂

yes

maximal, complemented

C₄²×#3866

C₂

yes

maximal, characteristic, complemented

#14558084

C₂

yes

maximal, characteristic

C₂×#10596953

C₂²

yes

—

C₂×C₄×#3866

—

C₂×C₄×#3866

C₂²

yes

—

C₄⋊#64438

—

#11924601

—

C₂×#10596966

C₂²

yes

—

#11924533

C₄

yes

characteristic, complemented

#11924525

C₄

yes

complemented

#11924511

—

#11924498

—

C₄²×D₃₇

C₂²

yes

characteristic

C₄⋊Q₈×C₃₇

C₄

yes

characteristic, complemented

C₂²×#3866

—

C₂×Q₈×C₃₇

C₂×C₄

yes

—

#10596966

—

#10596966

—

C₂×D₁₄₈

—

C₂×D₁₄₈

—

C₂×D₁₄₈

C₂×C₄

yes

—

C₂×C₄×D₃₇

—

C₂×C₄×D₃₇

D₄

yes

—

C₂×C₄×D₃₇

C₂³

yes

—

—

—

D₄

yes

complemented

#10596953

—

C₄⋊C₄×C₃₇

—

C₄²×C₃₇

C₂×C₄

yes

characteristic

#10596938

—

C₄×Dic₃₇

C₂×C₄

yes

characteristic

C₄×Dic₃₇

D₄

yes

—

C₂²×D₃₇

—

C₂²×D₃₇

C₂×D₄

yes

characteristic

C₂×#3866

—

C₂×#3866

—

Q₈×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

C₂²⋊C₄

yes

—

C₂×C₄×C₃₇

C₂²×C₄

yes

characteristic

C₂×Dic₃₇

C₂²⋊C₄

yes

—

C₂×Dic₃₇

C₂×D₄

yes

characteristic

D₁₄₈

—

D₁₄₈

—

D₁₄₈

—

C₄×D₃₇

—

C₄×D₃₇

—

C₄×D₃₇

—

C₄×D₃₇

C₂×D₄

yes

—

C₄×D₃₇

#41

yes

—

#64438

—

C₂²×C₃₇

C₂×C₂²⋊C₄

yes

characteristic

C₂×D₃₇

—

C₂×D₃₇

—

C₂×D₃₇

—

C₂×D₃₇

#121

yes

—

#3866

—

C₄×C₃₇

—

C₄×C₃₇

#104

yes

complemented

C₄×C₃₇

C₄×D₄

yes

—

Dic₃₇

—

Dic₃₇

yes

Dic₃₇

yes

—

C₂×C₃₇

yes

C₂×C₃₇

yes

—

D₃₇

—

D₃₇

—

#532

—

#423

—

C₂×#104

—

C₄³

—

C₃₇

#1962

yes

characteristic, complemented, Sylow

C₂×#41

—

C₂×M₁₆

—

C₄⋊Q₈

#3866

yes

characteristic, complemented

#127

—

#124

—

#121

—

C₄×D₄

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

—

C₄⋊C₈

—

#104

—

#41

—

#41

—

C₂×Q₈

C₂×#3866

yes

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

100

M₁₆

—

101

C₂×C₈

—

102

C₄⋊C₄

—

103

C₂²⋊C₄

—

104

C₄²

—

105

C₄²

—

106

C₄²

—

107

C₄²

—

108

C₄²

—

109

C₄²

C₂×#3866

yes

characteristic

110

C₂³

—

111

C₂³

—

112

Q₈

—

113

D₄

—

114

D₄

—

148

—

115

D₄

—

148

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

148

—

125

C₂×C₄

—

148

—

126

C₂×C₄

C₂²⋊#3866

yes

—

127

C₂×C₄

C₂²×#3866

yes

characteristic

128

C₈

—

148

—

129

C₂²

—

130

C₂²

—

131

C₂²

—

148

—

132

C₂²

—

148

—

133

C₂²

C₂×C₂²⋊#3866

yes

characteristic

134

C₄

—

135

C₄

—

136

C₄

—

137

C₄

—

138

C₄

—

139

C₄

#11924601

yes

complemented

140

C₄

D₄×#3866

yes

—

141

C₂

—

296

—

142

C₂

—

143

C₂

#14558485

yes

characteristic

144

C₂

C₂×#11924601

yes

—

	subgroup	G/H
center	C₂²	C₂×C₂²⋊#3866
commutator	C₂×C₄×C₃₇	C₂²×C₄
Frattini	C₂×C₄	C₂²×#3866
Fitting	C₄⋊Q₈×C₃₇	C₄
socle	C₂²×C₃₇	C₂×C₂²⋊C₄
prime radical	#14558084	C₂
cosocle	C₂×C₄×D₃₇	C₂³
hypercenter	C₄⋊Q₈	#3866
hypocenter	C₃₇	#1962
LCS	G ▷ C₂×C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

subgroup

G/H

center

C₂²

C₂×C₂²⋊#3866

commutator

C₂×C₄×C₃₇

C₂²×C₄

Frattini

C₂×C₄

C₂²×#3866

Fitting

C₄⋊Q₈×C₃₇

C₄

socle

C₂²×C₃₇

C₂×C₂²⋊C₄

prime radical

#14558084

C₂

cosocle

C₂×C₄×D₃₇

C₂³

C₄⋊Q₈

C₃₇

G ▷ C₂×C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

N	H ≅ G/N	type
1. #14558537	C₂	semidirect
2. C₂×#11924601	C₂	semidirect
3. C₄²×#3866	C₂	semidirect
11. #11924533	C₄	semidirect
12. #11924525	C₄	semidirect
16. C₄⋊Q₈×C₃₇	C₄	semidirect
30. C₄×#3866	D₄	semidirect
63. C₄×C₃₇	#104	semidirect
77. C₃₇	#1962	semidirect
80. C₄⋊Q₈	#3866	semidirect
139. C₄	#11924601	semidirect

H ≅ G/N

type

1. #14558537

C₂

semidirect

2. C₂×#11924601

C₂

semidirect

3. C₄²×#3866

C₂

semidirect

11. #11924533

C₄

semidirect

12. #11924525

C₄

semidirect

16. C₄⋊Q₈×C₃₇

C₄

semidirect

30. C₄×#3866

D₄

semidirect

63. C₄×C₃₇

#104

semidirect

77. C₃₇

#1962

semidirect

80. C₄⋊Q₈

#3866

semidirect

139. C₄

#11924601

semidirect

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

subgroup	index
#14558537	2
C₂×#11924601	2
C₄²×#3866	2
#14558084	2
#1962	37

subgroup

index

C₄²×#3866

Subgroups — #18010729

Conjugacy classes of subgroups

Complements

Maximal subgroups