Group #18010732 — subgroups

Subgroups — #18010732

Order 4736.

320 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 320 conjugacy classes of subgroups — too many to draw. Only its 70 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	#14558538	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#11924601	C₂	1	4	yes	maximal, complemented
3	#14558469	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	C₂²	1	2	yes	—
7	#11924603	—	2	2	no	—
8	#11924601	—	2	8	no	—
9	(C₂×#3866)⋊C₄	—	2	2	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	2	no	—
29	C₄×#3866	D₄	1	4	yes	complemented
30	#10596953	—	2	4	no	—
31	C₄⋊C₄×C₃₇	—	2	2	no	—
32	C₄²×C₃₇	C₂×C₄	1	1	yes	characteristic
33	#10596938	—	2	4	no	—
34	C₄×Dic₃₇	C₂×C₄	1	1	yes	characteristic
35	C₄×Dic₃₇	D₄	1	2	yes	—
36	C₂²×D₃₇	—	2	2	no	—
37	C₂²×D₃₇	C₂×D₄	1	1	yes	characteristic
38	C₂×#3866	—	2	4	no	—
39	C₂×#3866	—	4	4	no	—
40	Q₈×C₃₇	—	2	4	no	—
41	C₂×C₄×C₃₇	—	2	2	no	—
42	C₂×C₄×C₃₇	C₂²⋊C₄	1	2	yes	—
43	C₂×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
44	C₂×Dic₃₇	C₂²⋊C₄	1	2	yes	—
45	C₂×Dic₃₇	C₂×D₄	1	1	yes	characteristic
46	D₁₄₈	—	2	4	no	—
47	D₁₄₈	—	4	4	no	—
48	D₁₄₈	—	4	4	no	—
49	C₄×D₃₇	—	4	2	no	—
50	C₄×D₃₇	—	4	2	no	—
51	C₄×D₃₇	—	4	2	no	—
52	C₄×D₃₇	C₂×D₄	1	2	yes	—
53	C₄×D₃₇	#41	1	2	yes	—
54	#64438	—	4	2	no	—
55	C₂²×C₃₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
56	C₂×D₃₇	—	2	2	no	—
57	C₂×D₃₇	—	4	2	no	—
58	C₂×D₃₇	—	4	4	no	—
59	C₂×D₃₇	#121	1	2	yes	—
60	#3866	—	4	4	no	—
61	C₄×C₃₇	—	2	2	no	—
62	C₄×C₃₇	—	4	2	no	—
63	C₄×C₃₇	C₄×D₄	1	2	yes	—
64	Dic₃₇	—	2	2	no	—
65	Dic₃₇	#124	1	1	yes	characteristic
66	Dic₃₇	#127	1	1	yes	characteristic
67	#1963	—	37	1	no	maximal, Sylow
68	C₂×C₃₇	#390	1	1	yes	characteristic
69	C₂×C₃₇	#421	1	2	yes	—
70	D₃₇	—	2	2	no	—
71	D₃₇	—	8	2	no	—
72	#532	—	37	1	no	—
73	#423	—	37	1	no	—
74	C₂×#104	—	37	4	no	—
75	#376	—	37	1	no	—
76	C₃₇	#1963	1	1	yes	characteristic, complemented, Sylow
77	C₂×#41	—	37	2	no	—
78	C₂×M₁₆	—	37	2	no	—
79	C₄⋊Q₈	#3866	1	1	yes	characteristic, complemented
80	#127	—	37	1	no	—
81	#124	—	37	2	no	—
82	#121	—	74	2	no	—
83	C₄×D₄	—	74	2	no	—
84	C₂×C₄²	—	37	1	no	—
85	C₂×C₄²	—	37	2	no	—
86	C₄⋊C₈	—	74	2	no	—
87	#104	—	74	8	no	—
88	#95	—	74	2	no	—
89	#41	—	74	4	no	—
90	#41	—	74	4	no	—
91	C₂×Q₈	C₂×#3866	1	2	yes	—
92	C₂×D₄	—	37	2	no	—
93	C₂×D₄	—	74	2	no	—
94	C₂×D₄	—	74	2	no	—
95	C₂²×C₄	—	37	1	no	—
96	C₂²×C₄	—	37	2	no	—
97	C₂²×C₄	—	74	2	no	—
98	C₂²×C₄	—	74	2	no	—
99	M₁₆	—	74	4	no	—
100	C₂×C₈	—	74	2	no	—
101	C₄⋊C₄	—	2	2	no	—
102	C₂²⋊C₄	—	74	4	no	—
103	C₄²	—	37	2	no	—
104	C₄²	—	37	1	no	—
105	C₄²	—	37	4	no	—
106	C₄²	—	74	2	no	—
107	C₄²	C₂×#3866	1	1	yes	characteristic
108	C₂³	—	37	1	no	—
109	C₂³	—	74	2	no	—
110	Q₈	—	2	4	no	—
111	D₄	—	74	4	no	—
112	D₄	—	148	4	no	—
113	D₄	—	148	4	no	—
114	C₂×C₄	—	2	2	no	—
115	C₂×C₄	—	37	1	no	—
116	C₂×C₄	—	37	2	no	—
117	C₂×C₄	—	37	2	no	—
118	C₂×C₄	—	37	2	no	—
119	C₂×C₄	—	74	4	no	—
120	C₂×C₄	—	148	2	no	—
121	C₂×C₄	—	148	2	no	—
122	C₂×C₄	—	148	2	no	—
123	C₂×C₄	—	148	4	no	—
124	C₂×C₄	C₂²⋊#3866	1	2	yes	—
125	C₂×C₄	C₂²×#3866	1	1	yes	characteristic
126	C₈	—	148	2	no	—
127	C₂²	—	37	2	no	—
128	C₂²	—	74	2	no	—
129	C₂²	—	148	2	no	—
130	C₂²	—	148	4	no	—
131	C₂²	C₂×C₂²⋊#3866	1	1	yes	characteristic
132	C₄	—	2	2	no	—
133	C₄	—	4	2	no	—
134	C₄	—	37	1	no	—
135	C₄	—	37	1	no	—
136	C₄	—	74	2	no	—
137	C₄	—	148	4	no	—
138	C₄	D₄×#3866	1	2	yes	—
139	C₂	—	296	2	no	—
140	C₂	—	74	2	no	—
141	C₂	#14558485	1	1	yes	characteristic
142	C₂	#14558532	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#14558538

C₂

yes

maximal, characteristic, complemented

C₂×#11924601

C₂

yes

maximal, complemented

#14558469

C₂

yes

maximal, characteristic, complemented

#14558084

C₂

yes

maximal, characteristic

C₂×#10596953

C₂²

yes

—

C₂×C₄×#3866

C₂²

yes

—

—

—

—

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

—

C₄×C₃₇

C₄×D₄

yes

—

Dic₃₇

—

Dic₃₇

yes

Dic₃₇

yes

—

C₂×C₃₇

yes

C₂×C₃₇

yes

—

D₃₇

—

D₃₇

—

—

—

—

—

C₃₇

#1963

yes

characteristic, complemented, Sylow

C₂×#41

—

C₂×M₁₆

—

C₄⋊Q₈

#3866

yes

characteristic, complemented

#127

—

#124

—

#121

—

C₄×D₄

—

C₂×C₄²

—

C₂×C₄²

—

C₄⋊C₈

—

—

—

—

—

C₂×Q₈

C₂×#3866

yes

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

M₁₆

—

100

C₂×C₈

—

101

C₄⋊C₄

—

102

C₂²⋊C₄

—

103

C₄²

—

104

C₄²

—

105

C₄²

—

106

C₄²

—

107

C₄²

C₂×#3866

yes

characteristic

108

C₂³

—

109

C₂³

—

110

Q₈

—

111

D₄

—

112

D₄

—

148

—

113

D₄

—

148

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

148

—

121

C₂×C₄

—

148

—

122

C₂×C₄

—

148

—

123

C₂×C₄

—

148

—

124

C₂×C₄

C₂²⋊#3866

yes

—

125

C₂×C₄

C₂²×#3866

yes

characteristic

126

C₈

—

148

—

127

C₂²

—

128

C₂²

—

129

C₂²

—

148

—

130

C₂²

—

148

—

131

C₂²

C₂×C₂²⋊#3866

yes

characteristic

132

C₄

—

133

C₄

—

134

C₄

—

135

C₄

—

136

C₄

—

137

C₄

—

148

—

138

C₄

D₄×#3866

yes

—

139

C₂

—

296

—

140

C₂

—

141

C₂

#14558485

yes

characteristic

142

C₂

#14558532

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	#1963
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. #14558538	C₂	semidirect
2. C₂×#11924601	C₂	semidirect
3. #14558469	C₂	semidirect
11. #11924533	C₄	semidirect
12. #11924525	C₄	semidirect
16. C₄⋊Q₈×C₃₇	C₄	semidirect
29. C₄×#3866	D₄	semidirect
76. C₃₇	#1963	semidirect
79. C₄⋊Q₈	#3866	semidirect

H ≅ G/N

type

1. #14558538

C₂

semidirect

2. C₂×#11924601

C₂

semidirect

3. #14558469

C₂

semidirect

11. #11924533

C₄

semidirect

12. #11924525

C₄

semidirect

16. C₄⋊Q₈×C₃₇

C₄

semidirect

29. C₄×#3866

D₄

semidirect

76. C₃₇

#1963

semidirect

79. C₄⋊Q₈

#3866

semidirect

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

subgroup	index
#14558538	2
C₂×#11924601	2
#14558469	2
#14558084	2
#1963	37

subgroup

index

Subgroups — #18010732

Conjugacy classes of subgroups

Complements

Maximal subgroups