Group #20253974 — subgroups

Subgroups — #20253974

Order 6016.

368 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 368 conjugacy classes of subgroups — too many to draw. Only its 149 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	#15689680	C₂	1	2	yes	maximal
2	#15689528	C₂	1	1	yes	maximal, characteristic, complemented
3	#15689503	C₂	1	1	yes	maximal, characteristic, complemented
4	#15689474	C₂	1	2	yes	maximal, complemented
5	#15689364	C₂	1	4	yes	maximal
6	C₄×Dic₁₈₈	C₂	1	4	yes	maximal, complemented
7	C₂×C₄×C₈×C₄₇	C₂	1	1	yes	maximal, characteristic
8	C₂×Dic₁₈₈	C₂²	1	4	yes	—
9	#13094024	C₂²	1	2	yes	—
10	#13094024	C₂²	1	2	yes	—
11	#13094016	—	2	2	no	—
12	#13094003	—	2	2	no	—
13	#13094003	C₂²	1	4	yes	—
14	C₄×Dic₉₄	—	2	2	no	—
15	C₄×Dic₉₄	C₂²	1	4	yes	—
16	#13093983	—	2	2	no	—
17	#13093958	C₂²	1	4	yes	—
18	#13093949	C₂²	1	8	yes	—
19	C₂²×C₈×C₄₇	C₂²	1	2	yes	—
20	C₂×C₄²×C₄₇	C₂²	1	1	yes	characteristic
21	C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
22	C₄×C₈×C₄₇	C₂²	1	2	yes	—
23	C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
24	C₂²×C₄×C₄₇	D₄	1	1	yes	characteristic
25	C₂²×C₄×C₄₇	D₄	1	1	yes	characteristic
26	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
27	C₂×Dic₉₄	—	2	2	no	—
28	C₂×Dic₉₄	C₂³	1	4	yes	—
29	C₂×C₈×C₄₇	—	2	4	no	—
30	C₂×C₈×C₄₇	C₂³	1	2	yes	—
31	C₂×C₈×C₄₇	C₂³	1	2	yes	—
32	C₄²×C₄₇	D₄	1	1	yes	characteristic
33	C₄²×C₄₇	D₄	1	1	yes	characteristic
34	C₄²×C₄₇	C₂³	1	1	yes	characteristic
35	C₄²×C₄₇	C₂³	1	1	yes	characteristic
36	C₂²⋊Dic₄₇	—	2	4	no	—
37	C₄⋊Dic₄₇	—	2	2	no	—
38	C₄⋊Dic₄₇	—	2	2	no	—
39	C₄⋊Dic₄₇	—	2	2	no	—
40	C₄⋊Dic₄₇	C₂³	1	4	yes	—
41	Dic₄₇⋊C₄	—	2	4	no	—
42	Dic₄₇⋊C₄	—	2	4	no	—
43	C₄×Dic₄₇	—	2	4	no	—
44	Dic₁₈₈	—	2	8	no	—
45	C₂³×C₄₇	C₂×D₄	1	1	yes	characteristic
46	C₂×C₄×C₄₇	—	2	2	no	—
47	C₂×C₄×C₄₇	—	2	2	no	—
48	C₂×C₄×C₄₇	Q₁₆	1	4	yes	complemented
49	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
50	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
51	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
52	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
53	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
54	C₂×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
55	C₂×Dic₄₇	—	2	4	no	—
56	C₂×Dic₄₇	—	2	4	no	—
57	Dic₉₄	—	2	8	no	—
58	Dic₉₄	—	4	4	no	—
59	C₈×C₄₇	—	2	2	no	—
60	C₈×C₄₇	#41	1	4	yes	—
61	C₂²×C₄₇	—	2	1	no	—
62	C₂²×C₄₇	—	2	1	no	—
63	C₂²×C₄₇	C₂×Q₁₆	1	2	yes	—
64	C₂²×C₄₇	C₂²×D₄	1	1	yes	characteristic
65	C₄×C₄₇	—	2	1	no	—
66	C₄×C₄₇	—	2	1	no	—
67	C₄×C₄₇	C₂×Q₁₆	1	2	yes	—
68	C₄×C₄₇	C₂×Q₁₆	1	2	yes	—
69	C₄×C₄₇	#135	1	2	yes	—
70	C₄×C₄₇	C₂×#41	1	2	yes	—
71	Dic₄₇	—	4	4	no	—
72	Dic₄₇	—	4	4	no	—
73	#3170	—	47	1	no	maximal, Sylow
74	C₂×C₄₇	—	2	2	no	—
75	C₂×C₄₇	#532	1	1	yes	characteristic
76	C₂×C₄₇	C₂²×Q₁₆	1	1	yes	characteristic
77	C₂×C₄₇	C₂×#135	1	1	yes	characteristic
78	#533	—	47	2	no	—
79	#500	—	47	1	no	—
80	#494	—	47	1	no	—
81	#487	—	47	2	no	—
82	#467	—	47	4	no	—
83	C₄×Q₁₆	—	47	4	no	—
84	C₂×C₄×C₈	D₄₇	1	1	yes	characteristic
85	C₄₇	#3170	1	1	yes	characteristic, complemented, Sylow
86	C₂×Q₁₆	—	47	4	no	—
87	C₂²×C₈	C₂×D₄₇	1	2	yes	—
88	C₄⋊Q₈	—	47	2	no	—
89	C₄⋊Q₈	—	47	2	no	—
90	#125	—	94	2	no	—
91	C₂²⋊Q₈	—	47	4	no	—
92	C₂²⋊Q₈	—	94	2	no	—
93	C₄×Q₈	—	47	4	no	—
94	C₄×Q₈	—	94	2	no	—
95	#117	—	94	2	no	—
96	C₂×C₄²	C₂×D₄₇	1	1	yes	characteristic
97	#107	—	47	4	no	—
98	#103	—	47	8	no	—
99	C₄×C₈	C₂×D₄₇	1	2	yes	—
100	C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
101	C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
102	C₂×Q₈	—	47	4	no	—
103	C₂×Q₈	—	94	2	no	—
104	C₂²×C₄	D₁₈₈	1	1	yes	characteristic
105	C₂²×C₄	D₁₈₈	1	1	yes	characteristic
106	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
107	Q₁₆	—	94	8	no	—
108	C₂×C₈	—	2	4	no	—
109	C₂×C₈	C₂²×D₄₇	1	2	yes	—
110	C₂×C₈	C₂²×D₄₇	1	2	yes	—
111	C₄⋊C₄	—	47	4	no	—
112	C₄⋊C₄	—	94	4	no	—
113	C₄⋊C₄	—	94	4	no	—
114	C₄⋊C₄	—	94	2	no	—
115	C₄⋊C₄	—	94	2	no	—
116	C₄⋊C₄	—	94	2	no	—
117	C₂²⋊C₄	—	94	4	no	—
118	C₄²	—	94	4	no	—
119	C₄²	D₁₈₈	1	1	yes	characteristic
120	C₄²	D₁₈₈	1	1	yes	characteristic
121	C₄²	C₂²×D₄₇	1	1	yes	characteristic
122	C₄²	C₂²×D₄₇	1	1	yes	characteristic
123	C₂³	C₂×D₁₈₈	1	1	yes	characteristic
124	Q₈	—	94	8	no	—
125	Q₈	—	188	4	no	—
126	C₂×C₄	—	2	2	no	—
127	C₂×C₄	—	2	2	no	—
128	C₂×C₄	—	94	4	no	—
129	C₂×C₄	—	94	4	no	—
130	C₂×C₄	Dic₁₈₈	1	4	yes	complemented
131	C₂×C₄	C₂×D₁₈₈	1	1	yes	characteristic
132	C₂×C₄	C₂×D₁₈₈	1	1	yes	characteristic
133	C₂×C₄	C₂×D₁₈₈	1	1	yes	characteristic
134	C₂×C₄	C₂×D₁₈₈	1	1	yes	characteristic
135	C₂×C₄	C₂×D₁₈₈	1	1	yes	characteristic
136	C₂×C₄	C₂³×D₄₇	1	1	yes	characteristic
137	C₈	—	2	2	no	—
138	C₈	#10625350	1	4	yes	—
139	C₂²	—	2	1	no	—
140	C₂²	—	2	1	no	—
141	C₂²	C₂×Dic₁₈₈	1	2	yes	—
142	C₂²	C₂²×D₁₈₈	1	1	yes	characteristic
143	C₄	—	2	1	no	—
144	C₄	—	2	1	no	—
145	C₄	—	188	4	no	—
146	C₄	—	188	4	no	—
147	C₄	C₂×Dic₁₈₈	1	2	yes	—
148	C₄	C₂×Dic₁₈₈	1	2	yes	—
149	C₄	#13094046	1	2	yes	—
150	C₄	C₂×#10625350	1	2	yes	—
151	C₂	—	2	2	no	—
152	C₂	#15689673	1	1	yes	characteristic
153	C₂	C₂²×Dic₁₈₈	1	1	yes	characteristic
154	C₂	C₂×#13094046	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#15689680

C₂

yes

maximal

#15689528

C₂

yes

maximal, characteristic, complemented

#15689503

C₂

yes

maximal, characteristic, complemented

#15689474

C₂

yes

maximal, complemented

#15689364

C₂

yes

maximal

C₄×Dic₁₈₈

C₂

yes

maximal, complemented

C₂×C₄×C₈×C₄₇

C₂

yes

maximal, characteristic

C₂×Dic₁₈₈

C₂²

yes

—

#13094024

C₂²

yes

—

#13094024

C₂²

yes

—

#13094016

—

#13094003

—

#13094003

C₂²

yes

—

C₄×Dic₉₄

—

C₄×Dic₉₄

C₂²

yes

—

#13093983

—

#13093958

C₂²

yes

—

#13093949

C₂²

yes

—

C₂²×C₈×C₄₇

C₂²

yes

—

C₂×C₄²×C₄₇

C₂²

yes

characteristic

C₄×C₈×C₄₇

C₂²

yes

characteristic

C₄×C₈×C₄₇

C₂²

yes

—

C₄×C₈×C₄₇

C₂²

yes

characteristic

C₂²×C₄×C₄₇

D₄

yes

characteristic

C₂²×C₄×C₄₇

D₄

yes

characteristic

C₂²×C₄×C₄₇

C₂³

yes

characteristic

C₂×Dic₉₄

—

C₂×Dic₉₄

C₂³

yes

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

C₂³

yes

—

C₂×C₈×C₄₇

C₂³

yes

—

C₄²×C₄₇

D₄

yes

characteristic

C₄²×C₄₇

D₄

yes

characteristic

C₄²×C₄₇

C₂³

yes

characteristic

C₄²×C₄₇

C₂³

yes

characteristic

C₂²⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

C₂³

yes

—

Dic₄₇⋊C₄

—

Dic₄₇⋊C₄

—

C₄×Dic₄₇

—

Dic₁₈₈

—

C₂³×C₄₇

C₂×D₄

yes

characteristic

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

Q₁₆

yes

complemented

C₂×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂×C₄×C₄₇

C₂⁴

yes

characteristic

C₂×Dic₄₇

—

C₂×Dic₄₇

—

Dic₉₄

—

Dic₉₄

—

C₈×C₄₇

—

C₈×C₄₇

#41

yes

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

C₂×Q₁₆

yes

—

C₂²×C₄₇

C₂²×D₄

yes

characteristic

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

C₂×Q₁₆

yes

—

C₄×C₄₇

C₂×Q₁₆

yes

—

C₄×C₄₇

#135

yes

—

C₄×C₄₇

C₂×#41

yes

—

Dic₄₇

—

Dic₄₇

—

#3170

—

maximal, Sylow

C₂×C₄₇

—

C₂×C₄₇

#532

yes

characteristic

C₂×C₄₇

C₂²×Q₁₆

yes

C₂×C₄₇

yes

—

—

—

—

—

C₄×Q₁₆

—

C₂×C₄×C₈

D₄₇

yes

characteristic

C₄₇

#3170

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

C₂²×C₈

C₂×D₄₇

yes

—

C₄⋊Q₈

—

C₄⋊Q₈

—

#125

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

C₄×Q₈

—

C₄×Q₈

—

#117

—

C₂×C₄²

C₂×D₄₇

yes

characteristic

#107

—

#103

—

C₄×C₈

C₂×D₄₇

yes

—

100

C₄×C₈

C₂×D₄₇

yes

characteristic

101

C₄×C₈

C₂×D₄₇

yes

characteristic

102

C₂×Q₈

—

103

C₂×Q₈

—

104

C₂²×C₄

D₁₈₈

yes

characteristic

105

C₂²×C₄

D₁₈₈

yes

characteristic

106

C₂²×C₄

C₂²×D₄₇

yes

characteristic

107

Q₁₆

—

108

C₂×C₈

—

109

C₂×C₈

C₂²×D₄₇

yes

—

110

C₂×C₈

C₂²×D₄₇

yes

—

111

C₄⋊C₄

—

112

C₄⋊C₄

—

113

C₄⋊C₄

—

114

C₄⋊C₄

—

115

C₄⋊C₄

—

116

C₄⋊C₄

—

117

C₂²⋊C₄

—

118

C₄²

—

119

C₄²

D₁₈₈

yes

characteristic

120

C₄²

D₁₈₈

yes

characteristic

121

C₄²

C₂²×D₄₇

yes

characteristic

122

C₄²

C₂²×D₄₇

yes

characteristic

123

C₂³

C₂×D₁₈₈

yes

characteristic

124

Q₈

—

125

Q₈

—

188

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

Dic₁₈₈

yes

complemented

131

C₂×C₄

C₂×D₁₈₈

yes

characteristic

132

C₂×C₄

C₂×D₁₈₈

yes

characteristic

133

C₂×C₄

C₂×D₁₈₈

yes

characteristic

134

C₂×C₄

C₂×D₁₈₈

yes

characteristic

135

C₂×C₄

C₂×D₁₈₈

yes

characteristic

136

C₂×C₄

C₂³×D₄₇

yes

characteristic

137

C₈

—

138

C₈

#10625350

yes

—

139

C₂²

—

140

C₂²

—

141

C₂²

C₂×Dic₁₈₈

yes

—

142

C₂²

C₂²×D₁₈₈

yes

characteristic

143

C₄

—

144

C₄

—

145

C₄

—

188

—

146

C₄

—

188

—

147

C₄

C₂×Dic₁₈₈

yes

—

148

C₄

C₂×Dic₁₈₈

yes

—

149

C₄

#13094046

yes

—

150

C₄

C₂×#10625350

yes

—

151

C₂

—

152

C₂

#15689673

yes

characteristic

153

C₂

C₂²×Dic₁₈₈

yes

characteristic

154

C₂

C₂×#13094046

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₁₈₈
commutator	C₂×C₄×C₄₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₄₇
Fitting	C₂×C₄×C₈×C₄₇	C₂
socle	C₂²×C₄₇	C₂²×D₄
hypercenter	C₂×C₄×C₈	D₄₇
hypocenter	C₄₇	#3170
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈

subgroup

G/H

center

C₂×C₄

C₂×D₁₈₈

commutator

C₂×C₄×C₄₇

C₂⁴

Frattini

C₂×C₄

C₂³×D₄₇

Fitting

C₂×C₄×C₈×C₄₇

C₂

socle

C₂²×C₄₇

C₂²×D₄

hypercenter

C₂×C₄×C₈

D₄₇

hypocenter

C₄₇

#3170

LCS

G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈

N	H ≅ G/N	type
2. #15689528	C₂	semidirect
3. #15689503	C₂	semidirect
4. #15689474	C₂	semidirect
6. C₄×Dic₁₈₈	C₂	semidirect
48. C₂×C₄×C₄₇	Q₁₆	semidirect
85. C₄₇	#3170	semidirect
130. C₂×C₄	Dic₁₈₈	semidirect

H ≅ G/N

type

2. #15689528

C₂

semidirect

3. #15689503

C₂

semidirect

4. #15689474

C₂

semidirect

6. C₄×Dic₁₈₈

C₂

semidirect

48. C₂×C₄×C₄₇

Q₁₆

semidirect

85. C₄₇

#3170

semidirect

130. C₂×C₄

Dic₁₈₈

semidirect

Subgroups — #20253974

Conjugacy classes of subgroups

Complements