Group #20242884 — subgroups

Subgroups — #20242884

Order 6016.

160 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 160 conjugacy classes of subgroups — too many to draw. Only its 55 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	#15689410	C₂	1	1	yes	maximal, characteristic
2	#15689325	C₂	1	1	yes	maximal, characteristic
3	#15689202	C₂	1	1	yes	maximal, characteristic
4	#15688876	C₂	1	1	yes	maximal, characteristic, complemented
5	#15688872	C₂	1	1	yes	maximal, characteristic, complemented
6	#15688869	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄₇×#498	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#10625336	—	2	1	no	—
9	C₂×#10625324	—	2	1	no	—
10	C₄×D₁₈₈	C₂²	1	1	yes	characteristic
11	#13093955	—	2	1	no	—
12	C₄⋊(C₄₇⋊C₈)	C₂²	1	1	yes	characteristic
13	C₄⋊(C₄₇⋊C₈)	C₂²	1	1	yes	characteristic
14	C₄⋊(C₄₇⋊C₈)	C₂²	1	1	yes	characteristic
15	#13093948	—	2	1	no	—
16	#13093944	—	2	1	no	—
17	(C₂×D₄₇)⋊C₈	—	2	1	no	—
18	C₄⋊Q₈×C₄₇	C₂²	1	1	yes	characteristic
19	C₄⋊Q₈×C₄₇	C₂²	1	1	yes	characteristic
20	#106×C₄₇	—	2	1	no	—
21	#106×C₄₇	—	2	1	no	—
22	C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
23	C₂×Q₈×C₄₇	—	2	1	no	—
24	C₂×Q₈×C₄₇	—	2	1	no	—
25	C₂×D₁₈₈	D₄	1	1	yes	characteristic
26	C₂×C₄×D₄₇	—	2	1	no	—
27	C₂×C₈×C₄₇	D₄	1	1	yes	characteristic
28	C₂×C₈×C₄₇	D₄	1	1	yes	characteristic
29	C₄⋊C₄×C₄₇	—	2	1	no	—
30	C₄⋊C₄×C₄₇	—	2	1	no	—
31	C₄⋊C₄×C₄₇	—	2	1	no	—
32	C₄⋊C₄×C₄₇	—	2	1	no	—
33	C₄²×C₄₇	C₂³	1	1	yes	characteristic
34	#10625336	—	4	2	no	—
35	(C₂×D₄₇)⋊C₄	—	2	1	no	—
36	C₄⋊Dic₄₇	D₄	1	1	yes	characteristic
37	C₂×C₄₇⋊C₈	—	2	1	no	—
38	C₂×C₄₇⋊C₈	—	2	1	no	—
39	C₂×C₄₇⋊C₈	—	2	1	no	—
40	#10625324	—	4	2	no	—
41	C₂²×D₄₇	—	2	1	no	—
42	Q₈×C₄₇	—	4	1	no	—
43	Q₈×C₄₇	—	4	1	no	—
44	Q₈×C₄₇	—	8	1	no	—
45	C₂×C₄×C₄₇	—	4	1	no	—
46	C₂×C₄×C₄₇	—	4	1	no	—
47	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
48	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₄₇	#41	1	1	yes	characteristic
50	C₂×Dic₄₇	—	2	1	no	—
51	D₁₈₈	—	2	1	no	—
52	D₁₈₈	—	2	1	no	—
53	C₄×D₄₇	—	4	1	no	—
54	C₄×D₄₇	—	4	1	no	—
55	C₈×C₄₇	—	2	1	no	—
56	C₈×C₄₇	SD₁₆	1	2	yes	complemented
57	C₄₇⋊C₈	—	4	1	no	—
58	C₄₇⋊C₈	—	4	1	no	—
59	C₄₇⋊C₈	—	4	1	no	—
60	C₂²×C₄₇	#121	1	1	yes	characteristic
61	C₂×D₄₇	—	4	1	no	—
62	C₂×D₄₇	—	4	1	no	—
63	C₂×D₄₇	—	4	1	no	—
64	C₄×C₄₇	—	2	1	no	—
65	C₄×C₄₇	—	8	1	no	—
66	C₄×C₄₇	—	8	1	no	—
67	C₄×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
68	C₄×C₄₇	AGL(1,ℤ/8)	1	1	yes	characteristic
69	C₄×C₄₇	#137	1	1	yes	characteristic
70	C₄×C₄₇	#137	1	1	yes	characteristic
71	Dic₄₇	—	4	1	no	—
72	#1691	—	47	1	no	maximal, Sylow
73	C₂×C₄₇	#461	1	1	yes	characteristic
74	C₂×C₄₇	#468	1	1	yes	characteristic
75	C₂×C₄₇	#471	1	1	yes	characteristic
76	D₄₇	—	8	1	no	—
77	#498	D₄₇	1	1	yes	characteristic, complemented
78	#476	—	47	1	no	—
79	#461	—	47	1	no	—
80	#436	—	47	1	no	—
81	#335	—	47	1	no	—
82	#333	—	47	1	no	—
83	#332	—	47	1	no	—
84	C₄₇	#1691	1	1	yes	characteristic, complemented, Sylow
85	C₂×SD₁₆	—	94	1	no	—
86	C₂×M₁₆	—	94	1	no	—
87	C₄⋊Q₈	C₂×D₄₇	1	1	yes	characteristic
88	C₄⋊Q₈	C₂×D₄₇	1	1	yes	characteristic
89	C₄×D₄	—	47	1	no	—
90	#106	—	2	1	no	—
91	#106	—	2	1	no	—
92	#106	—	94	1	no	—
93	C₄⋊C₈	—	47	1	no	—
94	C₄⋊C₈	—	47	1	no	—
95	C₄⋊C₈	—	47	1	no	—
96	#103	—	94	1	no	—
97	#102	—	94	1	no	—
98	C₂²⋊C₈	—	94	1	no	—
99	C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
100	C₂×Q₈	—	2	1	no	—
101	C₂×Q₈	—	2	1	no	—
102	C₂×D₄	—	47	1	no	—
103	C₂²×C₄	—	94	1	no	—
104	SD₁₆	—	188	2	no	—
105	M₁₆	—	188	2	no	—
106	C₂×C₈	—	94	1	no	—
107	C₂×C₈	—	94	1	no	—
108	C₂×C₈	—	94	1	no	—
109	C₂×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
110	C₂×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
111	C₄⋊C₄	—	2	1	no	—
112	C₄⋊C₄	—	2	1	no	—
113	C₄⋊C₄	—	2	1	no	—
114	C₄⋊C₄	—	2	1	no	—
115	C₄⋊C₄	—	47	1	no	—
116	C₂²⋊C₄	—	94	1	no	—
117	C₄²	C₂²×D₄₇	1	1	yes	characteristic
118	C₂³	—	94	1	no	—
119	Q₈	—	4	1	no	—
120	Q₈	—	4	1	no	—
121	Q₈	—	8	1	no	—
122	D₄	—	94	1	no	—
123	D₄	—	94	1	no	—
124	C₂×C₄	—	4	1	no	—
125	C₂×C₄	—	4	1	no	—
126	C₂×C₄	—	94	1	no	—
127	C₂×C₄	—	188	1	no	—
128	C₂×C₄	—	188	1	no	—
129	C₂×C₄	D₄×D₄₇	1	1	yes	characteristic
130	C₂×C₄	#10625352	1	1	yes	characteristic
131	C₂×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
132	C₈	—	2	1	no	—
133	C₈	—	188	1	no	—
134	C₈	—	188	1	no	—
135	C₈	—	188	1	no	—
136	C₈	#10625336	1	2	yes	complemented
137	C₂²	—	188	1	no	—
138	C₂²	—	188	1	no	—
139	C₂²	—	188	1	no	—
140	C₂²	#13093995	1	1	yes	characteristic
141	C₄	—	2	1	no	—
142	C₄	—	8	1	no	—
143	C₄	—	8	1	no	—
144	C₄	—	188	1	no	—
145	C₄	C₂×#10625336	1	1	yes	characteristic
146	C₄	#13094051	1	1	yes	characteristic
147	C₄	#13094053	1	1	yes	characteristic
148	C₄	#13094055	1	1	yes	characteristic
149	C₂	—	376	1	no	—
150	C₂	#15689325	1	1	yes	characteristic
151	C₂	#15689366	1	1	yes	characteristic
152	C₂	#15689383	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#15689410

C₂

yes

maximal, characteristic

#15689325

C₂

yes

maximal, characteristic

#15689202

C₂

yes

maximal, characteristic

#15688876

C₂

yes

maximal, characteristic, complemented

#15688872

C₂

yes

maximal, characteristic, complemented

#15688869

C₂

yes

maximal, characteristic, complemented

C₄₇×#498

C₂

yes

maximal, characteristic, complemented

C₂×#10625336

—

C₂×#10625324

—

C₄×D₁₈₈

C₂²

yes

characteristic

#13093955

—

C₄⋊(C₄₇⋊C₈)

C₂²

yes

characteristic

C₄⋊(C₄₇⋊C₈)

C₂²

yes

characteristic

C₄⋊(C₄₇⋊C₈)

C₂²

yes

characteristic

#13093948

—

#13093944

—

(C₂×D₄₇)⋊C₈

—

C₄⋊Q₈×C₄₇

C₂²

yes

characteristic

C₄⋊Q₈×C₄₇

C₂²

yes

characteristic

#106×C₄₇

—

#106×C₄₇

—

C₄×C₈×C₄₇

C₂²

yes

characteristic

C₂×Q₈×C₄₇

—

C₂×Q₈×C₄₇

—

C₂×D₁₈₈

D₄

yes

characteristic

C₂×C₄×D₄₇

—

C₂×C₈×C₄₇

D₄

yes

characteristic

C₂×C₈×C₄₇

D₄

yes

characteristic

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄²×C₄₇

C₂³

yes

characteristic

#10625336

—

(C₂×D₄₇)⋊C₄

—

C₄⋊Dic₄₇

D₄

yes

characteristic

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

#10625324

—

C₂²×D₄₇

—

Q₈×C₄₇

—

Q₈×C₄₇

—

Q₈×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂×C₄×C₄₇

#41

yes

characteristic

C₂×Dic₄₇

—

D₁₈₈

—

D₁₈₈

—

C₄×D₄₇

—

C₄×D₄₇

—

C₈×C₄₇

—

C₈×C₄₇

SD₁₆

yes

complemented

C₄₇⋊C₈

—

C₄₇⋊C₈

—

C₄₇⋊C₈

—

C₂²×C₄₇

#121

yes

characteristic

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

C₂×SD₁₆

yes

C₄×C₄₇

yes

C₄×C₄₇

yes

C₄×C₄₇

yes

Dic₄₇

—

—

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

D₄₇

—

#498

D₄₇

yes

characteristic, complemented

—

—

—

—

—

—

C₄₇

#1691

yes

characteristic, complemented, Sylow

C₂×SD₁₆

—

C₂×M₁₆

—

C₄⋊Q₈

C₂×D₄₇

yes

characteristic

C₄⋊Q₈

C₂×D₄₇

yes

characteristic

C₄×D₄

—

#106

—

#106

—

#106

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

#103

—

#102

—

C₂²⋊C₈

—

C₄×C₈

C₂×D₄₇

yes

characteristic

100

C₂×Q₈

—

101

C₂×Q₈

—

102

C₂×D₄

—

103

C₂²×C₄

—

104

SD₁₆

—

188

—

105

M₁₆

—

188

—

106

C₂×C₈

—

107

C₂×C₈

—

108

C₂×C₈

—

109

C₂×C₈

C₂²⋊D₄₇

yes

characteristic

110

C₂×C₈

C₂²⋊D₄₇

yes

characteristic

111

C₄⋊C₄

—

112

C₄⋊C₄

—

113

C₄⋊C₄

—

114

C₄⋊C₄

—

115

C₄⋊C₄

—

116

C₂²⋊C₄

—

117

C₄²

C₂²×D₄₇

yes

characteristic

118

C₂³

—

119

Q₈

—

120

Q₈

—

121

Q₈

—

122

D₄

—

123

D₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

188

—

128

C₂×C₄

—

188

—

129

C₂×C₄

D₄×D₄₇

yes

characteristic

130

C₂×C₄

#10625352

yes

characteristic

131

C₂×C₄

C₂×C₂²⋊D₄₇

yes

characteristic

132

C₈

—

133

C₈

—

188

—

134

C₈

—

188

—

135

C₈

—

188

—

136

C₈

#10625336

yes

complemented

137

C₂²

—

188

—

138

C₂²

—

188

—

139

C₂²

—

188

—

140

C₂²

#13093995

yes

characteristic

141

C₄

—

142

C₄

—

143

C₄

—

144

C₄

—

188

—

145

C₄

C₂×#10625336

yes

characteristic

146

C₄

#13094051

yes

characteristic

147

C₄

#13094053

yes

characteristic

148

C₄

#13094055

yes

characteristic

149

C₂

—

376

—

150

C₂

#15689325

yes

characteristic

151

C₂

#15689366

yes

characteristic

152

C₂

#15689383

yes

characteristic

	subgroup	G/H
center	C₂²	#13093995
commutator	C₄²×C₄₇	C₂³
Frattini	C₄²	C₂²×D₄₇
Fitting	C₄₇×#498	C₂
socle	C₂²×C₄₇	#121
hypercenter	#498	D₄₇
hypocenter	C₄₇	#1691
LCS	G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #498

subgroup

G/H

center

C₂²

#13093995

commutator

C₄²×C₄₇

C₂³

Frattini

C₄²

C₂²×D₄₇

Fitting

C₄₇×#498

C₂

C₂²×C₄₇

D₄₇

C₄₇

G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #498

N	H ≅ G/N	type
4. #15688876	C₂	semidirect
5. #15688872	C₂	semidirect
6. #15688869	C₂	semidirect
7. C₄₇×#498	C₂	semidirect
56. C₈×C₄₇	SD₁₆	semidirect
77. #498	D₄₇	semidirect
84. C₄₇	#1691	semidirect
136. C₈	#10625336	semidirect

H ≅ G/N

type

4. #15688876

C₂

semidirect

5. #15688872

C₂

semidirect

6. #15688869

C₂

semidirect

7. C₄₇×#498

C₂

semidirect

56. C₈×C₄₇

SD₁₆

semidirect

77. #498

D₄₇

semidirect

84. C₄₇

#1691

semidirect

136. C₈

#10625336

semidirect

Subgroups — #20242884

Conjugacy classes of subgroups

Complements