Group #20243602 — subgroups

Subgroups — #20243602

Order 6016.

272 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 272 conjugacy classes of subgroups — too many to draw. Only its 89 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	#15689601	C₂	1	1	yes	maximal, characteristic
2	#15689163	C₂	1	2	yes	maximal
3	C₂×#13093943	C₂	1	1	yes	maximal, characteristic
4	#15688883	C₂	1	2	yes	maximal
5	C₄₇×#391	C₂	1	1	yes	maximal, characteristic
6	C₂×#10625329	C₂²	1	2	yes	—
7	Q₈×Dic₄₇	—	2	2	no	—
8	Q₈×Dic₄₇	—	2	2	no	—
9	#13093983	—	2	2	no	—
10	#13093983	C₂²	1	2	yes	—
11	#13093957	—	2	2	no	—
12	#13093955	—	2	2	no	—
13	#13093943	C₄	1	4	yes	complemented
14	C₂²×Q₈×C₄₇	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊C₄×C₄₇	C₂²	1	1	yes	characteristic
16	C₂×C₄²×C₄₇	C₂²	1	1	yes	characteristic
17	#95×C₄₇	—	2	2	no	—
18	C₂×Q₈×C₄₇	—	2	2	no	—
19	C₂×Q₈×C₄₇	—	2	2	no	—
20	C₂×Q₈×C₄₇	C₂×C₄	1	2	yes	—
21	C₂×Q₈×C₄₇	Q₈	1	2	yes	—
22	C₂²×C₄×C₄₇	—	2	1	no	—
23	C₂²×C₄×C₄₇	—	2	1	no	—
24	C₂²×C₄×C₄₇	D₄	1	2	yes	—
25	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
26	C₄⋊C₄×C₄₇	—	2	2	no	—
27	C₄²×C₄₇	—	2	2	no	—
28	C₂²⋊Dic₄₇	—	2	2	no	—
29	C₄⋊Dic₄₇	—	2	2	no	—
30	C₄⋊Dic₄₇	—	2	2	no	—
31	C₄⋊Dic₄₇	D₄	1	2	yes	—
32	C₄⋊Dic₄₇	D₄	1	2	yes	—
33	C₄×Dic₄₇	—	2	4	no	—
34	C₄×Dic₄₇	—	2	2	no	—
35	#10625329	—	2	2	no	—
36	#10625329	C₂×C₄	1	4	yes	—
37	C₂×C₄₇⋊C₈	—	2	2	no	—
38	C₂³×C₄₇	C₂×D₄	1	1	yes	characteristic
39	Q₈×C₄₇	—	2	4	no	—
40	Q₈×C₄₇	—	2	4	no	—
41	C₂×C₄×C₄₇	—	2	2	no	—
42	C₂×C₄×C₄₇	—	2	2	no	—
43	C₂×C₄×C₄₇	—	2	2	no	—
44	C₂×C₄×C₄₇	—	4	1	no	—
45	C₂×C₄×C₄₇	—	4	1	no	—
46	C₂×C₄×C₄₇	—	4	2	no	—
47	C₂×C₄×C₄₇	—	4	1	no	—
48	C₂×C₄×C₄₇	C₄⋊C₄	1	4	yes	—
49	C₂×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
50	C₂×C₄×C₄₇	C₂×D₄	1	2	yes	—
51	C₂×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
52	C₂×C₄×C₄₇	#41	1	2	yes	—
53	C₂×Dic₄₇	—	2	2	no	—
54	C₂×Dic₄₇	—	2	2	no	—
55	C₄₇⋊C₈	—	2	4	no	—
56	C₂²×C₄₇	—	2	1	no	—
57	C₂²×C₄₇	—	2	1	no	—
58	C₂²×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
59	C₂²×C₄₇	#120	1	1	yes	characteristic
60	C₂²×C₄₇	#123	1	1	yes	characteristic
61	C₄×C₄₇	—	2	4	no	—
62	C₄×C₄₇	—	4	2	no	—
63	C₄×C₄₇	—	8	1	no	—
64	C₄×C₄₇	C₄×D₄	1	2	yes	—
65	C₄×C₄₇	C₂²⋊Q₈	1	2	yes	—
66	Dic₄₇	—	4	2	no	—
67	Dic₄₇	—	4	2	no	—
68	#1857	—	47	1	no	maximal, Sylow
69	C₂×C₄₇	—	2	1	no	—
70	C₂×C₄₇	—	2	1	no	—
71	C₂×C₄₇	#385	1	1	yes	characteristic
72	C₂×C₄₇	#454	1	1	yes	characteristic
73	C₂×C₄₇	#456	1	1	yes	characteristic
74	#519	—	47	1	no	—
75	#428	—	47	2	no	—
76	C₂×#101	—	47	1	no	—
77	#391	D₄₇	1	1	yes	characteristic
78	#339	—	47	2	no	—
79	C₄₇	#1857	1	1	yes	characteristic, complemented, Sylow
80	C₂²×Q₈	C₂×D₄₇	1	1	yes	characteristic
81	C₂×M₁₆	—	47	2	no	—
82	C₄×Q₈	—	94	2	no	—
83	C₄×Q₈	—	94	2	no	—
84	#117	—	47	2	no	—
85	#117	—	94	2	no	—
86	C₂×C₄⋊C₄	C₂×D₄₇	1	1	yes	characteristic
87	C₂×C₄²	C₂×D₄₇	1	1	yes	characteristic
88	#107	—	94	2	no	—
89	#106	—	94	2	no	—
90	#101	—	47	4	no	—
91	#95	—	2	2	no	—
92	C₂×Q₈	—	2	2	no	—
93	C₂×Q₈	—	2	2	no	—
94	C₂×Q₈	Dic₉₄	1	2	yes	—
95	C₂×Q₈	C₄×D₄₇	1	2	yes	—
96	C₂²×C₄	—	2	1	no	—
97	C₂²×C₄	—	2	1	no	—
98	C₂²×C₄	C₂²⋊D₄₇	1	2	yes	—
99	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
100	M₁₆	—	47	4	no	—
101	M₁₆	—	94	2	no	—
102	C₂×C₈	—	94	2	no	—
103	C₄⋊C₄	—	2	2	no	—
104	C₄⋊C₄	—	47	2	no	—
105	C₄⋊C₄	—	47	2	no	—
106	C₄⋊C₄	—	94	2	no	—
107	C₄⋊C₄	—	94	2	no	—
108	C₂²⋊C₄	—	94	2	no	—
109	C₄²	—	2	2	no	—
110	C₄²	—	94	4	no	—
111	C₄²	—	94	2	no	—
112	C₂³	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
113	Q₈	—	2	4	no	—
114	Q₈	—	2	4	no	—
115	C₂×C₄	—	2	2	no	—
116	C₂×C₄	—	2	2	no	—
117	C₂×C₄	—	2	2	no	—
118	C₂×C₄	—	4	1	no	—
119	C₂×C₄	—	4	1	no	—
120	C₂×C₄	—	4	2	no	—
121	C₂×C₄	—	4	1	no	—
122	C₂×C₄	—	94	2	no	—
123	C₂×C₄	—	94	2	no	—
124	C₂×C₄	Dic₄₇⋊C₄	1	4	yes	—
125	C₂×C₄	C₂×Dic₉₄	1	1	yes	characteristic
126	C₂×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
127	C₂×C₄	D₄×D₄₇	1	2	yes	—
128	C₂×C₄	#10625352	1	2	yes	—
129	C₈	—	94	4	no	—
130	C₂²	—	2	1	no	—
131	C₂²	—	2	1	no	—
132	C₂²	C₂×Dic₄₇⋊C₄	1	1	yes	characteristic
133	C₂²	#13093992	1	1	yes	characteristic
134	C₂²	#13094005	1	1	yes	characteristic
135	C₄	—	2	4	no	—
136	C₄	—	4	2	no	—
137	C₄	—	8	1	no	—
138	C₄	—	188	2	no	—
139	C₄	—	188	2	no	—
140	C₄	(C₂²⋊D₄₇)⋊C₄	1	2	yes	—
141	C₄	#13094001	1	2	yes	—
142	C₂	—	2	1	no	—
143	C₂	—	2	1	no	—
144	C₂	#15688995	1	1	yes	characteristic
145	C₂	#15689295	1	1	yes	characteristic
146	C₂	#15689305	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#15689601

C₂

yes

maximal, characteristic

#15689163

C₂

yes

maximal

C₂×#13093943

C₂

yes

maximal, characteristic

#15688883

C₂

yes

maximal

C₄₇×#391

C₂

yes

maximal, characteristic

C₂×#10625329

C₂²

yes

—

Q₈×Dic₄₇

—

Q₈×Dic₄₇

—

#13093983

—

#13093983

C₂²

yes

—

#13093957

—

#13093955

—

#13093943

C₄

yes

complemented

C₂²×Q₈×C₄₇

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₄₇

C₂²

yes

characteristic

C₂×C₄²×C₄₇

C₂²

yes

characteristic

#95×C₄₇

—

C₂×Q₈×C₄₇

—

C₂×Q₈×C₄₇

—

C₂×Q₈×C₄₇

C₂×C₄

yes

—

C₂×Q₈×C₄₇

Q₈

yes

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

D₄

yes

—

C₂²×C₄×C₄₇

C₂³

yes

characteristic

C₄⋊C₄×C₄₇

—

C₄²×C₄₇

—

C₂²⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

D₄

yes

—

C₄⋊Dic₄₇

D₄

yes

—

C₄×Dic₄₇

—

C₄×Dic₄₇

—

#10625329

—

#10625329

C₂×C₄

yes

—

C₂×C₄₇⋊C₈

—

C₂³×C₄₇

C₂×D₄

yes

characteristic

Q₈×C₄₇

—

Q₈×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₄⋊C₄

yes

—

C₂×C₄×C₄₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₄₇

C₂×D₄

yes

—

C₂×C₄×C₄₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₄₇

#41

yes

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₄₇⋊C₈

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

C₂×C₄⋊C₄

yes

C₂²×C₄₇

yes

C₂²×C₄₇

yes

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

C₄×D₄

yes

—

C₄×C₄₇

C₂²⋊Q₈

yes

—

Dic₄₇

—

Dic₄₇

—

#1857

—

maximal, Sylow

C₂×C₄₇

—

C₂×C₄₇

—

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

—

—

—

D₄₇

yes

characteristic

#339

—

C₄₇

#1857

yes

characteristic, complemented, Sylow

C₂²×Q₈

C₂×D₄₇

yes

characteristic

C₂×M₁₆

—

C₄×Q₈

—

C₄×Q₈

—

#117

—

#117

—

C₂×C₄⋊C₄

C₂×D₄₇

yes

characteristic

C₂×C₄²

C₂×D₄₇

yes

—

—

—

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

Dic₉₄

yes

—

C₂×Q₈

C₄×D₄₇

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²⋊D₄₇

yes

—

C₂²×C₄

C₂²×D₄₇

yes

characteristic

100

M₁₆

—

101

M₁₆

—

102

C₂×C₈

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

106

C₄⋊C₄

—

107

C₄⋊C₄

—

108

C₂²⋊C₄

—

109

C₄²

—

110

C₄²

—

111

C₄²

—

112

C₂³

C₂×C₂²⋊D₄₇

yes

characteristic

113

Q₈

—

114

Q₈

—

115

C₂×C₄

—

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

Dic₄₇⋊C₄

yes

—

125

C₂×C₄

C₂×Dic₉₄

yes

characteristic

126

C₂×C₄

C₂×C₄×D₄₇

yes

characteristic

127

C₂×C₄

D₄×D₄₇

yes

—

128

C₂×C₄

#10625352

yes

—

129

C₈

—

130

C₂²

—

131

C₂²

—

132

C₂²

C₂×Dic₄₇⋊C₄

yes

characteristic

133

C₂²

#13093992

yes

characteristic

134

C₂²

#13094005

yes

characteristic

135

C₄

—

136

C₄

—

137

C₄

—

138

C₄

—

188

—

139

C₄

—

188

—

140

C₄

(C₂²⋊D₄₇)⋊C₄

yes

—

141

C₄

#13094001

yes

—

142

C₂

—

143

C₂

—

144

C₂

#15688995

yes

characteristic

145

C₂

#15689295

yes

characteristic

146

C₂

#15689305

yes

characteristic

	subgroup	G/H
center	C₂²	#13093992
commutator	C₂×C₄×C₄₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₄₇
Fitting	C₄₇×#391	C₂
socle	C₂²×C₄₇	#120
hypercenter	#391	D₄₇
hypocenter	C₄₇	#1857
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #391

subgroup

G/H

center

C₂²

#13093992

commutator

C₂×C₄×C₄₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₄₇

Fitting

C₄₇×#391

C₂

C₂²×C₄₇

D₄₇

C₄₇

G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #391

N	H ≅ G/N	type
13. #13093943	C₄	semidirect
79. C₄₇	#1857	semidirect

H ≅ G/N

type

13. #13093943

C₄

semidirect

79. C₄₇

#1857

semidirect

Subgroups — #20243602

Conjugacy classes of subgroups

Complements