Group #20242192 — subgroups

Subgroups — #20242192

Order 6016.

164 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 164 conjugacy classes of subgroups — too many to draw. Only its 73 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	Q₈×C₄₇⋊C₈	C₂	1	1	yes	maximal, characteristic
2	#15689247	C₂	1	1	yes	maximal, characteristic
3	C₄×#10625337	C₂	1	1	yes	maximal, characteristic
4	#15688875	C₂	1	1	yes	maximal, characteristic
5	#15688855	C₂	1	1	yes	maximal, characteristic
6	C₈×C₄₇⋊C₈	C₂	1	1	yes	maximal, characteristic
7	C₄₇×Q₈⋊C₈	C₂	1	1	yes	maximal, characteristic
8	C₂×#10625337	C₄	1	1	yes	characteristic
9	C₄×Dic₉₄	C₂²	1	1	yes	characteristic
10	#13093957	C₄	1	1	yes	characteristic
11	C₄⋊(C₄₇⋊C₈)	—	2	1	no	—
12	C₄⋊(C₄₇⋊C₈)	C₂²	1	1	yes	characteristic
13	Dic₄₇⋊C₈	—	2	1	no	—
14	#13093948	C₄	1	1	yes	characteristic
15	#13093947	C₄	1	1	yes	characteristic
16	C₄×C₄₇⋊C₈	—	2	1	no	—
17	C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
18	C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
19	C₈×Dic₄₇	—	2	1	no	—
20	C₄×Q₈×C₄₇	C₂²	1	1	yes	characteristic
21	C₄⋊C₈×C₄₇	C₂²	1	1	yes	characteristic
22	C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
23	C₂×Q₈×C₄₇	C₂×C₄	1	1	yes	characteristic
24	C₂×Dic₉₄	C₂×C₄	1	1	yes	characteristic
25	C₂×C₈×C₄₇	—	2	1	no	—
26	C₂×C₈×C₄₇	—	2	1	no	—
27	C₄⋊C₄×C₄₇	—	2	1	no	—
28	C₄⋊C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
29	C₄²×C₄₇	—	2	1	no	—
30	C₄²×C₄₇	C₂³	1	1	yes	characteristic
31	#10625337	C₈	1	4	yes	complemented
32	C₄⋊Dic₄₇	C₂×C₄	1	1	yes	characteristic
33	Dic₄₇⋊C₄	—	2	1	no	—
34	C₄×Dic₄₇	—	2	1	no	—
35	C₂×C₄₇⋊C₈	—	2	1	no	—
36	C₂×C₄₇⋊C₈	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄₇⋊C₈	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄₇⋊C₈	D₄	1	1	yes	characteristic
39	C₂×C₄₇⋊C₈	D₄	1	1	yes	characteristic
40	Q₈×C₄₇	—	2	1	no	—
41	Q₈×C₄₇	C₂×C₈	1	2	yes	—
42	C₂×C₄×C₄₇	—	2	1	no	—
43	C₂×C₄×C₄₇	—	2	1	no	—
44	C₂×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
46	C₂×C₄×C₄₇	#41	1	1	yes	characteristic
47	C₂×Dic₄₇	—	2	1	no	—
48	C₂×Dic₄₇	—	2	1	no	—
49	Dic₉₄	—	2	1	no	—
50	Dic₉₄	C₂×C₈	1	2	yes	—
51	C₈×C₄₇	—	2	2	no	—
52	C₈×C₄₇	—	4	1	no	—
53	C₄₇⋊C₈	—	2	1	no	—
54	C₄₇⋊C₈	—	2	1	no	—
55	C₄₇⋊C₈	—	4	1	no	—
56	C₄₇⋊C₈	C₂×C₈	1	2	yes	—
57	C₄₇⋊C₈	Q₁₆	1	2	yes	complemented
58	C₂²×C₄₇	C₄×D₄	1	1	yes	characteristic
59	C₄×C₄₇	—	2	2	no	—
60	C₄×C₄₇	—	2	1	no	—
61	C₄×C₄₇	—	4	1	no	—
62	C₄×C₄₇	C₂²×C₈	1	1	yes	characteristic
63	C₄×C₄₇	#131	1	1	yes	characteristic
64	C₄×C₄₇	C₂×Q₁₆	1	1	yes	characteristic
65	C₄×C₄₇	#135	1	1	yes	characteristic
66	Dic₄₇	—	2	2	no	—
67	Dic₄₇	—	4	1	no	—
68	C₈×Q₁₆	—	47	1	no	maximal, Sylow
69	C₂×C₄₇	C₈×D₄	1	1	yes	characteristic
70	C₂×C₄₇	C₄×Q₁₆	1	1	yes	characteristic
71	C₂×C₄₇	#443	1	1	yes	characteristic
72	C₈×Q₈	—	47	1	no	—
73	C₈×Q₈	—	47	1	no	—
74	C₄×Q₁₆	—	47	1	no	—
75	#335	—	47	1	no	—
76	Q₈⋊C₈	—	47	1	no	—
77	Q₈⋊C₈	D₄₇	1	1	yes	characteristic
78	C₈²	—	47	1	no	—
79	C₄₇	C₈×Q₁₆	1	1	yes	characteristic, complemented, Sylow
80	C₂×Q₁₆	—	47	1	no	—
81	C₄×Q₈	—	47	1	no	—
82	C₄×Q₈	C₂×D₄₇	1	1	yes	characteristic
83	#107	—	47	1	no	—
84	C₄⋊C₈	—	47	1	no	—
85	C₄⋊C₈	—	94	1	no	—
86	C₄⋊C₈	—	94	1	no	—
87	C₄⋊C₈	C₂×D₄₇	1	1	yes	characteristic
88	#103	—	47	1	no	—
89	#103	—	47	1	no	—
90	C₄×C₈	—	47	1	no	—
91	C₄×C₈	—	47	1	no	—
92	C₄×C₈	—	94	1	no	—
93	C₄×C₈	—	94	1	no	—
94	C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
95	C₂×Q₈	—	47	1	no	—
96	C₂×Q₈	C₄×D₄₇	1	1	yes	characteristic
97	Q₁₆	—	47	4	no	—
98	C₂×C₈	—	2	1	no	—
99	C₂×C₈	—	2	1	no	—
100	C₂×C₈	—	47	1	no	—
101	C₂×C₈	—	47	1	no	—
102	C₂×C₈	—	47	1	no	—
103	C₂×C₈	—	47	1	no	—
104	C₂×C₈	—	94	1	no	—
105	C₄⋊C₄	—	2	1	no	—
106	C₄⋊C₄	—	47	1	no	—
107	C₄⋊C₄	—	94	1	no	—
108	C₄⋊C₄	C₄×D₄₇	1	1	yes	characteristic
109	C₄²	—	2	1	no	—
110	C₄²	—	94	1	no	—
111	C₄²	C₂²×D₄₇	1	1	yes	characteristic
112	Q₈	—	2	1	no	—
113	Q₈	—	47	2	no	—
114	Q₈	—	94	1	no	—
115	Q₈	C₈×D₄₇	1	2	yes	—
116	C₂×C₄	—	2	1	no	—
117	C₂×C₄	—	2	1	no	—
118	C₂×C₄	—	94	1	no	—
119	C₂×C₄	—	94	1	no	—
120	C₂×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
121	C₂×C₄	D₄×D₄₇	1	1	yes	characteristic
122	C₂×C₄	#10625352	1	1	yes	characteristic
123	C₈	—	2	2	no	—
124	C₈	—	4	1	no	—
125	C₈	—	47	2	no	—
126	C₈	—	47	2	no	—
127	C₈	—	94	1	no	—
128	C₈	—	94	1	no	—
129	C₈	—	188	1	no	—
130	C₂²	(C₂²⋊D₄₇)⋊C₄	1	1	yes	characteristic
131	C₄	—	2	1	no	—
132	C₄	—	2	2	no	—
133	C₄	—	4	1	no	—
134	C₄	—	94	2	no	—
135	C₄	—	188	1	no	—
136	C₄	C₂×C₈×D₄₇	1	1	yes	characteristic
137	C₄	#13094032	1	1	yes	characteristic
138	C₄	Q₁₆×D₄₇	1	1	yes	characteristic
139	C₄	#13094045	1	1	yes	characteristic
140	C₂	(C₂²⋊D₄₇)⋊C₈	1	1	yes	characteristic
141	C₂	#15689215	1	1	yes	characteristic
142	C₂	#15689237	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

Q₈×C₄₇⋊C₈

C₂

yes

maximal, characteristic

#15689247

C₂

yes

maximal, characteristic

C₄×#10625337

C₂

yes

maximal, characteristic

#15688875

C₂

yes

maximal, characteristic

#15688855

C₂

yes

maximal, characteristic

C₈×C₄₇⋊C₈

C₂

yes

maximal, characteristic

C₄₇×Q₈⋊C₈

C₂

yes

maximal, characteristic

C₂×#10625337

C₄

yes

characteristic

C₄×Dic₉₄

C₂²

yes

characteristic

#13093957

C₄

yes

characteristic

C₄⋊(C₄₇⋊C₈)

—

C₄⋊(C₄₇⋊C₈)

C₂²

yes

characteristic

Dic₄₇⋊C₈

—

#13093948

C₄

yes

characteristic

#13093947

C₄

yes

characteristic

C₄×C₄₇⋊C₈

—

C₄×C₄₇⋊C₈

C₂²

yes

characteristic

C₄×C₄₇⋊C₈

C₂²

yes

characteristic

C₈×Dic₄₇

—

C₄×Q₈×C₄₇

C₂²

yes

characteristic

C₄⋊C₈×C₄₇

C₂²

yes

characteristic

C₄×C₈×C₄₇

C₂²

yes

characteristic

C₂×Q₈×C₄₇

C₂×C₄

yes

characteristic

C₂×Dic₉₄

C₂×C₄

yes

characteristic

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

C₂×C₄

yes

characteristic

C₄²×C₄₇

—

C₄²×C₄₇

C₂³

yes

characteristic

#10625337

C₈

yes

complemented

C₄⋊Dic₄₇

C₂×C₄

yes

characteristic

Dic₄₇⋊C₄

—

C₄×Dic₄₇

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

C₂×C₄

yes

characteristic

C₂×C₄₇⋊C₈

C₂×C₄

yes

characteristic

C₂×C₄₇⋊C₈

D₄

yes

characteristic

C₂×C₄₇⋊C₈

D₄

yes

characteristic

Q₈×C₄₇

—

Q₈×C₄₇

C₂×C₈

yes

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂×C₄×C₄₇

#41

yes

characteristic

C₂×Dic₄₇

—

C₂×Dic₄₇

—

Dic₉₄

—

Dic₉₄

C₂×C₈

yes

—

C₈×C₄₇

—

C₈×C₄₇

—

C₄₇⋊C₈

—

C₄₇⋊C₈

—

C₄₇⋊C₈

—

C₄₇⋊C₈

C₂×C₈

yes

—

C₄₇⋊C₈

Q₁₆

yes

complemented

C₂²×C₄₇

C₄×D₄

yes

characteristic

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

C₂²×C₈

yes

characteristic

C₄×C₄₇

#131

yes

characteristic

C₄×C₄₇

C₂×Q₁₆

yes

characteristic

C₄×C₄₇

#135

yes

characteristic

Dic₄₇

—

Dic₄₇

—

C₈×Q₁₆

—

maximal, Sylow

C₂×C₄₇

C₈×D₄

yes

characteristic

C₂×C₄₇

C₄×Q₁₆

yes

characteristic

C₂×C₄₇

#443

yes

characteristic

C₈×Q₈

—

C₈×Q₈

—

C₄×Q₁₆

—

#335

—

Q₈⋊C₈

—

Q₈⋊C₈

D₄₇

yes

characteristic

C₈²

—

C₄₇

C₈×Q₁₆

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

C₄×Q₈

—

C₄×Q₈

C₂×D₄₇

yes

characteristic

#107

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

C₂×D₄₇

yes

characteristic

#103

—

#103

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×D₄₇

yes

characteristic

C₂×Q₈

—

C₂×Q₈

C₄×D₄₇

yes

characteristic

Q₁₆

—

C₂×C₈

—

C₂×C₈

—

100

C₂×C₈

—

101

C₂×C₈

—

102

C₂×C₈

—

103

C₂×C₈

—

104

C₂×C₈

—

105

C₄⋊C₄

—

106

C₄⋊C₄

—

107

C₄⋊C₄

—

108

C₄⋊C₄

C₄×D₄₇

yes

characteristic

109

C₄²

—

110

C₄²

—

111

C₄²

C₂²×D₄₇

yes

characteristic

112

Q₈

—

113

Q₈

—

114

Q₈

—

115

Q₈

C₈×D₄₇

yes

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

C₂×C₄×D₄₇

yes

characteristic

121

C₂×C₄

D₄×D₄₇

yes

characteristic

122

C₂×C₄

#10625352

yes

characteristic

123

C₈

—

124

C₈

—

125

C₈

—

126

C₈

—

127

C₈

—

128

C₈

—

129

C₈

—

188

—

130

C₂²

(C₂²⋊D₄₇)⋊C₄

yes

characteristic

131

C₄

—

132

C₄

—

133

C₄

—

134

C₄

—

135

C₄

—

188

—

136

C₄

C₂×C₈×D₄₇

yes

characteristic

137

C₄

#13094032

yes

characteristic

138

C₄

Q₁₆×D₄₇

yes

characteristic

139

C₄

#13094045

yes

characteristic

140

C₂

(C₂²⋊D₄₇)⋊C₈

yes

characteristic

141

C₂

#15689215

yes

characteristic

142

C₂

#15689237

yes

characteristic

	subgroup	G/H
center	C₂×C₄	D₄×D₄₇
commutator	C₄×C₄₇	C₂²×C₈
Frattini	C₄²	C₂²×D₄₇
Fitting	C₄₇×Q₈⋊C₈	C₂
socle	C₂²×C₄₇	C₄×D₄
hypercenter	Q₈⋊C₈	D₄₇
hypocenter	C₄₇	C₈×Q₁₆
LCS	G ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ Q₈⋊C₈

subgroup

G/H

center

C₂×C₄

D₄×D₄₇

commutator

C₄×C₄₇

C₂²×C₈

Frattini

C₄²

C₂²×D₄₇

Fitting

C₄₇×Q₈⋊C₈

C₂

socle

C₂²×C₄₇

C₄×D₄

hypercenter

Q₈⋊C₈

D₄₇

hypocenter

C₄₇

C₈×Q₁₆

LCS

G ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂×C₄ ◁ C₄² ◁ Q₈⋊C₈

N	H ≅ G/N	type
31. #10625337	C₈	semidirect
57. C₄₇⋊C₈	Q₁₆	semidirect
79. C₄₇	C₈×Q₁₆	semidirect

H ≅ G/N

type

31. #10625337

C₈

semidirect

57. C₄₇⋊C₈

Q₁₆

semidirect

79. C₄₇

C₈×Q₁₆

semidirect

Subgroups — #20242192

Conjugacy classes of subgroups

Complements