Group #20242844 — subgroups

Subgroups — #20242844

Order 6016.

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

normal subgroups only

This group has 144 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	#15689473	C₂	1	1	yes	maximal, characteristic
2	#15689418	C₂	1	1	yes	maximal, characteristic
3	Q₈×C₄₇⋊C₈	C₂	1	1	yes	maximal, characteristic
4	#15688875	C₂	1	1	yes	maximal, characteristic
5	#15688871	C₂	1	1	yes	maximal, characteristic
6	#15688869	C₂	1	1	yes	maximal, characteristic
7	C₄₇×#475	C₂	1	1	yes	maximal, characteristic
8	#13094024	C₂²	1	1	yes	characteristic
9	#13093957	—	2	1	no	—
10	C₄⋊(C₄₇⋊C₈)	—	2	1	no	—
11	C₄⋊(C₄₇⋊C₈)	C₂²	1	1	yes	characteristic
12	C₄⋊(C₄₇⋊C₈)	C₂²	1	1	yes	characteristic
13	#13093948	—	2	1	no	—
14	#13093948	—	2	1	no	—
15	#13093948	—	2	1	no	—
16	C₄×C₄₇⋊C₈	—	2	1	no	—
17	C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
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	C₄⋊C₈×C₄₇	C₂²	1	1	yes	characteristic
22	#103×C₄₇	—	2	1	no	—
23	C₂×Q₈×C₄₇	—	2	1	no	—
24	C₂×Q₈×C₄₇	D₄	1	1	yes	characteristic
25	C₂×Dic₉₄	—	2	1	no	—
26	C₂×C₈×C₄₇	—	2	1	no	—
27	C₄⋊C₄×C₄₇	—	2	1	no	—
28	C₄⋊C₄×C₄₇	—	2	1	no	—
29	C₄⋊C₄×C₄₇	—	2	1	no	—
30	C₄⋊C₄×C₄₇	D₄	1	1	yes	characteristic
31	C₄²×C₄₇	—	2	1	no	—
32	C₄²×C₄₇	C₂³	1	1	yes	characteristic
33	C₄⋊Dic₄₇	—	2	1	no	—
34	C₄⋊Dic₄₇	—	2	1	no	—
35	C₂×C₄₇⋊C₈	—	2	1	no	—
36	C₂×C₄₇⋊C₈	—	2	1	no	—
37	C₂×C₄₇⋊C₈	D₄	1	1	yes	characteristic
38	C₂×C₄₇⋊C₈	D₄	1	1	yes	characteristic
39	Q₈×C₄₇	—	2	1	no	—
40	Q₈×C₄₇	—	4	1	no	—
41	Q₈×C₄₇	—	4	1	no	—
42	Q₈×C₄₇	Q₁₆	1	2	yes	—
43	C₂×C₄×C₄₇	—	2	1	no	—
44	C₂×C₄×C₄₇	—	2	1	no	—
45	C₂×C₄×C₄₇	—	4	1	no	—
46	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
47	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
48	C₂×C₄×C₄₇	#41	1	1	yes	characteristic
49	C₂×Dic₄₇	—	4	1	no	—
50	Dic₉₄	—	8	1	no	—
51	C₈×C₄₇	—	4	1	no	—
52	C₄₇⋊C₈	—	2	1	no	—
53	C₄₇⋊C₈	—	2	1	no	—
54	C₄₇⋊C₈	—	4	1	no	—
55	C₄₇⋊C₈	—	4	1	no	—
56	C₂²×C₄₇	#121	1	1	yes	characteristic
57	C₄×C₄₇	—	2	2	no	—
58	C₄×C₄₇	—	2	1	no	—
59	C₄×C₄₇	—	4	1	no	—
60	C₄×C₄₇	—	8	1	no	—
61	C₄×C₄₇	C₂×Q₁₆	1	1	yes	characteristic
62	C₄×C₄₇	#135	1	1	yes	characteristic
63	C₄×C₄₇	#135	1	1	yes	characteristic
64	C₄×C₄₇	#137	1	1	yes	characteristic
65	Dic₄₇	—	8	1	no	—
66	#1684	—	47	1	no	maximal, Sylow
67	C₂×C₄₇	#464	1	1	yes	characteristic
68	C₂×C₄₇	#467	1	1	yes	characteristic
69	C₂×C₄₇	#471	1	1	yes	characteristic
70	#487	—	47	1	no	—
71	#477	—	47	1	no	—
72	#475	D₄₇	1	1	yes	characteristic
73	C₈×Q₈	—	47	1	no	—
74	#335	—	47	1	no	—
75	#333	—	47	1	no	—
76	#332	—	47	1	no	—
77	C₄₇	#1684	1	1	yes	characteristic, complemented, Sylow
78	C₄⋊Q₈	—	47	1	no	—
79	C₄⋊Q₈	C₂×D₄₇	1	1	yes	characteristic
80	C₄×Q₈	C₂×D₄₇	1	1	yes	characteristic
81	#107	—	94	1	no	—
82	#106	—	2	1	no	—
83	C₄⋊C₈	—	47	1	no	—
84	C₄⋊C₈	—	47	1	no	—
85	C₄⋊C₈	—	94	1	no	—
86	C₄⋊C₈	C₂×D₄₇	1	1	yes	characteristic
87	#103	—	2	1	no	—
88	#103	—	94	1	no	—
89	#103	—	94	1	no	—
90	#103	—	94	1	no	—
91	C₄×C₈	—	47	1	no	—
92	C₄×C₈	—	94	1	no	—
93	C₂×Q₈	—	2	1	no	—
94	C₂×Q₈	—	94	1	no	—
95	C₂×Q₈	C₂²⋊D₄₇	1	1	yes	characteristic
96	C₂×C₈	—	2	1	no	—
97	C₂×C₈	—	47	1	no	—
98	C₂×C₈	—	47	1	no	—
99	C₂×C₈	—	94	1	no	—
100	C₂×C₈	—	94	1	no	—
101	C₄⋊C₄	—	2	1	no	—
102	C₄⋊C₄	—	2	1	no	—
103	C₄⋊C₄	—	2	1	no	—
104	C₄⋊C₄	—	94	1	no	—
105	C₄⋊C₄	—	94	1	no	—
106	C₄⋊C₄	C₂²⋊D₄₇	1	1	yes	characteristic
107	C₄²	—	2	1	no	—
108	C₄²	C₂²×D₄₇	1	1	yes	characteristic
109	Q₈	—	2	1	no	—
110	Q₈	—	4	1	no	—
111	Q₈	—	4	1	no	—
112	Q₈	—	376	1	no	—
113	Q₈	#10625337	1	2	yes	—
114	C₂×C₄	—	2	1	no	—
115	C₂×C₄	—	2	1	no	—
116	C₂×C₄	—	4	1	no	—
117	C₂×C₄	—	188	1	no	—
118	C₂×C₄	D₄×D₄₇	1	1	yes	characteristic
119	C₂×C₄	#10625352	1	1	yes	characteristic
120	C₂×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
121	C₈	—	4	1	no	—
122	C₈	—	94	1	no	—
123	C₈	—	94	1	no	—
124	C₈	—	188	1	no	—
125	C₈	—	188	1	no	—
126	C₂²	#13093996	1	1	yes	characteristic
127	C₄	—	2	2	no	—
128	C₄	—	2	1	no	—
129	C₄	—	4	1	no	—
130	C₄	—	8	1	no	—
131	C₄	—	376	1	no	—
132	C₄	C₂×#10625337	1	1	yes	characteristic
133	C₄	#13094044	1	1	yes	characteristic
134	C₄	#13094045	1	1	yes	characteristic
135	C₄	#13094053	1	1	yes	characteristic
136	C₂	#15689343	1	1	yes	characteristic
137	C₂	#15689362	1	1	yes	characteristic
138	C₂	#15689388	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#15689473

C₂

yes

maximal, characteristic

#15689418

C₂

yes

maximal, characteristic

Q₈×C₄₇⋊C₈

C₂

yes

maximal, characteristic

#15688875

C₂

yes

maximal, characteristic

#15688871

C₂

yes

maximal, characteristic

#15688869

C₂

yes

maximal, characteristic

C₄₇×#475

C₂

yes

maximal, characteristic

#13094024

C₂²

yes

characteristic

#13093957

—

C₄⋊(C₄₇⋊C₈)

—

C₄⋊(C₄₇⋊C₈)

C₂²

yes

characteristic

C₄⋊(C₄₇⋊C₈)

C₂²

yes

—

—

—

C₄×C₄₇⋊C₈

—

C₄×C₄₇⋊C₈

C₂²

yes

characteristic

C₄⋊Q₈×C₄₇

C₂²

yes

characteristic

C₄×Q₈×C₄₇

C₂²

yes

characteristic

#106×C₄₇

—

C₄⋊C₈×C₄₇

C₂²

yes

characteristic

#103×C₄₇

—

C₂×Q₈×C₄₇

—

C₂×Q₈×C₄₇

D₄

yes

characteristic

C₂×Dic₉₄

—

C₂×C₈×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

D₄

yes

characteristic

C₄²×C₄₇

—

C₄²×C₄₇

C₂³

yes

characteristic

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

D₄

yes

characteristic

C₂×C₄₇⋊C₈

D₄

yes

characteristic

Q₈×C₄₇

—

Q₈×C₄₇

—

Q₈×C₄₇

—

Q₈×C₄₇

Q₁₆

yes

—

C₂×C₄×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

—

Dic₉₄

—

C₈×C₄₇

—

C₄₇⋊C₈

—

C₄₇⋊C₈

—

C₄₇⋊C₈

—

C₄₇⋊C₈

—

C₂²×C₄₇

#121

yes

characteristic

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

C₂×Q₁₆

yes

C₄×C₄₇

yes

C₄×C₄₇

yes

C₄×C₄₇

yes

Dic₄₇

—

—

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

—

—

D₄₇

yes

characteristic

C₈×Q₈

—

#335

—

#333

—

#332

—

C₄₇

#1684

yes

characteristic, complemented, Sylow

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₄₇

yes

characteristic

C₄×Q₈

C₂×D₄₇

yes

characteristic

#107

—

#106

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

C₂×D₄₇

yes

—

—

—

—

C₄×C₈

—

C₄×C₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂²⋊D₄₇

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

100

C₂×C₈

—

101

C₄⋊C₄

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

106

C₄⋊C₄

C₂²⋊D₄₇

yes

characteristic

107

C₄²

—

108

C₄²

C₂²×D₄₇

yes

characteristic

109

Q₈

—

110

Q₈

—

111

Q₈

—

112

Q₈

—

376

—

113

Q₈

#10625337

yes

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

188

—

118

C₂×C₄

D₄×D₄₇

yes

characteristic

119

C₂×C₄

#10625352

yes

characteristic

120

C₂×C₄

C₂×C₂²⋊D₄₇

yes

characteristic

121

C₈

—

122

C₈

—

123

C₈

—

124

C₈

—

188

—

125

C₈

—

188

—

126

C₂²

#13093996

yes

characteristic

127

C₄

—

128

C₄

—

129

C₄

—

130

C₄

—

131

C₄

—

376

—

132

C₄

C₂×#10625337

yes

characteristic

133

C₄

#13094044

yes

characteristic

134

C₄

#13094045

yes

characteristic

135

C₄

#13094053

yes

characteristic

136

C₂

#15689343

yes

characteristic

137

C₂

#15689362

yes

characteristic

138

C₂

#15689388

yes

characteristic

	subgroup	G/H
center	C₂²	#13093996
commutator	C₄²×C₄₇	C₂³
Frattini	C₄²	C₂²×D₄₇
Fitting	C₄₇×#475	C₂
socle	C₂²×C₄₇	#121
hypercenter	#475	D₄₇
hypocenter	C₄₇	#1684
LCS	G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #475

subgroup

G/H

center

C₂²

#13093996

commutator

C₄²×C₄₇

C₂³

Frattini

C₄²

C₂²×D₄₇

Fitting

C₄₇×#475

C₂

C₂²×C₄₇

D₄₇

C₄₇

G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #475

N	H ≅ G/N	type
77. C₄₇	#1684	semidirect

H ≅ G/N

type

77. C₄₇

#1684

semidirect

Subgroups — #20242844

Conjugacy classes of subgroups

Complements