Group #20242971 — subgroups

Subgroups — #20242971

Order 6016.

140 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 140 conjugacy classes of subgroups — too many to draw. Only its 51 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	#15689474	C₂	1	1	yes	maximal, characteristic
2	#15689419	C₂	1	1	yes	maximal, characteristic
3	#15689409	C₂	1	1	yes	maximal, characteristic
4	#15689253	C₂	1	1	yes	maximal, characteristic
5	#15688871	C₂	1	1	yes	maximal, characteristic
6	#15688868	C₂	1	1	yes	maximal, characteristic
7	C₄₇×#334	C₂	1	1	yes	maximal, characteristic
8	#13094024	C₂²	1	1	yes	characteristic
9	#13094024	C₂²	1	1	yes	characteristic
10	C₄×Dic₉₄	C₂²	1	1	yes	characteristic
11	#13093958	—	2	1	no	—
12	#13093956	—	2	1	no	—
13	C₄⋊(C₄₇⋊C₈)	C₂²	1	1	yes	characteristic
14	Dic₄₇⋊C₈	—	2	1	no	—
15	#13093949	—	2	1	no	—
16	#13093949	—	2	1	no	—
17	#13093949	—	2	1	no	—
18	#13093949	—	2	1	no	—
19	#13093934	—	2	1	no	—
20	C₄⋊C₈×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×Dic₉₄	—	2	1	no	—
24	C₂×Dic₉₄	—	2	1	no	—
25	C₂×Dic₉₄	D₄	1	1	yes	characteristic
26	C₂×C₈×C₄₇	—	2	1	no	—
27	C₂×C₈×C₄₇	—	2	1	no	—
28	C₂×C₈×C₄₇	D₄	1	1	yes	characteristic
29	C₂×C₈×C₄₇	D₄	1	1	yes	characteristic
30	C₄²×C₄₇	C₂³	1	1	yes	characteristic
31	C₄⋊Dic₄₇	—	2	1	no	—
32	C₄⋊Dic₄₇	—	2	1	no	—
33	C₄⋊Dic₄₇	—	2	1	no	—
34	C₄⋊Dic₄₇	—	2	1	no	—
35	C₄⋊Dic₄₇	D₄	1	1	yes	characteristic
36	Dic₄₇⋊C₄	—	2	1	no	—
37	C₄×Dic₄₇	—	2	1	no	—
38	C₂×C₄₇⋊C₈	—	2	1	no	—
39	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
40	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
41	C₂×C₄×C₄₇	#41	1	1	yes	characteristic
42	C₂×Dic₄₇	—	2	1	no	—
43	C₂×Dic₄₇	—	2	1	no	—
44	C₂×Dic₄₇	—	4	1	no	—
45	C₂×Dic₄₇	—	4	1	no	—
46	Dic₉₄	—	2	1	no	—
47	Dic₉₄	—	2	1	no	—
48	Dic₉₄	—	4	1	no	—
49	Dic₉₄	—	4	1	no	—
50	Dic₉₄	—	8	1	no	—
51	C₈×C₄₇	—	2	1	no	—
52	C₈×C₄₇	—	2	1	no	—
53	C₈×C₄₇	—	4	1	no	—
54	C₈×C₄₇	—	4	1	no	—
55	C₄₇⋊C₈	—	4	1	no	—
56	C₂²×C₄₇	#121	1	1	yes	characteristic
57	C₄×C₄₇	—	2	1	no	—
58	C₄×C₄₇	#135	1	1	yes	characteristic
59	C₄×C₄₇	#137	1	1	yes	characteristic
60	C₄×C₄₇	#137	1	1	yes	characteristic
61	C₄×C₄₇	#137	1	1	yes	characteristic
62	Dic₄₇	—	4	1	no	—
63	Dic₄₇	—	4	1	no	—
64	Dic₄₇	—	8	1	no	—
65	Dic₄₇	—	8	1	no	—
66	#1704	—	47	1	no	maximal, Sylow
67	C₂×C₄₇	#464	1	1	yes	characteristic
68	C₂×C₄₇	#470	1	1	yes	characteristic
69	C₂×C₄₇	#471	1	1	yes	characteristic
70	#487	—	47	1	no	—
71	#477	—	47	1	no	—
72	#475	—	47	1	no	—
73	#446	—	47	1	no	—
74	#334	D₄₇	1	1	yes	characteristic
75	#333	—	47	1	no	—
76	#332	—	47	1	no	—
77	C₄₇	#1704	1	1	yes	characteristic, complemented, Sylow
78	C₄⋊Q₈	—	47	1	no	—
79	C₄⋊Q₈	—	47	1	no	—
80	C₄×Q₈	—	47	1	no	—
81	#107	—	94	1	no	—
82	#106	—	94	1	no	—
83	C₄⋊C₈	—	47	1	no	—
84	C₄⋊C₈	—	94	1	no	—
85	C₄⋊C₈	C₂×D₄₇	1	1	yes	characteristic
86	C₄⋊C₈	C₂×D₄₇	1	1	yes	characteristic
87	#103	—	94	1	no	—
88	#103	—	94	1	no	—
89	#103	—	94	1	no	—
90	#103	—	94	1	no	—
91	#97	—	94	1	no	—
92	C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
93	C₂×Q₈	—	47	1	no	—
94	C₂×Q₈	—	94	1	no	—
95	C₂×Q₈	—	94	1	no	—
96	C₂×C₈	—	2	1	no	—
97	C₂×C₈	—	2	1	no	—
98	C₂×C₈	—	94	1	no	—
99	C₂×C₈	D₁₈₈	1	1	yes	characteristic
100	C₂×C₈	D₁₈₈	1	1	yes	characteristic
101	C₄⋊C₄	—	47	1	no	—
102	C₄⋊C₄	—	94	1	no	—
103	C₄⋊C₄	—	94	1	no	—
104	C₄⋊C₄	—	94	1	no	—
105	C₄⋊C₄	—	94	1	no	—
106	C₄⋊C₄	—	94	1	no	—
107	C₄²	—	94	1	no	—
108	C₄²	C₂²×D₄₇	1	1	yes	characteristic
109	Q₈	—	94	1	no	—
110	Q₈	—	94	1	no	—
111	Q₈	—	376	1	no	—
112	Q₈	—	188	1	no	—
113	Q₈	—	188	1	no	—
114	C₂×C₄	—	94	1	no	—
115	C₂×C₄	—	94	1	no	—
116	C₂×C₄	—	188	1	no	—
117	C₂×C₄	—	188	1	no	—
118	C₂×C₄	C₂×D₁₈₈	1	1	yes	characteristic
119	C₂×C₄	D₄×D₄₇	1	1	yes	characteristic
120	C₂×C₄	#10625354	1	1	yes	characteristic
121	C₈	—	2	1	no	—
122	C₈	—	2	1	no	—
123	C₈	—	4	1	no	—
124	C₈	—	4	1	no	—
125	C₈	—	188	1	no	—
126	C₂²	#13093999	1	1	yes	characteristic
127	C₄	—	2	1	no	—
128	C₄	—	376	1	no	—
129	C₄	—	376	1	no	—
130	C₄	—	188	1	no	—
131	C₄	—	188	1	no	—
132	C₄	#13094046	1	1	yes	characteristic
133	C₄	#13094053	1	1	yes	characteristic
134	C₄	#13094053	1	1	yes	characteristic
135	C₄	#13094057	1	1	yes	characteristic
136	C₂	#15689346	1	1	yes	characteristic
137	C₂	#15689382	1	1	yes	characteristic
138	C₂	#15689387	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#15689474

C₂

yes

maximal, characteristic

#15689419

C₂

yes

maximal, characteristic

#15689409

C₂

yes

maximal, characteristic

#15689253

C₂

yes

maximal, characteristic

#15688871

C₂

yes

maximal, characteristic

#15688868

C₂

yes

maximal, characteristic

C₄₇×#334

C₂

yes

maximal, characteristic

#13094024

C₂²

yes

characteristic

#13094024

C₂²

yes

characteristic

C₄×Dic₉₄

C₂²

yes

characteristic

#13093958

—

#13093956

—

C₄⋊(C₄₇⋊C₈)

C₂²

yes

characteristic

Dic₄₇⋊C₈

—

—

—

—

—

—

C₄⋊C₈×C₄₇

C₂²

yes

characteristic

C₄⋊C₈×C₄₇

C₂²

yes

characteristic

C₄×C₈×C₄₇

C₂²

yes

characteristic

C₂×Dic₉₄

—

C₂×Dic₉₄

—

C₂×Dic₉₄

D₄

yes

characteristic

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

D₄

yes

characteristic

C₂×C₈×C₄₇

D₄

yes

characteristic

C₄²×C₄₇

C₂³

yes

characteristic

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

D₄

yes

characteristic

Dic₄₇⋊C₄

—

C₄×Dic₄₇

—

C₂×C₄₇⋊C₈

—

C₂×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂×C₄×C₄₇

#41

yes

characteristic

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

Dic₉₄

—

Dic₉₄

—

Dic₉₄

—

Dic₉₄

—

Dic₉₄

—

C₈×C₄₇

—

C₈×C₄₇

—

C₈×C₄₇

—

C₈×C₄₇

—

C₄₇⋊C₈

—

C₂²×C₄₇

#121

yes

characteristic

C₄×C₄₇

—

C₄×C₄₇

yes

C₄×C₄₇

yes

C₄×C₄₇

yes

C₄×C₄₇

yes

Dic₄₇

—

Dic₄₇

—

Dic₄₇

—

Dic₄₇

—

—

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

—

—

—

—

D₄₇

yes

characteristic

#333

—

#332

—

C₄₇

#1704

yes

characteristic, complemented, Sylow

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄×Q₈

—

#107

—

#106

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

C₂×D₄₇

yes

characteristic

C₄⋊C₈

C₂×D₄₇

yes

—

—

—

—

—

C₄×C₈

C₂×D₄₇

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

D₁₈₈

yes

characteristic

100

C₂×C₈

D₁₈₈

yes

characteristic

101

C₄⋊C₄

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

106

C₄⋊C₄

—

107

C₄²

—

108

C₄²

C₂²×D₄₇

yes

characteristic

109

Q₈

—

110

Q₈

—

111

Q₈

—

376

—

112

Q₈

—

188

—

113

Q₈

—

188

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

188

—

117

C₂×C₄

—

188

—

118

C₂×C₄

C₂×D₁₈₈

yes

characteristic

119

C₂×C₄

D₄×D₄₇

yes

characteristic

120

C₂×C₄

#10625354

yes

characteristic

121

C₈

—

122

C₈

—

123

C₈

—

124

C₈

—

125

C₈

—

188

—

126

C₂²

#13093999

yes

characteristic

127

C₄

—

128

C₄

—

376

—

129

C₄

—

376

—

130

C₄

—

188

—

131

C₄

—

188

—

132

C₄

#13094046

yes

characteristic

133

C₄

#13094053

yes

characteristic

134

C₄

#13094053

yes

characteristic

135

C₄

#13094057

yes

characteristic

136

C₂

#15689346

yes

characteristic

137

C₂

#15689382

yes

characteristic

138

C₂

#15689387

yes

characteristic

	subgroup	G/H
center	C₂²	#13093999
commutator	C₄²×C₄₇	C₂³
Frattini	C₄²	C₂²×D₄₇
Fitting	C₄₇×#334	C₂
socle	C₂²×C₄₇	#121
hypercenter	#334	D₄₇
hypocenter	C₄₇	#1704
LCS	G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #334

subgroup

G/H

center

C₂²

#13093999

commutator

C₄²×C₄₇

C₂³

Frattini

C₄²

C₂²×D₄₇

Fitting

C₄₇×#334

C₂

C₂²×C₄₇

D₄₇

C₄₇

G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #334

N	H ≅ G/N	type
77. C₄₇	#1704	semidirect

H ≅ G/N

type

77. C₄₇

#1704

semidirect

Subgroups — #20242971

Conjugacy classes of subgroups

Complements