Group #20252455 — subgroups

Subgroups — #20252455

Order 6016.

410 conjugacy classes of subgroups; only the 205 normal subgroups are shown.

normal subgroups only

This group has 410 conjugacy classes of subgroups — too many to draw. Only its 205 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	#15689201	C₂	1	4	yes	maximal, complemented
2	D₄×C₄₇⋊C₈	C₂	1	4	yes	maximal, complemented
3	#15689176	C₂	1	2	yes	maximal, complemented
4	#15689090	C₂	1	4	yes	maximal, complemented
5	C₄₇×#532	C₂	1	1	yes	maximal, characteristic
6	C₂×#10625329	C₂²	1	4	yes	—
7	C₂²×C₄₇⋊C₈	C₂²	1	4	yes	—
8	C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
9	C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
10	C₂²⋊(C₄₇⋊C₈)	C₂²	1	4	yes	—
11	C₂²⋊(C₄₇⋊C₈)	C₂²	1	8	yes	—
12	C₄×C₄₇⋊C₈	C₂²	1	4	yes	—
13	C₂×#41×C₄₇	C₂²	1	2	yes	—
14	C₄⋊Q₈×C₄₇	C₄	1	1	yes	characteristic
15	#127×C₄₇	C₄	1	1	yes	characteristic
16	#124×C₄₇	C₄	1	2	yes	—
17	#121×C₄₇	C₄	1	4	yes	—
18	C₄×D₄×C₄₇	C₂²	1	4	yes	—
19	C₂×C₄²×C₄₇	C₂²	1	1	yes	characteristic
20	#41×C₄₇	—	4	2	no	—
21	#41×C₄₇	—	4	2	no	—
22	C₂×Q₈×C₄₇	C₂×C₄	1	2	yes	—
23	C₂×D₄×C₄₇	C₂×C₄	1	4	yes	—
24	C₂×D₄×C₄₇	C₂×C₄	1	4	yes	—
25	C₂×D₄×C₄₇	C₂×C₄	1	2	yes	—
26	C₂²×C₄×C₄₇	C₂×C₄	1	2	yes	—
27	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
28	C₂²×C₄×C₄₇	C₂³	1	4	yes	—
29	C₄⋊C₄×C₄₇	C₂×C₄	1	4	yes	—
30	C₂²⋊C₄×C₄₇	C₂×C₄	1	8	yes	—
31	C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
32	C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
33	C₄²×C₄₇	C₂³	1	2	yes	—
34	#10625329	—	4	4	no	—
35	C₂×C₄₇⋊C₈	—	4	4	no	—
36	C₂×C₄₇⋊C₈	C₂³	1	4	yes	—
37	C₂×C₄₇⋊C₈	C₂³	1	4	yes	—
38	C₂³×C₄₇	C₂²×C₄	1	4	yes	—
39	C₂³×C₄₇	C₂²×C₄	1	1	yes	characteristic
40	Q₈×C₄₇	—	4	2	no	—
41	D₄×C₄₇	—	2	8	no	—
42	D₄×C₄₇	—	4	2	no	—
43	D₄×C₄₇	—	4	4	no	—
44	C₂×C₄×C₄₇	—	2	4	no	—
45	C₂×C₄×C₄₇	—	2	4	no	—
46	C₂×C₄×C₄₇	—	2	4	no	—
47	C₂×C₄×C₄₇	—	2	1	no	—
48	C₂×C₄×C₄₇	—	2	1	no	—
49	C₂×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
50	C₂×C₄×C₄₇	C₂²×C₄	1	2	yes	—
51	C₂×C₄×C₄₇	C₂²×C₄	1	4	yes	—
52	C₂×C₄×C₄₇	C₂²×C₄	1	2	yes	—
53	C₂×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
54	C₄₇⋊C₈	—	2	4	no	—
55	C₄₇⋊C₈	—	2	4	no	—
56	C₂²×C₄₇	—	2	1	no	—
57	C₂²×C₄₇	—	2	4	no	—
58	C₂²×C₄₇	—	2	2	no	—
59	C₂²×C₄₇	—	2	8	no	—
60	C₂²×C₄₇	C₂³×C₄	1	1	yes	characteristic
61	C₄×C₄₇	—	2	1	no	—
62	C₄×C₄₇	—	2	4	no	—
63	C₄×C₄₇	—	2	2	no	—
64	C₄×C₄₇	#131	1	4	yes	—
65	C₄×C₄₇	H(5,𝔽₂)	1	1	yes	characteristic
66	C₄×C₄₇	H(5,𝔽₂)	1	1	yes	characteristic
67	#2979	—	47	1	no	maximal, Sylow
68	C₂×C₄₇	—	4	1	no	—
69	C₂×C₄₇	—	4	4	no	—
70	C₂×C₄₇	#518	1	1	yes	characteristic
71	C₂×C₄₇	C₂×#131	1	2	yes	—
72	#532	D₄₇	1	1	yes	characteristic
73	#436	—	47	4	no	—
74	C₈×D₄	—	47	4	no	—
75	#431	—	47	2	no	—
76	#408	—	47	4	no	—
77	C₄₇	#2979	1	1	yes	characteristic, complemented, Sylow
78	C₂×#41	C₂×D₄₇	1	2	yes	—
79	C₂×M₁₆	—	47	4	no	—
80	C₂²×C₈	—	47	4	no	—
81	C₄⋊Q₈	Dic₄₇	1	1	yes	characteristic
82	#127	Dic₄₇	1	1	yes	characteristic
83	#124	Dic₄₇	1	2	yes	—
84	#121	Dic₄₇	1	4	yes	—
85	C₄×D₄	C₂×D₄₇	1	4	yes	—
86	C₂×C₄²	C₂×D₄₇	1	1	yes	characteristic
87	C₄⋊C₈	—	47	2	no	—
88	C₄⋊C₈	—	47	2	no	—
89	C₂²⋊C₈	—	47	4	no	—
90	C₂²⋊C₈	—	47	8	no	—
91	C₄×C₈	—	47	4	no	—
92	#41	—	4	2	no	—
93	#41	—	4	2	no	—
94	C₂×Q₈	C₂×Dic₄₇	1	2	yes	—
95	C₂×D₄	C₂×Dic₄₇	1	4	yes	—
96	C₂×D₄	C₂×Dic₄₇	1	4	yes	—
97	C₂×D₄	C₂×Dic₄₇	1	2	yes	—
98	C₂²×C₄	C₂×Dic₄₇	1	2	yes	—
99	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
100	C₂²×C₄	C₂²×D₄₇	1	4	yes	—
101	M₁₆	—	188	4	no	—
102	C₂×C₈	—	47	4	no	—
103	C₂×C₈	—	47	4	no	—
104	C₂×C₈	—	188	4	no	—
105	C₄⋊C₄	C₂×Dic₄₇	1	4	yes	—
106	C₂²⋊C₄	C₂×Dic₄₇	1	8	yes	—
107	C₄²	C₂×Dic₄₇	1	1	yes	characteristic
108	C₄²	C₂×Dic₄₇	1	1	yes	characteristic
109	C₄²	C₂²×D₄₇	1	2	yes	—
110	C₂³	C₂²×Dic₄₇	1	4	yes	—
111	C₂³	C₂²×Dic₄₇	1	1	yes	characteristic
112	Q₈	—	4	2	no	—
113	D₄	—	2	8	no	—
114	D₄	—	4	2	no	—
115	D₄	—	4	4	no	—
116	C₂×C₄	—	2	4	no	—
117	C₂×C₄	—	2	4	no	—
118	C₂×C₄	—	2	4	no	—
119	C₂×C₄	—	2	1	no	—
120	C₂×C₄	—	2	1	no	—
121	C₂×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
122	C₂×C₄	C₂²×Dic₄₇	1	2	yes	—
123	C₂×C₄	C₂²×Dic₄₇	1	4	yes	—
124	C₂×C₄	C₂²×Dic₄₇	1	2	yes	—
125	C₂×C₄	C₂³×D₄₇	1	1	yes	characteristic
126	C₈	—	94	4	no	—
127	C₈	—	94	4	no	—
128	C₂²	—	2	1	no	—
129	C₂²	—	2	4	no	—
130	C₂²	—	2	2	no	—
131	C₂²	—	2	8	no	—
132	C₂²	C₂³×Dic₄₇	1	1	yes	characteristic
133	C₄	—	2	1	no	—
134	C₄	—	2	4	no	—
135	C₄	—	2	2	no	—
136	C₄	#13094031	1	4	yes	—
137	C₄	#13094069	1	1	yes	characteristic
138	C₄	#13094069	1	1	yes	characteristic
139	C₂	—	4	1	no	—
140	C₂	—	4	4	no	—
141	C₂	#15689598	1	1	yes	characteristic
142	C₂	C₂×#13094031	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#15689201

C₂

yes

maximal, complemented

D₄×C₄₇⋊C₈

C₂

yes

maximal, complemented

#15689176

C₂

yes

maximal, complemented

#15689090

C₂

yes

maximal, complemented

C₄₇×#532

C₂

yes

maximal, characteristic

C₂×#10625329

C₂²

yes

—

C₂²×C₄₇⋊C₈

C₂²

yes

—

C₄⋊(C₄₇⋊C₈)

C₂²

yes

—

C₄⋊(C₄₇⋊C₈)

C₂²

yes

—

C₂²⋊(C₄₇⋊C₈)

C₂²

yes

—

C₂²⋊(C₄₇⋊C₈)

C₂²

yes

—

C₄×C₄₇⋊C₈

C₂²

yes

—

C₂×#41×C₄₇

C₂²

yes

—

C₄⋊Q₈×C₄₇

C₄

yes

characteristic

#127×C₄₇

C₄

yes

characteristic

#124×C₄₇

C₄

yes

—

#121×C₄₇

C₄

yes

—

C₄×D₄×C₄₇

C₂²

yes

—

C₂×C₄²×C₄₇

C₂²

yes

characteristic

#41×C₄₇

—

#41×C₄₇

—

C₂×Q₈×C₄₇

C₂×C₄

yes

—

C₂×D₄×C₄₇

C₂×C₄

yes

—

C₂×D₄×C₄₇

C₂×C₄

yes

—

C₂×D₄×C₄₇

C₂×C₄

yes

—

C₂²×C₄×C₄₇

C₂×C₄

yes

—

C₂²×C₄×C₄₇

C₂³

yes

characteristic

C₂²×C₄×C₄₇

C₂³

yes

—

C₄⋊C₄×C₄₇

C₂×C₄

yes

—

C₂²⋊C₄×C₄₇

C₂×C₄

yes

—

C₄²×C₄₇

C₂×C₄

yes

characteristic

C₄²×C₄₇

C₂×C₄

yes

characteristic

C₄²×C₄₇

C₂³

yes

—

#10625329

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

C₂³

yes

—

C₂×C₄₇⋊C₈

C₂³

yes

—

C₂³×C₄₇

C₂²×C₄

yes

—

C₂³×C₄₇

C₂²×C₄

yes

characteristic

Q₈×C₄₇

—

D₄×C₄₇

—

D₄×C₄₇

—

D₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₄₇

C₂²×C₄

yes

—

C₂×C₄×C₄₇

C₂²×C₄

yes

—

C₂×C₄×C₄₇

C₂²×C₄

yes

—

C₂×C₄×C₄₇

C₂⁴

yes

characteristic

C₄₇⋊C₈

—

C₄₇⋊C₈

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

C₂³×C₄

yes

characteristic

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

#131

yes

—

C₄×C₄₇

yes

C₄×C₄₇

yes

—

C₂×C₄₇

—

C₂×C₄₇

—

C₂×C₄₇

#518

yes

characteristic

C₂×C₄₇

C₂×#131

yes

—

#532

D₄₇

yes

characteristic

#436

—

C₈×D₄

—

#431

—

#408

—

C₄₇

#2979

yes

characteristic, complemented, Sylow

C₂×#41

C₂×D₄₇

yes

—

C₂×M₁₆

—

C₂²×C₈

—

C₄⋊Q₈

Dic₄₇

yes

characteristic

#127

Dic₄₇

yes

characteristic

#124

Dic₄₇

yes

—

#121

Dic₄₇

yes

—

C₄×D₄

C₂×D₄₇

yes

—

C₂×C₄²

C₂×D₄₇

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₂²⋊C₈

—

C₂²⋊C₈

—

C₄×C₈

—

#41

—

#41

—

C₂×Q₈

C₂×Dic₄₇

yes

—

C₂×D₄

C₂×Dic₄₇

yes

—

C₂×D₄

C₂×Dic₄₇

yes

—

C₂×D₄

C₂×Dic₄₇

yes

—

C₂²×C₄

C₂×Dic₄₇

yes

—

C₂²×C₄

C₂²×D₄₇

yes

characteristic

100

C₂²×C₄

C₂²×D₄₇

yes

—

101

M₁₆

—

188

—

102

C₂×C₈

—

103

C₂×C₈

—

104

C₂×C₈

—

188

—

105

C₄⋊C₄

C₂×Dic₄₇

yes

—

106

C₂²⋊C₄

C₂×Dic₄₇

yes

—

107

C₄²

C₂×Dic₄₇

yes

characteristic

108

C₄²

C₂×Dic₄₇

yes

characteristic

109

C₄²

C₂²×D₄₇

yes

—

110

C₂³

C₂²×Dic₄₇

yes

—

111

C₂³

C₂²×Dic₄₇

yes

characteristic

112

Q₈

—

113

D₄

—

114

D₄

—

115

D₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

C₂²×Dic₄₇

yes

characteristic

122

C₂×C₄

C₂²×Dic₄₇

yes

—

123

C₂×C₄

C₂²×Dic₄₇

yes

—

124

C₂×C₄

C₂²×Dic₄₇

yes

—

125

C₂×C₄

C₂³×D₄₇

yes

characteristic

126

C₈

—

127

C₈

—

128

C₂²

—

129

C₂²

—

130

C₂²

—

131

C₂²

—

132

C₂²

C₂³×Dic₄₇

yes

characteristic

133

C₄

—

134

C₄

—

135

C₄

—

136

C₄

#13094031

yes

—

137

C₄

#13094069

yes

characteristic

138

C₄

#13094069

yes

characteristic

139

C₂

—

140

C₂

—

141

C₂

#15689598

yes

characteristic

142

C₂

C₂×#13094031

yes

—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₄₇
commutator, socle	C₂²×C₄₇	C₂³×C₄
Fitting	C₄₇×#532	C₂
hypercenter	#532	D₄₇
hypocenter	C₄₇	#2979
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ #532

subgroup

G/H

center, Frattini

C₂×C₄

C₂³×D₄₇

commutator, socle

C₂²×C₄₇

C₂³×C₄

Fitting

C₄₇×#532

C₂

D₄₇

C₄₇

G ▷ C₂²×C₄₇ ▷ C₄₇

C₁ ◁ C₂×C₄ ◁ #532

N	H ≅ G/N	type
1. #15689201	C₂	semidirect
2. D₄×C₄₇⋊C₈	C₂	semidirect
3. #15689176	C₂	semidirect
4. #15689090	C₂	semidirect
77. C₄₇	#2979	semidirect

H ≅ G/N

type

1. #15689201

C₂

semidirect

2. D₄×C₄₇⋊C₈

C₂

semidirect

3. #15689176

C₂

semidirect

4. #15689090

C₂

semidirect

77. C₄₇

#2979

semidirect

Subgroups — #20252455

Conjugacy classes of subgroups

Complements