Group #20251870 — subgroups

Subgroups — #20251870

Order 6016.

704 conjugacy classes of subgroups; only the 249 normal subgroups are shown.

normal subgroups only

This group has 704 conjugacy classes of subgroups — too many to draw. Only its 249 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	C₂×#13094071	C₂	1	1	yes	maximal, characteristic
2	#15689890	C₂	1	1	yes	maximal, characteristic, complemented
3	#15689105	C₂	1	6	yes	maximal, complemented
4	C₂×#13093942	C₂	1	6	yes	maximal, complemented
5	C₄₇×#568	C₂	1	1	yes	maximal, characteristic, complemented
6	#13094071	—	2	4	no	—
7	#13094071	—	2	4	no	—
8	C₂×#10625350	C₂²	1	3	yes	—
9	C₂×#10625352	C₄	1	6	yes	—
10	C₂²×Dic₉₄	C₂²	1	3	yes	—
11	C₂×Q₈×D₄₇	C₄	1	2	yes	—
12	#13094031	—	2	4	no	—
13	C₂×#10625329	C₂²	1	6	yes	—
14	#13093942	C₂²	1	12	yes	—
15	#13093940	C₂²	1	4	yes	—
16	C₂×#41×C₄₇	C₂²	1	1	yes	characteristic
17	#131×C₄₇	—	2	4	no	—
18	C₂×M₁₆×C₄₇	C₂²	1	6	yes	—
19	#41×C₄₇	D₄	1	4	yes	—
20	#41×C₄₇	D₄	1	4	yes	—
21	C₂×Q₈×C₄₇	C₂×C₄	1	1	yes	characteristic
22	C₂×D₄×C₄₇	C₂×C₄	1	3	yes	—
23	C₂²×C₄×C₄₇	C₂³	1	3	yes	—
24	C₂×C₂²⋊D₄₇	C₂×C₄	1	6	yes	—
25	C₂²×Dic₄₇	C₂×C₄	1	6	yes	—
26	Q₈×D₄₇	—	2	8	no	—
27	#10625352	—	2	24	no	—
28	#10625350	—	2	6	no	—
29	#10625350	—	2	6	no	—
30	C₂×D₁₈₈	C₂³	1	1	yes	characteristic
31	C₂×C₄×D₄₇	C₂×C₄	1	6	yes	—
32	C₂×Dic₉₄	—	2	6	no	—
33	C₂×Dic₉₄	—	2	6	no	—
34	C₂×Dic₉₄	C₂×C₄	1	6	yes	—
35	C₂×Dic₉₄	C₂³	1	3	yes	—
36	M₁₆×C₄₇	—	2	6	no	—
37	M₁₆×C₄₇	C₂³	1	4	yes	—
38	C₂×C₈×C₄₇	—	2	6	no	—
39	#10625329	—	2	6	no	—
40	#10625329	C₂³	1	4	yes	—
41	C₂×C₄₇⋊C₈	—	2	6	no	—
42	C₂³×C₄₇	C₂²×C₄	1	3	yes	—
43	C₂²×D₄₇	C₂²×C₄	1	2	yes	—
44	Q₈×C₄₇	C₂²⋊C₄	1	4	yes	—
45	D₄×C₄₇	C₂²⋊C₄	1	12	yes	—
46	C₂×C₄×C₄₇	C₂²×C₄	1	3	yes	—
47	C₂×C₄×C₄₇	C₂×D₄	1	6	yes	—
48	C₂×C₄×C₄₇	C₂×D₄	1	6	yes	—
49	C₂×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
50	C₂²⋊D₄₇	—	2	12	no	—
51	C₂×Dic₄₇	—	2	12	no	—
52	C₂×Dic₄₇	C₂²×C₄	1	6	yes	—
53	D₁₈₈	—	2	1	no	—
54	D₁₈₈	—	2	1	no	—
55	C₄×D₄₇	—	2	12	no	—
56	Dic₉₄	—	2	12	no	—
57	Dic₉₄	—	2	3	no	—
58	Dic₉₄	—	2	3	no	—
59	C₈×C₄₇	—	2	4	no	—
60	C₄₇⋊C₈	—	2	4	no	—
61	C₂²×C₄₇	—	4	3	no	—
62	C₂²×C₄₇	C₂×C₂²⋊C₄	1	6	yes	—
63	C₂²×C₄₇	C₂³×C₄	1	1	yes	characteristic
64	C₂×D₄₇	—	2	2	no	—
65	C₂×D₄₇	—	4	2	no	—
66	C₄×C₄₇	C₂×C₂²⋊C₄	1	6	yes	—
67	C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
68	C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
69	Dic₄₇	—	2	6	no	—
70	#2889	—	47	1	no	maximal, Sylow
71	C₂×C₄₇	—	2	1	no	—
72	C₂×C₄₇	—	2	6	no	—
73	C₂×C₄₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
74	D₄₇	—	4	2	no	—
75	C₂×2₋¹⁺⁴	—	47	1	no	—
76	#568	—	47	1	no	—
77	#568	D₄₇	1	1	yes	characteristic, complemented
78	#413	—	47	6	no	—
79	C₂×#101	—	47	6	no	—
80	C₄₇	#2889	1	1	yes	characteristic, complemented, Sylow
81	2₋¹⁺⁴	—	94	4	no	—
82	2₋¹⁺⁴	—	94	4	no	—
83	C₂×#41	—	47	6	no	—
84	C₂×#41	—	47	3	no	—
85	C₂×#41	C₂×D₄₇	1	1	yes	characteristic
86	C₂²×Q₈	—	47	2	no	—
87	C₂²×Q₈	—	47	3	no	—
88	#131	—	2	4	no	—
89	#131	—	94	4	no	—
90	C₂×M₁₆	—	47	6	no	—
91	C₂×M₁₆	C₂×D₄₇	1	6	yes	—
92	#101	—	47	12	no	—
93	#100	—	47	4	no	—
94	#41	—	94	24	no	—
95	#41	—	94	6	no	—
96	#41	—	94	6	no	—
97	#41	D₁₈₈	1	4	yes	—
98	#41	C₂²⋊D₄₇	1	4	yes	—
99	C₂×Q₈	—	47	6	no	—
100	C₂×Q₈	—	47	3	no	—
101	C₂×Q₈	—	94	8	no	—
102	C₂×Q₈	—	94	6	no	—
103	C₂×Q₈	—	94	6	no	—
104	C₂×Q₈	C₄×D₄₇	1	1	yes	characteristic
105	C₂×D₄	—	47	6	no	—
106	C₂×D₄	—	47	1	no	—
107	C₂×D₄	C₄×D₄₇	1	3	yes	—
108	C₂²×C₄	—	47	6	no	—
109	C₂²×C₄	—	47	6	no	—
110	C₂²×C₄	C₂²×D₄₇	1	3	yes	—
111	M₁₆	—	2	6	no	—
112	M₁₆	—	47	4	no	—
113	M₁₆	—	94	6	no	—
114	M₁₆	C₂²×D₄₇	1	4	yes	—
115	C₂×C₈	—	2	6	no	—
116	C₂×C₈	—	94	6	no	—
117	C₂³	—	47	2	no	—
118	C₂³	C₂×C₄×D₄₇	1	3	yes	—
119	Q₈	—	94	12	no	—
120	Q₈	—	94	3	no	—
121	Q₈	—	94	3	no	—
122	Q₈	(C₂×D₄₇)⋊C₄	1	4	yes	—
123	D₄	—	94	12	no	—
124	D₄	—	94	1	no	—
125	D₄	—	94	1	no	—
126	D₄	(C₂×D₄₇)⋊C₄	1	12	yes	—
127	C₂×C₄	—	47	6	no	—
128	C₂×C₄	—	94	12	no	—
129	C₂×C₄	—	94	12	no	—
130	C₂×C₄	C₂×C₄×D₄₇	1	3	yes	—
131	C₂×C₄	C₂×D₁₈₈	1	6	yes	—
132	C₂×C₄	C₂×C₂²⋊D₄₇	1	6	yes	—
133	C₂×C₄	C₂³×D₄₇	1	1	yes	characteristic
134	C₈	—	2	4	no	—
135	C₈	—	94	4	no	—
136	C₂²	—	4	3	no	—
137	C₂²	—	94	2	no	—
138	C₂²	—	188	2	no	—
139	C₂²	C₂×(C₂×D₄₇)⋊C₄	1	6	yes	—
140	C₂²	C₂²×C₄×D₄₇	1	1	yes	characteristic
141	C₄	—	94	6	no	—
142	C₄	C₂×(C₂×D₄₇)⋊C₄	1	6	yes	—
143	C₄	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
144	C₄	C₂²×D₁₈₈	1	1	yes	characteristic
145	C₂	—	2	1	no	—
146	C₂	—	2	6	no	—
147	C₂	—	188	2	no	—
148	C₂	C₂²×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13094071

C₂

yes

maximal, characteristic

#15689890

C₂

yes

maximal, characteristic, complemented

#15689105

C₂

yes

maximal, complemented

C₂×#13093942

C₂

yes

maximal, complemented

C₄₇×#568

C₂

yes

maximal, characteristic, complemented

#13094071

—

#13094071

—

C₂×#10625350

C₂²

yes

—

C₂×#10625352

C₄

yes

—

C₂²×Dic₉₄

C₂²

yes

—

C₂×Q₈×D₄₇

C₄

yes

—

#13094031

—

C₂×#10625329

C₂²

yes

—

#13093942

C₂²

yes

—

#13093940

C₂²

yes

—

C₂×#41×C₄₇

C₂²

yes

characteristic

#131×C₄₇

—

C₂×M₁₆×C₄₇

C₂²

yes

—

#41×C₄₇

D₄

yes

—

#41×C₄₇

D₄

yes

—

C₂×Q₈×C₄₇

C₂×C₄

yes

characteristic

C₂×D₄×C₄₇

C₂×C₄

yes

—

C₂²×C₄×C₄₇

C₂³

yes

—

C₂×C₂²⋊D₄₇

C₂×C₄

yes

—

C₂²×Dic₄₇

C₂×C₄

yes

—

Q₈×D₄₇

—

#10625352

—

#10625350

—

#10625350

—

C₂×D₁₈₈

C₂³

yes

characteristic

C₂×C₄×D₄₇

C₂×C₄

yes

—

C₂×Dic₉₄

—

C₂×Dic₉₄

—

C₂×Dic₉₄

C₂×C₄

yes

—

C₂×Dic₉₄

C₂³

yes

—

M₁₆×C₄₇

—

M₁₆×C₄₇

C₂³

yes

—

C₂×C₈×C₄₇

—

#10625329

—

#10625329

C₂³

yes

—

C₂×C₄₇⋊C₈

—

C₂³×C₄₇

C₂²×C₄

yes

—

C₂²×D₄₇

C₂²×C₄

yes

—

Q₈×C₄₇

C₂²⋊C₄

yes

—

D₄×C₄₇

C₂²⋊C₄

yes

—

C₂×C₄×C₄₇

C₂²×C₄

yes

—

C₂×C₄×C₄₇

C₂×D₄

yes

—

C₂×C₄×C₄₇

C₂×D₄

yes

—

C₂×C₄×C₄₇

C₂⁴

yes

characteristic

C₂²⋊D₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

C₂²×C₄

yes

—

D₁₈₈

—

D₁₈₈

—

C₄×D₄₇

—

Dic₉₄

—

Dic₉₄

—

Dic₉₄

—

C₈×C₄₇

—

C₄₇⋊C₈

—

C₂²×C₄₇

—

C₂²×C₄₇

C₂×C₂²⋊C₄

yes

—

C₂²×C₄₇

C₂³×C₄

yes

characteristic

C₂×D₄₇

—

C₂×D₄₇

—

C₄×C₄₇

C₂×C₂²⋊C₄

yes

—

C₄×C₄₇

C₂²×D₄

yes

characteristic

C₄×C₄₇

C₂²×D₄

yes

characteristic

Dic₄₇

—

#2889

—

maximal, Sylow

C₂×C₄₇

—

C₂×C₄₇

—

C₂×C₄₇

C₂²×C₂²⋊C₄

yes

characteristic

D₄₇

—

C₂×2₋¹⁺⁴

—

#568

—

#568

D₄₇

yes

characteristic, complemented

#413

—

C₂×#101

—

C₄₇

#2889

yes

characteristic, complemented, Sylow

2₋¹⁺⁴

—

2₋¹⁺⁴

—

C₂×#41

—

C₂×#41

—

C₂×#41

C₂×D₄₇

yes

characteristic

C₂²×Q₈

—

C₂²×Q₈

—

#131

—

#131

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₄₇

yes

—

—

—

—

—

—

D₁₈₈

yes

—

#41

C₂²⋊D₄₇

yes

—

C₂×Q₈

—

100

C₂×Q₈

—

101

C₂×Q₈

—

102

C₂×Q₈

—

103

C₂×Q₈

—

104

C₂×Q₈

C₄×D₄₇

yes

characteristic

105

C₂×D₄

—

106

C₂×D₄

—

107

C₂×D₄

C₄×D₄₇

yes

—

108

C₂²×C₄

—

109

C₂²×C₄

—

110

C₂²×C₄

C₂²×D₄₇

yes

—

111

M₁₆

—

112

M₁₆

—

113

M₁₆

—

114

M₁₆

C₂²×D₄₇

yes

—

115

C₂×C₈

—

116

C₂×C₈

—

117

C₂³

—

118

C₂³

C₂×C₄×D₄₇

yes

—

119

Q₈

—

120

Q₈

—

121

Q₈

—

122

Q₈

(C₂×D₄₇)⋊C₄

yes

—

123

D₄

—

124

D₄

—

125

D₄

—

126

D₄

(C₂×D₄₇)⋊C₄

yes

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

C₂×C₄×D₄₇

yes

—

131

C₂×C₄

C₂×D₁₈₈

yes

—

132

C₂×C₄

C₂×C₂²⋊D₄₇

yes

—

133

C₂×C₄

C₂³×D₄₇

yes

characteristic

134

C₈

—

135

C₈

—

136

C₂²

—

137

C₂²

—

138

C₂²

—

188

—

139

C₂²

C₂×(C₂×D₄₇)⋊C₄

yes

—

140

C₂²

C₂²×C₄×D₄₇

yes

characteristic

141

C₄

—

142

C₄

C₂×(C₂×D₄₇)⋊C₄

yes

—

143

C₄

C₂²×C₂²⋊D₄₇

yes

characteristic

144

C₄

C₂²×D₁₈₈

yes

characteristic

145

C₂

—

146

C₂

—

147

C₂

—

188

—

148

C₂

C₂²×(C₂×D₄₇)⋊C₄

yes

characteristic

	subgroup	G/H
center	C₂	C₂²×(C₂×D₄₇)⋊C₄
commutator	C₂²×C₄₇	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₄₇
Fitting	C₄₇×#568	C₂
socle	C₂×C₄₇	C₂²×C₂²⋊C₄
hypercenter	#568	D₄₇
hypocenter	C₄₇	#2889
LCS	G ▷ C₂²×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂ ◁ C₂×#41 ◁ #568

subgroup

G/H

center

C₂

C₂²×(C₂×D₄₇)⋊C₄

commutator

C₂²×C₄₇

C₂³×C₄

Frattini

C₂×C₄

C₂³×D₄₇

Fitting

C₄₇×#568

C₂

socle

C₂×C₄₇

C₂²×C₂²⋊C₄

D₄₇

C₄₇

G ▷ C₂²×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂ ◁ C₂×#41 ◁ #568

N	H ≅ G/N	type
2. #15689890	C₂	semidirect
3. #15689105	C₂	semidirect
4. C₂×#13093942	C₂	semidirect
5. C₄₇×#568	C₂	semidirect
77. #568	D₄₇	semidirect
80. C₄₇	#2889	semidirect

H ≅ G/N

type

2. #15689890

C₂

semidirect

3. #15689105

C₂

semidirect

4. C₂×#13093942

C₂

semidirect

5. C₄₇×#568

C₂

semidirect

77. #568

D₄₇

semidirect

80. C₄₇

#2889

semidirect

Subgroups — #20251870

Conjugacy classes of subgroups

Complements