Group #20251866 — subgroups

Subgroups — #20251866

Order 6016.

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

normal subgroups only

This group has 756 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×#13094070	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×#13093940	C₂	1	6	yes	maximal, complemented
5	C₄₇×#568	C₂	1	1	yes	maximal, characteristic, complemented
6	#13094070	—	2	4	no	—
7	#13094070	—	2	4	no	—
8	C₂×#10625354	C₄	1	2	yes	—
9	C₂×#10625350	C₂²	1	3	yes	—
10	C₂²×D₁₈₈	C₂²	1	3	yes	—
11	C₂×D₄×D₄₇	C₄	1	6	yes	—
12	#13094031	—	2	4	no	—
13	C₂×#10625329	C₂²	1	6	yes	complemented
14	#13093942	C₂²	1	4	yes	complemented
15	#13093940	C₂²	1	12	yes	complemented
16	C₂×#41×C₄₇	C₂²	1	1	yes	characteristic
17	#131×C₄₇	—	2	4	no	—
18	C₂×M₁₆×C₄₇	C₂²	1	6	yes	complemented
19	C₂³×D₄₇	C₂×C₄	1	6	yes	—
20	#41×C₄₇	D₄	1	4	yes	—
21	#41×C₄₇	D₄	1	4	yes	—
22	C₂×Q₈×C₄₇	C₂×C₄	1	1	yes	characteristic
23	C₂×D₄×C₄₇	C₂×C₄	1	3	yes	—
24	C₂²×C₄×C₄₇	C₂³	1	3	yes	—
25	C₂×C₂²⋊D₄₇	C₂×C₄	1	6	yes	—
26	#10625354	—	2	8	no	—
27	D₄×D₄₇	—	2	24	no	—
28	#10625350	—	2	6	no	—
29	#10625350	—	2	6	no	—
30	C₂×D₁₈₈	—	2	6	no	—
31	C₂×D₁₈₈	—	2	6	no	—
32	C₂×D₁₈₈	C₂×C₄	1	6	yes	—
33	C₂×D₁₈₈	C₂³	1	3	yes	—
34	C₂×C₄×D₄₇	C₂×C₄	1	6	yes	—
35	C₂×Dic₉₄	C₂³	1	1	yes	characteristic
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	—	2	12	no	—
44	C₂²×D₄₇	—	8	6	no	—
45	C₂²×D₄₇	C₂²×C₄	1	6	yes	—
46	Q₈×C₄₇	C₂²⋊C₄	1	4	yes	—
47	D₄×C₄₇	C₂²⋊C₄	1	12	yes	—
48	C₂×C₄×C₄₇	C₂²×C₄	1	3	yes	—
49	C₂×C₄×C₄₇	C₂×D₄	1	6	yes	—
50	C₂×C₄×C₄₇	C₂×D₄	1	6	yes	—
51	C₂×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
52	C₂²⋊D₄₇	—	2	12	no	—
53	C₂×Dic₄₇	C₂²×C₄	1	2	yes	—
54	D₁₈₈	—	2	12	no	—
55	D₁₈₈	—	2	3	no	—
56	D₁₈₈	—	2	3	no	—
57	C₄×D₄₇	—	2	12	no	—
58	Dic₉₄	—	2	1	no	—
59	Dic₉₄	—	2	1	no	—
60	C₈×C₄₇	—	2	4	no	—
61	C₄₇⋊C₈	—	2	4	no	—
62	C₂²×C₄₇	—	4	3	no	—
63	C₂²×C₄₇	C₂×C₂²⋊C₄	1	6	yes	—
64	C₂²×C₄₇	C₂³×C₄	1	1	yes	characteristic
65	C₂×D₄₇	—	2	6	no	—
66	C₂×D₄₇	—	4	6	no	—
67	C₂×D₄₇	—	8	12	no	—
68	C₄×C₄₇	C₂×C₂²⋊C₄	1	6	yes	—
69	C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
70	C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
71	Dic₄₇	—	2	2	no	—
72	#2888	—	47	1	no	maximal, Sylow
73	C₂×C₄₇	—	2	1	no	—
74	C₂×C₄₇	—	2	6	no	—
75	C₂×C₄₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
76	D₄₇	—	4	6	no	—
77	C₂×H(5,𝔽₂)	—	47	1	no	—
78	#568	—	47	1	no	—
79	#568	D₄₇	1	1	yes	characteristic, complemented
80	#413	—	47	6	no	—
81	C₂×#100	—	47	6	no	—
82	C₄₇	#2888	1	1	yes	characteristic, complemented, Sylow
83	H(5,𝔽₂)	—	94	4	no	—
84	H(5,𝔽₂)	—	94	4	no	—
85	C₂×#41	—	47	2	no	—
86	C₂×#41	—	47	3	no	—
87	C₂×#41	C₂×D₄₇	1	1	yes	characteristic
88	C₂²×D₄	—	47	6	no	—
89	C₂²×D₄	—	47	3	no	—
90	#131	—	2	4	no	—
91	#131	—	94	4	no	—
92	C₂×M₁₆	—	47	6	no	—
93	C₂×M₁₆	C₂×D₄₇	1	6	yes	complemented
94	#101	—	47	4	no	—
95	#100	—	47	12	no	—
96	C₂⁴	—	47	6	no	—
97	#41	—	94	8	no	—
98	#41	—	94	6	no	—
99	#41	—	94	6	no	—
100	#41	D₁₈₈	1	4	yes	—
101	#41	C₂²⋊D₄₇	1	4	yes	—
102	C₂×Q₈	—	47	1	no	—
103	C₂×Q₈	C₄×D₄₇	1	1	yes	characteristic
104	C₂×D₄	—	47	6	no	—
105	C₂×D₄	—	47	6	no	—
106	C₂×D₄	—	47	3	no	—
107	C₂×D₄	—	94	24	no	—
108	C₂×D₄	—	94	6	no	—
109	C₂×D₄	—	94	6	no	—
110	C₂×D₄	C₄×D₄₇	1	3	yes	—
111	C₂²×C₄	—	47	6	no	—
112	C₂²×C₄	C₂²×D₄₇	1	3	yes	—
113	M₁₆	—	2	6	no	—
114	M₁₆	—	47	4	no	—
115	M₁₆	—	94	6	no	—
116	M₁₆	C₂²×D₄₇	1	4	yes	—
117	C₂×C₈	—	2	6	no	—
118	C₂×C₈	—	94	6	no	—
119	C₂³	—	47	6	no	—
120	C₂³	—	94	12	no	—
121	C₂³	—	376	6	no	—
122	C₂³	C₂×C₄×D₄₇	1	3	yes	—
123	Q₈	—	94	1	no	—
124	Q₈	—	94	1	no	—
125	Q₈	(C₂×D₄₇)⋊C₄	1	4	yes	—
126	D₄	—	94	12	no	—
127	D₄	—	94	12	no	—
128	D₄	—	94	3	no	—
129	D₄	—	94	3	no	—
130	D₄	(C₂×D₄₇)⋊C₄	1	12	yes	—
131	C₂×C₄	—	47	2	no	—
132	C₂×C₄	—	94	12	no	—
133	C₂×C₄	C₂×C₄×D₄₇	1	3	yes	—
134	C₂×C₄	C₂×D₁₈₈	1	6	yes	—
135	C₂×C₄	C₂×C₂²⋊D₄₇	1	6	yes	—
136	C₂×C₄	C₂³×D₄₇	1	1	yes	characteristic
137	C₈	—	2	4	no	—
138	C₈	—	94	4	no	—
139	C₂²	—	4	3	no	—
140	C₂²	—	94	6	no	—
141	C₂²	—	376	12	no	—
142	C₂²	—	188	6	no	—
143	C₂²	C₂×(C₂×D₄₇)⋊C₄	1	6	yes	—
144	C₂²	C₂²×C₄×D₄₇	1	1	yes	characteristic
145	C₄	—	94	2	no	—
146	C₄	C₂×(C₂×D₄₇)⋊C₄	1	6	yes	—
147	C₄	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
148	C₄	C₂²×D₁₈₈	1	1	yes	characteristic
149	C₂	—	2	1	no	—
150	C₂	—	2	6	no	—
151	C₂	—	188	6	no	—
152	C₂	C₂²×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13094070

C₂

yes

maximal, characteristic

#15689890

C₂

yes

maximal, characteristic, complemented

#15689105

C₂

yes

maximal, complemented

C₂×#13093940

C₂

yes

maximal, complemented

C₄₇×#568

C₂

yes

maximal, characteristic, complemented

#13094070

—

#13094070

—

C₂×#10625354

C₄

yes

—

C₂×#10625350

C₂²

yes

—

C₂²×D₁₈₈

C₂²

yes

—

C₂×D₄×D₄₇

C₄

yes

—

#13094031

—

C₂×#10625329

C₂²

yes

complemented

#13093942

C₂²

yes

complemented

#13093940

C₂²

yes

complemented

C₂×#41×C₄₇

C₂²

yes

characteristic

#131×C₄₇

—

C₂×M₁₆×C₄₇

C₂²

yes

complemented

C₂³×D₄₇

C₂×C₄

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

—

#10625354

—

D₄×D₄₇

—

#10625350

—

#10625350

—

C₂×D₁₈₈

—

C₂×D₁₈₈

—

C₂×D₁₈₈

C₂×C₄

yes

—

C₂×D₁₈₈

C₂³

yes

—

C₂×C₄×D₄₇

C₂×C₄

yes

—

C₂×Dic₉₄

C₂³

yes

characteristic

M₁₆×C₄₇

—

M₁₆×C₄₇

C₂³

yes

—

C₂×C₈×C₄₇

—

#10625329

—

#10625329

C₂³

yes

—

C₂×C₄₇⋊C₈

—

C₂³×C₄₇

C₂²×C₄

yes

—

C₂²×D₄₇

—

C₂²×D₄₇

—

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

C₂²×C₄

yes

—

D₁₈₈

—

D₁₈₈

—

D₁₈₈

—

C₄×D₄₇

—

Dic₉₄

—

Dic₉₄

—

C₈×C₄₇

—

C₄₇⋊C₈

—

C₂²×C₄₇

—

C₂²×C₄₇

C₂×C₂²⋊C₄

yes

—

C₂²×C₄₇

C₂³×C₄

yes

characteristic

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

—

C₄×C₄₇

C₂×C₂²⋊C₄

yes

—

C₄×C₄₇

C₂²×D₄

yes

characteristic

C₄×C₄₇

C₂²×D₄

yes

characteristic

Dic₄₇

—

#2888

—

maximal, Sylow

C₂×C₄₇

—

C₂×C₄₇

—

C₂×C₄₇

C₂²×C₂²⋊C₄

yes

characteristic

D₄₇

—

C₂×H(5,𝔽₂)

—

#568

—

#568

D₄₇

yes

characteristic, complemented

#413

—

C₂×#100

—

C₄₇

#2888

yes

characteristic, complemented, Sylow

H(5,𝔽₂)

—

H(5,𝔽₂)

—

C₂×#41

—

C₂×#41

—

C₂×#41

C₂×D₄₇

yes

characteristic

C₂²×D₄

—

C₂²×D₄

—

#131

—

#131

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₄₇

yes

complemented

#101

—

#100

—

C₂⁴

—

#41

—

#41

—

#41

—

100

#41

D₁₈₈

yes

—

101

#41

C₂²⋊D₄₇

yes

—

102

C₂×Q₈

—

103

C₂×Q₈

C₄×D₄₇

yes

characteristic

104

C₂×D₄

—

105

C₂×D₄

—

106

C₂×D₄

—

107

C₂×D₄

—

108

C₂×D₄

—

109

C₂×D₄

—

110

C₂×D₄

C₄×D₄₇

yes

—

111

C₂²×C₄

—

112

C₂²×C₄

C₂²×D₄₇

yes

—

113

M₁₆

—

114

M₁₆

—

115

M₁₆

—

116

M₁₆

C₂²×D₄₇

yes

—

117

C₂×C₈

—

118

C₂×C₈

—

119

C₂³

—

120

C₂³

—

121

C₂³

—

376

—

122

C₂³

C₂×C₄×D₄₇

yes

—

123

Q₈

—

124

Q₈

—

125

Q₈

(C₂×D₄₇)⋊C₄

yes

—

126

D₄

—

127

D₄

—

128

D₄

—

129

D₄

—

130

D₄

(C₂×D₄₇)⋊C₄

yes

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

C₂×C₄×D₄₇

yes

—

134

C₂×C₄

C₂×D₁₈₈

yes

—

135

C₂×C₄

C₂×C₂²⋊D₄₇

yes

—

136

C₂×C₄

C₂³×D₄₇

yes

characteristic

137

C₈

—

138

C₈

—

139

C₂²

—

140

C₂²

—

141

C₂²

—

376

—

142

C₂²

—

188

—

143

C₂²

C₂×(C₂×D₄₇)⋊C₄

yes

—

144

C₂²

C₂²×C₄×D₄₇

yes

characteristic

145

C₄

—

146

C₄

C₂×(C₂×D₄₇)⋊C₄

yes

—

147

C₄

C₂²×C₂²⋊D₄₇

yes

characteristic

148

C₄

C₂²×D₁₈₈

yes

characteristic

149

C₂

—

150

C₂

—

151

C₂

—

188

—

152

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	#2888
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×#13093940	C₂	semidirect
5. C₄₇×#568	C₂	semidirect
13. C₂×#10625329	C₂²	semidirect
14. #13093942	C₂²	semidirect
15. #13093940	C₂²	semidirect
18. C₂×M₁₆×C₄₇	C₂²	semidirect
79. #568	D₄₇	semidirect
82. C₄₇	#2888	semidirect
93. C₂×M₁₆	C₂×D₄₇	semidirect

H ≅ G/N

type

2. #15689890

C₂

semidirect

3. #15689105

C₂

semidirect

4. C₂×#13093940

C₂

semidirect

5. C₄₇×#568

C₂

semidirect

13. C₂×#10625329

C₂²

semidirect

14. #13093942

C₂²

semidirect

15. #13093940

C₂²

semidirect

18. C₂×M₁₆×C₄₇

C₂²

semidirect

79. #568

D₄₇

semidirect

82. C₄₇

#2888

semidirect

93. C₂×M₁₆

C₂×D₄₇

semidirect

Subgroups — #20251866

Conjugacy classes of subgroups

Complements