Group #20252519 — subgroups

Subgroups — #20252519

Order 6016.

394 conjugacy classes of subgroups; only the 179 normal subgroups are shown.

normal subgroups only

This group has 394 conjugacy classes of subgroups — too many to draw. Only its 179 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	C₄×#10625354	C₂	1	1	yes	maximal, characteristic
2	Q₈×C₄₇⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
3	#15689202	C₂	1	3	yes	maximal, complemented
4	#15689200	C₂	1	3	yes	maximal, complemented
5	#15689174	C₂	1	3	yes	maximal, complemented
6	#15689145	C₂	1	3	yes	maximal, complemented
7	C₈×Q₈×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#10625354	C₄	1	1	yes	characteristic
9	C₂×#10625324	C₂²	1	6	yes	—
10	Q₈×Dic₄₇	C₄	1	1	yes	characteristic
11	C₄×D₁₈₈	C₂²	1	3	yes	—
12	#13093986	C₄	1	3	yes	—
13	#13093982	C₄	1	3	yes	—
14	C₄²×D₄₇	C₂²	1	3	yes	—
15	C₄⋊(C₄₇⋊C₈)	C₂²	1	3	yes	—
16	Dic₄₇⋊C₈	C₂²	1	3	yes	—
17	Dic₄₇⋊C₈	C₂²	1	1	yes	characteristic
18	(C₂×D₄₇)⋊C₈	C₂²	1	3	yes	—
19	(C₂×D₄₇)⋊C₈	C₂²	1	3	yes	—
20	C₄×C₄₇⋊C₈	C₂²	1	3	yes	—
21	C₄×Q₈×C₄₇	C₂²	1	1	yes	characteristic
22	C₄⋊C₈×C₄₇	C₂²	1	3	yes	—
23	C₄×C₈×C₄₇	C₂²	1	3	yes	—
24	C₂×Q₈×C₄₇	C₂×C₄	1	1	yes	characteristic
25	#10625354	—	2	4	no	—
26	C₂×D₁₈₈	C₂×C₄	1	3	yes	—
27	C₂×C₄×D₄₇	C₂×C₄	1	3	yes	—
28	C₂×C₄×D₄₇	C₂×C₄	1	3	yes	—
29	C₂×C₄×D₄₇	C₂³	1	3	yes	—
30	C₂×C₈×C₄₇	C₂³	1	3	yes	—
31	C₂×C₈×C₄₇	C₂³	1	1	yes	characteristic
32	C₄⋊C₄×C₄₇	C₂×C₄	1	3	yes	—
33	C₄²×C₄₇	C₂³	1	3	yes	—
34	(C₂×D₄₇)⋊C₄	C₂×C₄	1	6	yes	—
35	C₄⋊Dic₄₇	C₂×C₄	1	3	yes	—
36	C₄×Dic₄₇	C₂×C₄	1	3	yes	—
37	C₄×Dic₄₇	C₂×C₄	1	3	yes	—
38	C₄×Dic₄₇	C₂³	1	1	yes	characteristic
39	C₂×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
40	C₂×C₄₇⋊C₈	C₂³	1	3	yes	—
41	#10625324	—	4	6	no	—
42	C₂²×D₄₇	C₂²×C₄	1	3	yes	—
43	Q₈×C₄₇	M₁₆	1	4	yes	complemented
44	C₂×C₄×C₄₇	C₂²×C₄	1	3	yes	—
45	C₂×C₄×C₄₇	C₂²×C₄	1	3	yes	—
46	C₂×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
47	C₂×Dic₄₇	C₂²×C₄	1	1	yes	characteristic
48	C₂×Dic₄₇	C₂²×C₄	1	3	yes	—
49	C₂×Dic₄₇	C₂²×C₄	1	1	yes	characteristic
50	D₁₈₈	—	2	6	no	—
51	C₄×D₄₇	—	2	6	no	—
52	C₄×D₄₇	—	2	3	no	—
53	C₄×D₄₇	—	2	3	no	—
54	C₄×D₄₇	—	2	6	no	—
55	C₈×C₄₇	—	2	3	no	—
56	C₈×C₄₇	—	2	1	no	—
57	C₄₇⋊C₈	—	2	3	no	—
58	C₄₇⋊C₈	—	2	1	no	—
59	C₂²×C₄₇	C₂³×C₄	1	1	yes	characteristic
60	C₂×D₄₇	—	2	3	no	—
61	C₂×D₄₇	—	2	3	no	—
62	C₂×D₄₇	—	2	3	no	—
63	C₄×C₄₇	—	2	3	no	—
64	C₄×C₄₇	C₂×M₁₆	1	6	yes	—
65	C₄×C₄₇	H(5,𝔽₂)	1	1	yes	characteristic
66	C₄×C₄₇	2₋¹⁺⁴	1	1	yes	characteristic
67	Dic₄₇	—	2	3	no	—
68	Dic₄₇	—	2	1	no	—
69	Dic₄₇	#131	1	2	yes	—
70	#2991	—	47	1	no	maximal, Sylow
71	C₂×C₄₇	#520	1	1	yes	characteristic
72	C₂×C₄₇	C₂²×M₁₆	1	1	yes	characteristic
73	C₂×C₄₇	C₂×#131	1	1	yes	characteristic
74	D₄₇	—	4	3	no	—
75	C₄×#41	—	47	1	no	—
76	C₈×Q₈	—	47	1	no	—
77	C₈×Q₈	D₄₇	1	1	yes	characteristic, complemented
78	#436	—	47	3	no	—
79	#436	—	47	3	no	—
80	#431	—	47	3	no	—
81	#423	—	47	3	no	—
82	C₄₇	#2991	1	1	yes	characteristic, complemented, Sylow
83	C₂×#41	—	47	1	no	—
84	C₂×M₁₆	—	47	6	no	—
85	C₄×Q₈	—	47	1	no	—
86	C₄×Q₈	C₂×D₄₇	1	1	yes	characteristic
87	C₄×D₄	—	47	3	no	—
88	C₄×D₄	—	47	3	no	—
89	#117	—	47	3	no	—
90	C₂×C₄²	—	47	3	no	—
91	C₄⋊C₈	—	47	3	no	—
92	C₄⋊C₈	—	47	1	no	—
93	C₄⋊C₈	—	47	3	no	—
94	C₄⋊C₈	C₂×D₄₇	1	3	yes	—
95	C₂²⋊C₈	—	47	3	no	—
96	C₂²⋊C₈	—	47	3	no	—
97	C₄×C₈	—	47	3	no	—
98	C₄×C₈	C₂×D₄₇	1	3	yes	—
99	#41	—	94	4	no	—
100	C₂×Q₈	C₄×D₄₇	1	1	yes	characteristic
101	C₂×D₄	—	47	3	no	—
102	C₂²×C₄	—	47	3	no	—
103	C₂²×C₄	—	47	3	no	—
104	C₂²×C₄	—	47	3	no	—
105	M₁₆	—	188	6	no	—
106	C₂×C₈	—	47	1	no	—
107	C₂×C₈	—	47	3	no	—
108	C₂×C₈	C₂²×D₄₇	1	1	yes	characteristic
109	C₂×C₈	C₂²×D₄₇	1	3	yes	—
110	C₄⋊C₄	—	47	3	no	—
111	C₄⋊C₄	C₄×D₄₇	1	3	yes	—
112	C₂²⋊C₄	—	47	6	no	—
113	C₄²	—	47	3	no	—
114	C₄²	—	47	3	no	—
115	C₄²	—	47	1	no	—
116	C₄²	C₂²×D₄₇	1	3	yes	—
117	C₂³	—	47	3	no	—
118	Q₈	#10625324	1	4	yes	complemented
119	D₄	—	94	6	no	—
120	C₂×C₄	—	47	1	no	—
121	C₂×C₄	—	47	1	no	—
122	C₂×C₄	—	47	3	no	—
123	C₂×C₄	—	94	6	no	—
124	C₂×C₄	—	94	6	no	—
125	C₂×C₄	—	94	3	no	—
126	C₂×C₄	—	94	3	no	—
127	C₂×C₄	C₂×C₄×D₄₇	1	3	yes	—
128	C₂×C₄	C₂×C₄×D₄₇	1	3	yes	—
129	C₂×C₄	C₂³×D₄₇	1	1	yes	characteristic
130	C₈	—	2	3	no	—
131	C₈	—	2	1	no	—
132	C₈	—	94	3	no	—
133	C₈	—	94	1	no	—
134	C₂²	—	94	3	no	—
135	C₂²	—	94	3	no	—
136	C₂²	—	94	3	no	—
137	C₂²	C₂²×C₄×D₄₇	1	1	yes	characteristic
138	C₄	—	2	3	no	—
139	C₄	—	47	2	no	—
140	C₄	—	94	3	no	—
141	C₄	—	94	1	no	—
142	C₄	C₂×#10625324	1	6	yes	—
143	C₄	#13094070	1	1	yes	characteristic
144	C₄	#13094072	1	1	yes	characteristic
145	C₂	—	188	3	no	—
146	C₂	#15689605	1	1	yes	characteristic
147	C₂	C₂²×#10625324	1	1	yes	characteristic
148	C₂	#131×D₄₇	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×#10625354

C₂

yes

maximal, characteristic

Q₈×C₄₇⋊C₈

C₂

yes

maximal, characteristic, complemented

#15689202

C₂

yes

maximal, complemented

#15689200

C₂

yes

maximal, complemented

#15689174

C₂

yes

maximal, complemented

#15689145

C₂

yes

maximal, complemented

C₈×Q₈×C₄₇

C₂

yes

maximal, characteristic, complemented

C₂×#10625354

C₄

yes

characteristic

C₂×#10625324

C₂²

yes

—

Q₈×Dic₄₇

C₄

yes

characteristic

C₄×D₁₈₈

C₂²

yes

—

#13093986

C₄

yes

—

#13093982

C₄

yes

—

C₄²×D₄₇

C₂²

yes

—

C₄⋊(C₄₇⋊C₈)

C₂²

yes

—

Dic₄₇⋊C₈

C₂²

yes

—

Dic₄₇⋊C₈

C₂²

yes

characteristic

(C₂×D₄₇)⋊C₈

C₂²

yes

—

(C₂×D₄₇)⋊C₈

C₂²

yes

—

C₄×C₄₇⋊C₈

C₂²

yes

—

C₄×Q₈×C₄₇

C₂²

yes

characteristic

C₄⋊C₈×C₄₇

C₂²

yes

—

C₄×C₈×C₄₇

C₂²

yes

—

C₂×Q₈×C₄₇

C₂×C₄

yes

characteristic

#10625354

—

C₂×D₁₈₈

C₂×C₄

yes

—

C₂×C₄×D₄₇

C₂×C₄

yes

—

C₂×C₄×D₄₇

C₂×C₄

yes

—

C₂×C₄×D₄₇

C₂³

yes

—

C₂×C₈×C₄₇

C₂³

yes

—

C₂×C₈×C₄₇

C₂³

yes

characteristic

C₄⋊C₄×C₄₇

C₂×C₄

yes

—

C₄²×C₄₇

C₂³

yes

—

(C₂×D₄₇)⋊C₄

C₂×C₄

yes

—

C₄⋊Dic₄₇

C₂×C₄

yes

—

C₄×Dic₄₇

C₂×C₄

yes

—

C₄×Dic₄₇

C₂×C₄

yes

—

C₄×Dic₄₇

C₂³

yes

characteristic

C₂×C₄₇⋊C₈

C₂³

yes

characteristic

C₂×C₄₇⋊C₈

C₂³

yes

—

#10625324

—

C₂²×D₄₇

C₂²×C₄

yes

—

Q₈×C₄₇

M₁₆

yes

complemented

C₂×C₄×C₄₇

C₂²×C₄

yes

—

C₂×C₄×C₄₇

C₂²×C₄

yes

—

C₂×C₄×C₄₇

C₂⁴

yes

characteristic

C₂×Dic₄₇

C₂²×C₄

yes

characteristic

C₂×Dic₄₇

C₂²×C₄

yes

—

C₂×Dic₄₇

C₂²×C₄

yes

characteristic

D₁₈₈

—

C₄×D₄₇

—

C₄×D₄₇

—

C₄×D₄₇

—

C₄×D₄₇

—

C₈×C₄₇

—

C₈×C₄₇

—

C₄₇⋊C₈

—

C₄₇⋊C₈

—

C₂²×C₄₇

C₂³×C₄

yes

characteristic

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

—

C₄×C₄₇

—

C₄×C₄₇

C₂×M₁₆

yes

—

C₄×C₄₇

H(5,𝔽₂)

yes

characteristic

C₄×C₄₇

2₋¹⁺⁴

yes

characteristic

Dic₄₇

—

Dic₄₇

—

Dic₄₇

#131

yes

—

—

C₂×C₄₇

yes

C₂×C₄₇

C₂²×M₁₆

yes

characteristic

C₂×C₄₇

C₂×#131

yes

characteristic

D₄₇

—

C₄×#41

—

C₈×Q₈

—

C₈×Q₈

D₄₇

yes

characteristic, complemented

—

—

—

—

C₄₇

#2991

yes

characteristic, complemented, Sylow

C₂×#41

—

C₂×M₁₆

—

C₄×Q₈

—

C₄×Q₈

C₂×D₄₇

yes

characteristic

C₄×D₄

—

C₄×D₄

—

#117

—

C₂×C₄²

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

C₂×D₄₇

yes

—

C₂²⋊C₈

—

C₂²⋊C₈

—

C₄×C₈

—

C₄×C₈

C₂×D₄₇

yes

—

#41

—

100

C₂×Q₈

C₄×D₄₇

yes

characteristic

101

C₂×D₄

—

102

C₂²×C₄

—

103

C₂²×C₄

—

104

C₂²×C₄

—

105

M₁₆

—

188

—

106

C₂×C₈

—

107

C₂×C₈

—

108

C₂×C₈

C₂²×D₄₇

yes

characteristic

109

C₂×C₈

C₂²×D₄₇

yes

—

110

C₄⋊C₄

—

111

C₄⋊C₄

C₄×D₄₇

yes

—

112

C₂²⋊C₄

—

113

C₄²

—

114

C₄²

—

115

C₄²

—

116

C₄²

C₂²×D₄₇

yes

—

117

C₂³

—

118

Q₈

#10625324

yes

complemented

119

D₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

C₂×C₄×D₄₇

yes

—

128

C₂×C₄

C₂×C₄×D₄₇

yes

—

129

C₂×C₄

C₂³×D₄₇

yes

characteristic

130

C₈

—

131

C₈

—

132

C₈

—

133

C₈

—

134

C₂²

—

135

C₂²

—

136

C₂²

—

137

C₂²

C₂²×C₄×D₄₇

yes

characteristic

138

C₄

—

139

C₄

—

140

C₄

—

141

C₄

—

142

C₄

C₂×#10625324

yes

—

143

C₄

#13094070

yes

characteristic

144

C₄

#13094072

yes

characteristic

145

C₂

—

188

—

146

C₂

#15689605

yes

characteristic

147

C₂

C₂²×#10625324

yes

characteristic

148

C₂

#131×D₄₇

yes

characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₄₇
commutator, socle	C₂²×C₄₇	C₂³×C₄
Fitting	C₈×Q₈×C₄₇	C₂
hypercenter	C₈×Q₈	D₄₇
hypocenter	C₄₇	#2991
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₈×Q₈

subgroup

G/H

center, Frattini

C₂×C₄

C₂³×D₄₇

commutator, socle

C₂²×C₄₇

C₂³×C₄

Fitting

C₈×Q₈×C₄₇

C₂

hypercenter

C₈×Q₈

D₄₇

C₄₇

G ▷ C₂²×C₄₇ ▷ C₄₇

C₁ ◁ C₂×C₄ ◁ C₈×Q₈

N	H ≅ G/N	type
2. Q₈×C₄₇⋊C₈	C₂	semidirect
3. #15689202	C₂	semidirect
4. #15689200	C₂	semidirect
5. #15689174	C₂	semidirect
6. #15689145	C₂	semidirect
7. C₈×Q₈×C₄₇	C₂	semidirect
43. Q₈×C₄₇	M₁₆	semidirect
77. C₈×Q₈	D₄₇	semidirect
82. C₄₇	#2991	semidirect
118. Q₈	#10625324	semidirect

H ≅ G/N

type

2. Q₈×C₄₇⋊C₈

C₂

semidirect

3. #15689202

C₂

semidirect

4. #15689200

C₂

semidirect

5. #15689174

C₂

semidirect

6. #15689145

C₂

semidirect

7. C₈×Q₈×C₄₇

C₂

semidirect

43. Q₈×C₄₇

M₁₆

semidirect

77. C₈×Q₈

D₄₇

semidirect

82. C₄₇

#2991

semidirect

118. Q₈

#10625324

semidirect

Subgroups — #20252519

Conjugacy classes of subgroups

Complements