Group #20256026 — subgroups

Subgroups — #20256026

Order 6016.

370 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 139 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	Q₈×D₁₈₈	C₂	1	1	yes	maximal, characteristic
2	#15689473	C₂	1	3	yes	maximal, complemented
3	#15689408	C₂	1	3	yes	maximal, complemented
4	#15689325	C₂	1	3	yes	maximal
5	Q₈×C₄₇⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄×#10625336	C₂	1	3	yes	maximal
7	C₄₇×#557	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×Q₈×D₄₇	—	2	1	no	—
9	C₂×#10625336	C₂²	1	3	yes	—
10	#13094024	C₂²	1	3	yes	—
11	#13094004	—	2	3	no	—
12	C₄×D₁₈₈	C₂²	1	3	yes	—
13	#13093955	C₂²	1	3	yes	—
14	C₄⋊(C₄₇⋊C₈)	C₂²	1	3	yes	—
15	#13093948	C₂²	1	3	yes	—
16	#13093948	C₂²	1	6	yes	—
17	#13093944	C₂²	1	1	yes	characteristic
18	C₄×C₄₇⋊C₈	C₂²	1	3	yes	—
19	C₄⋊Q₈×C₄₇	C₂²	1	3	yes	—
20	#125×C₄₇	—	2	3	no	—
21	C₄×Q₈×C₄₇	—	2	1	no	—
22	C₄×Q₈×C₄₇	C₂²	1	3	yes	—
23	C₄×Q₈×C₄₇	C₂²	1	1	yes	characteristic
24	C₂×Q₈×C₄₇	D₄	1	1	yes	characteristic
25	C₂×Q₈×C₄₇	C₂³	1	3	yes	—
26	Q₈×D₄₇	—	4	4	no	—
27	C₂×D₁₈₈	C₂³	1	1	yes	characteristic
28	C₂×C₄×D₄₇	—	2	3	no	—
29	C₂×Dic₉₄	—	2	3	no	—
30	C₄⋊C₄×C₄₇	—	2	3	no	—
31	C₄⋊C₄×C₄₇	—	2	3	no	—
32	C₄⋊C₄×C₄₇	—	2	3	no	—
33	C₄⋊C₄×C₄₇	D₄	1	3	yes	—
34	C₄⋊C₄×C₄₇	C₂³	1	1	yes	characteristic
35	C₄²×C₄₇	—	2	3	no	—
36	C₄²×C₄₇	C₂³	1	3	yes	—
37	#10625336	—	2	6	no	—
38	(C₂×D₄₇)⋊C₄	—	2	3	no	—
39	C₄⋊Dic₄₇	—	2	3	no	—
40	C₄⋊Dic₄₇	C₂³	1	3	yes	—
41	C₂×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
42	C₂×C₄₇⋊C₈	C₂³	1	3	yes	—
43	C₂²×D₄₇	—	2	1	no	—
44	Q₈×C₄₇	—	2	3	no	—
45	Q₈×C₄₇	—	2	3	no	—
46	Q₈×C₄₇	SD₁₆	1	4	yes	complemented
47	C₂×C₄×C₄₇	—	2	1	no	—
48	C₂×C₄×C₄₇	—	2	3	no	—
49	C₂×C₄×C₄₇	C₂×D₄	1	3	yes	—
50	C₂×C₄×C₄₇	C₂×D₄	1	3	yes	—
51	C₂×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
52	C₂×Dic₄₇	—	2	3	no	—
53	D₁₈₈	—	2	1	no	—
54	D₁₈₈	#41	1	2	yes	—
55	C₄×D₄₇	—	4	6	no	—
56	Dic₉₄	—	4	6	no	—
57	C₄₇⋊C₈	—	2	3	no	—
58	C₄₇⋊C₈	—	2	1	no	—
59	C₂²×C₄₇	C₂²×D₄	1	1	yes	characteristic
60	C₂×D₄₇	—	2	2	no	—
61	C₂×D₄₇	—	4	1	no	—
62	C₂×D₄₇	—	4	1	no	—
63	C₄×C₄₇	—	2	3	no	—
64	C₄×C₄₇	—	4	3	no	—
65	C₄×C₄₇	—	4	1	no	—
66	C₄×C₄₇	C₂×SD₁₆	1	6	yes	—
67	C₄×C₄₇	C₂×#41	1	1	yes	characteristic
68	C₄×C₄₇	2₋¹⁺⁴	1	1	yes	characteristic
69	Dic₄₇	—	4	3	no	—
70	#3359	—	47	1	no	maximal, Sylow
71	C₂×C₄₇	#548	1	1	yes	characteristic
72	C₂×C₄₇	C₂²×SD₁₆	1	1	yes	characteristic
73	C₂×C₄₇	#578	1	1	yes	characteristic
74	D₄₇	—	4	2	no	—
75	#557	D₄₇	1	1	yes	characteristic, complemented
76	D₄×Q₈	—	47	1	no	—
77	#487	—	47	3	no	—
78	#475	—	47	3	no	—
79	#461	—	47	3	no	—
80	C₈×Q₈	—	47	1	no	—
81	C₄×SD₁₆	—	47	3	no	—
82	C₄₇	#3359	1	1	yes	characteristic, complemented, Sylow
83	C₂²×Q₈	—	94	1	no	—
84	C₂×SD₁₆	—	47	3	no	—
85	C₄⋊Q₈	—	47	3	no	—
86	C₄⋊Q₈	C₂×D₄₇	1	3	yes	—
87	#125	—	2	3	no	—
88	C₂²⋊Q₈	—	94	3	no	—
89	C₄×Q₈	—	2	1	no	—
90	C₄×Q₈	C₂×D₄₇	1	3	yes	—
91	C₄×Q₈	C₂×D₄₇	1	1	yes	characteristic
92	C₄×D₄	—	47	3	no	—
93	#106	—	47	3	no	—
94	C₄⋊C₈	—	47	3	no	—
95	#103	—	47	3	no	—
96	#103	—	47	6	no	—
97	#102	—	47	1	no	—
98	C₄×C₈	—	47	3	no	—
99	C₂×Q₈	—	94	3	no	—
100	C₂×Q₈	—	188	4	no	—
101	C₂×Q₈	C₂²⋊D₄₇	1	1	yes	characteristic
102	C₂×Q₈	C₂²×D₄₇	1	3	yes	—
103	C₂×D₄	—	47	1	no	—
104	C₂²×C₄	—	94	3	no	—
105	SD₁₆	—	94	6	no	—
106	C₂×C₈	—	47	1	no	—
107	C₂×C₈	—	47	3	no	—
108	C₄⋊C₄	—	2	3	no	—
109	C₄⋊C₄	—	2	3	no	—
110	C₄⋊C₄	—	2	3	no	—
111	C₄⋊C₄	—	47	3	no	—
112	C₄⋊C₄	—	94	3	no	—
113	C₄⋊C₄	C₂²⋊D₄₇	1	3	yes	—
114	C₄⋊C₄	C₂²×D₄₇	1	1	yes	characteristic
115	C₂²⋊C₄	—	94	3	no	—
116	C₄²	—	2	3	no	—
117	C₄²	C₂²×D₄₇	1	3	yes	—
118	C₂³	—	94	1	no	—
119	Q₈	—	2	3	no	—
120	Q₈	—	2	3	no	—
121	Q₈	—	188	6	no	—
122	Q₈	#10625336	1	4	yes	complemented
123	D₄	—	47	2	no	—
124	D₄	—	94	1	no	—
125	C₂×C₄	—	2	1	no	—
126	C₂×C₄	—	2	3	no	—
127	C₂×C₄	—	94	3	no	—
128	C₂×C₄	—	188	6	no	—
129	C₂×C₄	C₂×C₂²⋊D₄₇	1	3	yes	—
130	C₂×C₄	C₂×C₂²⋊D₄₇	1	3	yes	—
131	C₂×C₄	C₂³×D₄₇	1	1	yes	characteristic
132	C₈	—	94	3	no	—
133	C₈	—	94	1	no	—
134	C₂²	—	94	2	no	—
135	C₂²	—	188	1	no	—
136	C₂²	—	188	1	no	—
137	C₂²	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
138	C₄	—	2	3	no	—
139	C₄	—	4	3	no	—
140	C₄	—	4	1	no	—
141	C₄	—	188	3	no	—
142	C₄	C₂×#10625336	1	6	yes	—
143	C₄	#41×D₄₇	1	1	yes	characteristic
144	C₄	#13094071	1	1	yes	characteristic
145	C₂	—	188	2	no	—
146	C₂	#15689780	1	1	yes	characteristic
147	C₂	C₂²×#10625336	1	1	yes	characteristic
148	C₂	#15689938	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

Q₈×D₁₈₈

C₂

yes

maximal, characteristic

#15689473

C₂

yes

maximal, complemented

#15689408

C₂

yes

maximal, complemented

#15689325

C₂

yes

maximal

Q₈×C₄₇⋊C₈

C₂

yes

maximal, characteristic, complemented

C₄×#10625336

C₂

yes

maximal

C₄₇×#557

C₂

yes

maximal, characteristic, complemented

C₂×Q₈×D₄₇

—

C₂×#10625336

C₂²

yes

—

#13094024

C₂²

yes

—

#13094004

—

C₄×D₁₈₈

C₂²

yes

—

#13093955

C₂²

yes

—

C₄⋊(C₄₇⋊C₈)

C₂²

yes

—

#13093948

C₂²

yes

—

#13093948

C₂²

yes

—

#13093944

C₂²

yes

characteristic

C₄×C₄₇⋊C₈

C₂²

yes

—

C₄⋊Q₈×C₄₇

C₂²

yes

—

#125×C₄₇

—

C₄×Q₈×C₄₇

—

C₄×Q₈×C₄₇

C₂²

yes

—

C₄×Q₈×C₄₇

C₂²

yes

characteristic

C₂×Q₈×C₄₇

D₄

yes

characteristic

C₂×Q₈×C₄₇

C₂³

yes

—

Q₈×D₄₇

—

C₂×D₁₈₈

C₂³

yes

characteristic

C₂×C₄×D₄₇

—

C₂×Dic₉₄

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

D₄

yes

—

C₄⋊C₄×C₄₇

C₂³

yes

characteristic

C₄²×C₄₇

—

C₄²×C₄₇

C₂³

yes

—

#10625336

—

(C₂×D₄₇)⋊C₄

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

C₂³

yes

—

C₂×C₄₇⋊C₈

C₂³

yes

characteristic

C₂×C₄₇⋊C₈

C₂³

yes

—

C₂²×D₄₇

—

Q₈×C₄₇

—

Q₈×C₄₇

—

Q₈×C₄₇

SD₁₆

yes

complemented

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂×D₄

yes

—

C₂×C₄×C₄₇

C₂×D₄

yes

—

C₂×C₄×C₄₇

C₂⁴

yes

characteristic

C₂×Dic₄₇

—

D₁₈₈

—

D₁₈₈

#41

yes

—

C₄×D₄₇

—

Dic₉₄

—

C₄₇⋊C₈

—

C₄₇⋊C₈

—

C₂²×C₄₇

C₂²×D₄

yes

characteristic

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

C₂×SD₁₆

yes

—

C₄×C₄₇

C₂×#41

yes

characteristic

C₄×C₄₇

2₋¹⁺⁴

yes

characteristic

Dic₄₇

—

—

C₂×C₄₇

yes

C₂×C₄₇

C₂²×SD₁₆

yes

characteristic

C₂×C₄₇

#578

yes

characteristic

D₄₇

—

#557

D₄₇

yes

characteristic, complemented

D₄×Q₈

—

#487

—

#475

—

#461

—

C₈×Q₈

—

C₄×SD₁₆

—

C₄₇

#3359

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₂×SD₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₄₇

yes

—

#125

—

C₂²⋊Q₈

—

C₄×Q₈

—

C₄×Q₈

C₂×D₄₇

yes

—

C₄×Q₈

C₂×D₄₇

yes

characteristic

C₄×D₄

—

#106

—

C₄⋊C₈

—

#103

—

#103

—

#102

—

C₄×C₈

—

C₂×Q₈

—

100

C₂×Q₈

—

188

—

101

C₂×Q₈

C₂²⋊D₄₇

yes

characteristic

102

C₂×Q₈

C₂²×D₄₇

yes

—

103

C₂×D₄

—

104

C₂²×C₄

—

105

SD₁₆

—

106

C₂×C₈

—

107

C₂×C₈

—

108

C₄⋊C₄

—

109

C₄⋊C₄

—

110

C₄⋊C₄

—

111

C₄⋊C₄

—

112

C₄⋊C₄

—

113

C₄⋊C₄

C₂²⋊D₄₇

yes

—

114

C₄⋊C₄

C₂²×D₄₇

yes

characteristic

115

C₂²⋊C₄

—

116

C₄²

—

117

C₄²

C₂²×D₄₇

yes

—

118

C₂³

—

119

Q₈

—

120

Q₈

—

121

Q₈

—

188

—

122

Q₈

#10625336

yes

complemented

123

D₄

—

124

D₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

188

—

129

C₂×C₄

C₂×C₂²⋊D₄₇

yes

—

130

C₂×C₄

C₂×C₂²⋊D₄₇

yes

—

131

C₂×C₄

C₂³×D₄₇

yes

characteristic

132

C₈

—

133

C₈

—

134

C₂²

—

135

C₂²

—

188

—

136

C₂²

—

188

—

137

C₂²

C₂²×C₂²⋊D₄₇

yes

characteristic

138

C₄

—

139

C₄

—

140

C₄

—

141

C₄

—

188

—

142

C₄

C₂×#10625336

yes

—

143

C₄

#41×D₄₇

yes

characteristic

144

C₄

#13094071

yes

characteristic

145

C₂

—

188

—

146

C₂

#15689780

yes

characteristic

147

C₂

C₂²×#10625336

yes

characteristic

148

C₂

#15689938

yes

characteristic

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₄₇
commutator	C₂×C₄×C₄₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₄₇
Fitting	C₄₇×#557	C₂
socle	C₂²×C₄₇	C₂²×D₄
hypercenter	#557	D₄₇
hypocenter	C₄₇	#3359
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ #557

subgroup

G/H

center

C₂²

C₂²×C₂²⋊D₄₇

commutator

C₂×C₄×C₄₇

C₂⁴

Frattini

C₂×C₄

C₂³×D₄₇

Fitting

C₄₇×#557

C₂

socle

C₂²×C₄₇

C₂²×D₄

D₄₇

C₄₇

G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂² ◁ C₄×Q₈ ◁ #557

N	H ≅ G/N	type
2. #15689473	C₂	semidirect
3. #15689408	C₂	semidirect
5. Q₈×C₄₇⋊C₈	C₂	semidirect
7. C₄₇×#557	C₂	semidirect
46. Q₈×C₄₇	SD₁₆	semidirect
75. #557	D₄₇	semidirect
82. C₄₇	#3359	semidirect
122. Q₈	#10625336	semidirect

H ≅ G/N

type

2. #15689473

C₂

semidirect

3. #15689408

C₂

semidirect

5. Q₈×C₄₇⋊C₈

C₂

semidirect

7. C₄₇×#557

C₂

semidirect

46. Q₈×C₄₇

SD₁₆

semidirect

75. #557

D₄₇

semidirect

82. C₄₇

#3359

semidirect

122. Q₈

#10625336

semidirect

Subgroups — #20256026

Conjugacy classes of subgroups

Complements