Group #20256055 — subgroups

Subgroups — #20256055

Order 6016.

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

normal subgroups only

This group has 392 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×Dic₉₄	C₂	1	1	yes	maximal, characteristic, complemented
2	#15689469	C₂	1	3	yes	maximal, complemented
3	#15689413	C₂	1	3	yes	maximal, complemented
4	#15689320	C₂	1	3	yes	maximal, complemented
5	Q₈×C₄₇⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄×#10625335	C₂	1	3	yes	maximal, complemented
7	C₄₇×#550	C₂	1	1	yes	maximal, characteristic
8	C₂×#10625335	C₂²	1	3	yes	—
9	#13094024	C₂²	1	3	yes	—
10	#13094023	—	2	3	no	—
11	Q₈×Dic₄₇	—	2	1	no	—
12	C₄×Dic₉₄	C₂²	1	3	yes	—
13	#13093955	C₂²	1	3	yes	—
14	C₄⋊(C₄₇⋊C₈)	C₂²	1	3	yes	—
15	#13093947	C₂²	1	1	yes	characteristic
16	#13093945	C₂²	1	6	yes	—
17	#13093945	C₂²	1	3	yes	—
18	C₄×C₄₇⋊C₈	C₂²	1	3	yes	—
19	C₂×#41×C₄₇	—	2	1	no	—
20	#127×C₄₇	C₂²	1	3	yes	—
21	#121×C₄₇	—	2	3	no	—
22	C₄×Q₈×C₄₇	C₂²	1	1	yes	characteristic
23	C₄×D₄×C₄₇	C₂²	1	3	yes	—
24	#41×C₄₇	—	4	4	no	—
25	C₂×Q₈×C₄₇	D₄	1	1	yes	characteristic
26	C₂×D₄×C₄₇	—	2	3	no	—
27	C₂×D₄×C₄₇	—	2	3	no	—
28	C₂×D₄×C₄₇	C₂³	1	3	yes	—
29	C₂²×C₄×C₄₇	—	2	3	no	—
30	C₂×Dic₉₄	—	2	3	no	—
31	C₂×Dic₉₄	C₂³	1	1	yes	characteristic
32	C₄⋊C₄×C₄₇	D₄	1	3	yes	—
33	C₄⋊C₄×C₄₇	C₂³	1	1	yes	characteristic
34	C₂²⋊C₄×C₄₇	—	2	3	no	—
35	C₄²×C₄₇	C₂³	1	3	yes	—
36	#10625335	—	2	6	no	—
37	C₄⋊Dic₄₇	—	2	3	no	—
38	C₄⋊Dic₄₇	C₂³	1	3	yes	—
39	Dic₄₇⋊C₄	—	2	3	no	—
40	C₄×Dic₄₇	—	2	3	no	—
41	C₂×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
42	C₂×C₄₇⋊C₈	C₂³	1	3	yes	—
43	C₂³×C₄₇	—	2	3	no	—
44	Q₈×C₄₇	SD₁₆	1	4	yes	complemented
45	D₄×C₄₇	—	2	3	no	—
46	D₄×C₄₇	—	2	3	no	—
47	D₄×C₄₇	—	4	6	no	—
48	D₄×C₄₇	—	8	3	no	—
49	C₂×C₄×C₄₇	—	2	1	no	—
50	C₂×C₄×C₄₇	—	4	6	no	—
51	C₂×C₄×C₄₇	C₂×D₄	1	3	yes	—
52	C₂×C₄×C₄₇	C₂×D₄	1	3	yes	—
53	C₂×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
54	C₂×Dic₄₇	—	2	1	no	—
55	C₂×Dic₄₇	—	2	3	no	—
56	Dic₉₄	—	2	1	no	—
57	Dic₉₄	—	8	3	no	—
58	Dic₉₄	#41	1	2	yes	complemented
59	C₄₇⋊C₈	—	2	3	no	—
60	C₄₇⋊C₈	—	2	1	no	—
61	C₂²×C₄₇	—	4	3	no	—
62	C₂²×C₄₇	—	4	3	no	—
63	C₂²×C₄₇	—	4	3	no	—
64	C₂²×C₄₇	C₂²×D₄	1	1	yes	characteristic
65	C₄×C₄₇	—	2	3	no	—
66	C₄×C₄₇	—	4	1	no	—
67	C₄×C₄₇	C₂×SD₁₆	1	6	yes	—
68	C₄×C₄₇	C₂×#41	1	1	yes	characteristic
69	C₄×C₄₇	2₋¹⁺⁴	1	1	yes	characteristic
70	Dic₄₇	—	2	2	no	—
71	Dic₄₇	—	4	3	no	—
72	#3362	—	47	1	no	maximal, Sylow
73	C₂×C₄₇	—	8	3	no	—
74	C₂×C₄₇	#548	1	1	yes	characteristic
75	C₂×C₄₇	C₂²×SD₁₆	1	1	yes	characteristic
76	C₂×C₄₇	#576	1	1	yes	characteristic
77	Q₈²	—	47	1	no	—
78	#550	D₄₇	1	1	yes	characteristic
79	#486	—	47	3	no	—
80	#476	—	47	3	no	—
81	#460	—	47	3	no	—
82	C₈×Q₈	—	47	1	no	—
83	C₄×SD₁₆	—	47	3	no	—
84	C₄₇	#3362	1	1	yes	characteristic, complemented, Sylow
85	C₂×#41	—	2	1	no	—
86	C₂×SD₁₆	—	47	3	no	—
87	C₄⋊Q₈	—	47	3	no	—
88	C₄⋊Q₈	—	94	3	no	—
89	#127	C₂×D₄₇	1	3	yes	—
90	#121	—	2	3	no	—
91	C₄×Q₈	—	47	3	no	—
92	C₄×Q₈	—	94	1	no	—
93	C₄×Q₈	C₂×D₄₇	1	1	yes	characteristic
94	C₄×D₄	C₂×D₄₇	1	3	yes	—
95	#106	—	47	3	no	—
96	C₄⋊C₈	—	47	3	no	—
97	#103	—	47	1	no	—
98	#102	—	47	3	no	—
99	#102	—	47	6	no	—
100	C₄×C₈	—	47	3	no	—
101	#41	—	4	4	no	—
102	C₂×Q₈	—	47	1	no	—
103	C₂×Q₈	—	94	3	no	—
104	C₂×Q₈	C₂²⋊D₄₇	1	1	yes	characteristic
105	C₂×D₄	—	2	3	no	—
106	C₂×D₄	—	2	3	no	—
107	C₂×D₄	C₂²×D₄₇	1	3	yes	—
108	C₂²×C₄	—	2	3	no	—
109	SD₁₆	—	94	6	no	—
110	C₂×C₈	—	47	1	no	—
111	C₂×C₈	—	47	3	no	—
112	C₄⋊C₄	—	47	3	no	—
113	C₄⋊C₄	—	94	3	no	—
114	C₄⋊C₄	—	94	3	no	—
115	C₄⋊C₄	C₂²⋊D₄₇	1	3	yes	—
116	C₄⋊C₄	C₂²×D₄₇	1	1	yes	characteristic
117	C₂²⋊C₄	—	2	3	no	—
118	C₄²	—	94	3	no	—
119	C₄²	C₂²×D₄₇	1	3	yes	—
120	C₂³	—	2	3	no	—
121	Q₈	—	47	2	no	—
122	Q₈	—	94	1	no	—
123	Q₈	—	376	3	no	—
124	Q₈	#10625335	1	4	yes	complemented
125	D₄	—	2	3	no	—
126	D₄	—	2	3	no	—
127	D₄	—	4	6	no	—
128	D₄	—	8	3	no	—
129	C₂×C₄	—	2	1	no	—
130	C₂×C₄	—	4	6	no	—
131	C₂×C₄	—	94	1	no	—
132	C₂×C₄	—	94	3	no	—
133	C₂×C₄	C₂×C₂²⋊D₄₇	1	3	yes	—
134	C₂×C₄	C₂×C₂²⋊D₄₇	1	3	yes	—
135	C₂×C₄	C₂³×D₄₇	1	1	yes	characteristic
136	C₈	—	94	3	no	—
137	C₈	—	94	1	no	—
138	C₂²	—	4	3	no	—
139	C₂²	—	4	3	no	—
140	C₂²	—	4	3	no	—
141	C₂²	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
142	C₄	—	2	3	no	—
143	C₄	—	4	1	no	—
144	C₄	—	94	2	no	—
145	C₄	—	188	3	no	—
146	C₄	C₂×#10625335	1	6	yes	—
147	C₄	#41×D₄₇	1	1	yes	characteristic
148	C₄	#13094071	1	1	yes	characteristic
149	C₂	—	8	3	no	—
150	C₂	#15689780	1	1	yes	characteristic
151	C₂	C₂²×#10625335	1	1	yes	characteristic
152	C₂	#15689926	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

Q₈×Dic₉₄

C₂

yes

maximal, characteristic, complemented

#15689469

C₂

yes

maximal, complemented

#15689413

C₂

yes

maximal, complemented

#15689320

C₂

yes

maximal, complemented

Q₈×C₄₇⋊C₈

C₂

yes

maximal, characteristic, complemented

C₄×#10625335

C₂

yes

maximal, complemented

C₄₇×#550

C₂

yes

maximal, characteristic

C₂×#10625335

C₂²

yes

—

#13094024

C₂²

yes

—

#13094023

—

Q₈×Dic₄₇

—

C₄×Dic₉₄

C₂²

yes

—

#13093955

C₂²

yes

—

C₄⋊(C₄₇⋊C₈)

C₂²

yes

—

#13093947

C₂²

yes

characteristic

#13093945

C₂²

yes

—

#13093945

C₂²

yes

—

C₄×C₄₇⋊C₈

C₂²

yes

—

C₂×#41×C₄₇

—

#127×C₄₇

C₂²

yes

—

#121×C₄₇

—

C₄×Q₈×C₄₇

C₂²

yes

characteristic

C₄×D₄×C₄₇

C₂²

yes

—

#41×C₄₇

—

C₂×Q₈×C₄₇

D₄

yes

characteristic

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

C₂³

yes

—

C₂²×C₄×C₄₇

—

C₂×Dic₉₄

—

C₂×Dic₉₄

C₂³

yes

characteristic

C₄⋊C₄×C₄₇

D₄

yes

—

C₄⋊C₄×C₄₇

C₂³

yes

characteristic

C₂²⋊C₄×C₄₇

—

C₄²×C₄₇

C₂³

yes

—

#10625335

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

C₂³

yes

—

Dic₄₇⋊C₄

—

C₄×Dic₄₇

—

C₂×C₄₇⋊C₈

C₂³

yes

characteristic

C₂×C₄₇⋊C₈

C₂³

yes

—

C₂³×C₄₇

—

Q₈×C₄₇

SD₁₆

yes

complemented

D₄×C₄₇

—

D₄×C₄₇

—

D₄×C₄₇

—

D₄×C₄₇

—

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

—

C₂×Dic₄₇

—

Dic₉₄

—

Dic₉₄

—

Dic₉₄

#41

yes

complemented

C₄₇⋊C₈

—

C₄₇⋊C₈

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

C₂²×D₄

yes

characteristic

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

C₂×SD₁₆

yes

—

C₄×C₄₇

C₂×#41

yes

characteristic

C₄×C₄₇

2₋¹⁺⁴

yes

characteristic

Dic₄₇

—

Dic₄₇

—

#3362

—

maximal, Sylow

C₂×C₄₇

—

C₂×C₄₇

#548

yes

characteristic

C₂×C₄₇

C₂²×SD₁₆

yes

characteristic

C₂×C₄₇

#576

yes

characteristic

Q₈²

—

#550

D₄₇

yes

—

—

—

C₈×Q₈

—

C₄×SD₁₆

—

C₄₇

#3362

yes

characteristic, complemented, Sylow

C₂×#41

—

C₂×SD₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

—

#127

C₂×D₄₇

yes

—

#121

—

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

C₂×D₄₇

yes

characteristic

C₄×D₄

C₂×D₄₇

yes

—

#106

—

C₄⋊C₈

—

#103

—

#102

—

#102

—

100

C₄×C₈

—

101

#41

—

102

C₂×Q₈

—

103

C₂×Q₈

—

104

C₂×Q₈

C₂²⋊D₄₇

yes

characteristic

105

C₂×D₄

—

106

C₂×D₄

—

107

C₂×D₄

C₂²×D₄₇

yes

—

108

C₂²×C₄

—

109

SD₁₆

—

110

C₂×C₈

—

111

C₂×C₈

—

112

C₄⋊C₄

—

113

C₄⋊C₄

—

114

C₄⋊C₄

—

115

C₄⋊C₄

C₂²⋊D₄₇

yes

—

116

C₄⋊C₄

C₂²×D₄₇

yes

characteristic

117

C₂²⋊C₄

—

118

C₄²

—

119

C₄²

C₂²×D₄₇

yes

—

120

C₂³

—

121

Q₈

—

122

Q₈

—

123

Q₈

—

376

—

124

Q₈

#10625335

yes

complemented

125

D₄

—

126

D₄

—

127

D₄

—

128

D₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

C₂×C₂²⋊D₄₇

yes

—

134

C₂×C₄

C₂×C₂²⋊D₄₇

yes

—

135

C₂×C₄

C₂³×D₄₇

yes

characteristic

136

C₈

—

137

C₈

—

138

C₂²

—

139

C₂²

—

140

C₂²

—

141

C₂²

C₂²×C₂²⋊D₄₇

yes

characteristic

142

C₄

—

143

C₄

—

144

C₄

—

145

C₄

—

188

—

146

C₄

C₂×#10625335

yes

—

147

C₄

#41×D₄₇

yes

characteristic

148

C₄

#13094071

yes

characteristic

149

C₂

—

150

C₂

#15689780

yes

characteristic

151

C₂

C₂²×#10625335

yes

characteristic

152

C₂

#15689926

yes

characteristic

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₄₇
commutator	C₂×C₄×C₄₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₄₇
Fitting	C₄₇×#550	C₂
socle	C₂²×C₄₇	C₂²×D₄
hypercenter	#550	D₄₇
hypocenter	C₄₇	#3362
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ #550

subgroup

G/H

center

C₂²

C₂²×C₂²⋊D₄₇

commutator

C₂×C₄×C₄₇

C₂⁴

Frattini

C₂×C₄

C₂³×D₄₇

Fitting

C₄₇×#550

C₂

socle

C₂²×C₄₇

C₂²×D₄

D₄₇

C₄₇

G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂² ◁ C₄×Q₈ ◁ #550

N	H ≅ G/N	type
1. Q₈×Dic₉₄	C₂	semidirect
2. #15689469	C₂	semidirect
3. #15689413	C₂	semidirect
4. #15689320	C₂	semidirect
5. Q₈×C₄₇⋊C₈	C₂	semidirect
6. C₄×#10625335	C₂	semidirect
44. Q₈×C₄₇	SD₁₆	semidirect
58. Dic₉₄	#41	semidirect
84. C₄₇	#3362	semidirect
124. Q₈	#10625335	semidirect

H ≅ G/N

type

1. Q₈×Dic₉₄

C₂

semidirect

2. #15689469

C₂

semidirect

3. #15689413

C₂

semidirect

4. #15689320

C₂

semidirect

5. Q₈×C₄₇⋊C₈

C₂

semidirect

6. C₄×#10625335

C₂

semidirect

44. Q₈×C₄₇

SD₁₆

semidirect

58. Dic₉₄

#41

semidirect

84. C₄₇

#3362

semidirect

124. Q₈

#10625335

semidirect

Subgroups — #20256055

Conjugacy classes of subgroups

Complements