Group #20256059 — subgroups

Subgroups — #20256059

Order 6016.

392 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 392 conjugacy classes of subgroups — too many to draw. Only its 141 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	#15689795	C₂	1	1	yes	maximal, characteristic
2	#15689471	C₂	1	3	yes	maximal, complemented
3	#15689410	C₂	1	3	yes	maximal, complemented
4	#15689323	C₂	1	3	yes	maximal, complemented
5	Q₈×C₄₇⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄×#10625336	C₂	1	3	yes	maximal, complemented
7	Q₈²×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#10625354	—	2	1	no	—
9	C₂×#10625336	C₂²	1	3	yes	—
10	#13094021	C₂²	1	3	yes	—
11	#13093999	—	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	1	yes	characteristic
16	#13093944	C₂²	1	6	yes	—
17	#13093944	C₂²	1	3	yes	—
18	C₄×C₄₇⋊C₈	C₂²	1	3	yes	—
19	C₄⋊Q₈×C₄₇	—	2	3	no	—
20	C₄⋊Q₈×C₄₇	C₂²	1	3	yes	—
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	—	2	3	no	—
25	C₂×Q₈×C₄₇	D₄	1	1	yes	characteristic
26	C₂×Q₈×C₄₇	C₂³	1	1	yes	characteristic
27	#10625354	—	4	4	no	—
28	C₂×D₁₈₈	—	2	3	no	—
29	C₂×D₁₈₈	—	2	3	no	—
30	C₂×D₁₈₈	C₂³	1	3	yes	—
31	C₂×C₄×D₄₇	—	2	3	no	—
32	C₄⋊C₄×C₄₇	—	2	3	no	—
33	C₄⋊C₄×C₄₇	—	2	3	no	—
34	C₄⋊C₄×C₄₇	D₄	1	3	yes	—
35	C₄⋊C₄×C₄₇	C₂³	1	3	yes	—
36	C₄²×C₄₇	—	2	3	no	—
37	C₄²×C₄₇	C₂³	1	3	yes	—
38	#10625336	—	2	6	no	—
39	(C₂×D₄₇)⋊C₄	—	2	3	no	—
40	C₄⋊Dic₄₇	C₂³	1	1	yes	characteristic
41	C₂×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
42	C₂×C₄₇⋊C₈	C₂³	1	3	yes	—
43	C₂²×D₄₇	—	2	3	no	—
44	Q₈×C₄₇	—	2	1	no	—
45	Q₈×C₄₇	—	8	3	no	—
46	Q₈×C₄₇	SD₁₆	1	4	yes	complemented
47	Q₈×C₄₇	#41	1	2	yes	complemented
48	C₂×C₄×C₄₇	—	2	1	no	—
49	C₂×C₄×C₄₇	—	2	3	no	—
50	C₂×C₄×C₄₇	C₂×D₄	1	3	yes	—
51	C₂×C₄×C₄₇	C₂×D₄	1	3	yes	—
52	C₂×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
53	C₂×Dic₄₇	—	2	1	no	—
54	D₁₈₈	—	2	3	no	—
55	D₁₈₈	—	2	3	no	—
56	D₁₈₈	—	4	6	no	—
57	D₁₈₈	—	8	3	no	—
58	C₄×D₄₇	—	4	6	no	—
59	C₄₇⋊C₈	—	2	3	no	—
60	C₄₇⋊C₈	—	2	1	no	—
61	C₂²×C₄₇	C₂²×D₄	1	1	yes	characteristic
62	C₂×D₄₇	—	4	3	no	—
63	C₂×D₄₇	—	4	3	no	—
64	C₂×D₄₇	—	4	3	no	—
65	C₄×C₄₇	—	2	2	no	—
66	C₄×C₄₇	—	2	3	no	—
67	C₄×C₄₇	—	4	3	no	—
68	C₄×C₄₇	C₂×SD₁₆	1	6	yes	—
69	C₄×C₄₇	C₂×#41	1	1	yes	characteristic
70	C₄×C₄₇	2₋¹⁺⁴	1	1	yes	characteristic
71	Dic₄₇	—	4	1	no	—
72	#3362	—	47	1	no	maximal, Sylow
73	C₂×C₄₇	#548	1	1	yes	characteristic
74	C₂×C₄₇	C₂²×SD₁₆	1	1	yes	characteristic
75	C₂×C₄₇	#576	1	1	yes	characteristic
76	D₄₇	—	8	3	no	—
77	Q₈²	D₄₇	1	1	yes	characteristic, complemented
78	#550	—	47	1	no	—
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	—	94	1	no	—
86	C₂×SD₁₆	—	47	3	no	—
87	C₄⋊Q₈	—	2	3	no	—
88	C₄⋊Q₈	C₂×D₄₇	1	3	yes	—
89	#127	—	47	3	no	—
90	#121	—	94	3	no	—
91	C₄×Q₈	—	2	1	no	—
92	C₄×Q₈	C₂×D₄₇	1	3	yes	—
93	C₄×Q₈	C₂×D₄₇	1	1	yes	characteristic
94	C₄×D₄	—	47	3	no	—
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	—	188	4	no	—
102	C₂×Q₈	—	2	3	no	—
103	C₂×Q₈	C₂²⋊D₄₇	1	1	yes	characteristic
104	C₂×Q₈	C₂²×D₄₇	1	1	yes	characteristic
105	C₂×D₄	—	47	3	no	—
106	C₂×D₄	—	94	3	no	—
107	C₂×D₄	—	94	3	no	—
108	C₂²×C₄	—	94	3	no	—
109	SD₁₆	—	94	6	no	—
110	C₂×C₈	—	47	1	no	—
111	C₂×C₈	—	47	3	no	—
112	C₄⋊C₄	—	2	3	no	—
113	C₄⋊C₄	—	2	3	no	—
114	C₄⋊C₄	—	47	1	no	—
115	C₄⋊C₄	C₂²⋊D₄₇	1	3	yes	—
116	C₄⋊C₄	C₂²×D₄₇	1	3	yes	—
117	C₂²⋊C₄	—	94	3	no	—
118	C₄²	—	2	3	no	—
119	C₄²	C₂²×D₄₇	1	3	yes	—
120	C₂³	—	94	3	no	—
121	Q₈	—	2	1	no	—
122	Q₈	—	8	3	no	—
123	Q₈	#10625336	1	4	yes	complemented
124	Q₈	#10625354	1	2	yes	complemented
125	D₄	—	94	3	no	—
126	D₄	—	94	3	no	—
127	D₄	—	376	3	no	—
128	D₄	—	188	6	no	—
129	C₂×C₄	—	2	1	no	—
130	C₂×C₄	—	2	3	no	—
131	C₂×C₄	—	94	1	no	—
132	C₂×C₄	—	188	6	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	—	188	3	no	—
139	C₂²	—	188	3	no	—
140	C₂²	—	188	3	no	—
141	C₂²	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
142	C₄	—	2	2	no	—
143	C₄	—	2	3	no	—
144	C₄	—	4	3	no	—
145	C₄	—	188	1	no	—
146	C₄	C₂×#10625336	1	6	yes	—
147	C₄	C₂×#10625354	1	1	yes	characteristic
148	C₄	#13094072	1	1	yes	characteristic
149	C₂	—	376	3	no	—
150	C₂	#15689784	1	1	yes	characteristic
151	C₂	C₂²×#10625336	1	1	yes	characteristic
152	C₂	#15689926	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#15689795

C₂

yes

maximal, characteristic

#15689471

C₂

yes

maximal, complemented

#15689410

C₂

yes

maximal, complemented

#15689323

C₂

yes

maximal, complemented

Q₈×C₄₇⋊C₈

C₂

yes

maximal, characteristic, complemented

C₄×#10625336

C₂

yes

maximal, complemented

Q₈²×C₄₇

C₂

yes

maximal, characteristic, complemented

C₂×#10625354

—

C₂×#10625336

C₂²

yes

—

#13094021

C₂²

yes

—

#13093999

—

C₄×D₁₈₈

C₂²

yes

—

#13093955

C₂²

yes

—

C₄⋊(C₄₇⋊C₈)

C₂²

yes

—

#13093948

C₂²

yes

characteristic

#13093944

C₂²

yes

—

#13093944

C₂²

yes

—

C₄×C₄₇⋊C₈

C₂²

yes

—

C₄⋊Q₈×C₄₇

—

C₄⋊Q₈×C₄₇

C₂²

yes

—

C₄×Q₈×C₄₇

—

C₄×Q₈×C₄₇

C₂²

yes

—

C₄×Q₈×C₄₇

C₂²

yes

characteristic

C₂×Q₈×C₄₇

—

C₂×Q₈×C₄₇

D₄

yes

characteristic

C₂×Q₈×C₄₇

C₂³

yes

characteristic

#10625354

—

C₂×D₁₈₈

—

C₂×D₁₈₈

—

C₂×D₁₈₈

C₂³

yes

—

C₂×C₄×D₄₇

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

D₄

yes

—

C₄⋊C₄×C₄₇

C₂³

yes

—

C₄²×C₄₇

—

C₄²×C₄₇

C₂³

yes

—

#10625336

—

(C₂×D₄₇)⋊C₄

—

C₄⋊Dic₄₇

C₂³

yes

characteristic

C₂×C₄₇⋊C₈

C₂³

yes

characteristic

C₂×C₄₇⋊C₈

C₂³

yes

—

C₂²×D₄₇

—

Q₈×C₄₇

—

Q₈×C₄₇

—

Q₈×C₄₇

SD₁₆

yes

complemented

Q₈×C₄₇

#41

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

—

D₁₈₈

—

D₁₈₈

—

C₄×D₄₇

—

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

#576

yes

characteristic

D₄₇

—

Q₈²

D₄₇

yes

characteristic, complemented

—

—

—

—

C₈×Q₈

—

C₄×SD₁₆

—

C₄₇

#3362

yes

characteristic, complemented, Sylow

C₂×#41

—

C₂×SD₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₄₇

yes

—

#127

—

#121

—

C₄×Q₈

—

C₄×Q₈

C₂×D₄₇

yes

—

C₄×Q₈

C₂×D₄₇

yes

characteristic

C₄×D₄

—

#106

—

C₄⋊C₈

—

#103

—

#102

—

#102

—

100

C₄×C₈

—

101

#41

—

188

—

102

C₂×Q₈

—

103

C₂×Q₈

C₂²⋊D₄₇

yes

characteristic

104

C₂×Q₈

C₂²×D₄₇

yes

characteristic

105

C₂×D₄

—

106

C₂×D₄

—

107

C₂×D₄

—

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

—

117

C₂²⋊C₄

—

118

C₄²

—

119

C₄²

C₂²×D₄₇

yes

—

120

C₂³

—

121

Q₈

—

122

Q₈

—

123

Q₈

#10625336

yes

complemented

124

Q₈

#10625354

yes

complemented

125

D₄

—

126

D₄

—

127

D₄

—

376

—

128

D₄

—

188

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

188

—

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

—

188

—

139

C₂²

—

188

—

140

C₂²

—

188

—

141

C₂²

C₂²×C₂²⋊D₄₇

yes

characteristic

142

C₄

—

143

C₄

—

144

C₄

—

145

C₄

—

188

—

146

C₄

C₂×#10625336

yes

—

147

C₄

C₂×#10625354

yes

characteristic

148

C₄

#13094072

yes

characteristic

149

C₂

—

376

—

150

C₂

#15689784

yes

characteristic

151

C₂

C₂²×#10625336

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	Q₈²×C₄₇	C₂
socle	C₂²×C₄₇	C₂²×D₄
hypercenter	Q₈²	D₄₇
hypocenter	C₄₇	#3362
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ Q₈²

subgroup

G/H

center

C₂²

C₂²×C₂²⋊D₄₇

commutator

C₂×C₄×C₄₇

C₂⁴

Frattini

C₂×C₄

C₂³×D₄₇

Fitting

Q₈²×C₄₇

C₂

socle

C₂²×C₄₇

C₂²×D₄

hypercenter

Q₈²

D₄₇

hypocenter

C₄₇

#3362

LCS

G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂² ◁ C₄×Q₈ ◁ Q₈²

N	H ≅ G/N	type
2. #15689471	C₂	semidirect
3. #15689410	C₂	semidirect
4. #15689323	C₂	semidirect
5. Q₈×C₄₇⋊C₈	C₂	semidirect
6. C₄×#10625336	C₂	semidirect
7. Q₈²×C₄₇	C₂	semidirect
46. Q₈×C₄₇	SD₁₆	semidirect
47. Q₈×C₄₇	#41	semidirect
77. Q₈²	D₄₇	semidirect
84. C₄₇	#3362	semidirect
123. Q₈	#10625336	semidirect
124. Q₈	#10625354	semidirect

H ≅ G/N

type

2. #15689471

C₂

semidirect

3. #15689410

C₂

semidirect

4. #15689323

C₂

semidirect

5. Q₈×C₄₇⋊C₈

C₂

semidirect

6. C₄×#10625336

C₂

semidirect

7. Q₈²×C₄₇

C₂

semidirect

46. Q₈×C₄₇

SD₁₆

semidirect

47. Q₈×C₄₇

#41

semidirect

77. Q₈²

D₄₇

semidirect

84. C₄₇

#3362

semidirect

123. Q₈

#10625336

semidirect

124. Q₈

#10625354

semidirect

Subgroups — #20256059

Conjugacy classes of subgroups

Complements