Group #20253920 — subgroups

Subgroups — #20253920

Order 6016.

340 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 340 conjugacy classes of subgroups — too many to draw. Only its 149 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	#15689751	C₂	1	2	yes	maximal
2	#15689531	C₂	1	4	yes	maximal, complemented
3	#15689523	C₂	1	2	yes	maximal, complemented
4	#15689520	C₂	1	2	yes	maximal, complemented
5	#15689166	C₂	1	2	yes	maximal
6	C₂×#13093956	C₂	1	1	yes	maximal, characteristic
7	#15689078	C₂	1	1	yes	maximal, characteristic
8	C₄₇×#427	C₂	1	1	yes	maximal, characteristic
9	C₂×#10625329	C₂²	1	1	yes	characteristic
10	#13094024	—	2	2	no	—
11	#13094023	C₂²	1	4	yes	—
12	#13094016	—	2	2	no	—
13	#13094015	C₂²	1	4	yes	—
14	#13094001	—	2	4	no	—
15	#13093983	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊Dic₄₇	C₂²	1	2	yes	—
17	#13093957	C₂²	1	4	yes	—
18	#13093956	C₂²	1	2	yes	—
19	#13093956	C₂²	1	2	yes	—
20	#13093955	C₂²	1	4	yes	—
21	#13093934	C₂²	1	2	yes	—
22	C₈×Dic₄₇	C₂²	1	2	yes	—
23	C₂²×C₈×C₄₇	C₂²	1	1	yes	characteristic
24	#117×C₄₇	C₂²	1	2	yes	—
25	#107×C₄₇	C₂²	1	2	yes	—
26	#106×C₄₇	C₂²	1	1	yes	characteristic
27	#106×C₄₇	C₂²	1	1	yes	characteristic
28	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
29	C₂²×Dic₄₇	—	2	2	no	—
30	C₂×Dic₉₄	—	2	4	no	—
31	C₂×C₈×C₄₇	Q₈	1	4	yes	complemented
32	C₂×C₈×C₄₇	C₂³	1	1	yes	characteristic
33	C₂×C₈×C₄₇	C₂³	1	1	yes	characteristic
34	C₄⋊C₄×C₄₇	C₂³	1	2	yes	—
35	C₄⋊C₄×C₄₇	C₂³	1	2	yes	—
36	C₂²⋊C₄×C₄₇	—	2	2	no	—
37	C₄²×C₄₇	—	2	2	no	—
38	C₂²⋊Dic₄₇	D₄	1	2	yes	—
39	C₄⋊Dic₄₇	—	2	2	no	—
40	C₄⋊Dic₄₇	—	2	4	no	—
41	C₄⋊Dic₄₇	D₄	1	2	yes	—
42	C₄⋊Dic₄₇	C₂³	1	4	yes	—
43	Dic₄₇⋊C₄	—	2	4	no	—
44	Dic₄₇⋊C₄	—	2	4	no	—
45	C₄×Dic₄₇	C₂³	1	2	yes	—
46	#10625329	Q₈	1	4	yes	complemented
47	C₂×C₄₇⋊C₈	C₂³	1	2	yes	—
48	C₂³×C₄₇	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₄₇	—	2	2	no	—
50	C₂×C₄×C₄₇	—	2	2	no	—
51	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
52	C₂×C₄×C₄₇	C₂×Q₈	1	2	yes	—
53	C₂×C₄×C₄₇	C₂×Q₈	1	2	yes	—
54	C₂×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
55	C₂×Dic₄₇	—	2	4	no	—
56	C₂×Dic₄₇	—	8	2	no	—
57	C₂×Dic₄₇	C₂×D₄	1	4	yes	—
58	Dic₉₄	—	8	4	no	—
59	C₈×C₄₇	C₂×Q₈	1	2	yes	—
60	C₈×C₄₇	C₂×Q₈	1	2	yes	—
61	C₄₇⋊C₈	C₂×Q₈	1	4	yes	—
62	C₂²×C₄₇	—	2	2	no	—
63	C₂²×C₄₇	C₄⋊Q₈	1	2	yes	—
64	C₂²×C₄₇	C₂²×D₄	1	1	yes	characteristic
65	C₄×C₄₇	—	4	2	no	—
66	C₄×C₄₇	—	4	2	no	—
67	C₄×C₄₇	C₄⋊Q₈	1	2	yes	—
68	C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
69	C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
70	Dic₄₇	—	2	4	no	—
71	Dic₄₇	—	4	4	no	—
72	#3163	—	47	1	no	maximal, Sylow
73	C₂×C₄₇	—	2	2	no	—
74	C₂×C₄₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
75	C₂×C₄₇	#577	1	2	yes	—
76	#544	—	47	2	no	—
77	#501	—	47	4	no	—
78	#499	—	47	2	no	—
79	#498	—	47	2	no	—
80	#428	—	47	2	no	—
81	#427	D₄₇	1	1	yes	characteristic
82	C₂×#106	—	47	1	no	—
83	#405	—	47	1	no	—
84	C₄₇	#3163	1	1	yes	characteristic, complemented, Sylow
85	C₂×M₁₆	—	47	1	no	—
86	C₂²×C₈	C₂×D₄₇	1	1	yes	characteristic
87	C₄⋊Q₈	—	47	4	no	—
88	C₄⋊Q₈	—	94	2	no	—
89	#125	—	47	4	no	—
90	#125	—	94	2	no	—
91	C₂²⋊Q₈	—	94	4	no	—
92	#117	—	47	1	no	—
93	#117	C₂×D₄₇	1	2	yes	—
94	C₂×C₄⋊C₄	—	47	2	no	—
95	#107	—	47	4	no	—
96	#107	C₂×D₄₇	1	2	yes	—
97	#106	—	47	2	no	—
98	#106	—	47	2	no	—
99	#106	—	47	4	no	—
100	#106	C₂×D₄₇	1	1	yes	characteristic
101	#106	C₂×D₄₇	1	1	yes	characteristic
102	#97	—	47	2	no	—
103	C₄×C₈	—	47	2	no	—
104	C₂×Q₈	—	94	4	no	—
105	C₂²×C₄	—	94	2	no	—
106	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
107	M₁₆	—	47	4	no	—
108	C₂×C₈	—	47	2	no	—
109	C₂×C₈	Dic₉₄	1	4	yes	complemented
110	C₂×C₈	C₂²×D₄₇	1	1	yes	characteristic
111	C₂×C₈	C₂²×D₄₇	1	1	yes	characteristic
112	C₄⋊C₄	—	47	2	no	—
113	C₄⋊C₄	—	47	4	no	—
114	C₄⋊C₄	—	94	4	no	—
115	C₄⋊C₄	—	94	4	no	—
116	C₄⋊C₄	—	94	4	no	—
117	C₄⋊C₄	—	94	2	no	—
118	C₄⋊C₄	C₂²×D₄₇	1	2	yes	—
119	C₄⋊C₄	C₂²×D₄₇	1	2	yes	—
120	C₂²⋊C₄	—	2	2	no	—
121	C₂²⋊C₄	—	47	2	no	—
122	C₄²	—	2	2	no	—
123	C₄²	—	47	2	no	—
124	C₂³	D₄×D₄₇	1	1	yes	characteristic
125	Q₈	—	376	4	no	—
126	C₂×C₄	—	2	2	no	—
127	C₂×C₄	—	2	2	no	—
128	C₂×C₄	—	47	4	no	—
129	C₂×C₄	—	94	4	no	—
130	C₂×C₄	—	376	2	no	—
131	C₂×C₄	C₂×Dic₉₄	1	2	yes	—
132	C₂×C₄	D₄×D₄₇	1	1	yes	characteristic
133	C₂×C₄	Q₈×D₄₇	1	2	yes	—
134	C₂×C₄	C₂³×D₄₇	1	1	yes	characteristic
135	C₈	—	47	4	no	—
136	C₈	C₂×Dic₉₄	1	2	yes	—
137	C₈	C₂×Dic₉₄	1	2	yes	—
138	C₂²	—	2	2	no	—
139	C₂²	#13094023	1	2	yes	—
140	C₂²	C₂×D₄×D₄₇	1	1	yes	characteristic
141	C₄	—	4	2	no	—
142	C₄	—	4	2	no	—
143	C₄	—	94	4	no	—
144	C₄	—	188	4	no	—
145	C₄	#13094023	1	2	yes	—
146	C₄	C₂×Q₈×D₄₇	1	1	yes	characteristic
147	C₄	C₂²×Dic₉₄	1	1	yes	characteristic
148	C₂	—	2	2	no	—
149	C₂	C₂×#13094023	1	1	yes	characteristic
150	C₂	#15689930	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#15689751

C₂

yes

maximal

#15689531

C₂

yes

maximal, complemented

#15689523

C₂

yes

maximal, complemented

#15689520

C₂

yes

maximal, complemented

#15689166

C₂

yes

maximal

C₂×#13093956

C₂

yes

maximal, characteristic

#15689078

C₂

yes

maximal, characteristic

C₄₇×#427

C₂

yes

maximal, characteristic

C₂×#10625329

C₂²

yes

characteristic

#13094024

—

#13094023

C₂²

yes

—

#13094016

—

#13094015

C₂²

yes

—

#13094001

—

#13093983

C₂²

yes

characteristic

C₂×C₄⋊Dic₄₇

C₂²

yes

—

#13093957

C₂²

yes

—

#13093956

C₂²

yes

—

#13093956

C₂²

yes

—

#13093955

C₂²

yes

—

#13093934

C₂²

yes

—

C₈×Dic₄₇

C₂²

yes

—

C₂²×C₈×C₄₇

C₂²

yes

characteristic

#117×C₄₇

C₂²

yes

—

#107×C₄₇

C₂²

yes

—

#106×C₄₇

C₂²

yes

characteristic

#106×C₄₇

C₂²

yes

characteristic

C₂²×C₄×C₄₇

C₂³

yes

characteristic

C₂²×Dic₄₇

—

C₂×Dic₉₄

—

C₂×C₈×C₄₇

Q₈

yes

complemented

C₂×C₈×C₄₇

C₂³

yes

characteristic

C₂×C₈×C₄₇

C₂³

yes

characteristic

C₄⋊C₄×C₄₇

C₂³

yes

—

C₄⋊C₄×C₄₇

C₂³

yes

—

C₂²⋊C₄×C₄₇

—

C₄²×C₄₇

—

C₂²⋊Dic₄₇

D₄

yes

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

D₄

yes

—

C₄⋊Dic₄₇

C₂³

yes

—

Dic₄₇⋊C₄

—

Dic₄₇⋊C₄

—

C₄×Dic₄₇

C₂³

yes

—

#10625329

Q₈

yes

complemented

C₂×C₄₇⋊C₈

C₂³

yes

—

C₂³×C₄₇

C₂×D₄

yes

characteristic

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂×C₄×C₄₇

C₂×Q₈

yes

—

C₂×C₄×C₄₇

C₂×Q₈

yes

—

C₂×C₄×C₄₇

C₂⁴

yes

characteristic

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

C₂×D₄

yes

—

Dic₉₄

—

C₈×C₄₇

C₂×Q₈

yes

—

C₈×C₄₇

C₂×Q₈

yes

—

C₄₇⋊C₈

C₂×Q₈

yes

—

C₂²×C₄₇

—

C₂²×C₄₇

C₄⋊Q₈

yes

—

C₂²×C₄₇

C₂²×D₄

yes

characteristic

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

C₄⋊Q₈

yes

—

C₄×C₄₇

C₂²×Q₈

yes

characteristic

C₄×C₄₇

C₂²×Q₈

yes

characteristic

Dic₄₇

—

Dic₄₇

—

#3163

—

maximal, Sylow

C₂×C₄₇

—

C₂×C₄₇

C₂×C₄⋊Q₈

yes

characteristic

C₂×C₄₇

#577

yes

—

—

—

—

—

—

D₄₇

yes

characteristic

C₂×#106

—

#405

—

C₄₇

#3163

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

C₂²×C₈

C₂×D₄₇

yes

characteristic

C₄⋊Q₈

—

C₄⋊Q₈

—

#125

—

#125

—

C₂²⋊Q₈

—

#117

—

#117

C₂×D₄₇

yes

—

C₂×C₄⋊C₄

—

#107

—

#107

C₂×D₄₇

yes

—

#106

—

#106

—

#106

—

100

#106

C₂×D₄₇

yes

characteristic

101

#106

C₂×D₄₇

yes

characteristic

102

#97

—

103

C₄×C₈

—

104

C₂×Q₈

—

105

C₂²×C₄

—

106

C₂²×C₄

C₂²×D₄₇

yes

characteristic

107

M₁₆

—

108

C₂×C₈

—

109

C₂×C₈

Dic₉₄

yes

complemented

110

C₂×C₈

C₂²×D₄₇

yes

characteristic

111

C₂×C₈

C₂²×D₄₇

yes

characteristic

112

C₄⋊C₄

—

113

C₄⋊C₄

—

114

C₄⋊C₄

—

115

C₄⋊C₄

—

116

C₄⋊C₄

—

117

C₄⋊C₄

—

118

C₄⋊C₄

C₂²×D₄₇

yes

—

119

C₄⋊C₄

C₂²×D₄₇

yes

—

120

C₂²⋊C₄

—

121

C₂²⋊C₄

—

122

C₄²

—

123

C₄²

—

124

C₂³

D₄×D₄₇

yes

characteristic

125

Q₈

—

376

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

376

—

131

C₂×C₄

C₂×Dic₉₄

yes

—

132

C₂×C₄

D₄×D₄₇

yes

characteristic

133

C₂×C₄

Q₈×D₄₇

yes

—

134

C₂×C₄

C₂³×D₄₇

yes

characteristic

135

C₈

—

136

C₈

C₂×Dic₉₄

yes

—

137

C₈

C₂×Dic₉₄

yes

—

138

C₂²

—

139

C₂²

#13094023

yes

—

140

C₂²

C₂×D₄×D₄₇

yes

characteristic

141

C₄

—

142

C₄

—

143

C₄

—

144

C₄

—

188

—

145

C₄

#13094023

yes

—

146

C₄

C₂×Q₈×D₄₇

yes

characteristic

147

C₄

C₂²×Dic₉₄

yes

characteristic

148

C₂

—

149

C₂

C₂×#13094023

yes

characteristic

150

C₂

#15689930

yes

—

	subgroup	G/H
center	C₂²	C₂×D₄×D₄₇
commutator	C₂×C₄×C₄₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₄₇
Fitting	C₄₇×#427	C₂
socle	C₂²×C₄₇	C₂²×D₄
hypercenter	#427	D₄₇
hypocenter	C₄₇	#3163
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #427

subgroup

G/H

center

C₂²

C₂×D₄×D₄₇

commutator

C₂×C₄×C₄₇

C₂⁴

Frattini

C₂×C₄

C₂³×D₄₇

Fitting

C₄₇×#427

C₂

socle

C₂²×C₄₇

C₂²×D₄

D₄₇

C₄₇

G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #427

N	H ≅ G/N	type
2. #15689531	C₂	semidirect
3. #15689523	C₂	semidirect
4. #15689520	C₂	semidirect
31. C₂×C₈×C₄₇	Q₈	semidirect
46. #10625329	Q₈	semidirect
84. C₄₇	#3163	semidirect
109. C₂×C₈	Dic₉₄	semidirect

H ≅ G/N

type

2. #15689531

C₂

semidirect

3. #15689523

C₂

semidirect

4. #15689520

C₂

semidirect

31. C₂×C₈×C₄₇

Q₈

semidirect

46. #10625329

Q₈

semidirect

84. C₄₇

#3163

semidirect

109. C₂×C₈

Dic₉₄

semidirect

Subgroups — #20253920

Conjugacy classes of subgroups

Complements