Group #20253706 — subgroups

Subgroups — C₂×#15689480 #20253706

Order 6016.

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

normal subgroups only

This group has 412 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	C₂×#13094012	C₂	1	1	yes	maximal, characteristic
2	#15689480	C₂	1	8	yes	maximal, direct product factor
3	C₂×#13093947	C₂	1	2	yes	maximal, complemented
4	C₂×#13093944	C₂	1	2	yes	maximal
5	C₂×C₄×C₄₇⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#125×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×Dic₉₄	C₂²	1	1	yes	characteristic
8	C₂²×D₁₈₈	C₂²	1	1	yes	characteristic
9	C₂²×C₄₇⋊C₈	C₂²	1	2	yes	—
10	#13094012	—	2	2	no	—
11	#13094012	C₂²	1	4	yes	—
12	C₂×(C₂×D₄₇)⋊C₄	—	2	2	no	—
13	#13093947	C₂²	1	8	yes	complemented
14	#13093944	C₂²	1	8	yes	—
15	C₄×C₄₇⋊C₈	C₂²	1	4	yes	complemented
16	#125×C₄₇	—	2	2	no	—
17	#125×C₄₇	C₂²	1	4	yes	complemented
18	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
19	C₂×C₄⋊C₄×C₄₇	C₂²	1	2	yes	—
20	C₂×C₄²×C₄₇	C₂²	1	1	yes	characteristic
21	C₂³×D₄₇	—	2	1	no	—
22	C₂²×C₄×C₄₇	—	2	2	no	—
23	C₂²×C₄×C₄₇	D₄	1	2	yes	—
24	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
25	C₂²×Dic₄₇	—	2	1	no	—
26	C₂×D₁₈₈	—	2	4	no	—
27	C₂×D₁₈₈	—	2	1	no	—
28	C₂×D₁₈₈	C₂³	1	2	yes	—
29	C₂×Dic₉₄	—	2	4	no	—
30	C₂×Dic₉₄	—	2	1	no	—
31	C₂×Dic₉₄	C₂³	1	2	yes	—
32	C₄⋊C₄×C₄₇	—	2	8	no	—
33	C₄⋊C₄×C₄₇	—	2	2	no	—
34	C₄⋊C₄×C₄₇	C₂³	1	4	yes	—
35	C₄²×C₄₇	D₄	1	2	yes	—
36	C₄²×C₄₇	C₂³	1	2	yes	—
37	(C₂×D₄₇)⋊C₄	—	2	8	no	—
38	C₂×C₄₇⋊C₈	—	2	4	no	—
39	C₂×C₄₇⋊C₈	C₂³	1	4	yes	—
40	C₂³×C₄₇	C₂×D₄	1	1	yes	characteristic
41	C₂²×D₄₇	—	2	2	no	—
42	C₂²×D₄₇	—	4	2	no	—
43	C₂²×D₄₇	—	4	4	no	—
44	C₂×C₄×C₄₇	—	2	4	no	—
45	C₂×C₄×C₄₇	—	2	4	no	—
46	C₂×C₄×C₄₇	—	4	4	no	—
47	C₂×C₄×C₄₇	C₂×D₄	1	4	yes	—
48	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₄₇	#41	1	4	yes	—
50	C₂×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
51	C₂×Dic₄₇	—	2	2	no	—
52	C₂×Dic₄₇	—	4	2	no	—
53	D₁₈₈	—	2	4	no	—
54	D₁₈₈	—	4	2	no	—
55	Dic₉₄	—	2	4	no	—
56	Dic₉₄	—	4	2	no	—
57	C₄₇⋊C₈	—	2	4	no	—
58	C₂²×C₄₇	#124	1	2	yes	—
59	C₂²×C₄₇	#135	1	4	yes	—
60	C₂²×C₄₇	C₂²×D₄	1	1	yes	characteristic
61	C₂×D₄₇	—	4	2	no	—
62	C₂×D₄₇	—	4	4	no	—
63	C₂×D₄₇	—	8	4	no	—
64	C₄×C₄₇	—	2	4	no	—
65	C₄×C₄₇	—	4	4	no	—
66	C₄×C₄₇	#124	1	2	yes	—
67	C₄×C₄₇	C₂×#41	1	2	yes	—
68	Dic₄₇	—	4	2	no	—
69	C₂×#488	—	47	1	no	maximal, Sylow
70	C₂×C₄₇	#488	1	4	yes	complemented
71	C₂×C₄₇	C₂×#124	1	1	yes	characteristic
72	C₂×C₄₇	C₂×#135	1	2	yes	—
73	D₄₇	—	8	2	no	—
74	C₂×#125	D₄₇	1	1	yes	characteristic, complemented
75	C₂×#124	—	47	1	no	—
76	#488	—	47	8	no	—
77	C₂×#103	—	47	2	no	—
78	C₂×#102	—	47	2	no	—
79	C₂×C₄×C₈	—	47	1	no	—
80	C₄₇	C₂×#488	1	1	yes	characteristic, complemented, Sylow
81	C₂²×Q₈	—	47	1	no	—
82	C₂²×D₄	—	47	1	no	—
83	C₂²×C₈	—	47	2	no	—
84	#125	—	2	2	no	—
85	#125	C₂×D₄₇	1	4	yes	complemented
86	#124	—	47	4	no	—
87	#124	—	94	2	no	—
88	C₂×C₄⋊C₄	—	2	2	no	—
89	C₂×C₄⋊C₄	C₂×D₄₇	1	2	yes	—
90	C₂×C₂²⋊C₄	—	94	2	no	—
91	C₂×C₄²	C₂×D₄₇	1	1	yes	characteristic
92	#103	—	47	8	no	—
93	#102	—	47	8	no	—
94	C₄×C₈	—	47	4	no	—
95	C₂⁴	—	94	1	no	—
96	C₂×Q₈	—	47	2	no	—
97	C₂×Q₈	—	94	1	no	—
98	C₂×Q₈	—	94	4	no	—
99	C₂×D₄	—	47	2	no	—
100	C₂×D₄	—	94	4	no	—
101	C₂×D₄	—	94	1	no	—
102	C₂²×C₄	—	2	2	no	—
103	C₂²×C₄	—	94	1	no	—
104	C₂²×C₄	C₂²⋊D₄₇	1	2	yes	—
105	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
106	C₂×C₈	—	47	4	no	—
107	C₂×C₈	—	94	4	no	—
108	C₄⋊C₄	—	2	8	no	—
109	C₄⋊C₄	—	2	2	no	—
110	C₄⋊C₄	C₂²×D₄₇	1	4	yes	—
111	C₂²⋊C₄	—	94	8	no	—
112	C₄²	C₂²⋊D₄₇	1	2	yes	—
113	C₄²	C₂²×D₄₇	1	2	yes	—
114	C₂³	—	94	2	no	—
115	C₂³	—	188	2	no	—
116	C₂³	—	188	4	no	—
117	C₂³	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
118	Q₈	—	94	4	no	—
119	Q₈	—	188	2	no	—
120	D₄	—	94	4	no	—
121	D₄	—	188	2	no	—
122	C₂×C₄	—	2	4	no	—
123	C₂×C₄	—	2	4	no	—
124	C₂×C₄	—	4	4	no	—
125	C₂×C₄	—	94	2	no	—
126	C₂×C₄	—	188	2	no	—
127	C₂×C₄	#10625354	1	4	yes	—
128	C₂×C₄	C₂×C₂²⋊D₄₇	1	4	yes	—
129	C₂×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
130	C₂×C₄	C₂³×D₄₇	1	1	yes	characteristic
131	C₈	—	94	4	no	—
132	C₂²	—	376	4	no	—
133	C₂²	—	188	2	no	—
134	C₂²	—	188	4	no	—
135	C₂²	#13094013	1	2	yes	—
136	C₂²	#13094044	1	4	yes	—
137	C₂²	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
138	C₄	—	2	4	no	—
139	C₄	—	4	4	no	—
140	C₄	—	188	2	no	—
141	C₄	#13094013	1	2	yes	—
142	C₄	C₂×#10625354	1	2	yes	—
143	C₂	—	376	2	no	—
144	C₂	#15689480	1	4	yes	direct product factor
145	C₂	C₂×#13094013	1	1	yes	characteristic
146	C₂	C₂×#13094044	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13094012

C₂

yes

maximal, characteristic

#15689480

C₂

yes

maximal, direct product factor

C₂×#13093947

C₂

yes

maximal, complemented

C₂×#13093944

C₂

yes

maximal

C₂×C₄×C₄₇⋊C₈

C₂

yes

maximal, characteristic, complemented

C₂×#125×C₄₇

C₂

yes

maximal, characteristic, complemented

C₂²×Dic₉₄

C₂²

yes

characteristic

C₂²×D₁₈₈

C₂²

yes

characteristic

C₂²×C₄₇⋊C₈

C₂²

yes

—

#13094012

—

#13094012

C₂²

yes

—

C₂×(C₂×D₄₇)⋊C₄

—

#13093947

C₂²

yes

complemented

#13093944

C₂²

yes

—

C₄×C₄₇⋊C₈

C₂²

yes

complemented

#125×C₄₇

—

#125×C₄₇

C₂²

yes

complemented

C₂×C₄⋊C₄×C₄₇

—

C₂×C₄⋊C₄×C₄₇

C₂²

yes

—

C₂×C₄²×C₄₇

C₂²

yes

characteristic

C₂³×D₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

D₄

yes

—

C₂²×C₄×C₄₇

C₂³

yes

characteristic

C₂²×Dic₄₇

—

C₂×D₁₈₈

—

C₂×D₁₈₈

—

C₂×D₁₈₈

C₂³

yes

—

C₂×Dic₉₄

—

C₂×Dic₉₄

—

C₂×Dic₉₄

C₂³

yes

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

C₂³

yes

—

C₄²×C₄₇

D₄

yes

—

C₄²×C₄₇

C₂³

yes

—

(C₂×D₄₇)⋊C₄

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

C₂³

yes

—

C₂³×C₄₇

C₂×D₄

yes

characteristic

C₂²×D₄₇

—

C₂²×D₄₇

—

C₂²×D₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂×D₄

yes

—

C₂×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂×C₄×C₄₇

#41

yes

—

C₂×C₄×C₄₇

C₂⁴

yes

characteristic

C₂×Dic₄₇

—

C₂×Dic₄₇

—

D₁₈₈

—

D₁₈₈

—

Dic₉₄

—

Dic₉₄

—

C₄₇⋊C₈

—

C₂²×C₄₇

#124

yes

—

C₂²×C₄₇

#135

yes

—

C₂²×C₄₇

C₂²×D₄

yes

characteristic

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

#124

yes

—

C₄×C₄₇

C₂×#41

yes

—

Dic₄₇

—

—

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

—

D₄₇

—

C₂×#125

D₄₇

yes

characteristic, complemented

—

—

—

—

C₂×C₄×C₈

—

C₄₇

C₂×#488

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₂²×D₄

—

C₂²×C₈

—

#125

—

#125

C₂×D₄₇

yes

complemented

#124

—

#124

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₄₇

yes

—

C₂×C₂²⋊C₄

—

C₂×C₄²

C₂×D₄₇

yes

characteristic

#103

—

#102

—

C₄×C₈

—

C₂⁴

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×D₄

—

100

C₂×D₄

—

101

C₂×D₄

—

102

C₂²×C₄

—

103

C₂²×C₄

—

104

C₂²×C₄

C₂²⋊D₄₇

yes

—

105

C₂²×C₄

C₂²×D₄₇

yes

characteristic

106

C₂×C₈

—

107

C₂×C₈

—

108

C₄⋊C₄

—

109

C₄⋊C₄

—

110

C₄⋊C₄

C₂²×D₄₇

yes

—

111

C₂²⋊C₄

—

112

C₄²

C₂²⋊D₄₇

yes

—

113

C₄²

C₂²×D₄₇

yes

—

114

C₂³

—

115

C₂³

—

188

—

116

C₂³

—

188

—

117

C₂³

C₂×C₂²⋊D₄₇

yes

characteristic

118

Q₈

—

119

Q₈

—

188

—

120

D₄

—

121

D₄

—

188

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

188

—

127

C₂×C₄

#10625354

yes

—

128

C₂×C₄

C₂×C₂²⋊D₄₇

yes

—

129

C₂×C₄

C₂×C₂²⋊D₄₇

yes

characteristic

130

C₂×C₄

C₂³×D₄₇

yes

characteristic

131

C₈

—

132

C₂²

—

376

—

133

C₂²

—

188

—

134

C₂²

—

188

—

135

C₂²

#13094013

yes

—

136

C₂²

#13094044

yes

—

137

C₂²

C₂²×C₂²⋊D₄₇

yes

characteristic

138

C₄

—

139

C₄

—

140

C₄

—

188

—

141

C₄

#13094013

yes

—

142

C₄

C₂×#10625354

yes

—

143

C₂

—

376

—

144

C₂

#15689480

yes

direct product factor

145

C₂

C₂×#13094013

yes

characteristic

146

C₂

C₂×#13094044

yes

—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₄₇
commutator	C₂×C₄×C₄₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₄₇
Fitting	C₂×#125×C₄₇	C₂
socle	C₂³×C₄₇	C₂×D₄
hypercenter	C₂×#125	D₄₇
hypocenter	C₄₇	C₂×#488
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#125

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₄₇

commutator

C₂×C₄×C₄₇

C₂⁴

Frattini

C₂×C₄

C₂³×D₄₇

Fitting

C₂×#125×C₄₇

C₂

socle

C₂³×C₄₇

C₂×D₄

D₄₇

C₄₇

G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#125

N	H ≅ G/N	type
2. #15689480	C₂	direct (N × H)
3. C₂×#13093947	C₂	semidirect
5. C₂×C₄×C₄₇⋊C₈	C₂	semidirect
6. C₂×#125×C₄₇	C₂	semidirect
13. #13093947	C₂²	semidirect
15. C₄×C₄₇⋊C₈	C₂²	semidirect
17. #125×C₄₇	C₂²	semidirect
70. C₂×C₄₇	#488	semidirect
74. C₂×#125	D₄₇	semidirect
80. C₄₇	C₂×#488	semidirect
85. #125	C₂×D₄₇	semidirect
144. C₂	#15689480	direct (N × H)

H ≅ G/N

type

2. #15689480

C₂

direct (N × H)

3. C₂×#13093947

C₂

semidirect

5. C₂×C₄×C₄₇⋊C₈

C₂

semidirect

6. C₂×#125×C₄₇

C₂

semidirect

13. #13093947

C₂²

semidirect

15. C₄×C₄₇⋊C₈

C₂²

semidirect

17. #125×C₄₇

C₂²

semidirect

70. C₂×C₄₇

#488

semidirect

74. C₂×#125

D₄₇

semidirect

80. C₄₇

C₂×#488

semidirect

85. #125

C₂×D₄₇

semidirect

144. C₂

#15689480

direct (N × H)

Subgroups — C₂×#15689480 #20253706

Conjugacy classes of subgroups

Complements

Subgroups — C2×#15689480 #20253706

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#15689480 #20253706