Group #20245774 — subgroups

Subgroups — C₂×#15688988 #20245774

Order 6016.

712 conjugacy classes of subgroups; only the 349 normal subgroups are shown.

normal subgroups only

This group has 712 conjugacy classes of subgroups — too many to draw. Only its 349 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	C₂²×C₄⋊Dic₄₇	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄×Dic₄₇	C₂	1	1	yes	maximal, characteristic
3	#15688988	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₄₇⋊#95	C₂	1	4	yes	maximal
5	C₂²×C₄⋊C₄×C₄₇	C₂	1	1	yes	maximal, characteristic
6	C₂³×Dic₄₇	C₂²	1	2	yes	—
7	C₂³×Dic₄₇	C₂²	1	2	yes	—
8	C₂×C₄⋊Dic₄₇	—	2	4	no	—
9	C₂×C₄⋊Dic₄₇	C₂²	1	4	yes	—
10	C₂×C₄×Dic₄₇	C₄	1	8	yes	complemented
11	C₂×C₄×Dic₄₇	C₂²	1	4	yes	—
12	C₄₇⋊#95	C₂²	1	16	yes	—
13	C₂³×C₄×C₄₇	C₂²	1	2	yes	—
14	C₂³×C₄×C₄₇	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊C₄×C₄₇	—	2	4	no	—
16	C₂×C₄⋊C₄×C₄₇	C₂²	1	4	yes	—
17	C₂⁴×C₄₇	C₂³	1	1	yes	characteristic
18	C₂²×C₄×C₄₇	—	2	4	no	—
19	C₂²×C₄×C₄₇	—	2	8	no	—
20	C₂²×C₄×C₄₇	C₂×C₄	1	4	yes	—
21	C₂²×C₄×C₄₇	D₄	1	4	yes	—
22	C₂²×C₄×C₄₇	Q₈	1	4	yes	—
23	C₂²×C₄×C₄₇	C₂³	1	4	yes	—
24	C₂²×C₄×C₄₇	C₂³	1	2	yes	—
25	C₂²×Dic₄₇	—	2	4	no	—
26	C₂²×Dic₄₇	—	2	4	no	—
27	C₂²×Dic₄₇	—	2	4	no	—
28	C₂²×Dic₄₇	—	2	8	no	—
29	C₂²×Dic₄₇	C₂×C₄	1	8	yes	—
30	C₂²×Dic₄₇	C₂³	1	4	yes	—
31	C₂²×Dic₄₇	C₂³	1	4	yes	—
32	C₄⋊C₄×C₄₇	—	2	8	no	—
33	C₄⋊Dic₄₇	—	2	8	no	—
34	C₄×Dic₄₇	C₂×C₄	1	16	yes	complemented
35	C₂³×C₄₇	C₂²×C₄	1	2	yes	—
36	C₂³×C₄₇	C₂×D₄	1	2	yes	—
37	C₂³×C₄₇	C₂×Q₈	1	2	yes	—
38	C₂³×C₄₇	#41	1	4	yes	—
39	C₂³×C₄₇	#41	1	4	yes	—
40	C₂³×C₄₇	C₂⁴	1	1	yes	characteristic
41	C₂×C₄×C₄₇	—	2	4	no	—
42	C₂×C₄×C₄₇	—	2	8	no	—
43	C₂×C₄×C₄₇	—	4	8	no	—
44	C₂×C₄×C₄₇	C₄⋊C₄	1	16	yes	—
45	C₂×C₄×C₄₇	C₂²×C₄	1	4	yes	—
46	C₂×C₄×C₄₇	C₂×D₄	1	4	yes	—
47	C₂×C₄×C₄₇	C₂×Q₈	1	4	yes	—
48	C₂×Dic₄₇	—	2	16	no	—
49	C₂×Dic₄₇	—	2	4	no	—
50	C₂×Dic₄₇	—	2	4	no	—
51	C₂×Dic₄₇	—	2	4	no	—
52	C₂×Dic₄₇	—	2	8	no	—
53	C₂×Dic₄₇	—	4	8	no	—
54	C₂×Dic₄₇	C₂²×C₄	1	8	yes	—
55	C₂²×C₄₇	C₂×C₄⋊C₄	1	4	yes	—
56	C₂²×C₄₇	#117	1	8	yes	—
57	C₂²×C₄₇	#124	1	4	yes	—
58	C₂²×C₄₇	#124	1	4	yes	—
59	C₂²×C₄₇	#125	1	8	yes	—
60	C₂²×C₄₇	C₂³×C₄	1	1	yes	characteristic
61	C₂²×C₄₇	C₂²×D₄	1	1	yes	characteristic
62	C₂²×C₄₇	C₂²×Q₈	1	1	yes	characteristic
63	C₂²×C₄₇	C₂×#41	1	2	yes	—
64	C₂²×C₄₇	C₂×#41	1	2	yes	—
65	C₄×C₄₇	—	4	4	no	—
66	C₄×C₄₇	C₂×C₄⋊C₄	1	8	yes	—
67	Dic₄₇	—	2	8	no	—
68	Dic₄₇	—	4	4	no	—
69	C₂×#382	—	47	1	no	maximal, Sylow
70	C₂×C₄₇	#382	1	8	yes	complemented
71	C₂×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
72	C₂×C₄₇	C₂×#117	1	2	yes	—
73	C₂×C₄₇	C₂×#124	1	1	yes	characteristic
74	C₂×C₄₇	C₂×#124	1	1	yes	characteristic
75	C₂×C₄₇	C₂×#125	1	2	yes	—
76	C₂²×C₄⋊C₄	—	47	1	no	—
77	C₂²×C₄⋊C₄	D₄₇	1	1	yes	characteristic
78	C₂²×C₄²	—	47	1	no	—
79	#382	—	47	8	no	—
80	C₂×#95	—	47	4	no	—
81	C₄₇	C₂×#382	1	1	yes	characteristic, complemented, Sylow
82	C₂³×C₄	—	47	2	no	—
83	C₂³×C₄	—	47	2	no	—
84	C₂³×C₄	C₂×D₄₇	1	2	yes	—
85	C₂³×C₄	C₂×D₄₇	1	1	yes	characteristic
86	C₂×C₄⋊C₄	—	2	4	no	—
87	C₂×C₄⋊C₄	—	47	4	no	—
88	C₂×C₄⋊C₄	—	94	4	no	—
89	C₂×C₄⋊C₄	C₂×D₄₇	1	4	yes	—
90	C₂×C₄²	—	47	8	no	—
91	C₂×C₄²	—	47	4	no	—
92	#95	—	47	16	no	—
93	C₂⁴	C₂²×D₄₇	1	1	yes	characteristic
94	C₂²×C₄	—	2	8	no	—
95	C₂²×C₄	—	2	4	no	—
96	C₂²×C₄	—	47	8	no	—
97	C₂²×C₄	—	47	4	no	—
98	C₂²×C₄	—	47	4	no	—
99	C₂²×C₄	—	94	4	no	—
100	C₂²×C₄	—	94	4	no	—
101	C₂²×C₄	—	94	8	no	—
102	C₂²×C₄	—	94	4	no	—
103	C₂²×C₄	Dic₉₄	1	4	yes	—
104	C₂²×C₄	C₄×D₄₇	1	4	yes	—
105	C₂²×C₄	C₂²⋊D₄₇	1	4	yes	—
106	C₂²×C₄	C₂²×D₄₇	1	4	yes	—
107	C₂²×C₄	C₂²×D₄₇	1	2	yes	—
108	C₄⋊C₄	—	2	8	no	—
109	C₄⋊C₄	—	94	8	no	—
110	C₄²	—	47	16	no	—
111	C₂³	C₂×Dic₉₄	1	2	yes	—
112	C₂³	C₂×C₄×D₄₇	1	2	yes	—
113	C₂³	#10625352	1	4	yes	—
114	C₂³	#10625354	1	4	yes	—
115	C₂³	C₂×C₂²⋊D₄₇	1	2	yes	—
116	C₂³	C₂³×D₄₇	1	1	yes	characteristic
117	C₂×C₄	—	2	4	no	—
118	C₂×C₄	—	2	8	no	—
119	C₂×C₄	—	4	8	no	—
120	C₂×C₄	—	47	8	no	—
121	C₂×C₄	—	94	16	no	—
122	C₂×C₄	—	94	4	no	—
123	C₂×C₄	—	94	4	no	—
124	C₂×C₄	—	94	4	no	—
125	C₂×C₄	—	94	8	no	—
126	C₂×C₄	—	188	8	no	—
127	C₂×C₄	Dic₄₇⋊C₄	1	16	yes	—
128	C₂×C₄	C₂×Dic₉₄	1	4	yes	—
129	C₂×C₄	C₂×C₄×D₄₇	1	4	yes	—
130	C₂×C₄	C₂×C₂²⋊D₄₇	1	4	yes	—
131	C₂²	C₂×Dic₄₇⋊C₄	1	4	yes	—
132	C₂²	#13093982	1	8	yes	—
133	C₂²	#13094011	1	4	yes	—
134	C₂²	#13094013	1	4	yes	—
135	C₂²	#13094015	1	8	yes	—
136	C₂²	C₂²×C₄×D₄₇	1	1	yes	characteristic
137	C₂²	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
138	C₂²	C₂²×Dic₉₄	1	1	yes	characteristic
139	C₂²	C₂×#10625352	1	2	yes	—
140	C₂²	C₂×#10625354	1	2	yes	—
141	C₄	—	4	4	no	—
142	C₄	—	94	8	no	—
143	C₄	—	188	4	no	—
144	C₄	C₂×Dic₄₇⋊C₄	1	8	yes	—
145	C₂	#15688988	1	8	yes	direct product factor
146	C₂	C₂²×Dic₄₇⋊C₄	1	1	yes	characteristic
147	C₂	C₂×#13093982	1	2	yes	—
148	C₂	C₂×#13094011	1	1	yes	characteristic
149	C₂	C₂×#13094013	1	1	yes	characteristic
150	C₂	C₂×#13094015	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×C₄⋊Dic₄₇

C₂

yes

maximal, characteristic

C₂²×C₄×Dic₄₇

C₂

yes

maximal, characteristic

#15688988

C₂

yes

maximal, direct product factor

C₂×C₄₇⋊#95

C₂

yes

maximal

C₂²×C₄⋊C₄×C₄₇

C₂

yes

maximal, characteristic

C₂³×Dic₄₇

C₂²

yes

—

C₂³×Dic₄₇

C₂²

yes

—

C₂×C₄⋊Dic₄₇

—

C₂×C₄⋊Dic₄₇

C₂²

yes

—

C₂×C₄×Dic₄₇

C₄

yes

complemented

C₂×C₄×Dic₄₇

C₂²

yes

—

C₄₇⋊#95

C₂²

yes

—

C₂³×C₄×C₄₇

C₂²

yes

—

C₂³×C₄×C₄₇

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₄₇

—

C₂×C₄⋊C₄×C₄₇

C₂²

yes

—

C₂⁴×C₄₇

C₂³

yes

characteristic

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

C₂×C₄

yes

—

C₂²×C₄×C₄₇

D₄

yes

—

C₂²×C₄×C₄₇

Q₈

yes

—

C₂²×C₄×C₄₇

C₂³

yes

—

C₂²×C₄×C₄₇

C₂³

yes

—

C₂²×Dic₄₇

—

C₂²×Dic₄₇

—

C₂²×Dic₄₇

—

C₂²×Dic₄₇

—

C₂²×Dic₄₇

C₂×C₄

yes

—

C₂²×Dic₄₇

C₂³

yes

—

C₂²×Dic₄₇

C₂³

yes

—

C₄⋊C₄×C₄₇

—

C₄⋊Dic₄₇

—

C₄×Dic₄₇

C₂×C₄

yes

complemented

C₂³×C₄₇

C₂²×C₄

yes

—

C₂³×C₄₇

C₂×D₄

yes

—

C₂³×C₄₇

C₂×Q₈

yes

—

C₂³×C₄₇

#41

yes

—

C₂³×C₄₇

#41

yes

—

C₂³×C₄₇

C₂⁴

yes

characteristic

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₄⋊C₄

yes

—

C₂×C₄×C₄₇

C₂²×C₄

yes

—

C₂×C₄×C₄₇

C₂×D₄

yes

—

C₂×C₄×C₄₇

C₂×Q₈

yes

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

C₂²×C₄

yes

—

C₂²×C₄₇

C₂×C₄⋊C₄

yes

—

C₂²×C₄₇

#117

yes

—

C₂²×C₄₇

#124

yes

—

C₂²×C₄₇

#124

yes

—

C₂²×C₄₇

#125

yes

—

C₂²×C₄₇

C₂³×C₄

yes

characteristic

C₂²×C₄₇

C₂²×D₄

yes

characteristic

C₂²×C₄₇

C₂²×Q₈

yes

characteristic

C₂²×C₄₇

C₂×#41

yes

—

C₂²×C₄₇

C₂×#41

yes

—

C₄×C₄₇

—

C₄×C₄₇

C₂×C₄⋊C₄

yes

—

Dic₄₇

—

Dic₄₇

—

—

C₂×C₄₇

yes

C₂×C₄₇

C₂²×C₄⋊C₄

yes

characteristic

C₂×C₄₇

C₂×#117

yes

—

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

—

C₂²×C₄⋊C₄

—

C₂²×C₄⋊C₄

D₄₇

yes

characteristic

C₂²×C₄²

—

#382

—

C₂×#95

—

C₄₇

C₂×#382

yes

characteristic, complemented, Sylow

C₂³×C₄

—

C₂³×C₄

—

C₂³×C₄

C₂×D₄₇

yes

—

C₂³×C₄

C₂×D₄₇

yes

characteristic

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₄₇

yes

—

C₂×C₄²

—

C₂×C₄²

—

#95

—

C₂⁴

C₂²×D₄₇

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

100

C₂²×C₄

—

101

C₂²×C₄

—

102

C₂²×C₄

—

103

C₂²×C₄

Dic₉₄

yes

—

104

C₂²×C₄

C₄×D₄₇

yes

—

105

C₂²×C₄

C₂²⋊D₄₇

yes

—

106

C₂²×C₄

C₂²×D₄₇

yes

—

107

C₂²×C₄

C₂²×D₄₇

yes

—

108

C₄⋊C₄

—

109

C₄⋊C₄

—

110

C₄²

—

111

C₂³

C₂×Dic₉₄

yes

—

112

C₂³

C₂×C₄×D₄₇

yes

—

113

C₂³

#10625352

yes

—

114

C₂³

#10625354

yes

—

115

C₂³

C₂×C₂²⋊D₄₇

yes

—

116

C₂³

C₂³×D₄₇

yes

characteristic

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

188

—

127

C₂×C₄

Dic₄₇⋊C₄

yes

—

128

C₂×C₄

C₂×Dic₉₄

yes

—

129

C₂×C₄

C₂×C₄×D₄₇

yes

—

130

C₂×C₄

C₂×C₂²⋊D₄₇

yes

—

131

C₂²

C₂×Dic₄₇⋊C₄

yes

—

132

C₂²

#13093982

yes

—

133

C₂²

#13094011

yes

—

134

C₂²

#13094013

yes

—

135

C₂²

#13094015

yes

—

136

C₂²

C₂²×C₄×D₄₇

yes

characteristic

137

C₂²

C₂²×C₂²⋊D₄₇

yes

characteristic

138

C₂²

C₂²×Dic₉₄

yes

characteristic

139

C₂²

C₂×#10625352

yes

—

140

C₂²

C₂×#10625354

yes

—

141

C₄

—

142

C₄

—

143

C₄

—

188

—

144

C₄

C₂×Dic₄₇⋊C₄

yes

—

145

C₂

#15688988

yes

direct product factor

146

C₂

C₂²×Dic₄₇⋊C₄

yes

characteristic

147

C₂

C₂×#13093982

yes

—

148

C₂

C₂×#13094011

yes

characteristic

149

C₂

C₂×#13094013

yes

characteristic

150

C₂

C₂×#13094015

yes

—

	subgroup	G/H
center	C₂⁴	C₂²×D₄₇
commutator	C₂²×C₄₇	C₂³×C₄
Frattini	C₂³	C₂³×D₄₇
Fitting	C₂²×C₄⋊C₄×C₄₇	C₂
socle	C₂⁴×C₄₇	C₂³
hypercenter	C₂²×C₄⋊C₄	D₄₇
hypocenter	C₄₇	C₂×#382
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄⋊C₄

subgroup

G/H

center

C₂⁴

C₂²×D₄₇

commutator

C₂²×C₄₇

C₂³×C₄

Frattini

C₂³

C₂³×D₄₇

Fitting

C₂²×C₄⋊C₄×C₄₇

C₂

socle

C₂⁴×C₄₇

C₂³

hypercenter

C₂²×C₄⋊C₄

D₄₇

C₄₇

G ▷ C₂²×C₄₇ ▷ C₄₇

C₁ ◁ C₂⁴ ◁ C₂²×C₄⋊C₄

N	H ≅ G/N	type
3. #15688988	C₂	direct (N × H)
10. C₂×C₄×Dic₄₇	C₄	semidirect
34. C₄×Dic₄₇	C₂×C₄	semidirect
70. C₂×C₄₇	#382	semidirect
81. C₄₇	C₂×#382	semidirect
145. C₂	#15688988	direct (N × H)

H ≅ G/N

type

3. #15688988

C₂

direct (N × H)

10. C₂×C₄×Dic₄₇

C₄

semidirect

34. C₄×Dic₄₇

C₂×C₄

semidirect

70. C₂×C₄₇

#382

semidirect

81. C₄₇

C₂×#382

semidirect

145. C₂

#15688988

direct (N × H)

Subgroups — C₂×#15688988 #20245774

Conjugacy classes of subgroups

Complements

Subgroups — C2×#15688988 #20245774

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#15688988 #20245774