Group #20245772 — subgroups

Subgroups — #382×D₄₇ #20245772

Order 6016.

712 conjugacy classes of subgroups; only the 269 normal subgroups are shown.

normal subgroups only

This group has 712 conjugacy classes of subgroups — too many to draw. Only its 269 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×C₄⋊C₄×D₄₇	C₂	1	2	yes	maximal
2	C₂×C₄²×D₄₇	C₂	1	1	yes	maximal, characteristic
3	#15688989	C₂	1	1	yes	maximal, characteristic, complemented
4	#15688988	C₂	1	2	yes	maximal, complemented
5	#15688987	C₂	1	4	yes	maximal, complemented
6	#95×D₄₇	C₂	1	4	yes	maximal
7	C₄₇×#382	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×C₄×D₄₇	C₂²	1	4	yes	—
9	C₂²×C₄×D₄₇	C₂²	1	2	yes	—
10	C₂²×C₄×D₄₇	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊Dic₄₇	C₂²	1	2	yes	—
12	C₂×Dic₄₇⋊C₄	C₂²	1	4	yes	—
13	C₄⋊C₄×D₄₇	—	2	8	no	—
14	C₂×C₄×Dic₄₇	C₂²	1	2	yes	—
15	C₂×C₄×Dic₄₇	C₂²	1	1	yes	characteristic
16	C₄²×D₄₇	C₄	1	8	yes	complemented
17	C₄₇⋊#95	C₂²	1	8	yes	—
18	C₄₇⋊#95	C₂²	1	4	yes	—
19	C₂×C₄⋊C₄×C₄₇	C₂²	1	2	yes	—
20	C₂×C₄²×C₄₇	C₂²	1	1	yes	characteristic
21	#95×C₄₇	C₂²	1	4	yes	—
22	C₂³×D₄₇	C₂³	1	1	yes	characteristic
23	C₂²×C₄×C₄₇	C₂³	1	4	yes	—
24	C₂²×C₄×C₄₇	C₂³	1	2	yes	—
25	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
26	C₂²×Dic₄₇	C₂³	1	4	yes	—
27	C₂²×Dic₄₇	C₂³	1	2	yes	—
28	C₂²×Dic₄₇	C₂³	1	1	yes	characteristic
29	C₂×C₄×D₄₇	—	2	8	no	—
30	C₂×C₄×D₄₇	—	2	8	no	—
31	C₂×C₄×D₄₇	—	2	16	no	—
32	C₂×C₄×D₄₇	C₂×C₄	1	8	yes	—
33	C₂×C₄×D₄₇	C₂×C₄	1	4	yes	—
34	C₂×C₄×D₄₇	D₄	1	4	yes	—
35	C₂×C₄×D₄₇	Q₈	1	4	yes	—
36	C₄⋊C₄×C₄₇	—	2	4	no	—
37	C₄²×C₄₇	C₂×C₄	1	4	yes	complemented
38	C₄⋊Dic₄₇	—	2	4	no	—
39	Dic₄₇⋊C₄	—	2	8	no	—
40	C₄×Dic₄₇	C₂×C₄	1	4	yes	complemented
41	C₄×Dic₄₇	C₂×C₄	1	8	yes	complemented
42	C₂³×C₄₇	C₂⁴	1	1	yes	characteristic
43	C₂²×D₄₇	C₂²×C₄	1	2	yes	—
44	C₂²×D₄₇	C₂×D₄	1	2	yes	—
45	C₂²×D₄₇	C₂×Q₈	1	2	yes	—
46	C₂²×D₄₇	#41	1	8	yes	—
47	C₂×C₄×C₄₇	—	2	4	no	—
48	C₂×C₄×C₄₇	—	2	4	no	—
49	C₂×C₄×C₄₇	—	2	8	no	—
50	C₂×C₄×C₄₇	C₂²×C₄	1	4	yes	—
51	C₂×C₄×C₄₇	C₂²×C₄	1	2	yes	—
52	C₂×C₄×C₄₇	C₂×D₄	1	2	yes	—
53	C₂×C₄×C₄₇	C₂×Q₈	1	2	yes	—
54	C₂×Dic₄₇	—	2	4	no	—
55	C₂×Dic₄₇	—	2	4	no	—
56	C₂×Dic₄₇	—	2	8	no	—
57	C₂×Dic₄₇	C₂²×C₄	1	4	yes	—
58	C₂×Dic₄₇	C₂²×C₄	1	2	yes	—
59	C₂×Dic₄₇	C₂×D₄	1	2	yes	—
60	C₂×Dic₄₇	C₂×Q₈	1	2	yes	—
61	C₄×D₄₇	—	2	16	no	—
62	C₄×D₄₇	—	4	16	no	—
63	C₄×D₄₇	C₄⋊C₄	1	16	yes	—
64	C₂²×C₄₇	C₂³×C₄	1	1	yes	characteristic
65	C₂²×C₄₇	C₂²×D₄	1	1	yes	characteristic
66	C₂²×C₄₇	C₂²×Q₈	1	1	yes	characteristic
67	C₂²×C₄₇	C₂×#41	1	4	yes	—
68	C₂×D₄₇	C₂×C₄⋊C₄	1	4	yes	—
69	C₂×D₄₇	#117	1	8	yes	—
70	C₂×D₄₇	#124	1	8	yes	—
71	C₂×D₄₇	#125	1	8	yes	—
72	C₄×C₄₇	—	2	4	no	—
73	C₄×C₄₇	—	4	4	no	—
74	C₄×C₄₇	C₂×C₄⋊C₄	1	4	yes	—
75	Dic₄₇	—	2	4	no	—
76	Dic₄₇	—	4	4	no	—
77	Dic₄₇	C₂×C₄⋊C₄	1	4	yes	—
78	C₂×#382	—	47	1	no	maximal, Sylow
79	C₂×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
80	C₂×C₄₇	C₂×#117	1	2	yes	—
81	C₂×C₄₇	C₂×#124	1	2	yes	—
82	C₂×C₄₇	C₂×#125	1	2	yes	—
83	D₄₇	#382	1	8	yes	direct product factor
84	C₂²×C₄⋊C₄	—	47	2	no	—
85	C₂²×C₄²	—	47	1	no	—
86	#382	—	47	2	no	—
87	#382	—	47	1	no	—
88	#382	—	47	4	no	—
89	#382	D₄₇	1	1	yes	characteristic, direct product factor
90	C₂×#95	—	47	4	no	—
91	C₄₇	C₂×#382	1	1	yes	characteristic, complemented, Sylow
92	C₂³×C₄	—	47	1	no	—
93	C₂³×C₄	—	47	2	no	—
94	C₂³×C₄	—	47	4	no	—
95	C₂×C₄⋊C₄	—	47	4	no	—
96	C₂×C₄⋊C₄	—	47	2	no	—
97	C₂×C₄⋊C₄	—	94	8	no	—
98	C₂×C₄⋊C₄	C₂×D₄₇	1	2	yes	—
99	C₂×C₄²	—	47	2	no	—
100	C₂×C₄²	—	47	1	no	—
101	C₂×C₄²	—	47	8	no	—
102	C₂×C₄²	C₂×D₄₇	1	1	yes	characteristic
103	#95	—	47	8	no	—
104	#95	—	47	4	no	—
105	#95	C₂×D₄₇	1	4	yes	—
106	C₂⁴	—	47	1	no	—
107	C₂²×C₄	—	47	1	no	—
108	C₂²×C₄	—	47	4	no	—
109	C₂²×C₄	—	47	4	no	—
110	C₂²×C₄	—	47	4	no	—
111	C₂²×C₄	—	47	2	no	—
112	C₂²×C₄	—	47	8	no	—
113	C₂²×C₄	—	47	4	no	—
114	C₂²×C₄	—	94	8	no	—
115	C₂²×C₄	—	94	8	no	—
116	C₂²×C₄	—	94	16	no	—
117	C₂²×C₄	C₂²×D₄₇	1	4	yes	—
118	C₂²×C₄	C₂²×D₄₇	1	2	yes	—
119	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
120	C₄⋊C₄	—	2	4	no	—
121	C₄⋊C₄	—	94	8	no	—
122	C₄⋊C₄	—	94	4	no	—
123	C₄²	—	47	8	no	—
124	C₄²	—	47	4	no	—
125	C₄²	C₄×D₄₇	1	4	yes	complemented
126	C₂³	—	47	2	no	—
127	C₂³	—	47	2	no	—
128	C₂³	—	47	2	no	—
129	C₂³	—	47	8	no	—
130	C₂³	C₂³×D₄₇	1	1	yes	characteristic
131	C₂×C₄	—	2	4	no	—
132	C₂×C₄	—	2	4	no	—
133	C₂×C₄	—	2	8	no	—
134	C₂×C₄	—	47	2	no	—
135	C₂×C₄	—	47	2	no	—
136	C₂×C₄	—	47	2	no	—
137	C₂×C₄	—	47	16	no	—
138	C₂×C₄	—	47	4	no	—
139	C₂×C₄	—	94	16	no	—
140	C₂×C₄	—	94	4	no	—
141	C₂×C₄	—	94	4	no	—
142	C₂×C₄	—	94	8	no	—
143	C₂×C₄	—	188	16	no	—
144	C₂×C₄	C₂×C₄×D₄₇	1	4	yes	—
145	C₂×C₄	C₂×C₄×D₄₇	1	2	yes	—
146	C₂×C₄	D₄×D₄₇	1	2	yes	—
147	C₂×C₄	Q₈×D₄₇	1	2	yes	—
148	C₂²	—	47	4	no	—
149	C₂²	—	47	8	no	—
150	C₂²	—	47	8	no	—
151	C₂²	—	47	8	no	—
152	C₂²	C₂²×C₄×D₄₇	1	1	yes	characteristic
153	C₂²	C₂×D₄×D₄₇	1	1	yes	characteristic
154	C₂²	C₂×Q₈×D₄₇	1	1	yes	characteristic
155	C₂²	#41×D₄₇	1	4	yes	—
156	C₄	—	2	4	no	—
157	C₄	—	4	4	no	—
158	C₄	—	47	4	no	—
159	C₄	—	94	4	no	—
160	C₄	—	188	4	no	—
161	C₄	C₄⋊C₄×D₄₇	1	4	yes	—
162	C₂	—	47	8	no	—
163	C₂	C₂×C₄⋊C₄×D₄₇	1	1	yes	characteristic
164	C₂	#117×D₄₇	1	2	yes	—
165	C₂	#124×D₄₇	1	2	yes	—
166	C₂	#125×D₄₇	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊C₄×D₄₇

C₂

yes

maximal

C₂×C₄²×D₄₇

C₂

yes

maximal, characteristic

#15688989

C₂

yes

maximal, characteristic, complemented

#15688988

C₂

yes

maximal, complemented

#15688987

C₂

yes

maximal, complemented

#95×D₄₇

C₂

yes

maximal

C₄₇×#382

C₂

yes

maximal, characteristic, complemented

C₂²×C₄×D₄₇

C₂²

yes

—

C₂²×C₄×D₄₇

C₂²

yes

—

C₂²×C₄×D₄₇

C₂²

yes

characteristic

C₂×C₄⋊Dic₄₇

C₂²

yes

—

C₂×Dic₄₇⋊C₄

C₂²

yes

—

C₄⋊C₄×D₄₇

—

C₂×C₄×Dic₄₇

C₂²

yes

—

C₂×C₄×Dic₄₇

C₂²

yes

characteristic

C₄²×D₄₇

C₄

yes

complemented

C₄₇⋊#95

C₂²

yes

—

C₄₇⋊#95

C₂²

yes

—

C₂×C₄⋊C₄×C₄₇

C₂²

yes

—

C₂×C₄²×C₄₇

C₂²

yes

characteristic

#95×C₄₇

C₂²

yes

—

C₂³×D₄₇

C₂³

yes

characteristic

C₂²×C₄×C₄₇

C₂³

yes

—

C₂²×C₄×C₄₇

C₂³

yes

—

C₂²×C₄×C₄₇

C₂³

yes

characteristic

C₂²×Dic₄₇

C₂³

yes

—

C₂²×Dic₄₇

C₂³

yes

—

C₂²×Dic₄₇

C₂³

yes

characteristic

C₂×C₄×D₄₇

—

C₂×C₄×D₄₇

—

C₂×C₄×D₄₇

—

C₂×C₄×D₄₇

C₂×C₄

yes

—

C₂×C₄×D₄₇

C₂×C₄

yes

—

C₂×C₄×D₄₇

D₄

yes

—

C₂×C₄×D₄₇

Q₈

yes

—

C₄⋊C₄×C₄₇

—

C₄²×C₄₇

C₂×C₄

yes

complemented

C₄⋊Dic₄₇

—

Dic₄₇⋊C₄

—

C₄×Dic₄₇

C₂×C₄

yes

complemented

C₄×Dic₄₇

C₂×C₄

yes

complemented

C₂³×C₄₇

C₂⁴

yes

characteristic

C₂²×D₄₇

C₂²×C₄

yes

—

C₂²×D₄₇

C₂×D₄

yes

—

C₂²×D₄₇

C₂×Q₈

yes

—

C₂²×D₄₇

#41

yes

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂²×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

C₂²×C₄

yes

—

C₂×Dic₄₇

C₂²×C₄

yes

—

C₂×Dic₄₇

C₂×D₄

yes

—

C₂×Dic₄₇

C₂×Q₈

yes

—

C₄×D₄₇

—

C₄×D₄₇

—

C₄×D₄₇

C₄⋊C₄

yes

—

C₂²×C₄₇

C₂³×C₄

yes

characteristic

C₂²×C₄₇

C₂²×D₄

yes

characteristic

C₂²×C₄₇

C₂²×Q₈

yes

characteristic

C₂²×C₄₇

C₂×#41

yes

—

C₂×D₄₇

C₂×C₄⋊C₄

yes

—

C₂×D₄₇

#117

yes

—

C₂×D₄₇

#124

yes

—

C₂×D₄₇

#125

yes

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

C₂×C₄⋊C₄

yes

—

Dic₄₇

—

Dic₄₇

—

Dic₄₇

C₂×C₄⋊C₄

yes

—

C₂×#382

—

maximal, Sylow

C₂×C₄₇

C₂²×C₄⋊C₄

yes

characteristic

C₂×C₄₇

C₂×#117

yes

—

C₂×C₄₇

C₂×#124

yes

—

C₂×C₄₇

C₂×#125

yes

—

D₄₇

#382

yes

direct product factor

C₂²×C₄⋊C₄

—

C₂²×C₄²

—

—

—

—

D₄₇

yes

characteristic, direct product factor

C₂×#95

—

C₄₇

C₂×#382

yes

characteristic, complemented, Sylow

C₂³×C₄

—

C₂³×C₄

—

C₂³×C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₄₇

yes

—

C₂×C₄²

—

100

C₂×C₄²

—

101

C₂×C₄²

—

102

C₂×C₄²

C₂×D₄₇

yes

characteristic

103

#95

—

104

#95

—

105

#95

C₂×D₄₇

yes

—

106

C₂⁴

—

107

C₂²×C₄

—

108

C₂²×C₄

—

109

C₂²×C₄

—

110

C₂²×C₄

—

111

C₂²×C₄

—

112

C₂²×C₄

—

113

C₂²×C₄

—

114

C₂²×C₄

—

115

C₂²×C₄

—

116

C₂²×C₄

—

117

C₂²×C₄

C₂²×D₄₇

yes

—

118

C₂²×C₄

C₂²×D₄₇

yes

—

119

C₂²×C₄

C₂²×D₄₇

yes

characteristic

120

C₄⋊C₄

—

121

C₄⋊C₄

—

122

C₄⋊C₄

—

123

C₄²

—

124

C₄²

—

125

C₄²

C₄×D₄₇

yes

complemented

126

C₂³

—

127

C₂³

—

128

C₂³

—

129

C₂³

—

130

C₂³

C₂³×D₄₇

yes

characteristic

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

188

—

144

C₂×C₄

C₂×C₄×D₄₇

yes

—

145

C₂×C₄

C₂×C₄×D₄₇

yes

—

146

C₂×C₄

D₄×D₄₇

yes

—

147

C₂×C₄

Q₈×D₄₇

yes

—

148

C₂²

—

149

C₂²

—

150

C₂²

—

151

C₂²

—

152

C₂²

C₂²×C₄×D₄₇

yes

characteristic

153

C₂²

C₂×D₄×D₄₇

yes

characteristic

154

C₂²

C₂×Q₈×D₄₇

yes

characteristic

155

C₂²

#41×D₄₇

yes

—

156

C₄

—

157

C₄

—

158

C₄

—

159

C₄

—

160

C₄

—

188

—

161

C₄

C₄⋊C₄×D₄₇

yes

—

162

C₂

—

163

C₂

C₂×C₄⋊C₄×D₄₇

yes

characteristic

164

C₂

#117×D₄₇

yes

—

165

C₂

#124×D₄₇

yes

—

166

C₂

#125×D₄₇

yes

—

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

subgroup

G/H

center, Frattini

C₂³

C₂³×D₄₇

commutator

C₂²×C₄₇

C₂³×C₄

Fitting

C₄₇×#382

C₂

socle

C₂³×C₄₇

C₂⁴

D₄₇

C₄₇

G ▷ C₂²×C₄₇ ▷ C₄₇

C₁ ◁ C₂³ ◁ #382

N	H ≅ G/N	type
3. #15688989	C₂	semidirect
4. #15688988	C₂	semidirect
5. #15688987	C₂	semidirect
7. C₄₇×#382	C₂	semidirect
16. C₄²×D₄₇	C₄	semidirect
37. C₄²×C₄₇	C₂×C₄	semidirect
40. C₄×Dic₄₇	C₂×C₄	semidirect
41. C₄×Dic₄₇	C₂×C₄	semidirect
83. D₄₇	#382	direct (N × H)
89. #382	D₄₇	direct (N × H)
91. C₄₇	C₂×#382	semidirect
125. C₄²	C₄×D₄₇	semidirect

H ≅ G/N

type

3. #15688989

C₂

semidirect

4. #15688988

C₂

semidirect

5. #15688987

C₂

semidirect

7. C₄₇×#382

C₂

semidirect

16. C₄²×D₄₇

C₄

semidirect

37. C₄²×C₄₇

C₂×C₄

semidirect

40. C₄×Dic₄₇

C₂×C₄

semidirect

41. C₄×Dic₄₇

C₂×C₄

semidirect

83. D₄₇

#382

direct (N × H)

89. #382

D₄₇

direct (N × H)

91. C₄₇

C₂×#382

semidirect

125. C₄²

C₄×D₄₇

semidirect

Subgroups — #382×D₄₇ #20245772

Conjugacy classes of subgroups

Complements

Subgroups — #382×D47 #20245772

Conjugacy classes of subgroups

Complements

Subgroups — #382×D₄₇ #20245772