Group #20245870 — subgroups

Subgroups — #20245870

Order 6016.

656 conjugacy classes of subgroups; only the 277 normal subgroups are shown.

normal subgroups only

This group has 656 conjugacy classes of subgroups — too many to draw. Only its 277 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×#13093983	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄×Dic₄₇	C₂	1	1	yes	maximal, characteristic
3	#15689019	C₂	1	4	yes	maximal, complemented
4	#15688995	C₂	1	4	yes	maximal
5	C₄×Dic₄₇⋊C₄	C₂	1	4	yes	maximal, complemented
6	C₂×#117×C₄₇	C₂	1	1	yes	maximal, characteristic
7	C₂³×Dic₄₇	C₂²	1	2	yes	—
8	#13093983	C₄	1	8	yes	complemented
9	C₂×C₄⋊Dic₄₇	C₂²	1	2	yes	—
10	C₂×Dic₄₇⋊C₄	C₂²	1	8	yes	—
11	C₂×C₂²⋊Dic₄₇	C₂²	1	2	yes	—
12	C₂×C₄×Dic₄₇	—	2	4	no	—
13	C₂×C₄×Dic₄₇	C₂²	1	2	yes	—
14	C₂×C₄×Dic₄₇	C₂²	1	2	yes	—
15	C₂×C₄×Dic₄₇	C₂²	1	2	yes	—
16	C₄₇⋊#95	C₂²	1	8	yes	—
17	C₂³×C₄×C₄₇	C₂²	1	1	yes	characteristic
18	#117×C₄₇	—	2	4	no	—
19	C₂×C₄⋊C₄×C₄₇	C₂²	1	2	yes	—
20	C₂×C₂²⋊C₄×C₄₇	C₂²	1	2	yes	—
21	C₂×C₄²×C₄₇	C₂²	1	2	yes	—
22	C₂⁴×C₄₇	C₂³	1	1	yes	characteristic
23	C₂²×C₄×C₄₇	C₂×C₄	1	4	yes	—
24	C₂²×C₄×C₄₇	D₄	1	4	yes	—
25	C₂²×C₄×C₄₇	Q₈	1	4	yes	—
26	C₂²×C₄×C₄₇	C₂³	1	4	yes	—
27	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
28	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
29	C₂²×Dic₄₇	—	2	4	no	—
30	C₂²×Dic₄₇	—	2	8	no	—
31	C₂²×Dic₄₇	C₂³	1	4	yes	—
32	C₂²×Dic₄₇	C₂³	1	4	yes	—
33	C₄⋊C₄×C₄₇	—	2	4	no	—
34	C₂²⋊C₄×C₄₇	—	2	4	no	—
35	C₄²×C₄₇	—	2	4	no	—
36	C₂²⋊Dic₄₇	C₂×C₄	1	8	yes	—
37	C₄⋊Dic₄₇	C₂×C₄	1	8	yes	complemented
38	Dic₄₇⋊C₄	—	2	16	no	—
39	C₄×Dic₄₇	—	2	4	no	—
40	C₄×Dic₄₇	—	2	4	no	—
41	C₄×Dic₄₇	C₂×C₄	1	8	yes	complemented
42	C₂³×C₄₇	—	2	4	no	—
43	C₂³×C₄₇	C₂²×C₄	1	2	yes	—
44	C₂³×C₄₇	C₂×D₄	1	2	yes	—
45	C₂³×C₄₇	C₂×Q₈	1	2	yes	—
46	C₂³×C₄₇	C₂⁴	1	1	yes	characteristic
47	C₂×C₄×C₄₇	—	2	4	no	—
48	C₂×C₄×C₄₇	—	2	8	no	—
49	C₂×C₄×C₄₇	C₄⋊C₄	1	16	yes	complemented
50	C₂×C₄×C₄₇	C₂²×C₄	1	2	yes	—
51	C₂×C₄×C₄₇	C₂²×C₄	1	2	yes	—
52	C₂×C₄×C₄₇	C₂×D₄	1	2	yes	—
53	C₂×C₄×C₄₇	C₂×D₄	1	2	yes	—
54	C₂×C₄×C₄₇	C₂×Q₈	1	2	yes	—
55	C₂×C₄×C₄₇	C₂×Q₈	1	2	yes	—
56	C₂×Dic₄₇	—	2	16	no	—
57	C₂×Dic₄₇	—	2	4	no	—
58	C₂×Dic₄₇	—	2	8	no	—
59	C₂×Dic₄₇	—	2	4	no	—
60	C₂×Dic₄₇	C₂²×C₄	1	8	yes	—
61	C₂×Dic₄₇	#41	1	8	yes	—
62	C₂²×C₄₇	—	2	4	no	—
63	C₂²×C₄₇	—	2	4	no	—
64	C₂²×C₄₇	—	2	4	no	—
65	C₂²×C₄₇	C₂×C₄⋊C₄	1	4	yes	—
66	C₂²×C₄₇	C₂³×C₄	1	1	yes	characteristic
67	C₂²×C₄₇	C₂²×D₄	1	1	yes	characteristic
68	C₂²×C₄₇	C₂²×Q₈	1	1	yes	characteristic
69	C₂²×C₄₇	C₂×#41	1	4	yes	—
70	C₄×C₄₇	—	4	4	no	—
71	C₄×C₄₇	C₂×C₄⋊C₄	1	4	yes	—
72	C₄×C₄₇	C₂×C₄⋊C₄	1	4	yes	—
73	Dic₄₇	—	2	8	no	—
74	Dic₄₇	—	2	8	no	—
75	#2296	—	47	1	no	maximal, Sylow
76	C₂×C₄₇	—	2	4	no	—
77	C₂×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
78	C₂×C₄₇	C₄×#41	1	2	yes	—
79	C₂×C₄₇	#525	1	2	yes	—
80	C₂×C₄₇	#533	1	2	yes	—
81	C₂×#117	—	47	1	no	—
82	C₂×#117	D₄₇	1	1	yes	characteristic
83	C₂²×C₄²	—	47	1	no	—
84	#389	—	47	4	no	—
85	#385	—	47	4	no	—
86	C₄×C₄⋊C₄	—	47	4	no	—
87	C₄₇	#2296	1	1	yes	characteristic, complemented, Sylow
88	C₂³×C₄	—	47	2	no	—
89	C₂³×C₄	C₂×D₄₇	1	1	yes	characteristic
90	#117	—	2	4	no	—
91	#117	—	47	8	no	—
92	C₂×C₄⋊C₄	—	47	2	no	—
93	C₂×C₄⋊C₄	—	47	8	no	—
94	C₂×C₄⋊C₄	C₂×D₄₇	1	2	yes	—
95	C₂×C₂²⋊C₄	—	47	2	no	—
96	C₂×C₂²⋊C₄	C₂×D₄₇	1	2	yes	—
97	C₂×C₄²	—	47	2	no	—
98	C₂×C₄²	—	47	2	no	—
99	C₂×C₄²	—	47	2	no	—
100	C₂×C₄²	—	94	4	no	—
101	C₂×C₄²	C₂×D₄₇	1	2	yes	—
102	#95	—	47	8	no	—
103	C₂⁴	C₂²×D₄₇	1	1	yes	characteristic
104	C₂²×C₄	—	47	4	no	—
105	C₂²×C₄	—	47	4	no	—
106	C₂²×C₄	—	94	8	no	—
107	C₂²×C₄	—	94	4	no	—
108	C₂²×C₄	Dic₉₄	1	4	yes	—
109	C₂²×C₄	C₄×D₄₇	1	4	yes	—
110	C₂²×C₄	C₂²⋊D₄₇	1	4	yes	—
111	C₂²×C₄	C₂²×D₄₇	1	4	yes	—
112	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
113	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
114	C₄⋊C₄	—	2	4	no	—
115	C₄⋊C₄	—	47	8	no	—
116	C₄⋊C₄	—	94	16	no	—
117	C₂²⋊C₄	—	2	4	no	—
118	C₂²⋊C₄	—	47	8	no	—
119	C₄²	—	2	4	no	—
120	C₄²	—	47	8	no	—
121	C₄²	—	94	4	no	—
122	C₄²	—	94	4	no	—
123	C₂³	—	2	4	no	—
124	C₂³	C₂×Dic₉₄	1	2	yes	—
125	C₂³	C₂×C₄×D₄₇	1	2	yes	—
126	C₂³	C₂×C₂²⋊D₄₇	1	2	yes	—
127	C₂³	C₂³×D₄₇	1	1	yes	characteristic
128	C₂×C₄	—	2	4	no	—
129	C₂×C₄	—	2	8	no	—
130	C₂×C₄	—	47	8	no	—
131	C₂×C₄	—	47	8	no	—
132	C₂×C₄	—	94	16	no	—
133	C₂×C₄	—	94	4	no	—
134	C₂×C₄	—	94	8	no	—
135	C₂×C₄	—	94	4	no	—
136	C₂×C₄	Dic₄₇⋊C₄	1	16	yes	complemented
137	C₂×C₄	C₂×Dic₉₄	1	2	yes	—
138	C₂×C₄	C₂×Dic₉₄	1	2	yes	—
139	C₂×C₄	C₂×C₄×D₄₇	1	2	yes	—
140	C₂×C₄	C₂×C₄×D₄₇	1	2	yes	—
141	C₂×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
142	C₂×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
143	C₂²	—	2	4	no	—
144	C₂²	—	2	4	no	—
145	C₂²	—	2	4	no	—
146	C₂²	C₂×Dic₄₇⋊C₄	1	4	yes	—
147	C₂²	C₂²×C₄×D₄₇	1	1	yes	characteristic
148	C₂²	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
149	C₂²	C₂²×Dic₉₄	1	1	yes	characteristic
150	C₂²	#41×D₄₇	1	4	yes	—
151	C₄	—	4	4	no	—
152	C₄	—	94	8	no	—
153	C₄	—	94	8	no	—
154	C₄	C₂×Dic₄₇⋊C₄	1	4	yes	—
155	C₄	C₂×Dic₄₇⋊C₄	1	4	yes	—
156	C₂	—	2	4	no	—
157	C₂	C₂²×Dic₄₇⋊C₄	1	1	yes	characteristic
158	C₂	#15689591	1	2	yes	—
159	C₂	#15689630	1	2	yes	—
160	C₂	#15689676	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13093983

C₂

yes

maximal, characteristic

C₂²×C₄×Dic₄₇

C₂

yes

maximal, characteristic

#15689019

C₂

yes

maximal, complemented

#15688995

C₂

yes

maximal

C₄×Dic₄₇⋊C₄

C₂

yes

maximal, complemented

C₂×#117×C₄₇

C₂

yes

maximal, characteristic

C₂³×Dic₄₇

C₂²

yes

—

#13093983

C₄

yes

complemented

C₂×C₄⋊Dic₄₇

C₂²

yes

—

C₂×Dic₄₇⋊C₄

C₂²

yes

—

C₂×C₂²⋊Dic₄₇

C₂²

yes

—

C₂×C₄×Dic₄₇

—

C₂×C₄×Dic₄₇

C₂²

yes

—

C₂×C₄×Dic₄₇

C₂²

yes

—

C₂×C₄×Dic₄₇

C₂²

yes

—

C₄₇⋊#95

C₂²

yes

—

C₂³×C₄×C₄₇

C₂²

yes

characteristic

#117×C₄₇

—

C₂×C₄⋊C₄×C₄₇

C₂²

yes

—

C₂×C₂²⋊C₄×C₄₇

C₂²

yes

—

C₂×C₄²×C₄₇

C₂²

yes

—

C₂⁴×C₄₇

C₂³

yes

characteristic

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

characteristic

C₂²×C₄×C₄₇

C₂³

yes

characteristic

C₂²×Dic₄₇

—

C₂²×Dic₄₇

—

C₂²×Dic₄₇

C₂³

yes

—

C₂²×Dic₄₇

C₂³

yes

—

C₄⋊C₄×C₄₇

—

C₂²⋊C₄×C₄₇

—

C₄²×C₄₇

—

C₂²⋊Dic₄₇

C₂×C₄

yes

—

C₄⋊Dic₄₇

C₂×C₄

yes

complemented

Dic₄₇⋊C₄

—

C₄×Dic₄₇

—

C₄×Dic₄₇

—

C₄×Dic₄₇

C₂×C₄

yes

complemented

C₂³×C₄₇

—

C₂³×C₄₇

C₂²×C₄

yes

—

C₂³×C₄₇

C₂×D₄

yes

—

C₂³×C₄₇

C₂×Q₈

yes

—

C₂³×C₄₇

C₂⁴

yes

characteristic

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₄⋊C₄

yes

complemented

C₂×C₄×C₄₇

C₂²×C₄

yes

—

C₂×C₄×C₄₇

C₂²×C₄

yes

—

C₂×C₄×C₄₇

C₂×D₄

yes

—

C₂×C₄×C₄₇

C₂×D₄

yes

—

C₂×C₄×C₄₇

C₂×Q₈

yes

—

C₂×C₄×C₄₇

C₂×Q₈

yes

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

C₂²×C₄

yes

—

C₂×Dic₄₇

#41

yes

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

C₂×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

—

C₄×C₄₇

—

C₄×C₄₇

C₂×C₄⋊C₄

yes

—

C₄×C₄₇

C₂×C₄⋊C₄

yes

—

Dic₄₇

—

Dic₄₇

—

#2296

—

maximal, Sylow

C₂×C₄₇

—

C₂×C₄₇

C₂²×C₄⋊C₄

yes

characteristic

C₂×C₄₇

C₄×#41

yes

—

C₂×C₄₇

#525

yes

—

C₂×C₄₇

#533

yes

—

C₂×#117

—

C₂×#117

D₄₇

yes

characteristic

C₂²×C₄²

—

#389

—

#385

—

C₄×C₄⋊C₄

—

C₄₇

#2296

yes

characteristic, complemented, Sylow

C₂³×C₄

—

C₂³×C₄

C₂×D₄₇

yes

characteristic

#117

—

#117

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₄₇

yes

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

C₂×D₄₇

yes

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

—

100

C₂×C₄²

—

101

C₂×C₄²

C₂×D₄₇

yes

—

102

#95

—

103

C₂⁴

C₂²×D₄₇

yes

characteristic

104

C₂²×C₄

—

105

C₂²×C₄

—

106

C₂²×C₄

—

107

C₂²×C₄

—

108

C₂²×C₄

Dic₉₄

yes

—

109

C₂²×C₄

C₄×D₄₇

yes

—

110

C₂²×C₄

C₂²⋊D₄₇

yes

—

111

C₂²×C₄

C₂²×D₄₇

yes

—

112

C₂²×C₄

C₂²×D₄₇

yes

characteristic

113

C₂²×C₄

C₂²×D₄₇

yes

characteristic

114

C₄⋊C₄

—

115

C₄⋊C₄

—

116

C₄⋊C₄

—

117

C₂²⋊C₄

—

118

C₂²⋊C₄

—

119

C₄²

—

120

C₄²

—

121

C₄²

—

122

C₄²

—

123

C₂³

—

124

C₂³

C₂×Dic₉₄

yes

—

125

C₂³

C₂×C₄×D₄₇

yes

—

126

C₂³

C₂×C₂²⋊D₄₇

yes

—

127

C₂³

C₂³×D₄₇

yes

characteristic

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

Dic₄₇⋊C₄

yes

complemented

137

C₂×C₄

C₂×Dic₉₄

yes

—

138

C₂×C₄

C₂×Dic₉₄

yes

—

139

C₂×C₄

C₂×C₄×D₄₇

yes

—

140

C₂×C₄

C₂×C₄×D₄₇

yes

—

141

C₂×C₄

C₂×C₂²⋊D₄₇

yes

—

142

C₂×C₄

C₂×C₂²⋊D₄₇

yes

—

143

C₂²

—

144

C₂²

—

145

C₂²

—

146

C₂²

C₂×Dic₄₇⋊C₄

yes

—

147

C₂²

C₂²×C₄×D₄₇

yes

characteristic

148

C₂²

C₂²×C₂²⋊D₄₇

yes

characteristic

149

C₂²

C₂²×Dic₉₄

yes

characteristic

150

C₂²

#41×D₄₇

yes

—

151

C₄

—

152

C₄

—

153

C₄

—

154

C₄

C₂×Dic₄₇⋊C₄

yes

—

155

C₄

C₂×Dic₄₇⋊C₄

yes

—

156

C₂

—

157

C₂

C₂²×Dic₄₇⋊C₄

yes

characteristic

158

C₂

#15689591

yes

—

159

C₂

#15689630

yes

—

160

C₂

#15689676

yes

—

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄₇
commutator	C₂²×C₄₇	C₂³×C₄
Frattini	C₂³	C₂³×D₄₇
Fitting	C₂×#117×C₄₇	C₂
socle	C₂³×C₄₇	C₂⁴
hypercenter	C₂×#117	D₄₇
hypocenter	C₄₇	#2296
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#117

subgroup

G/H

center

C₂²×C₄

C₂²×D₄₇

commutator

C₂²×C₄₇

C₂³×C₄

Frattini

C₂³

C₂³×D₄₇

Fitting

C₂×#117×C₄₇

C₂

socle

C₂³×C₄₇

C₂⁴

D₄₇

C₄₇

G ▷ C₂²×C₄₇ ▷ C₄₇

C₁ ◁ C₂²×C₄ ◁ C₂×#117

N	H ≅ G/N	type
3. #15689019	C₂	semidirect
5. C₄×Dic₄₇⋊C₄	C₂	semidirect
8. #13093983	C₄	semidirect
37. C₄⋊Dic₄₇	C₂×C₄	semidirect
41. C₄×Dic₄₇	C₂×C₄	semidirect
49. C₂×C₄×C₄₇	C₄⋊C₄	semidirect
87. C₄₇	#2296	semidirect
136. C₂×C₄	Dic₄₇⋊C₄	semidirect

H ≅ G/N

type

3. #15689019

C₂

semidirect

5. C₄×Dic₄₇⋊C₄

C₂

semidirect

8. #13093983

C₄

semidirect

37. C₄⋊Dic₄₇

C₂×C₄

semidirect

41. C₄×Dic₄₇

C₂×C₄

semidirect

49. C₂×C₄×C₄₇

C₄⋊C₄

semidirect

87. C₄₇

#2296

semidirect

136. C₂×C₄

Dic₄₇⋊C₄

semidirect

Subgroups — #20245870

Conjugacy classes of subgroups

Complements