Group #20249329 — subgroups

Subgroups — #20249329

Order 6016.

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

normal subgroups only

This group has 500 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	#15689049	C₂	1	4	yes	maximal, complemented
3	#15689034	C₂	1	4	yes	maximal, complemented
4	#15689028	C₂	1	2	yes	maximal, complemented
5	#15689024	C₂	1	2	yes	maximal
6	#15688969	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₄⋊Q₈×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×Dic₉₄	C₂²	1	1	yes	characteristic
9	C₂²×D₁₈₈	C₂²	1	1	yes	characteristic
10	#13094012	—	2	4	no	—
11	C₂×(C₂×D₄₇)⋊C₄	C₂²	1	2	yes	—
12	C₂×(C₂×D₄₇)⋊C₄	C₂²	1	2	yes	—
13	C₂×(C₂×D₄₇)⋊C₄	C₂²	1	8	yes	—
14	C₂×C₄×Dic₄₇	C₂²	1	2	yes	—
15	C₄₇⋊#95	C₂²	1	2	yes	—
16	C₄₇⋊#95	C₂²	1	2	yes	—
17	C₄₇⋊#95	C₂²	1	8	yes	—
18	C₂²×Q₈×C₄₇	C₂²	1	2	yes	—
19	C₄⋊Q₈×C₄₇	—	2	4	no	—
20	C₂×C₄⋊C₄×C₄₇	C₂²	1	2	yes	—
21	C₂×C₄⋊C₄×C₄₇	C₂²	1	2	yes	—
22	C₂×C₄²×C₄₇	C₂²	1	1	yes	characteristic
23	C₂³×D₄₇	C₂³	1	2	yes	—
24	C₂×Q₈×C₄₇	—	2	4	no	—
25	C₂×Q₈×C₄₇	—	4	4	no	—
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×C₄×C₄₇	C₂³	1	1	yes	characteristic
30	C₂²×Dic₄₇	C₂³	1	4	yes	—
31	C₂²×Dic₄₇	C₂³	1	2	yes	—
32	C₂×D₁₈₈	—	2	2	no	—
33	C₂×D₁₈₈	—	2	4	no	—
34	C₂×Dic₉₄	—	2	2	no	—
35	C₂×Dic₉₄	—	2	4	no	—
36	C₄⋊C₄×C₄₇	—	2	4	no	—
37	C₄⋊C₄×C₄₇	—	2	4	no	—
38	C₄²×C₄₇	D₄	1	4	yes	—
39	(C₂×D₄₇)⋊C₄	—	2	4	no	—
40	(C₂×D₄₇)⋊C₄	—	2	4	no	—
41	(C₂×D₄₇)⋊C₄	—	4	8	no	—
42	C₄×Dic₄₇	—	2	4	no	—
43	C₂³×C₄₇	C₂⁴	1	1	yes	characteristic
44	C₂²×D₄₇	—	2	2	no	—
45	C₂²×D₄₇	—	2	4	no	—
46	C₂²×D₄₇	—	2	4	no	—
47	C₂²×D₄₇	—	2	4	no	—
48	Q₈×C₄₇	—	4	8	no	—
49	C₂×C₄×C₄₇	—	2	2	no	—
50	C₂×C₄×C₄₇	—	2	2	no	—
51	C₂×C₄×C₄₇	—	2	4	no	—
52	C₂×C₄×C₄₇	—	2	8	no	—
53	C₂×C₄×C₄₇	C₂×D₄	1	2	yes	—
54	C₂×C₄×C₄₇	C₂×D₄	1	2	yes	—
55	C₂×C₄×C₄₇	C₂×D₄	1	2	yes	—
56	C₂×C₄×C₄₇	#41	1	4	yes	—
57	C₂×Dic₄₇	—	2	2	no	—
58	C₂×Dic₄₇	—	2	4	no	—
59	C₂×Dic₄₇	—	2	4	no	—
60	C₂×Dic₄₇	—	2	4	no	—
61	C₂×Dic₄₇	—	2	4	no	—
62	D₁₈₈	—	4	4	no	—
63	Dic₉₄	—	4	4	no	—
64	C₂²×C₄₇	C₂²×D₄	1	1	yes	characteristic
65	C₂²×C₄₇	C₂×#41	1	2	yes	—
66	C₂²×C₄₇	H(5,𝔽₂)	1	2	yes	—
67	C₂²×C₄₇	2₋¹⁺⁴	1	2	yes	—
68	C₂×D₄₇	—	4	2	no	—
69	C₂×D₄₇	—	4	2	no	—
70	C₂×D₄₇	—	4	2	no	—
71	C₂×D₄₇	—	4	8	no	—
72	C₄×C₄₇	—	2	2	no	—
73	C₄×C₄₇	—	2	2	no	—
74	C₄×C₄₇	—	4	4	no	—
75	C₄×C₄₇	#124	1	4	yes	—
76	Dic₄₇	—	4	2	no	—
77	Dic₄₇	—	4	4	no	—
78	#2640	—	47	1	no	maximal, Sylow
79	C₂×C₄₇	C₂×#124	1	1	yes	characteristic
80	C₂×C₄₇	#535	1	1	yes	characteristic
81	C₂×C₄₇	#536	1	1	yes	characteristic
82	C₂×C₄₇	#542	1	4	yes	—
83	D₄₇	—	8	2	no	—
84	C₂×C₄⋊Q₈	D₄₇	1	1	yes	characteristic, complemented
85	C₂×#124	—	47	1	no	—
86	#397	—	47	4	no	—
87	#393	—	47	4	no	—
88	#391	—	47	2	no	—
89	#390	—	47	2	no	—
90	#376	—	47	1	no	—
91	C₄₇	#2640	1	1	yes	characteristic, complemented, Sylow
92	C₂²×Q₈	—	47	1	no	—
93	C₂²×Q₈	C₂×D₄₇	1	2	yes	—
94	C₂²×D₄	—	47	1	no	—
95	C₄⋊Q₈	—	2	4	no	—
96	#124	—	94	4	no	—
97	C₂×C₄⋊C₄	C₂×D₄₇	1	2	yes	—
98	C₂×C₄⋊C₄	C₂×D₄₇	1	2	yes	—
99	C₂×C₂²⋊C₄	—	47	2	no	—
100	C₂×C₂²⋊C₄	—	47	2	no	—
101	C₂×C₂²⋊C₄	—	47	8	no	—
102	C₂×C₄²	—	47	2	no	—
103	C₂×C₄²	C₂×D₄₇	1	1	yes	characteristic
104	#95	—	47	2	no	—
105	#95	—	47	2	no	—
106	#95	—	47	8	no	—
107	C₂⁴	—	47	2	no	—
108	C₂×Q₈	—	2	4	no	—
109	C₂×Q₈	—	4	4	no	—
110	C₂×Q₈	—	94	4	no	—
111	C₂×Q₈	—	94	2	no	—
112	C₂×D₄	—	94	4	no	—
113	C₂×D₄	—	94	2	no	—
114	C₂²×C₄	—	47	4	no	—
115	C₂²×C₄	—	47	2	no	—
116	C₂²×C₄	C₂²×D₄₇	1	4	yes	—
117	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
118	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
119	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
120	C₄⋊C₄	—	2	4	no	—
121	C₄⋊C₄	—	2	4	no	—
122	C₂²⋊C₄	—	94	4	no	—
123	C₂²⋊C₄	—	94	4	no	—
124	C₂²⋊C₄	—	188	8	no	—
125	C₄²	—	94	4	no	—
126	C₄²	C₂²⋊D₄₇	1	4	yes	—
127	C₂³	—	94	2	no	—
128	C₂³	—	94	4	no	—
129	C₂³	—	94	4	no	—
130	C₂³	—	94	4	no	—
131	C₂³	C₂³×D₄₇	1	1	yes	characteristic
132	Q₈	—	4	8	no	—
133	Q₈	—	188	4	no	—
134	D₄	—	188	4	no	—
135	C₂×C₄	—	2	2	no	—
136	C₂×C₄	—	2	2	no	—
137	C₂×C₄	—	2	4	no	—
138	C₂×C₄	—	2	8	no	—
139	C₂×C₄	—	94	2	no	—
140	C₂×C₄	—	94	4	no	—
141	C₂×C₄	—	94	4	no	—
142	C₂×C₄	—	94	4	no	—
143	C₂×C₄	—	94	4	no	—
144	C₂×C₄	#10625354	1	4	yes	—
145	C₂×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
146	C₂×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
147	C₂×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
148	C₂²	—	188	8	no	—
149	C₂²	—	188	2	no	—
150	C₂²	—	188	2	no	—
151	C₂²	—	188	2	no	—
152	C₂²	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
153	C₂²	C₂×#10625354	1	2	yes	—
154	C₂²	#13094069	1	2	yes	—
155	C₂²	#13094072	1	2	yes	—
156	C₄	—	2	2	no	—
157	C₄	—	2	2	no	—
158	C₄	—	4	4	no	—
159	C₄	—	188	4	no	—
160	C₄	—	188	2	no	—
161	C₄	#13094013	1	4	yes	—
162	C₂	—	376	2	no	—
163	C₂	C₂×#13094013	1	1	yes	characteristic
164	C₂	#15689690	1	1	yes	characteristic
165	C₂	#15689697	1	1	yes	characteristic
166	C₂	#15689747	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13094012

C₂

yes

maximal, characteristic

#15689049

C₂

yes

maximal, complemented

#15689034

C₂

yes

maximal, complemented

#15689028

C₂

yes

maximal, complemented

#15689024

C₂

yes

maximal

#15688969

C₂

yes

maximal, characteristic, complemented

C₂×C₄⋊Q₈×C₄₇

C₂

yes

maximal, characteristic, complemented

C₂²×Dic₉₄

C₂²

yes

characteristic

C₂²×D₁₈₈

C₂²

yes

characteristic

#13094012

—

C₂×(C₂×D₄₇)⋊C₄

C₂²

yes

—

C₂×(C₂×D₄₇)⋊C₄

C₂²

yes

—

C₂×(C₂×D₄₇)⋊C₄

C₂²

yes

—

C₂×C₄×Dic₄₇

C₂²

yes

—

C₄₇⋊#95

C₂²

yes

—

C₄₇⋊#95

C₂²

yes

—

C₄₇⋊#95

C₂²

yes

—

C₂²×Q₈×C₄₇

C₂²

yes

—

C₄⋊Q₈×C₄₇

—

C₂×C₄⋊C₄×C₄₇

C₂²

yes

—

C₂×C₄⋊C₄×C₄₇

C₂²

yes

—

C₂×C₄²×C₄₇

C₂²

yes

characteristic

C₂³×D₄₇

C₂³

yes

—

C₂×Q₈×C₄₇

—

C₂×Q₈×C₄₇

—

C₂²×C₄×C₄₇

C₂³

yes

—

C₂²×C₄×C₄₇

C₂³

yes

characteristic

C₂²×C₄×C₄₇

C₂³

yes

characteristic

C₂²×C₄×C₄₇

C₂³

yes

characteristic

C₂²×Dic₄₇

C₂³

yes

—

C₂²×Dic₄₇

C₂³

yes

—

C₂×D₁₈₈

—

C₂×D₁₈₈

—

C₂×Dic₉₄

—

C₂×Dic₉₄

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄²×C₄₇

D₄

yes

—

(C₂×D₄₇)⋊C₄

—

(C₂×D₄₇)⋊C₄

—

(C₂×D₄₇)⋊C₄

—

C₄×Dic₄₇

—

C₂³×C₄₇

C₂⁴

yes

characteristic

C₂²×D₄₇

—

C₂²×D₄₇

—

C₂²×D₄₇

—

C₂²×D₄₇

—

Q₈×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂×D₄

yes

—

C₂×C₄×C₄₇

C₂×D₄

yes

—

C₂×C₄×C₄₇

C₂×D₄

yes

—

C₂×C₄×C₄₇

#41

yes

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

D₁₈₈

—

Dic₉₄

—

C₂²×C₄₇

C₂²×D₄

yes

characteristic

C₂²×C₄₇

C₂×#41

yes

—

C₂²×C₄₇

H(5,𝔽₂)

yes

—

C₂²×C₄₇

2₋¹⁺⁴

yes

—

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

#124

yes

—

Dic₄₇

—

Dic₄₇

—

—

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

—

D₄₇

—

C₂×C₄⋊Q₈

D₄₇

yes

characteristic, complemented

—

—

—

—

—

—

C₄₇

#2640

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₂²×Q₈

C₂×D₄₇

yes

—

C₂²×D₄

—

C₄⋊Q₈

—

#124

—

C₂×C₄⋊C₄

C₂×D₄₇

yes

—

C₂×C₄⋊C₄

C₂×D₄₇

yes

—

C₂×C₂²⋊C₄

—

100

C₂×C₂²⋊C₄

—

101

C₂×C₂²⋊C₄

—

102

C₂×C₄²

—

103

C₂×C₄²

C₂×D₄₇

yes

characteristic

104

#95

—

105

#95

—

106

#95

—

107

C₂⁴

—

108

C₂×Q₈

—

109

C₂×Q₈

—

110

C₂×Q₈

—

111

C₂×Q₈

—

112

C₂×D₄

—

113

C₂×D₄

—

114

C₂²×C₄

—

115

C₂²×C₄

—

116

C₂²×C₄

C₂²×D₄₇

yes

—

117

C₂²×C₄

C₂²×D₄₇

yes

characteristic

118

C₂²×C₄

C₂²×D₄₇

yes

characteristic

119

C₂²×C₄

C₂²×D₄₇

yes

characteristic

120

C₄⋊C₄

—

121

C₄⋊C₄

—

122

C₂²⋊C₄

—

123

C₂²⋊C₄

—

124

C₂²⋊C₄

—

188

—

125

C₄²

—

126

C₄²

C₂²⋊D₄₇

yes

—

127

C₂³

—

128

C₂³

—

129

C₂³

—

130

C₂³

—

131

C₂³

C₂³×D₄₇

yes

characteristic

132

Q₈

—

133

Q₈

—

188

—

134

D₄

—

188

—

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

—

144

C₂×C₄

#10625354

yes

—

145

C₂×C₄

C₂×C₂²⋊D₄₇

yes

—

146

C₂×C₄

C₂×C₂²⋊D₄₇

yes

—

147

C₂×C₄

C₂×C₂²⋊D₄₇

yes

—

148

C₂²

—

188

—

149

C₂²

—

188

—

150

C₂²

—

188

—

151

C₂²

—

188

—

152

C₂²

C₂²×C₂²⋊D₄₇

yes

characteristic

153

C₂²

C₂×#10625354

yes

—

154

C₂²

#13094069

yes

—

155

C₂²

#13094072

yes

—

156

C₄

—

157

C₄

—

158

C₄

—

159

C₄

—

188

—

160

C₄

—

188

—

161

C₄

#13094013

yes

—

162

C₂

—

376

—

163

C₂

C₂×#13094013

yes

characteristic

164

C₂

#15689690

yes

characteristic

165

C₂

#15689697

yes

characteristic

166

C₂

#15689747

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₄₇
commutator, socle	C₂³×C₄₇	C₂⁴
Fitting	C₂×C₄⋊Q₈×C₄₇	C₂
hypercenter	C₂×C₄⋊Q₈	D₄₇
hypocenter	C₄₇	#2640
LCS	G ▷ C₂³×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄⋊Q₈

subgroup

G/H

center, Frattini

C₂³

C₂³×D₄₇

commutator, socle

C₂³×C₄₇

C₂⁴

Fitting

C₂×C₄⋊Q₈×C₄₇

C₂

hypercenter

C₂×C₄⋊Q₈

D₄₇

C₄₇

G ▷ C₂³×C₄₇ ▷ C₄₇

C₁ ◁ C₂³ ◁ C₂×C₄⋊Q₈

N	H ≅ G/N	type
2. #15689049	C₂	semidirect
3. #15689034	C₂	semidirect
4. #15689028	C₂	semidirect
6. #15688969	C₂	semidirect
7. C₂×C₄⋊Q₈×C₄₇	C₂	semidirect
84. C₂×C₄⋊Q₈	D₄₇	semidirect
91. C₄₇	#2640	semidirect

H ≅ G/N

type

2. #15689049

C₂

semidirect

3. #15689034

C₂

semidirect

4. #15689028

C₂

semidirect

6. #15688969

C₂

semidirect

7. C₂×C₄⋊Q₈×C₄₇

C₂

semidirect

84. C₂×C₄⋊Q₈

D₄₇

semidirect

91. C₄₇

#2640

semidirect

Subgroups — #20249329

Conjugacy classes of subgroups

Complements