Group #15850627 — subgroups

Subgroups — #15850627

Order 3328.

706 conjugacy classes of subgroups; only the 243 normal subgroups are shown.

normal subgroups only

This group has 706 conjugacy classes of subgroups — too many to draw. Only its 243 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	#13125422	C₂	1	4	yes	maximal
2	#13125350	C₂	1	2	yes	maximal
3	#13114454	C₂	1	2	yes	maximal, complemented
4	#13114446	C₂	1	2	yes	maximal, complemented
5	#13114433	C₂	1	4	yes	maximal, complemented
6	C₂×C₈²×C₁₃	C₂	1	1	yes	maximal, characteristic
7	#11719175	C₂²	1	4	yes	—
8	C₂×C₄×C₈×C₁₃	C₂²	1	2	yes	—
9	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
10	#11718777	—	2	8	no	—
11	#11718767	—	2	4	no	—
12	#11718360	C₂²	1	4	yes	—
13	C₄×Dic₅₂	—	2	8	no	—
14	#11718344	—	2	4	no	—
15	#11718343	C₂²	1	4	yes	—
16	#11718343	C₂²	1	4	yes	—
17	#11718342	C₂²	1	8	yes	—
18	#11718341	C₂²	1	4	yes	—
19	C₈²×C₁₃	C₂²	1	2	yes	—
20	C₈²×C₁₃	C₂²	1	1	yes	characteristic
21	C₈²×C₁₃	C₂²	1	1	yes	characteristic
22	C₂²×C₈×C₁₃	D₄	1	4	yes	—
23	C₂²×C₈×C₁₃	D₄	1	2	yes	—
24	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
25	#88180	—	2	4	no	—
26	#88178	—	2	8	no	—
27	C₂×Dic₅₂	—	2	8	no	—
28	#88124	—	2	4	no	—
29	#88123	C₂³	1	8	yes	—
30	C₄×Dic₂₆	—	2	8	no	—
31	C₄×C₈×C₁₃	D₄	1	2	yes	—
32	C₄×C₈×C₁₃	D₄	1	4	yes	—
33	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
34	C₄×C₈×C₁₃	C₂³	1	2	yes	—
35	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
36	C₄×C₈×C₁₃	C₂³	1	2	yes	—
37	#88058	—	2	8	no	—
38	#88058	—	2	8	no	—
39	#88057	—	2	8	no	—
40	#88056	—	2	16	no	—
41	#19604	—	13	1	no	maximal, Sylow
42	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
43	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
44	C₂×Dic₂₆	—	2	8	no	—
45	C₂×C₈×C₁₃	—	2	8	no	—
46	C₂×C₈×C₁₃	Q₁₆	1	8	yes	complemented
47	C₂×C₈×C₁₃	C₂×D₄	1	4	yes	—
48	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
49	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
50	C₂×C₈×C₁₃	C₂×D₄	1	4	yes	—
51	C₄²×C₁₃	C₂×D₄	1	2	yes	—
52	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
53	C₄²×C₁₃	C₂⁴	1	1	yes	characteristic
54	C₂²⋊Dic₁₃	—	4	4	no	—
55	C₄⋊Dic₁₃	—	2	8	no	—
56	C₄⋊Dic₁₃	—	2	8	no	—
57	#6101	—	4	8	no	—
58	C₄×Dic₁₃	—	4	4	no	—
59	Dic₅₂	—	4	16	no	—
60	#3170	—	13	4	no	—
61	#3160	—	13	2	no	—
62	#1713	—	13	2	no	—
63	#1710	—	13	2	no	—
64	#1707	—	13	4	no	—
65	C₂×C₈²	D₁₃	1	1	yes	characteristic
66	C₂³×C₁₃	#127	1	1	yes	characteristic
67	C₂×C₄×C₁₃	—	2	4	no	—
68	C₂×C₄×C₁₃	#127	1	2	yes	—
69	C₂×C₄×C₁₃	#127	1	1	yes	characteristic
70	C₂×C₄×C₁₃	C₂×Q₁₆	1	4	yes	—
71	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
72	C₂×C₄×C₁₃	C₂²×D₄	1	2	yes	—
73	C₂×Dic₁₃	—	4	8	no	—
74	Dic₂₆	—	4	16	no	—
75	C₈×C₁₃	—	2	4	no	—
76	C₈×C₁₃	C₂×Q₁₆	1	4	yes	—
77	C₈×C₁₃	C₂×Q₁₆	1	4	yes	—
78	C₈×C₁₃	#135	1	8	yes	—
79	#533	—	13	4	no	—
80	#500	—	13	4	no	—
81	#500	—	13	4	no	—
82	#499	—	26	4	no	—
83	#498	—	13	4	no	—
84	#494	—	13	4	no	—
85	#487	—	13	8	no	—
86	#467	—	26	8	no	—
87	C₄×Q₁₆	—	26	8	no	—
88	#427	—	26	4	no	—
89	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
90	C₂×C₄×C₈	C₂×D₁₃	1	2	yes	—
91	C₈²	C₂×D₁₃	1	2	yes	—
92	C₈²	C₂×D₁₃	1	1	yes	characteristic
93	C₈²	C₂×D₁₃	1	1	yes	characteristic
94	C₂²×C₁₃	—	2	2	no	—
95	C₂²×C₁₃	#494	1	2	yes	—
96	C₂²×C₁₃	C₂×#127	1	1	yes	characteristic
97	C₄×C₁₃	—	2	2	no	—
98	C₄×C₁₃	#494	1	2	yes	—
99	C₄×C₁₃	C₂²×Q₁₆	1	2	yes	—
100	C₄×C₁₃	C₂×#135	1	4	yes	—
101	Dic₁₃	—	8	8	no	—
102	C₂×Q₁₆	—	26	8	no	—
103	C₂²×C₈	D₅₂	1	4	yes	—
104	C₂²×C₈	D₅₂	1	2	yes	—
105	C₄⋊Q₈	—	13	8	no	—
106	#125	—	26	4	no	—
107	C₂²⋊Q₈	—	26	8	no	—
108	C₄×Q₈	—	26	8	no	—
109	#117	—	26	4	no	—
110	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
111	#107	—	26	8	no	—
112	#107	—	26	8	no	—
113	#106	—	26	8	no	—
114	#103	—	26	16	no	—
115	C₄×C₈	D₅₂	1	4	yes	—
116	C₄×C₈	D₅₂	1	2	yes	—
117	C₄×C₈	C₂²×D₁₃	1	2	yes	—
118	C₄×C₈	C₂²×D₁₃	1	2	yes	—
119	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
120	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
121	C₂×C₁₃	—	2	2	no	—
122	C₂×C₁₃	C₂×#494	1	1	yes	characteristic
123	C₂×C₁₃	#3147	1	2	yes	—
124	C₂×Q₈	—	26	8	no	—
125	C₂²×C₄	C₂×D₅₂	1	2	yes	—
126	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
127	Q₁₆	—	52	16	no	—
128	C₂×C₈	—	2	8	no	—
129	C₂×C₈	Dic₅₂	1	8	yes	complemented
130	C₂×C₈	C₂×D₅₂	1	4	yes	—
131	C₂×C₈	C₂×D₅₂	1	4	yes	—
132	C₂×C₈	C₂×D₅₂	1	2	yes	—
133	C₂×C₈	C₂×D₅₂	1	2	yes	—
134	C₄⋊C₄	—	26	8	no	—
135	C₄⋊C₄	—	26	8	no	—
136	C₄⋊C₄	—	52	8	no	—
137	C₂²⋊C₄	—	52	4	no	—
138	C₄²	—	52	4	no	—
139	C₄²	C₂×D₅₂	1	1	yes	characteristic
140	C₄²	C₂×D₅₂	1	2	yes	—
141	C₄²	C₂³×D₁₃	1	1	yes	characteristic
142	C₁₃	#19604	1	1	yes	characteristic, complemented, Sylow
143	C₂³	#88128	1	1	yes	characteristic
144	Q₈	—	52	16	no	—
145	C₂×C₄	—	2	4	no	—
146	C₂×C₄	—	52	8	no	—
147	C₂×C₄	#88128	1	2	yes	—
148	C₂×C₄	#88128	1	1	yes	characteristic
149	C₂×C₄	C₂×Dic₅₂	1	4	yes	—
150	C₂×C₄	C₂²×D₅₂	1	2	yes	—
151	C₂×C₄	C₂²×D₅₂	1	1	yes	characteristic
152	C₈	—	2	4	no	—
153	C₈	#88158	1	8	yes	—
154	C₈	C₂×Dic₅₂	1	4	yes	—
155	C₈	C₂×Dic₅₂	1	4	yes	—
156	C₂²	—	2	2	no	—
157	C₂²	#11718360	1	2	yes	—
158	C₂²	C₂×#88128	1	1	yes	characteristic
159	C₄	—	2	2	no	—
160	C₄	—	104	8	no	—
161	C₄	#11718360	1	2	yes	—
162	C₄	C₂×#88158	1	4	yes	—
163	C₄	C₂²×Dic₅₂	1	2	yes	—
164	C₂	—	2	2	no	—
165	C₂	C₂×#11718360	1	1	yes	characteristic
166	C₂	#13125264	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#13125422

C₂

yes

maximal

#13125350

C₂

yes

maximal

#13114454

C₂

yes

maximal, complemented

#13114446

C₂

yes

maximal, complemented

#13114433

C₂

yes

maximal, complemented

C₂×C₈²×C₁₃

C₂

yes

maximal, characteristic

#11719175

C₂²

yes

—

C₂×C₄×C₈×C₁₃

C₂²

yes

—

C₂×C₄×C₈×C₁₃

C₂²

yes

—

—

C₂²

yes

—

C₄×Dic₅₂

—

#11718344

—

#11718343

C₂²

yes

—

#11718343

C₂²

yes

—

#11718342

C₂²

yes

—

#11718341

C₂²

yes

—

C₈²×C₁₃

C₂²

yes

—

C₈²×C₁₃

C₂²

yes

characteristic

C₈²×C₁₃

C₂²

yes

characteristic

C₂²×C₈×C₁₃

D₄

yes

—

C₂²×C₈×C₁₃

D₄

yes

—

C₂×C₄²×C₁₃

C₂³

yes

characteristic

#88180

—

#88178

—

C₂×Dic₅₂

—

#88124

—

#88123

C₂³

yes

—

C₄×Dic₂₆

—

C₄×C₈×C₁₃

D₄

yes

—

C₄×C₈×C₁₃

D₄

yes

—

C₄×C₈×C₁₃

C₂³

yes

characteristic

C₄×C₈×C₁₃

C₂³

yes

—

C₄×C₈×C₁₃

C₂³

yes

characteristic

C₄×C₈×C₁₃

C₂³

yes

—

—

—

—

—

—

C₂²×C₄×C₁₃

C₂×D₄

yes

—

C₂²×C₄×C₁₃

C₂×D₄

yes

characteristic

C₂×Dic₂₆

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

Q₁₆

yes

complemented

C₂×C₈×C₁₃

C₂×D₄

yes

—

C₂×C₈×C₁₃

C₂×D₄

yes

—

C₂×C₈×C₁₃

C₂×D₄

yes

—

C₂×C₈×C₁₃

C₂×D₄

yes

—

C₄²×C₁₃

C₂×D₄

yes

—

C₄²×C₁₃

C₂×D₄

yes

characteristic

C₄²×C₁₃

C₂⁴

yes

characteristic

C₂²⋊Dic₁₃

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

#6101

—

C₄×Dic₁₃

—

Dic₅₂

—

—

—

—

—

—

C₂×C₈²

D₁₃

yes

characteristic

C₂³×C₁₃

#127

yes

characteristic

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

#127

yes

—

C₂×C₄×C₁₃

#127

yes

characteristic

C₂×C₄×C₁₃

C₂×Q₁₆

yes

—

C₂×C₄×C₁₃

C₂²×D₄

yes

characteristic

C₂×C₄×C₁₃

C₂²×D₄

yes

—

C₂×Dic₁₃

—

Dic₂₆

—

C₈×C₁₃

—

C₈×C₁₃

C₂×Q₁₆

yes

—

C₈×C₁₃

C₂×Q₁₆

yes

—

C₈×C₁₃

#135

yes

—

—

—

—

—

—

—

—

—

C₄×Q₁₆

—

#427

—

C₂×C₄×C₈

C₂×D₁₃

yes

characteristic

C₂×C₄×C₈

C₂×D₁₃

yes

—

C₈²

C₂×D₁₃

yes

—

C₈²

C₂×D₁₃

yes

characteristic

C₈²

C₂×D₁₃

yes

characteristic

C₂²×C₁₃

—

C₂²×C₁₃

#494

yes

—

C₂²×C₁₃

C₂×#127

yes

characteristic

C₄×C₁₃

—

C₄×C₁₃

#494

yes

—

C₄×C₁₃

C₂²×Q₁₆

yes

—

100

C₄×C₁₃

C₂×#135

yes

—

101

Dic₁₃

—

102

C₂×Q₁₆

—

103

C₂²×C₈

D₅₂

yes

—

104

C₂²×C₈

D₅₂

yes

—

105

C₄⋊Q₈

—

106

#125

—

107

C₂²⋊Q₈

—

108

C₄×Q₈

—

109

#117

—

110

C₂×C₄²

C₂²×D₁₃

yes

characteristic

111

#107

—

112

#107

—

113

#106

—

114

#103

—

115

C₄×C₈

D₅₂

yes

—

116

C₄×C₈

D₅₂

yes

—

117

C₄×C₈

C₂²×D₁₃

yes

—

118

C₄×C₈

C₂²×D₁₃

yes

—

119

C₄×C₈

C₂²×D₁₃

yes

characteristic

120

C₄×C₈

C₂²×D₁₃

yes

characteristic

121

C₂×C₁₃

—

122

C₂×C₁₃

C₂×#494

yes

characteristic

123

C₂×C₁₃

#3147

yes

—

124

C₂×Q₈

—

125

C₂²×C₄

C₂×D₅₂

yes

—

126

C₂²×C₄

C₂×D₅₂

yes

characteristic

127

Q₁₆

—

128

C₂×C₈

—

129

C₂×C₈

Dic₅₂

yes

complemented

130

C₂×C₈

C₂×D₅₂

yes

—

131

C₂×C₈

C₂×D₅₂

yes

—

132

C₂×C₈

C₂×D₅₂

yes

—

133

C₂×C₈

C₂×D₅₂

yes

—

134

C₄⋊C₄

—

135

C₄⋊C₄

—

136

C₄⋊C₄

—

137

C₂²⋊C₄

—

138

C₄²

—

139

C₄²

C₂×D₅₂

yes

characteristic

140

C₄²

C₂×D₅₂

yes

—

141

C₄²

C₂³×D₁₃

yes

characteristic

142

C₁₃

#19604

yes

characteristic, complemented, Sylow

143

C₂³

#88128

yes

characteristic

144

Q₈

—

145

C₂×C₄

—

146

C₂×C₄

—

147

C₂×C₄

#88128

yes

—

148

C₂×C₄

#88128

yes

characteristic

149

C₂×C₄

C₂×Dic₅₂

yes

—

150

C₂×C₄

C₂²×D₅₂

yes

—

151

C₂×C₄

C₂²×D₅₂

yes

characteristic

152

C₈

—

153

C₈

#88158

yes

—

154

C₈

C₂×Dic₅₂

yes

—

155

C₈

C₂×Dic₅₂

yes

—

156

C₂²

—

157

C₂²

#11718360

yes

—

158

C₂²

C₂×#88128

yes

characteristic

159

C₄

—

160

C₄

—

104

—

161

C₄

#11718360

yes

—

162

C₄

C₂×#88158

yes

—

163

C₄

C₂²×Dic₅₂

yes

—

164

C₂

—

165

C₂

C₂×#11718360

yes

characteristic

166

C₂

#13125264

yes

—

	subgroup	G/H
center	C₂×C₄	#88128
commutator, cosocle	C₄²×C₁₃	C₂⁴
Frattini	C₄²	C₂³×D₁₃
Fitting	C₂×C₈²×C₁₃	C₂
socle	C₂²×C₁₃	C₂×#127
prime radical	C₂³×C₁₃	#127
hypercenter	C₂×C₈²	D₁₃
hypocenter	C₁₃	#19604
LCS	G ▷ C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₈²

subgroup

G/H

center

C₂×C₄

#88128

commutator, cosocle

C₄²×C₁₃

C₂⁴

Frattini

C₄²

C₂³×D₁₃

Fitting

C₂×C₈²×C₁₃

C₂

C₂²×C₁₃

C₂³×C₁₃

C₂×C₈²

D₁₃

hypocenter

C₁₃

#19604

LCS

G ▷ C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₈²

N	H ≅ G/N	type
3. #13114454	C₂	semidirect
4. #13114446	C₂	semidirect
5. #13114433	C₂	semidirect
46. C₂×C₈×C₁₃	Q₁₆	semidirect
129. C₂×C₈	Dic₅₂	semidirect
142. C₁₃	#19604	semidirect

H ≅ G/N

type

3. #13114454

C₂

semidirect

4. #13114446

C₂

semidirect

5. #13114433

C₂

semidirect

46. C₂×C₈×C₁₃

Q₁₆

semidirect

129. C₂×C₈

Dic₅₂

semidirect

142. C₁₃

#19604

semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#13125422	2
#13125350	2
#13114454	2
#13114446	2
#13114433	2
C₂×C₈²×C₁₃	2
#19604	13

subgroup

index

C₂×C₈²×C₁₃

Subgroups — #15850627

Conjugacy classes of subgroups

Complements

Maximal subgroups