Group #15779301 — subgroups

Subgroups — #15779301

Order 3328.

310 conjugacy classes of subgroups; only the 155 normal subgroups are shown.

normal subgroups only

This group has 310 conjugacy classes of subgroups — too many to draw. Only its 155 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×#6090	C₂	1	1	yes	maximal, characteristic
2	#13115000	C₂	1	1	yes	maximal, characteristic
3	#13112927	C₂	1	4	yes	maximal
4	C₁₃×#1846	C₂	1	1	yes	maximal, characteristic
5	C₂×C₁₃×#107	C₂²	1	2	yes	—
6	C₂×C₁₃×#106	—	2	1	no	—
7	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
8	C₁₃×#382	—	2	1	no	—
9	C₂×#88058	C₂²	1	2	yes	—
10	C₂×#88057	—	2	1	no	—
11	C₂²×#6090	C₂²	1	2	yes	—
12	#11718597	—	2	1	no	—
13	C₄×#6090	C₄	1	4	yes	complemented
14	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
15	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
16	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
17	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
18	C₂×C₄⋊Dic₁₃	—	2	2	no	—
19	C₁₃×#107	—	2	4	no	—
20	C₁₃×#106	—	4	2	no	—
21	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
22	C₄×C₈×C₁₃	Q₈	1	2	yes	—
23	C₁₃×#95	—	4	2	no	—
24	#88071	—	4	2	no	—
25	#88058	—	2	4	no	—
26	#88057	—	4	2	no	—
27	C₂×#6090	—	2	2	no	—
28	C₂×#6090	—	2	2	no	—
29	C₂×#6090	C₂×C₄	1	4	yes	—
30	#12889	—	13	1	no	maximal, Sylow
31	C₂²×C₄×C₁₃	—	4	2	no	—
32	C₂²×C₄×C₁₃	D₈	1	2	yes	—
33	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
34	C₂²×Dic₁₃	—	4	2	no	—
35	C₂×C₈×C₁₃	C₄⋊C₄	1	2	yes	—
36	C₂×C₈×C₁₃	SD₁₆	1	2	yes	—
37	C₂×C₈×C₁₃	SD₁₆	1	2	yes	—
38	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
39	C₂×C₈×C₁₃	C₂×Q₈	1	1	yes	characteristic
40	C₂×C₈×C₁₃	#41	1	2	yes	—
41	C₂×C₈×C₁₃	#41	1	2	yes	—
42	C₁₃×C₄⋊C₄	—	4	2	no	—
43	C₁₃×C₄⋊C₄	—	4	2	no	—
44	C₄²×C₁₃	C₄⋊C₄	1	2	yes	—
45	C₄²×C₁₃	Q₁₆	1	2	yes	—
46	C₄⋊Dic₁₃	—	4	2	no	—
47	C₄⋊Dic₁₃	—	4	2	no	—
48	#6090	—	2	4	no	—
49	C₂×C₄×C₁₆	—	13	1	no	—
50	#1846	—	13	1	no	—
51	#1846	D₁₃	1	1	yes	characteristic
52	#1380	—	13	4	no	—
53	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
54	C₂×C₄×C₁₃	—	2	4	no	—
55	C₂×C₄×C₁₃	—	8	2	no	—
56	C₂×C₄×C₁₃	—	8	4	no	—
57	C₂×C₄×C₁₃	#107	1	4	yes	—
58	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
59	C₂×C₄×C₁₃	#124	1	1	yes	characteristic
60	C₂×C₄×C₁₃	#124	1	1	yes	characteristic
61	C₂×C₄×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
62	C₂×C₄×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
63	C₂×C₄×C₁₃	C₂×Q₁₆	1	1	yes	characteristic
64	C₂×Dic₁₃	—	8	2	no	—
65	C₂×Dic₁₃	—	8	4	no	—
66	C₈×C₁₃	#106	1	4	yes	—
67	C₈×C₁₃	#117	1	2	yes	—
68	C₈×C₁₃	#125	1	2	yes	—
69	C₂²×C₁₆	—	13	2	no	—
70	C₂×#107	—	13	2	no	—
71	C₂×#107	C₂×D₁₃	1	2	yes	—
72	C₂×#106	—	2	1	no	—
73	C₂×#106	—	26	1	no	—
74	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
75	#382	—	2	1	no	—
76	#382	—	26	1	no	—
77	C₄×C₁₆	—	13	4	no	—
78	C₂²×C₁₃	C₂×#107	1	1	yes	characteristic
79	C₂²×C₁₃	#486	1	1	yes	characteristic
80	C₂²×C₁₃	#486	1	1	yes	characteristic
81	C₂²×C₁₃	#508	1	4	yes	—
82	C₄×C₁₃	—	2	4	no	—
83	C₄×C₁₃	—	16	2	no	—
84	C₄×C₁₃	#382	1	1	yes	characteristic
85	C₄×C₁₃	C₂×#106	1	1	yes	characteristic
86	C₄×C₁₃	#487	1	1	yes	characteristic
87	C₄×C₁₃	#487	1	1	yes	characteristic
88	Dic₁₃	—	16	2	no	—
89	C₂²×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
90	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
91	C₂×C₄⋊C₄	—	2	2	no	—
92	C₂×C₄⋊C₄	—	26	2	no	—
93	C₂×C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
94	C₂×C₁₆	—	13	4	no	—
95	C₂×C₁₆	—	26	2	no	—
96	C₂×C₁₆	—	26	2	no	—
97	#107	—	2	4	no	—
98	#107	—	26	4	no	—
99	#106	—	4	2	no	—
100	#106	—	52	2	no	—
101	C₄×C₈	Dic₂₆	1	2	yes	—
102	C₄×C₈	C₄×D₁₃	1	2	yes	—
103	#95	—	4	2	no	—
104	#95	—	52	2	no	—
105	C₂×C₁₃	#1834	1	1	yes	characteristic
106	C₂×C₁₃	#2158	1	2	yes	—
107	C₂×C₁₃	#2242	1	2	yes	—
108	C₂×C₁₃	#2242	1	2	yes	—
109	C₂²×C₄	—	4	2	no	—
110	C₂²×C₄	—	52	2	no	—
111	C₂²×C₄	#6104	1	2	yes	—
112	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
113	C₂×C₈	#6101	1	2	yes	—
114	C₂×C₈	#6105	1	2	yes	—
115	C₂×C₈	#6106	1	2	yes	—
116	C₂×C₈	C₂×Dic₂₆	1	1	yes	characteristic
117	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
118	C₂×C₈	#6129	1	2	yes	—
119	C₂×C₈	#6131	1	2	yes	—
120	C₄⋊C₄	—	4	2	no	—
121	C₄⋊C₄	—	4	2	no	—
122	C₄⋊C₄	—	52	2	no	—
123	C₄⋊C₄	—	52	2	no	—
124	C₄²	#6101	1	2	yes	—
125	C₄²	#6107	1	2	yes	—
126	C₁₆	—	26	4	no	—
127	C₁₃	#12889	1	1	yes	characteristic, complemented, Sylow
128	C₂³	C₂×#6104	1	1	yes	characteristic
129	C₂×C₄	—	2	4	no	—
130	C₂×C₄	—	8	2	no	—
131	C₂×C₄	—	8	4	no	—
132	C₂×C₄	—	104	2	no	—
133	C₂×C₄	—	104	4	no	—
134	C₂×C₄	#88047	1	4	yes	—
135	C₂×C₄	C₂×#6101	1	1	yes	characteristic
136	C₂×C₄	C₂×#6105	1	1	yes	characteristic
137	C₂×C₄	#88190	1	1	yes	characteristic
138	C₂×C₄	C₂×#6106	1	1	yes	characteristic
139	C₂×C₄	C₂×#6107	1	1	yes	characteristic
140	C₂×C₄	#88201	1	1	yes	characteristic
141	C₈	#88048	1	4	yes	—
142	C₈	#88144	1	2	yes	—
143	C₈	#88146	1	2	yes	—
144	C₂²	C₂×#88047	1	1	yes	characteristic
145	C₂²	#11718731	1	1	yes	characteristic
146	C₂²	#11718749	1	1	yes	characteristic
147	C₂²	#11718855	1	4	yes	—
148	C₄	—	2	4	no	—
149	C₄	—	16	2	no	—
150	C₄	—	208	2	no	—
151	C₄	C₂×#88048	1	1	yes	characteristic
152	C₄	#11718638	1	1	yes	characteristic
153	C₄	#11718742	1	1	yes	characteristic
154	C₄	#11718754	1	1	yes	characteristic
155	C₂	#13114963	1	1	yes	characteristic
156	C₂	#13116618	1	2	yes	—
157	C₂	#13117067	1	2	yes	—
158	C₂	#13117070	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄×#6090

C₂

yes

maximal, characteristic

#13115000

C₂

yes

maximal, characteristic

#13112927

C₂

yes

maximal

C₁₃×#1846

C₂

yes

maximal, characteristic

C₂×C₁₃×#107

C₂²

yes

—

C₂×C₁₃×#106

—

C₂×C₄×C₈×C₁₃

C₂²

yes

characteristic

C₁₃×#382

—

C₂×#88058

C₂²

yes

—

C₂×#88057

—

C₂²×#6090

C₂²

yes

—

#11718597

—

C₄×#6090

C₄

yes

complemented

C₂²×C₈×C₁₃

D₄

yes

characteristic

C₂²×C₈×C₁₃

C₂³

yes

characteristic

C₂×C₁₃×C₄⋊C₄

—

C₂×C₄²×C₁₃

D₄

yes

characteristic

C₂×C₄⋊Dic₁₃

—

C₁₃×#107

—

C₁₃×#106

—

C₄×C₈×C₁₃

C₂×C₄

yes

—

C₄×C₈×C₁₃

Q₈

yes

—

C₁₃×#95

—

—

—

—

—

—

C₂×C₄

yes

—

#12889

—

maximal, Sylow

C₂²×C₄×C₁₃

—

C₂²×C₄×C₁₃

D₈

yes

—

C₂²×C₄×C₁₃

C₂×D₄

yes

characteristic

C₂²×Dic₁₃

—

C₂×C₈×C₁₃

C₄⋊C₄

yes

—

C₂×C₈×C₁₃

SD₁₆

yes

—

C₂×C₈×C₁₃

SD₁₆

yes

—

C₂×C₈×C₁₃

C₂²×C₄

yes

characteristic

C₂×C₈×C₁₃

C₂×Q₈

yes

characteristic

C₂×C₈×C₁₃

#41

yes

—

C₂×C₈×C₁₃

#41

yes

—

C₁₃×C₄⋊C₄

—

C₁₃×C₄⋊C₄

—

C₄²×C₁₃

C₄⋊C₄

yes

—

C₄²×C₁₃

Q₁₆

yes

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

#6090

—

C₂×C₄×C₁₆

—

#1846

—

#1846

D₁₃

yes

characteristic

#1380

—

C₂³×C₁₃

C₂×D₈

yes

characteristic

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

#107

yes

—

C₂×C₄×C₁₃

C₂×C₄⋊C₄

yes

C₂×C₄×C₁₃

yes

C₂×C₄×C₁₃

yes

C₂×C₄×C₁₃

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₁₃

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₁₃

C₂×Q₁₆

yes

characteristic

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₈×C₁₃

#106

yes

—

C₈×C₁₃

#117

yes

—

C₈×C₁₃

#125

yes

—

C₂²×C₁₆

—

C₂×#107

—

C₂×#107

C₂×D₁₃

yes

—

C₂×#106

—

C₂×#106

—

C₂×C₄×C₈

C₂×D₁₃

yes

characteristic

#382

—

#382

—

C₄×C₁₆

—

C₂²×C₁₃

yes

C₂²×C₁₃

yes

C₂²×C₁₃

yes

C₂²×C₁₃

yes

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

yes

C₄×C₁₃

yes

C₄×C₁₃

yes

C₄×C₁₃

yes

Dic₁₃

—

C₂²×C₈

C₂²⋊D₁₃

yes

characteristic

C₂²×C₈

C₂²×D₁₃

yes

characteristic

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂²⋊D₁₃

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

#107

—

#107

—

#106

—

100

#106

—

101

C₄×C₈

Dic₂₆

yes

—

102

C₄×C₈

C₄×D₁₃

yes

—

103

#95

—

104

#95

—

105

C₂×C₁₃

#1834

yes

characteristic

106

C₂×C₁₃

#2158

yes

—

107

C₂×C₁₃

#2242

yes

—

108

C₂×C₁₃

#2242

yes

—

109

C₂²×C₄

—

110

C₂²×C₄

—

111

C₂²×C₄

#6104

yes

—

112

C₂²×C₄

C₂×C₂²⋊D₁₃

yes

characteristic

113

C₂×C₈

#6101

yes

—

114

C₂×C₈

#6105

yes

—

115

C₂×C₈

#6106

yes

—

116

C₂×C₈

C₂×Dic₂₆

yes

characteristic

117

C₂×C₈

C₂×C₄×D₁₃

yes

characteristic

118

C₂×C₈

#6129

yes

—

119

C₂×C₈

#6131

yes

—

120

C₄⋊C₄

—

121

C₄⋊C₄

—

122

C₄⋊C₄

—

123

C₄⋊C₄

—

124

C₄²

#6101

yes

—

125

C₄²

#6107

yes

—

126

C₁₆

—

127

C₁₃

#12889

yes

characteristic, complemented, Sylow

128

C₂³

C₂×#6104

yes

characteristic

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

104

—

133

C₂×C₄

—

104

—

134

C₂×C₄

#88047

yes

—

135

C₂×C₄

C₂×#6101

yes

characteristic

136

C₂×C₄

C₂×#6105

yes

characteristic

137

C₂×C₄

#88190

yes

characteristic

138

C₂×C₄

C₂×#6106

yes

characteristic

139

C₂×C₄

C₂×#6107

yes

characteristic

140

C₂×C₄

#88201

yes

characteristic

141

C₈

#88048

yes

—

142

C₈

#88144

yes

—

143

C₈

#88146

yes

—

144

C₂²

C₂×#88047

yes

characteristic

145

C₂²

#11718731

yes

characteristic

146

C₂²

#11718749

yes

characteristic

147

C₂²

#11718855

yes

—

148

C₄

—

149

C₄

—

150

C₄

—

208

—

151

C₄

C₂×#88048

yes

characteristic

152

C₄

#11718638

yes

characteristic

153

C₄

#11718742

yes

characteristic

154

C₄

#11718754

yes

characteristic

155

C₂

#13114963

yes

characteristic

156

C₂

#13116618

yes

—

157

C₂

#13117067

yes

—

158

C₂

#13117070

yes

—

	subgroup	G/H
center	C₂³	C₂×#6104
commutator	C₂×C₈×C₁₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₁₃
Fitting	C₁₃×#1846	C₂
socle, prime radical	C₂³×C₁₃	C₂×D₈
cosocle	C₂²×C₈×C₁₃	C₂³
hypercenter	#1846	D₁₃
hypocenter	C₁₃	#12889
LCS	G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1846

subgroup

G/H

center

C₂³

C₂×#6104

commutator

C₂×C₈×C₁₃

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₁₃

Fitting

C₁₃×#1846

C₂

socle, prime radical

C₂³×C₁₃

C₂×D₈

cosocle

C₂²×C₈×C₁₃

C₂³

D₁₃

C₁₃

G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1846

N	H ≅ G/N	type
13. C₄×#6090	C₄	semidirect
127. C₁₃	#12889	semidirect

H ≅ G/N

type

13. C₄×#6090

C₄

semidirect

127. C₁₃

#12889

semidirect

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

subgroup	index
C₂×C₄×#6090	2
#13115000	2
#13112927	2
C₁₃×#1846	2
#12889	13

subgroup

index

C₂×C₄×#6090

#13115000

#13112927

C₁₃×#1846

#12889

Subgroups — #15779301

Conjugacy classes of subgroups

Complements

Maximal subgroups