Group #15790181 — subgroups

Subgroups — #15790181

Order 3328.

1550 conjugacy classes of subgroups; only the 651 normal subgroups are shown.

normal subgroups only

This group has 1550 conjugacy classes of subgroups — too many to draw. Only its 651 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×#11718596	C₂	1	3	yes	maximal
2	C₂×#11718597	C₂	1	3	yes	maximal
3	#13113101	C₂	1	8	yes	maximal
4	C₂×C₁₃×#376	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₄²×C₁₃	C₂²	1	3	yes	—
6	C₂²×C₄⋊Dic₁₃	C₂²	1	1	yes	characteristic
7	C₂²×C₄⋊Dic₁₃	C₂²	1	3	yes	—
8	C₂²×C₄⋊Dic₁₃	C₂²	1	12	yes	—
9	C₁₃×#376	C₄	1	8	yes	complemented
10	C₂×C₁₃×#95	C₂²	1	4	yes	—
11	C₂×#88071	C₂²	1	12	yes	—
12	#11718597	—	2	12	no	—
13	#11718596	—	2	12	no	—
14	C₂³×C₄×C₁₃	C₂³	1	6	yes	—
15	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
16	C₂³×Dic₁₃	C₂³	1	6	yes	—
17	C₂³×Dic₁₃	C₂³	1	2	yes	—
18	C₂×C₄²×C₁₃	C₂×C₄	1	12	yes	—
19	C₂×C₄²×C₁₃	D₄	1	12	yes	—
20	C₂×C₄²×C₁₃	Q₈	1	12	yes	—
21	C₂×C₄⋊Dic₁₃	—	2	24	no	—
22	C₂×C₄⋊Dic₁₃	—	2	12	no	—
23	C₂×C₄⋊Dic₁₃	—	2	6	no	—
24	C₂×C₄⋊Dic₁₃	—	2	6	no	—
25	C₂×C₄⋊Dic₁₃	—	4	24	no	—
26	C₁₃×#95	C₂×C₄	1	16	yes	—
27	#88071	—	2	24	no	—
28	#14354	—	13	1	no	maximal, Sylow
29	C₂⁴×C₁₃	C₂⁴	1	1	yes	characteristic
30	C₂²×C₄×C₁₃	—	2	12	no	—
31	C₂²×C₄×C₁₃	—	2	12	no	—
32	C₂²×C₄×C₁₃	C₂²×C₄	1	12	yes	—
33	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
34	C₂²×C₄×C₁₃	C₂×D₄	1	12	yes	—
35	C₂²×C₄×C₁₃	C₂×D₄	1	6	yes	—
36	C₂²×C₄×C₁₃	C₂×Q₈	1	12	yes	—
37	C₂²×C₄×C₁₃	C₂×Q₈	1	6	yes	—
38	C₂²×Dic₁₃	—	2	6	no	—
39	C₂²×Dic₁₃	—	2	6	no	—
40	C₂²×Dic₁₃	—	2	12	no	—
41	C₂²×Dic₁₃	—	2	24	no	—
42	C₂²×Dic₁₃	—	2	8	no	—
43	C₄²×C₁₃	C₄⋊C₄	1	48	yes	complemented
44	C₄⋊Dic₁₃	—	4	48	no	—
45	C₄⋊Dic₁₃	—	4	12	no	—
46	C₄⋊Dic₁₃	—	4	4	no	—
47	C₂×#384	—	13	3	no	—
48	C₂×#382	—	13	3	no	—
49	C₂×#376	D₁₃	1	1	yes	characteristic
50	#1444	—	13	8	no	—
51	C₂³×C₁₃	C₂³×C₄	1	1	yes	characteristic
52	C₂³×C₁₃	C₂²×D₄	1	3	yes	—
53	C₂³×C₁₃	C₂²×Q₈	1	3	yes	—
54	C₂³×C₁₃	H(5,𝔽₂)	1	4	yes	—
55	C₂³×C₁₃	2₋¹⁺⁴	1	4	yes	—
56	C₂×C₄×C₁₃	—	2	24	no	—
57	C₂×C₄×C₁₃	—	2	12	no	—
58	C₂×C₄×C₁₃	—	2	12	no	—
59	C₂×C₄×C₁₃	—	2	24	no	—
60	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	24	yes	—
61	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	12	yes	—
62	C₂×C₄×C₁₃	#127	1	4	yes	—
63	C₂×C₄×C₁₃	C₄⋊Q₈	1	12	yes	—
64	C₂×Dic₁₃	—	4	6	no	—
65	C₂×Dic₁₃	—	4	2	no	—
66	C₂×Dic₁₃	—	4	24	no	—
67	C₂×Dic₁₃	—	4	12	no	—
68	C₂×Dic₁₃	—	4	12	no	—
69	C₂²×C₄⋊C₄	—	13	1	no	—
70	C₂²×C₄⋊C₄	—	13	3	no	—
71	C₂²×C₄⋊C₄	—	13	12	no	—
72	C₂²×C₄²	C₂×D₁₃	1	3	yes	—
73	#384	—	26	12	no	—
74	#382	—	26	12	no	—
75	#376	Dic₁₃	1	8	yes	complemented
76	C₂×#95	—	13	12	no	—
77	C₂×#95	C₂×D₁₃	1	4	yes	—
78	C₂²×C₁₃	C₂²×C₄⋊C₄	1	3	yes	—
79	C₂²×C₁₃	#520	1	4	yes	—
80	C₂²×C₁₃	C₂×#127	1	1	yes	characteristic
81	C₂²×C₁₃	C₂×C₄⋊Q₈	1	3	yes	—
82	C₂²×C₁₃	#535	1	6	yes	—
83	C₂²×C₁₃	#536	1	6	yes	—
84	C₂²×C₁₃	#544	1	12	yes	—
85	C₄×C₁₃	—	2	24	no	—
86	C₄×C₁₃	#384	1	8	yes	—
87	Dic₁₃	—	8	6	no	—
88	Dic₁₃	—	8	2	no	—
89	C₂³×C₄	—	13	6	no	—
90	C₂³×C₄	—	13	2	no	—
91	C₂³×C₄	C₂²×D₁₃	1	6	yes	—
92	C₂³×C₄	C₂²×D₁₃	1	1	yes	characteristic
93	C₂×C₄⋊C₄	—	26	6	no	—
94	C₂×C₄⋊C₄	—	26	6	no	—
95	C₂×C₄⋊C₄	—	26	24	no	—
96	C₂×C₄⋊C₄	—	26	12	no	—
97	C₂×C₄⋊C₄	—	52	24	no	—
98	C₂×C₄²	Dic₂₆	1	12	yes	—
99	C₂×C₄²	D₅₂	1	12	yes	—
100	C₂×C₄²	C₂×Dic₁₃	1	12	yes	—
101	#95	—	26	24	no	—
102	#95	C₂×Dic₁₃	1	16	yes	—
103	C₂×C₁₃	C₂×#384	1	1	yes	characteristic
104	C₂×C₁₃	#2317	1	6	yes	—
105	C₂×C₁₃	#2657	1	1	yes	characteristic
106	C₂×C₁₃	#2658	1	1	yes	characteristic
107	C₂×C₁₃	#2660	1	6	yes	—
108	C₂⁴	C₂³×D₁₃	1	1	yes	characteristic
109	C₂²×C₄	—	2	12	no	—
110	C₂²×C₄	—	2	12	no	—
111	C₂²×C₄	—	26	6	no	—
112	C₂²×C₄	—	26	6	no	—
113	C₂²×C₄	—	26	12	no	—
114	C₂²×C₄	—	26	24	no	—
115	C₂²×C₄	—	26	8	no	—
116	C₂²×C₄	C₂×Dic₂₆	1	12	yes	—
117	C₂²×C₄	C₂×Dic₂₆	1	6	yes	—
118	C₂²×C₄	C₂×D₅₂	1	12	yes	—
119	C₂²×C₄	C₂×D₅₂	1	6	yes	—
120	C₂²×C₄	C₂²×Dic₁₃	1	12	yes	—
121	C₂²×C₄	C₂²×Dic₁₃	1	2	yes	—
122	C₄⋊C₄	—	52	48	no	—
123	C₄⋊C₄	—	52	4	no	—
124	C₄⋊C₄	—	52	12	no	—
125	C₄²	C₄⋊Dic₁₃	1	48	yes	complemented
126	C₁₃	#14354	1	1	yes	characteristic, complemented, Sylow
127	C₂³	C₂²×Dic₂₆	1	3	yes	—
128	C₂³	C₂²×D₅₂	1	3	yes	—
129	C₂³	#88256	1	4	yes	—
130	C₂³	#88257	1	4	yes	—
131	C₂³	C₂³×Dic₁₃	1	1	yes	characteristic
132	C₂×C₄	—	2	24	no	—
133	C₂×C₄	—	2	12	no	—
134	C₂×C₄	—	2	12	no	—
135	C₂×C₄	—	2	24	no	—
136	C₂×C₄	—	52	6	no	—
137	C₂×C₄	—	52	2	no	—
138	C₂×C₄	—	52	24	no	—
139	C₂×C₄	—	52	12	no	—
140	C₂×C₄	—	52	12	no	—
141	C₂×C₄	#88123	1	12	yes	—
142	C₂×C₄	#88128	1	4	yes	—
143	C₂×C₄	C₂×C₄⋊Dic₁₃	1	24	yes	—
144	C₂×C₄	C₂×C₄⋊Dic₁₃	1	12	yes	—
145	C₂²	C₂×#88123	1	3	yes	—
146	C₂²	C₂×#88128	1	1	yes	characteristic
147	C₂²	#11719224	1	12	yes	—
148	C₂²	#11719230	1	6	yes	—
149	C₂²	#11719231	1	6	yes	—
150	C₂²	C₂²×C₄⋊Dic₁₃	1	3	yes	—
151	C₂²	#11719532	1	4	yes	—
152	C₄	—	2	24	no	—
153	C₄	—	104	6	no	—
154	C₄	—	104	2	no	—
155	C₄	#11718596	1	8	yes	—
156	C₂	C₂×#11718596	1	1	yes	characteristic
157	C₂	#13117459	1	6	yes	—
158	C₂	#13120942	1	1	yes	characteristic
159	C₂	#13120946	1	1	yes	characteristic
160	C₂	#13120955	1	6	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#11718596

C₂

yes

maximal

C₂×#11718597

C₂

yes

maximal

#13113101

C₂

yes

maximal

C₂×C₁₃×#376

C₂

yes

maximal, characteristic

C₂²×C₄²×C₁₃

C₂²

yes

—

C₂²×C₄⋊Dic₁₃

C₂²

yes

characteristic

C₂²×C₄⋊Dic₁₃

C₂²

yes

—

C₂²×C₄⋊Dic₁₃

C₂²

yes

—

C₁₃×#376

C₄

yes

complemented

C₂×C₁₃×#95

C₂²

yes

—

C₂×#88071

C₂²

yes

—

#11718597

—

#11718596

—

C₂³×C₄×C₁₃

C₂³

yes

—

C₂³×C₄×C₁₃

C₂³

yes

characteristic

C₂³×Dic₁₃

C₂³

yes

—

C₂³×Dic₁₃

C₂³

yes

—

C₂×C₄²×C₁₃

C₂×C₄

yes

—

C₂×C₄²×C₁₃

D₄

yes

—

C₂×C₄²×C₁₃

Q₈

yes

—

C₂×C₄⋊Dic₁₃

—

C₂×C₄⋊Dic₁₃

—

C₂×C₄⋊Dic₁₃

—

C₂×C₄⋊Dic₁₃

—

C₂×C₄⋊Dic₁₃

—

C₁₃×#95

C₂×C₄

yes

—

#88071

—

#14354

—

maximal, Sylow

C₂⁴×C₁₃

C₂⁴

yes

characteristic

C₂²×C₄×C₁₃

—

C₂²×C₄×C₁₃

—

C₂²×C₄×C₁₃

C₂²×C₄

yes

—

C₂²×C₄×C₁₃

C₂²×C₄

yes

—

C₂²×C₄×C₁₃

C₂×D₄

yes

—

C₂²×C₄×C₁₃

C₂×D₄

yes

—

C₂²×C₄×C₁₃

C₂×Q₈

yes

—

C₂²×C₄×C₁₃

C₂×Q₈

yes

—

C₂²×Dic₁₃

—

C₂²×Dic₁₃

—

C₂²×Dic₁₃

—

C₂²×Dic₁₃

—

C₂²×Dic₁₃

—

C₄²×C₁₃

C₄⋊C₄

yes

complemented

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

C₂×#384

—

C₂×#382

—

C₂×#376

D₁₃

yes

characteristic

#1444

—

C₂³×C₁₃

C₂³×C₄

yes

characteristic

C₂³×C₁₃

C₂²×D₄

yes

—

C₂³×C₁₃

C₂²×Q₈

yes

—

C₂³×C₁₃

H(5,𝔽₂)

yes

—

C₂³×C₁₃

2₋¹⁺⁴

yes

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

C₂×C₄⋊C₄

yes

—

C₂×C₄×C₁₃

C₂×C₄⋊C₄

yes

—

C₂×C₄×C₁₃

#127

yes

—

C₂×C₄×C₁₃

C₄⋊Q₈

yes

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂²×C₄⋊C₄

—

C₂²×C₄⋊C₄

—

C₂²×C₄⋊C₄

—

C₂²×C₄²

C₂×D₁₃

yes

—

#384

—

#382

—

#376

Dic₁₃

yes

complemented

C₂×#95

—

C₂×#95

C₂×D₁₃

yes

—

C₂²×C₁₃

C₂²×C₄⋊C₄

yes

—

C₂²×C₁₃

#520

yes

—

C₂²×C₁₃

C₂×#127

yes

characteristic

C₂²×C₁₃

C₂×C₄⋊Q₈

yes

—

C₂²×C₁₃

#535

yes

—

C₂²×C₁₃

#536

yes

—

C₂²×C₁₃

#544

yes

—

C₄×C₁₃

—

C₄×C₁₃

#384

yes

—

Dic₁₃

—

Dic₁₃

—

C₂³×C₄

—

C₂³×C₄

—

C₂³×C₄

C₂²×D₁₃

yes

—

C₂³×C₄

C₂²×D₁₃

yes

characteristic

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

Dic₂₆

yes

—

C₂×C₄²

D₅₂

yes

—

100

C₂×C₄²

C₂×Dic₁₃

yes

—

101

#95

—

102

#95

C₂×Dic₁₃

yes

—

103

C₂×C₁₃

C₂×#384

yes

characteristic

104

C₂×C₁₃

#2317

yes

—

105

C₂×C₁₃

#2657

yes

characteristic

106

C₂×C₁₃

#2658

yes

characteristic

107

C₂×C₁₃

#2660

yes

—

108

C₂⁴

C₂³×D₁₃

yes

characteristic

109

C₂²×C₄

—

110

C₂²×C₄

—

111

C₂²×C₄

—

112

C₂²×C₄

—

113

C₂²×C₄

—

114

C₂²×C₄

—

115

C₂²×C₄

—

116

C₂²×C₄

C₂×Dic₂₆

yes

—

117

C₂²×C₄

C₂×Dic₂₆

yes

—

118

C₂²×C₄

C₂×D₅₂

yes

—

119

C₂²×C₄

C₂×D₅₂

yes

—

120

C₂²×C₄

C₂²×Dic₁₃

yes

—

121

C₂²×C₄

C₂²×Dic₁₃

yes

—

122

C₄⋊C₄

—

123

C₄⋊C₄

—

124

C₄⋊C₄

—

125

C₄²

C₄⋊Dic₁₃

yes

complemented

126

C₁₃

#14354

yes

characteristic, complemented, Sylow

127

C₂³

C₂²×Dic₂₆

yes

—

128

C₂³

C₂²×D₅₂

yes

—

129

C₂³

#88256

yes

—

130

C₂³

#88257

yes

—

131

C₂³

C₂³×Dic₁₃

yes

characteristic

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

#88123

yes

—

142

C₂×C₄

#88128

yes

—

143

C₂×C₄

C₂×C₄⋊Dic₁₃

yes

—

144

C₂×C₄

C₂×C₄⋊Dic₁₃

yes

—

145

C₂²

C₂×#88123

yes

—

146

C₂²

C₂×#88128

yes

characteristic

147

C₂²

#11719224

yes

—

148

C₂²

#11719230

yes

—

149

C₂²

#11719231

yes

—

150

C₂²

C₂²×C₄⋊Dic₁₃

yes

—

151

C₂²

#11719532

yes

—

152

C₄

—

153

C₄

—

104

—

154

C₄

—

104

—

155

C₄

#11718596

yes

—

156

C₂

C₂×#11718596

yes

characteristic

157

C₂

#13117459

yes

—

158

C₂

#13120942

yes

characteristic

159

C₂

#13120946

yes

characteristic

160

C₂

#13120955

yes

—

	subgroup	G/H
center, Frattini	C₂⁴	C₂³×D₁₃
commutator	C₂³×C₁₃	C₂³×C₄
Fitting	C₂×C₁₃×#376	C₂
socle, prime radical, cosocle	C₂⁴×C₁₃	C₂⁴
hypercenter	C₂×#376	D₁₃
hypocenter	C₁₃	#14354
LCS	G ▷ C₂³×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂⁴ ◁ C₂×#376

subgroup

G/H

center, Frattini

C₂⁴

C₂³×D₁₃

commutator

C₂³×C₁₃

C₂³×C₄

Fitting

C₂×C₁₃×#376

C₂

socle, prime radical, cosocle

C₂⁴×C₁₃

C₂⁴

D₁₃

C₁₃

G ▷ C₂³×C₁₃ ▷ C₁₃

C₁ ◁ C₂⁴ ◁ C₂×#376

N	H ≅ G/N	type
9. C₁₃×#376	C₄	semidirect
43. C₄²×C₁₃	C₄⋊C₄	semidirect
75. #376	Dic₁₃	semidirect
125. C₄²	C₄⋊Dic₁₃	semidirect
126. C₁₃	#14354	semidirect

H ≅ G/N

type

9. C₁₃×#376

C₄

semidirect

43. C₄²×C₁₃

C₄⋊C₄

semidirect

75. #376

Dic₁₃

semidirect

125. C₄²

C₄⋊Dic₁₃

semidirect

126. C₁₃

#14354

semidirect

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

subgroup	index
C₂×#11718596	2
C₂×#11718597	2
#13113101	2
C₂×C₁₃×#376	2
#14354	13

subgroup

index

C₂×C₁₃×#376

Subgroups — #15790181

Conjugacy classes of subgroups

Complements

Maximal subgroups