Group #15794063 — subgroups

Subgroups — #15794063

Order 3328.

1182 conjugacy classes of subgroups; only the 411 normal subgroups are shown.

normal subgroups only

This group has 1182 conjugacy classes of subgroups — too many to draw. Only its 411 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×#11718598	C₂	1	2	yes	maximal
2	#13113089	C₂	1	12	yes	maximal
3	C₂×C₁₃×#384	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₁₃×C₄⋊C₄	C₂²	1	6	yes	—
5	C₂²×C₄²×C₁₃	C₂²	1	1	yes	characteristic
6	C₂²×C₄⋊Dic₁₃	C₂²	1	4	yes	—
7	C₁₃×#384	C₄	1	8	yes	complemented
8	C₂×#88071	C₂²	1	12	yes	—
9	C₂×#88071	C₂²	1	12	yes	—
10	#11718598	—	2	8	no	—
11	C₂³×C₄×C₁₃	C₂³	1	4	yes	—
12	C₂³×C₄×C₁₃	C₂³	1	3	yes	—
13	C₂³×Dic₁₃	C₂³	1	8	yes	—
14	C₂×C₁₃×C₄⋊C₄	—	4	6	no	—
15	C₂×C₁₃×C₄⋊C₄	—	4	6	no	—
16	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	24	yes	—
17	C₂×C₄²×C₁₃	C₂×C₄	1	4	yes	—
18	C₂×C₄²×C₁₃	D₄	1	4	yes	—
19	C₂×C₄²×C₁₃	Q₈	1	4	yes	—
20	C₂×C₄⋊Dic₁₃	—	4	12	no	—
21	#88071	—	2	24	no	—
22	#88071	—	4	12	no	—
23	#14676	—	13	1	no	maximal, Sylow
24	C₂⁴×C₁₃	C₂⁴	1	1	yes	characteristic
25	C₂²×C₄×C₁₃	—	2	6	no	—
26	C₂²×C₄×C₁₃	—	2	6	no	—
27	C₂²×C₄×C₁₃	—	2	12	no	—
28	C₂²×C₄×C₁₃	—	2	12	no	—
29	C₂²×C₄×C₁₃	C₂²×C₄	1	8	yes	—
30	C₂²×C₄×C₁₃	C₂²×C₄	1	6	yes	—
31	C₂²×C₄×C₁₃	C₂×D₄	1	6	yes	—
32	C₂²×C₄×C₁₃	C₂×Q₈	1	6	yes	—
33	C₂²×Dic₁₃	—	2	8	no	—
34	C₂²×Dic₁₃	—	2	24	no	—
35	C₂²×Dic₁₃	—	2	24	no	—
36	C₁₃×C₄⋊C₄	—	4	24	no	—
37	C₄²×C₁₃	C₄⋊C₄	1	16	yes	complemented
38	C₄⋊Dic₁₃	—	4	16	no	—
39	C₂×#384	D₁₃	1	1	yes	characteristic
40	C₂×#383	—	13	2	no	—
41	#1441	—	13	12	no	—
42	C₂³×C₁₃	C₂³×C₄	1	1	yes	characteristic
43	C₂³×C₁₃	C₂²×D₄	1	1	yes	characteristic
44	C₂³×C₁₃	C₂²×Q₈	1	1	yes	characteristic
45	C₂³×C₁₃	H(5,𝔽₂)	1	6	yes	—
46	C₂³×C₁₃	2₋¹⁺⁴	1	6	yes	—
47	C₂×C₄×C₁₃	—	2	6	no	—
48	C₂×C₄×C₁₃	—	2	6	no	—
49	C₂×C₄×C₁₃	—	2	12	no	—
50	C₂×C₄×C₁₃	—	2	12	no	—
51	C₂×C₄×C₁₃	—	2	24	no	—
52	C₂×C₄×C₁₃	—	4	4	no	—
53	C₂×C₄×C₁₃	—	4	12	no	—
54	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	12	yes	—
55	C₂×Dic₁₃	—	4	24	no	—
56	C₂×Dic₁₃	—	4	8	no	—
57	C₂×Dic₁₃	—	4	24	no	—
58	C₂²×C₄⋊C₄	—	13	4	no	—
59	C₂²×C₄⋊C₄	C₂×D₁₃	1	6	yes	—
60	C₂²×C₄²	C₂×D₁₃	1	1	yes	characteristic
61	#384	Dic₁₃	1	8	yes	complemented
62	#383	—	26	8	no	—
63	C₂×#95	—	13	12	no	—
64	C₂×#95	—	13	12	no	—
65	C₂²×C₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
66	C₂²×C₁₃	#518	1	3	yes	—
67	C₂²×C₁₃	#519	1	3	yes	—
68	C₂²×C₁₃	#537	1	6	yes	—
69	C₂²×C₁₃	#544	1	6	yes	—
70	C₂²×C₁₃	#560	1	4	yes	—
71	C₂²×C₁₃	#563	1	12	yes	—
72	C₄×C₁₃	—	2	12	no	—
73	C₄×C₁₃	—	4	8	no	—
74	Dic₁₃	—	8	8	no	—
75	C₂³×C₄	—	13	8	no	—
76	C₂³×C₄	C₂²×D₁₃	1	4	yes	—
77	C₂³×C₄	C₂²×D₁₃	1	3	yes	—
78	C₂×C₄⋊C₄	—	4	6	no	—
79	C₂×C₄⋊C₄	—	4	6	no	—
80	C₂×C₄⋊C₄	—	52	12	no	—
81	C₂×C₄⋊C₄	C₂×Dic₁₃	1	24	yes	—
82	C₂×C₄²	Dic₂₆	1	4	yes	—
83	C₂×C₄²	D₅₂	1	4	yes	—
84	C₂×C₄²	C₂×Dic₁₃	1	4	yes	—
85	#95	—	26	24	no	—
86	#95	—	52	12	no	—
87	C₂×C₁₃	#2316	1	3	yes	—
88	C₂×C₁₃	#2380	1	1	yes	characteristic
89	C₂×C₁₃	#2381	1	3	yes	—
90	C₂×C₁₃	#2823	1	4	yes	—
91	C₂×C₁₃	#2843	1	4	yes	—
92	C₂⁴	C₂³×D₁₃	1	1	yes	characteristic
93	C₂²×C₄	—	2	6	no	—
94	C₂²×C₄	—	2	6	no	—
95	C₂²×C₄	—	2	12	no	—
96	C₂²×C₄	—	2	12	no	—
97	C₂²×C₄	—	26	8	no	—
98	C₂²×C₄	—	26	24	no	—
99	C₂²×C₄	—	26	24	no	—
100	C₂²×C₄	C₂×Dic₂₆	1	6	yes	—
101	C₂²×C₄	C₂×D₅₂	1	6	yes	—
102	C₂²×C₄	C₂²×Dic₁₃	1	8	yes	—
103	C₂²×C₄	C₂²×Dic₁₃	1	6	yes	—
104	C₄⋊C₄	—	4	24	no	—
105	C₄⋊C₄	—	52	16	no	—
106	C₄²	C₄⋊Dic₁₃	1	16	yes	complemented
107	C₁₃	#14676	1	1	yes	characteristic, complemented, Sylow
108	C₂³	C₂²×Dic₂₆	1	1	yes	characteristic
109	C₂³	C₂²×D₅₂	1	1	yes	characteristic
110	C₂³	#88251	1	6	yes	—
111	C₂³	#88254	1	6	yes	—
112	C₂³	C₂³×Dic₁₃	1	1	yes	characteristic
113	C₂×C₄	—	2	12	no	—
114	C₂×C₄	—	2	12	no	—
115	C₂×C₄	—	2	6	no	—
116	C₂×C₄	—	2	6	no	—
117	C₂×C₄	—	2	24	no	—
118	C₂×C₄	—	4	4	no	—
119	C₂×C₄	—	4	12	no	—
120	C₂×C₄	—	52	8	no	—
121	C₂×C₄	—	52	24	no	—
122	C₂×C₄	—	52	24	no	—
123	C₂×C₄	C₂×C₄⋊Dic₁₃	1	12	yes	—
124	C₂²	#11719205	1	6	yes	—
125	C₂²	#11719215	1	6	yes	—
126	C₂²	#11719425	1	4	yes	—
127	C₂²	#11719440	1	12	yes	—
128	C₂²	C₂²×C₄⋊Dic₁₃	1	1	yes	characteristic
129	C₂²	#11719503	1	3	yes	—
130	C₂²	#11719517	1	3	yes	—
131	C₄	—	2	12	no	—
132	C₄	—	4	8	no	—
133	C₄	—	104	8	no	—
134	C₂	#13117454	1	3	yes	—
135	C₂	#13118013	1	1	yes	characteristic
136	C₂	#13118017	1	3	yes	—
137	C₂	#13122910	1	4	yes	—
138	C₂	#13123105	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#11718598

C₂

yes

maximal

#13113089

C₂

yes

maximal

C₂×C₁₃×#384

C₂

yes

maximal, characteristic

C₂²×C₁₃×C₄⋊C₄

C₂²

yes

—

C₂²×C₄²×C₁₃

C₂²

yes

characteristic

C₂²×C₄⋊Dic₁₃

C₂²

yes

—

C₁₃×#384

C₄

yes

complemented

C₂×#88071

C₂²

yes

—

C₂×#88071

C₂²

yes

—

#11718598

—

C₂³×C₄×C₁₃

C₂³

yes

—

C₂³×C₄×C₁₃

C₂³

yes

—

C₂³×Dic₁₃

C₂³

yes

—

C₂×C₁₃×C₄⋊C₄

—

C₂×C₁₃×C₄⋊C₄

—

C₂×C₁₃×C₄⋊C₄

C₂×C₄

yes

—

C₂×C₄²×C₁₃

C₂×C₄

yes

—

C₂×C₄²×C₁₃

D₄

yes

—

C₂×C₄²×C₁₃

Q₈

yes

—

C₂×C₄⋊Dic₁₃

—

—

—

—

C₂⁴×C₁₃

C₂⁴

yes

characteristic

C₂²×C₄×C₁₃

—

C₂²×C₄×C₁₃

—

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×Q₈

yes

—

C₂²×Dic₁₃

—

C₂²×Dic₁₃

—

C₂²×Dic₁₃

—

C₁₃×C₄⋊C₄

—

C₄²×C₁₃

C₄⋊C₄

yes

complemented

C₄⋊Dic₁₃

—

C₂×#384

D₁₃

yes

characteristic

C₂×#383

—

#1441

—

C₂³×C₁₃

C₂³×C₄

yes

characteristic

C₂³×C₁₃

C₂²×D₄

yes

characteristic

C₂³×C₁₃

C₂²×Q₈

yes

characteristic

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×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

C₂×C₄⋊C₄

yes

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂²×C₄⋊C₄

—

C₂²×C₄⋊C₄

C₂×D₁₃

yes

—

C₂²×C₄²

C₂×D₁₃

yes

characteristic

#384

Dic₁₃

yes

complemented

#383

—

C₂×#95

—

C₂×#95

—

C₂²×C₁₃

C₂²×C₄⋊C₄

yes

characteristic

C₂²×C₁₃

#518

yes

—

C₂²×C₁₃

#519

yes

—

C₂²×C₁₃

#537

yes

—

C₂²×C₁₃

#544

yes

—

C₂²×C₁₃

#560

yes

—

C₂²×C₁₃

#563

yes

—

C₄×C₁₃

—

C₄×C₁₃

—

Dic₁₃

—

C₂³×C₄

—

C₂³×C₄

C₂²×D₁₃

yes

—

C₂³×C₄

C₂²×D₁₃

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×Dic₁₃

yes

—

C₂×C₄²

Dic₂₆

yes

—

C₂×C₄²

D₅₂

yes

—

C₂×C₄²

C₂×Dic₁₃

yes

—

#95

—

#95

—

C₂×C₁₃

#2316

yes

—

C₂×C₁₃

#2380

yes

characteristic

C₂×C₁₃

#2381

yes

—

C₂×C₁₃

#2823

yes

—

C₂×C₁₃

#2843

yes

—

C₂⁴

C₂³×D₁₃

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

100

C₂²×C₄

C₂×Dic₂₆

yes

—

101

C₂²×C₄

C₂×D₅₂

yes

—

102

C₂²×C₄

C₂²×Dic₁₃

yes

—

103

C₂²×C₄

C₂²×Dic₁₃

yes

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

106

C₄²

C₄⋊Dic₁₃

yes

complemented

107

C₁₃

#14676

yes

characteristic, complemented, Sylow

108

C₂³

C₂²×Dic₂₆

yes

characteristic

109

C₂³

C₂²×D₅₂

yes

characteristic

110

C₂³

#88251

yes

—

111

C₂³

#88254

yes

—

112

C₂³

C₂³×Dic₁₃

yes

characteristic

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

C₂×C₄⋊Dic₁₃

yes

—

124

C₂²

#11719205

yes

—

125

C₂²

#11719215

yes

—

126

C₂²

#11719425

yes

—

127

C₂²

#11719440

yes

—

128

C₂²

C₂²×C₄⋊Dic₁₃

yes

characteristic

129

C₂²

#11719503

yes

—

130

C₂²

#11719517

yes

—

131

C₄

—

132

C₄

—

133

C₄

—

104

—

134

C₂

#13117454

yes

—

135

C₂

#13118013

yes

characteristic

136

C₂

#13118017

yes

—

137

C₂

#13122910

yes

—

138

C₂

#13123105

yes

—

	subgroup	G/H
center, Frattini	C₂⁴	C₂³×D₁₃
commutator	C₂³×C₁₃	C₂³×C₄
Fitting	C₂×C₁₃×#384	C₂
socle, prime radical, cosocle	C₂⁴×C₁₃	C₂⁴
hypercenter	C₂×#384	D₁₃
hypocenter	C₁₃	#14676
LCS	G ▷ C₂³×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂⁴ ◁ C₂×#384

subgroup

G/H

center, Frattini

C₂⁴

C₂³×D₁₃

commutator

C₂³×C₁₃

C₂³×C₄

Fitting

C₂×C₁₃×#384

C₂

socle, prime radical, cosocle

C₂⁴×C₁₃

C₂⁴

D₁₃

C₁₃

G ▷ C₂³×C₁₃ ▷ C₁₃

C₁ ◁ C₂⁴ ◁ C₂×#384

N	H ≅ G/N	type
7. C₁₃×#384	C₄	semidirect
37. C₄²×C₁₃	C₄⋊C₄	semidirect
61. #384	Dic₁₃	semidirect
106. C₄²	C₄⋊Dic₁₃	semidirect
107. C₁₃	#14676	semidirect

H ≅ G/N

type

7. C₁₃×#384

C₄

semidirect

37. C₄²×C₁₃

C₄⋊C₄

semidirect

61. #384

Dic₁₃

semidirect

106. C₄²

C₄⋊Dic₁₃

semidirect

107. C₁₃

#14676

semidirect

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

subgroup	index
C₂×#11718598	2
#13113089	2
C₂×C₁₃×#384	2
#14676	13

subgroup

index

C₂×#11718598

#13113089

C₂×C₁₃×#384

#14676

Subgroups — #15794063

Conjugacy classes of subgroups

Complements

Maximal subgroups