Group #15889473 — subgroups

Subgroups — #15889473

Order 3328.

912 conjugacy classes of subgroups; only the 215 normal subgroups are shown.

normal subgroups only

This group has 912 conjugacy classes of subgroups — too many to draw. Only its 215 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	C₄×#88128	C₂	1	1	yes	maximal, characteristic
2	#13115132	C₂	1	12	yes	maximal, complemented
3	#13114979	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₃×#1839	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×#6094	C₂²	1	4	yes	—
6	C₂×#88128	C₄	1	1	yes	characteristic
7	C₂×C₄×D₅₂	C₂²	1	6	yes	—
8	C₄³×C₁₃	C₂²	1	1	yes	characteristic
9	C₂×#88060	C₂²	1	12	yes	—
10	#11718600	C₄	1	6	yes	—
11	#11718596	C₄	1	1	yes	characteristic
12	C₁₃×#336	C₂²	1	6	yes	—
13	#11718116	C₂²	1	6	yes	—
14	C₂²×D₅₂	C₂×C₄	1	6	yes	—
15	C₂²×C₄×D₁₃	C₂³	1	4	yes	—
16	C₂²×C₈×C₁₃	C₂³	1	4	yes	—
17	C₂×C₄²×C₁₃	C₂×C₄	1	3	yes	—
18	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
19	C₂×C₄²×C₁₃	C₂³	1	3	yes	—
20	C₂×#6103	C₂×C₄	1	12	yes	—
21	C₂×C₄⋊Dic₁₃	C₂×C₄	1	6	yes	—
22	C₂²×#1055	C₂³	1	4	yes	—
23	C₂×#6094	—	4	12	no	—
24	#88128	—	2	4	no	—
25	C₄×D₅₂	—	4	6	no	—
26	C₄×D₅₂	—	4	6	no	—
27	#88060	—	4	12	no	—
28	#23357	—	13	1	no	maximal, Sylow
29	C₂³×D₁₃	C₂²×C₄	1	4	yes	—
30	C₂²×C₄×C₁₃	C₂²×C₄	1	3	yes	—
31	C₂²×C₄×C₁₃	C₂²×C₄	1	3	yes	—
32	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
33	C₂²×Dic₁₃	C₂²×C₄	1	4	yes	—
34	C₂×D₅₂	—	2	12	no	—
35	C₂×D₅₂	—	4	6	no	—
36	C₂×D₅₂	—	4	6	no	—
37	C₂×C₄×D₁₃	—	2	12	no	—
38	C₂×C₄×D₁₃	—	2	12	no	—
39	C₂×C₈×C₁₃	—	2	12	no	—
40	C₄²×C₁₃	—	2	6	no	—
41	C₄²×C₁₃	—	4	3	no	—
42	C₄²×C₁₃	M₁₆	1	4	yes	complemented
43	#6103	—	4	12	no	—
44	C₄⋊Dic₁₃	—	4	6	no	—
45	C₂×#1055	—	2	12	no	—
46	#6094	—	4	16	no	—
47	C₄×#127	—	13	1	no	—
48	#1870	—	13	12	no	—
49	#1839	—	13	1	no	—
50	#1839	D₁₃	1	1	yes	characteristic, complemented
51	C₂³×C₁₃	C₂³×C₄	1	1	yes	characteristic
52	C₂²×D₁₃	—	2	12	no	—
53	C₂²×D₁₃	—	2	4	no	—
54	C₂²×D₁₃	—	2	12	no	—
55	C₂×C₄×C₁₃	—	2	3	no	—
56	C₂×C₄×C₁₃	—	2	3	no	—
57	C₂×C₄×C₁₃	—	2	3	no	—
58	C₂×C₄×C₁₃	—	2	6	no	—
59	C₂×C₄×C₁₃	C₂×M₁₆	1	6	yes	—
60	C₂×C₄×C₁₃	H(5,𝔽₂)	1	6	yes	—
61	C₂×Dic₁₃	—	2	12	no	—
62	D₅₂	—	4	24	no	—
63	C₄×D₁₃	—	4	4	no	—
64	C₄×D₁₃	—	4	12	no	—
65	C₈×C₁₃	—	4	4	no	—
66	#1055	—	4	4	no	—
67	C₂²×M₁₆	—	13	4	no	—
68	C₂×#127	—	13	1	no	—
69	C₂×C₄×D₄	—	13	6	no	—
70	C₂×C₂²⋊C₈	—	13	12	no	—
71	#390	—	13	6	no	—
72	#384	—	13	1	no	—
73	C₄³	C₂×D₁₃	1	1	yes	characteristic
74	#336	—	13	6	no	—
75	#336	C₂×D₁₃	1	6	yes	—
76	C₂²×C₁₃	#518	1	3	yes	—
77	C₂²×C₁₃	C₂²×M₁₆	1	1	yes	characteristic
78	C₂²×C₁₃	#568	1	3	yes	—
79	C₂×D₁₃	—	4	4	no	—
80	C₂×D₁₃	—	4	12	no	—
81	C₂×D₁₃	—	4	12	no	—
82	C₄×C₁₃	—	2	6	no	—
83	C₄×C₁₃	—	4	3	no	—
84	C₄×C₁₃	#560	1	4	yes	—
85	Dic₁₃	—	4	4	no	—
86	C₂²×D₄	—	13	6	no	—
87	C₂³×C₄	—	13	4	no	—
88	C₂×M₁₆	—	52	12	no	—
89	C₂²×C₈	—	13	4	no	—
90	C₂²×C₈	C₂²×D₁₃	1	4	yes	—
91	#127	—	26	4	no	—
92	C₄×D₄	—	52	6	no	—
93	C₄×D₄	—	52	6	no	—
94	C₂×C₄⋊C₄	—	13	6	no	—
95	C₂×C₂²⋊C₄	—	13	12	no	—
96	C₂×C₄²	C₄×D₁₃	1	3	yes	—
97	C₂×C₄²	C₄×D₁₃	1	1	yes	characteristic
98	C₂×C₄²	C₂²×D₁₃	1	3	yes	—
99	C₂²⋊C₈	—	52	12	no	—
100	C₂×C₁₃	#2380	1	1	yes	characteristic
101	C₂×C₁₃	#2975	1	3	yes	—
102	C₂×C₁₃	#2980	1	3	yes	—
103	D₁₃	—	8	4	no	—
104	C₂⁴	—	13	4	no	—
105	C₂×D₄	—	26	12	no	—
106	C₂×D₄	—	52	6	no	—
107	C₂×D₄	—	52	6	no	—
108	C₂²×C₄	—	13	4	no	—
109	C₂²×C₄	—	26	12	no	—
110	C₂²×C₄	—	26	12	no	—
111	C₂²×C₄	C₂×C₄×D₁₃	1	3	yes	—
112	C₂²×C₄	C₂×C₄×D₁₃	1	3	yes	—
113	C₂²×C₄	C₂³×D₁₃	1	1	yes	characteristic
114	M₁₆	—	52	16	no	—
115	C₂×C₈	—	2	12	no	—
116	C₂×C₈	—	26	12	no	—
117	C₄⋊C₄	—	52	6	no	—
118	C₂²⋊C₄	—	52	12	no	—
119	C₄²	—	2	6	no	—
120	C₄²	—	4	3	no	—
121	C₄²	#6094	1	4	yes	complemented
122	C₁₃	#23357	1	1	yes	characteristic, complemented, Sylow
123	C₂³	—	26	4	no	—
124	C₂³	—	26	12	no	—
125	C₂³	—	26	12	no	—
126	C₂³	C₂²×C₄×D₁₃	1	1	yes	characteristic
127	D₄	—	52	24	no	—
128	C₂×C₄	—	2	3	no	—
129	C₂×C₄	—	2	3	no	—
130	C₂×C₄	—	2	6	no	—
131	C₂×C₄	—	2	3	no	—
132	C₂×C₄	—	26	12	no	—
133	C₂×C₄	—	52	4	no	—
134	C₂×C₄	—	52	12	no	—
135	C₂×C₄	C₂×#6094	1	6	yes	—
136	C₂×C₄	#88256	1	6	yes	—
137	C₈	—	4	4	no	—
138	C₈	—	52	4	no	—
139	C₂²	—	52	12	no	—
140	C₂²	—	52	4	no	—
141	C₂²	—	52	12	no	—
142	C₂²	#11719226	1	3	yes	—
143	C₂²	C₂²×#6094	1	1	yes	characteristic
144	C₂²	#11719450	1	3	yes	—
145	C₄	—	2	6	no	—
146	C₄	—	4	3	no	—
147	C₄	—	52	4	no	—
148	C₄	#11719416	1	4	yes	—
149	C₂	—	104	4	no	—
150	C₂	#13118012	1	1	yes	characteristic
151	C₂	#13123876	1	3	yes	—
152	C₂	#13123906	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×#88128

C₂

yes

maximal, characteristic

#13115132

C₂

yes

maximal, complemented

#13114979

C₂

yes

maximal, characteristic, complemented

C₁₃×#1839

C₂

yes

maximal, characteristic, complemented

C₂²×#6094

C₂²

yes

—

C₂×#88128

C₄

yes

characteristic

C₂×C₄×D₅₂

C₂²

yes

—

C₄³×C₁₃

C₂²

yes

characteristic

C₂×#88060

C₂²

yes

—

#11718600

C₄

yes

—

#11718596

C₄

yes

characteristic

C₁₃×#336

C₂²

yes

—

#11718116

C₂²

yes

—

C₂²×D₅₂

C₂×C₄

yes

—

C₂²×C₄×D₁₃

C₂³

yes

—

C₂²×C₈×C₁₃

C₂³

yes

—

C₂×C₄²×C₁₃

C₂×C₄

yes

—

C₂×C₄²×C₁₃

C₂×C₄

yes

characteristic

C₂×C₄²×C₁₃

C₂³

yes

—

C₂×#6103

C₂×C₄

yes

—

C₂×C₄⋊Dic₁₃

C₂×C₄

yes

—

C₂²×#1055

C₂³

yes

—

C₂×#6094

—

#88128

—

C₄×D₅₂

—

C₄×D₅₂

—

#88060

—

#23357

—

maximal, Sylow

C₂³×D₁₃

C₂²×C₄

yes

—

C₂²×C₄×C₁₃

C₂²×C₄

yes

—

C₂²×C₄×C₁₃

C₂²×C₄

yes

—

C₂²×C₄×C₁₃

C₂⁴

yes

characteristic

C₂²×Dic₁₃

C₂²×C₄

yes

—

C₂×D₅₂

—

C₂×D₅₂

—

C₂×D₅₂

—

C₂×C₄×D₁₃

—

C₂×C₄×D₁₃

—

C₂×C₈×C₁₃

—

C₄²×C₁₃

—

C₄²×C₁₃

—

C₄²×C₁₃

M₁₆

yes

complemented

#6103

—

C₄⋊Dic₁₃

—

—

—

—

—

—

D₁₃

yes

characteristic, complemented

C₂³×C₁₃

C₂³×C₄

yes

characteristic

C₂²×D₁₃

—

C₂²×D₁₃

—

C₂²×D₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

C₂×M₁₆

yes

—

C₂×C₄×C₁₃

H(5,𝔽₂)

yes

—

C₂×Dic₁₃

—

D₅₂

—

C₄×D₁₃

—

C₄×D₁₃

—

C₈×C₁₃

—

#1055

—

C₂²×M₁₆

—

C₂×#127

—

C₂×C₄×D₄

—

C₂×C₂²⋊C₈

—

#390

—

#384

—

C₄³

C₂×D₁₃

yes

characteristic

#336

—

#336

C₂×D₁₃

yes

—

C₂²×C₁₃

#518

yes

—

C₂²×C₁₃

C₂²×M₁₆

yes

characteristic

C₂²×C₁₃

#568

yes

—

C₂×D₁₃

—

C₂×D₁₃

—

C₂×D₁₃

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

#560

yes

—

Dic₁₃

—

C₂²×D₄

—

C₂³×C₄

—

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

C₂²×D₁₃

yes

—

#127

—

C₄×D₄

—

C₄×D₄

—

C₂×C₄⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₄²

C₄×D₁₃

yes

—

C₂×C₄²

C₄×D₁₃

yes

characteristic

C₂×C₄²

C₂²×D₁₃

yes

—

C₂²⋊C₈

—

100

C₂×C₁₃

#2380

yes

characteristic

101

C₂×C₁₃

#2975

yes

—

102

C₂×C₁₃

#2980

yes

—

103

D₁₃

—

104

C₂⁴

—

105

C₂×D₄

—

106

C₂×D₄

—

107

C₂×D₄

—

108

C₂²×C₄

—

109

C₂²×C₄

—

110

C₂²×C₄

—

111

C₂²×C₄

C₂×C₄×D₁₃

yes

—

112

C₂²×C₄

C₂×C₄×D₁₃

yes

—

113

C₂²×C₄

C₂³×D₁₃

yes

characteristic

114

M₁₆

—

115

C₂×C₈

—

116

C₂×C₈

—

117

C₄⋊C₄

—

118

C₂²⋊C₄

—

119

C₄²

—

120

C₄²

—

121

C₄²

#6094

yes

complemented

122

C₁₃

#23357

yes

characteristic, complemented, Sylow

123

C₂³

—

124

C₂³

—

125

C₂³

—

126

C₂³

C₂²×C₄×D₁₃

yes

characteristic

127

D₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

C₂×#6094

yes

—

136

C₂×C₄

#88256

yes

—

137

C₈

—

138

C₈

—

139

C₂²

—

140

C₂²

—

141

C₂²

—

142

C₂²

#11719226

yes

—

143

C₂²

C₂²×#6094

yes

characteristic

144

C₂²

#11719450

yes

—

145

C₄

—

146

C₄

—

147

C₄

—

148

C₄

#11719416

yes

—

149

C₂

—

104

—

150

C₂

#13118012

yes

characteristic

151

C₂

#13123876

yes

—

152

C₂

#13123906

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₁₃
commutator, socle	C₂³×C₁₃	C₂³×C₄
Fitting	C₁₃×#1839	C₂
prime radical	C₂×#88128	C₄
cosocle	C₂²×C₄×C₁₃	C₂⁴
hypercenter	#1839	D₁₃
hypocenter	C₁₃	#23357
LCS	G ▷ C₂³×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₄ ◁ #1839

subgroup

G/H

center, Frattini

C₂²×C₄

C₂³×D₁₃

commutator, socle

C₂³×C₁₃

C₂³×C₄

Fitting

C₁₃×#1839

C₂

prime radical

C₂×#88128

C₄

cosocle

C₂²×C₄×C₁₃

C₂⁴

D₁₃

C₁₃

G ▷ C₂³×C₁₃ ▷ C₁₃

C₁ ◁ C₂²×C₄ ◁ #1839

N	H ≅ G/N	type
2. #13115132	C₂	semidirect
3. #13114979	C₂	semidirect
4. C₁₃×#1839	C₂	semidirect
42. C₄²×C₁₃	M₁₆	semidirect
50. #1839	D₁₃	semidirect
121. C₄²	#6094	semidirect
122. C₁₃	#23357	semidirect

H ≅ G/N

type

2. #13115132

C₂

semidirect

3. #13114979

C₂

semidirect

4. C₁₃×#1839

C₂

semidirect

42. C₄²×C₁₃

M₁₆

semidirect

50. #1839

D₁₃

semidirect

121. C₄²

#6094

semidirect

122. C₁₃

#23357

semidirect

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

subgroup	index
C₄×#88128	2
#13115132	2
#13114979	2
C₁₃×#1839	2
#23357	13

subgroup

index

C₁₃×#1839

Subgroups — #15889473

Conjugacy classes of subgroups

Complements

Maximal subgroups