Group #15765988 — subgroups

Subgroups — #15765988

Order 3328.

484 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 484 conjugacy classes of subgroups — too many to draw. Only its 93 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	#13115659	C₂	1	3	yes	maximal
2	#13112717	C₂	1	3	yes	maximal, complemented
3	C₁₃×#2844	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#88128	C₂²	1	1	yes	characteristic
5	C₁₃×#400	—	2	3	no	—
6	C₁₃×#395	—	2	3	no	—
7	C₁₃×#382	C₂²	1	3	yes	—
8	C₂×#88049	—	2	3	no	—
9	C₂×#88049	—	2	3	no	—
10	C₂×#88043	C₂²	1	3	yes	—
11	C₂²×Q₈×C₁₃	—	2	1	no	—
12	C₂²×D₅₂	—	2	3	no	—
13	C₂×C₁₃×C₄⋊C₄	—	2	3	no	—
14	C₂×C₁₃×C₄⋊C₄	D₄	1	3	yes	—
15	C₂×C₁₃×C₄⋊C₄	D₄	1	3	yes	—
16	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
17	C₂²×#1055	—	2	3	no	—
18	#88128	—	2	3	no	—
19	#88128	Q₈	1	2	yes	complemented
20	C₁₃×#95	—	2	3	no	—
21	C₁₃×#95	—	2	1	no	—
22	C₁₃×#95	—	2	6	no	—
23	#88049	—	4	6	no	—
24	#88049	—	4	6	no	—
25	#88043	—	2	6	no	—
26	#10358	—	13	1	no	maximal, Sylow
27	C₂³×D₁₃	—	4	1	no	—
28	C₂×Q₈×C₁₃	—	8	3	no	—
29	C₂²×C₄×C₁₃	—	2	3	no	—
30	C₂²×C₄×C₁₃	—	2	3	no	—
31	C₂²×C₄×C₁₃	C₂×D₄	1	3	yes	—
32	C₂×D₅₂	—	2	6	no	—
33	C₂×D₅₂	—	4	3	no	—
34	C₂×D₅₂	—	4	3	no	—
35	C₂×D₅₂	—	4	3	no	—
36	C₂×D₅₂	—	4	3	no	—
37	C₂×D₅₂	—	4	3	no	—
38	C₁₃×C₄⋊C₄	—	4	3	no	—
39	C₁₃×C₄⋊C₄	—	4	3	no	—
40	C₁₃×C₄⋊C₄	—	8	3	no	—
41	C₄²×C₁₃	C₂×Q₈	1	1	yes	characteristic
42	C₄²×C₁₃	#41	1	3	yes	—
43	C₂×#1055	—	2	6	no	—
44	C₂×#1055	—	4	3	no	—
45	C₂×#1055	—	4	3	no	—
46	#2844	D₁₃	1	1	yes	characteristic, complemented
47	#1961	—	13	3	no	—
48	#1295	—	13	3	no	—
49	C₂³×C₁₃	#120	1	1	yes	characteristic
50	C₂²×D₁₃	—	4	2	no	—
51	C₂²×D₁₃	—	8	3	no	—
52	C₂²×D₁₃	—	8	3	no	—
53	Q₈×C₁₃	—	8	4	no	—
54	C₂×C₄×C₁₃	—	2	3	no	—
55	C₂×C₄×C₁₃	—	2	3	no	—
56	C₂×C₄×C₁₃	—	4	3	no	—
57	C₂×C₄×C₁₃	—	4	3	no	—
58	C₂×C₄×C₁₃	—	4	6	no	—
59	C₂×C₄×C₁₃	—	4	6	no	—
60	C₂×C₄×C₁₃	C₂²⋊Q₈	1	3	yes	—
61	C₂×C₄×C₁₃	#124	1	3	yes	—
62	D₅₂	—	4	6	no	—
63	D₅₂	—	4	6	no	—
64	D₅₂	—	8	3	no	—
65	D₅₂	—	8	3	no	—
66	#1055	—	4	6	no	—
67	C₂×#127	—	13	1	no	—
68	C₂×#102	—	26	3	no	—
69	C₂×#102	—	26	3	no	—
70	C₂×#97	—	13	3	no	—
71	#400	—	2	3	no	—
72	#395	—	2	3	no	—
73	#382	C₂×D₁₃	1	3	yes	—
74	C₂²×C₁₃	#393	1	1	yes	characteristic
75	C₂²×C₁₃	#453	1	3	yes	—
76	C₂²×C₁₃	#454	1	3	yes	—
77	C₂×D₁₃	—	8	6	no	—
78	C₂×D₁₃	—	16	1	no	—
79	C₂×D₁₃	—	16	3	no	—
80	C₄×C₁₃	—	2	3	no	—
81	C₄×C₁₃	—	2	3	no	—
82	C₄×C₁₃	—	8	3	no	—
83	C₄×C₁₃	—	8	3	no	—
84	C₂²×Q₈	—	2	1	no	—
85	C₂²×D₄	—	26	3	no	—
86	C₂²×C₈	—	26	3	no	—
87	#127	—	13	2	no	—
88	#127	—	26	3	no	—
89	C₂×C₄⋊C₄	—	2	3	no	—
90	C₂×C₄⋊C₄	C₂²⋊D₁₃	1	3	yes	—
91	C₂×C₄⋊C₄	C₂²⋊D₁₃	1	3	yes	—
92	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
93	#102	—	52	6	no	—
94	#102	—	52	6	no	—
95	#97	—	26	6	no	—
96	#95	—	2	3	no	—
97	#95	—	2	1	no	—
98	#95	—	2	6	no	—
99	C₂×C₁₃	#1655	1	1	yes	characteristic
100	C₂×C₁₃	#1657	1	3	yes	—
101	C₂×C₁₃	#2027	1	3	yes	—
102	D₁₃	—	16	2	no	—
103	C₂⁴	—	52	1	no	—
104	C₂×Q₈	—	8	3	no	—
105	C₂×D₄	—	26	6	no	—
106	C₂×D₄	—	52	3	no	—
107	C₂×D₄	—	52	3	no	—
108	C₂×D₄	—	52	3	no	—
109	C₂×D₄	—	52	3	no	—
110	C₂×D₄	—	52	3	no	—
111	C₂²×C₄	—	2	3	no	—
112	C₂²×C₄	—	2	3	no	—
113	C₂²×C₄	C₂×C₂²⋊D₁₃	1	3	yes	—
114	C₂×C₈	—	26	6	no	—
115	C₂×C₈	—	52	3	no	—
116	C₂×C₈	—	52	3	no	—
117	C₄⋊C₄	—	4	3	no	—
118	C₄⋊C₄	—	4	3	no	—
119	C₄⋊C₄	—	8	3	no	—
120	C₄²	Q₈×D₁₃	1	1	yes	characteristic
121	C₄²	#6131	1	3	yes	—
122	C₁₃	#10358	1	1	yes	characteristic, complemented, Sylow
123	C₂³	—	52	2	no	—
124	C₂³	—	104	3	no	—
125	C₂³	—	104	3	no	—
126	C₂³	#88207	1	1	yes	characteristic
127	Q₈	—	8	4	no	—
128	D₄	—	52	6	no	—
129	D₄	—	52	6	no	—
130	D₄	—	104	3	no	—
131	D₄	—	104	3	no	—
132	C₂×C₄	—	2	3	no	—
133	C₂×C₄	—	2	3	no	—
134	C₂×C₄	—	4	3	no	—
135	C₂×C₄	—	4	3	no	—
136	C₂×C₄	—	4	6	no	—
137	C₂×C₄	—	4	6	no	—
138	C₂×C₄	#88200	1	3	yes	—
139	C₂×C₄	#88201	1	3	yes	—
140	C₈	—	52	6	no	—
141	C₂²	—	104	6	no	—
142	C₂²	—	208	1	no	—
143	C₂²	—	208	3	no	—
144	C₂²	#11718892	1	1	yes	characteristic
145	C₂²	#11718902	1	3	yes	—
146	C₂²	#11718906	1	3	yes	—
147	C₄	—	2	3	no	—
148	C₄	—	2	3	no	—
149	C₄	—	8	3	no	—
150	C₄	—	8	3	no	—
151	C₂	—	208	2	no	—
152	C₂	#13114125	1	1	yes	characteristic
153	C₂	#13114134	1	3	yes	—
154	C₂	#13115989	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#13115659

C₂

yes

maximal

#13112717

C₂

yes

maximal, complemented

C₁₃×#2844

C₂

yes

maximal, characteristic, complemented

C₂×#88128

C₂²

yes

characteristic

C₁₃×#400

—

C₁₃×#395

—

C₁₃×#382

C₂²

yes

—

C₂×#88049

—

C₂×#88049

—

C₂×#88043

C₂²

yes

—

C₂²×Q₈×C₁₃

—

C₂²×D₅₂

—

C₂×C₁₃×C₄⋊C₄

—

C₂×C₁₃×C₄⋊C₄

D₄

yes

—

C₂×C₁₃×C₄⋊C₄

D₄

yes

—

C₂×C₄²×C₁₃

C₂³

yes

characteristic

C₂²×#1055

—

#88128

—

#88128

Q₈

yes

complemented

C₁₃×#95

—

C₁₃×#95

—

C₁₃×#95

—

—

—

—

—

C₂³×D₁₃

—

C₂×Q₈×C₁₃

—

C₂²×C₄×C₁₃

—

C₂²×C₄×C₁₃

—

C₂²×C₄×C₁₃

C₂×D₄

yes

—

C₂×D₅₂

—

C₂×D₅₂

—

C₂×D₅₂

—

C₂×D₅₂

—

C₂×D₅₂

—

C₂×D₅₂

—

C₁₃×C₄⋊C₄

—

C₁₃×C₄⋊C₄

—

C₁₃×C₄⋊C₄

—

C₄²×C₁₃

C₂×Q₈

yes

characteristic

C₄²×C₁₃

#41

yes

—

—

—

—

D₁₃

yes

characteristic, complemented

#1961

—

#1295

—

C₂³×C₁₃

#120

yes

characteristic

C₂²×D₁₃

—

C₂²×D₁₃

—

C₂²×D₁₃

—

Q₈×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

C₂²⋊Q₈

yes

—

C₂×C₄×C₁₃

#124

yes

—

D₅₂

—

D₅₂

—

D₅₂

—

D₅₂

—

—

—

—

—

C₂×#97

—

#400

—

#395

—

#382

C₂×D₁₃

yes

—

C₂²×C₁₃

#393

yes

characteristic

C₂²×C₁₃

#453

yes

—

C₂²×C₁₃

#454

yes

—

C₂×D₁₃

—

C₂×D₁₃

—

C₂×D₁₃

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

—

C₂²×Q₈

—

C₂²×D₄

—

C₂²×C₈

—

#127

—

#127

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂²⋊D₁₃

yes

—

C₂×C₄⋊C₄

C₂²⋊D₁₃

yes

—

C₂×C₄²

C₂²×D₁₃

yes

—

—

—

—

—

—

C₂×C₁₃

#1655

yes

characteristic

100

C₂×C₁₃

#1657

yes

—

101

C₂×C₁₃

#2027

yes

—

102

D₁₃

—

103

C₂⁴

—

104

C₂×Q₈

—

105

C₂×D₄

—

106

C₂×D₄

—

107

C₂×D₄

—

108

C₂×D₄

—

109

C₂×D₄

—

110

C₂×D₄

—

111

C₂²×C₄

—

112

C₂²×C₄

—

113

C₂²×C₄

C₂×C₂²⋊D₁₃

yes

—

114

C₂×C₈

—

115

C₂×C₈

—

116

C₂×C₈

—

117

C₄⋊C₄

—

118

C₄⋊C₄

—

119

C₄⋊C₄

—

120

C₄²

Q₈×D₁₃

yes

characteristic

121

C₄²

#6131

yes

—

122

C₁₃

#10358

yes

characteristic, complemented, Sylow

123

C₂³

—

124

C₂³

—

104

—

125

C₂³

—

104

—

126

C₂³

#88207

yes

characteristic

127

Q₈

—

128

D₄

—

129

D₄

—

130

D₄

—

104

—

131

D₄

—

104

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

#88200

yes

—

139

C₂×C₄

#88201

yes

—

140

C₈

—

141

C₂²

—

104

—

142

C₂²

—

208

—

143

C₂²

—

208

—

144

C₂²

#11718892

yes

characteristic

145

C₂²

#11718902

yes

—

146

C₂²

#11718906

yes

—

147

C₄

—

148

C₄

—

149

C₄

—

150

C₄

—

151

C₂

—

208

—

152

C₂

#13114125

yes

characteristic

153

C₂

#13114134

yes

—

154

C₂

#13115989

yes

—

	subgroup	G/H
center	C₂³	#88207
commutator, cosocle	C₂×C₄²×C₁₃	C₂³
Frattini	C₂×C₄²	C₂²×D₁₃
Fitting	C₁₃×#2844	C₂
socle	C₂³×C₁₃	#120
prime radical	C₂×#88128	C₂²
hypercenter	#2844	D₁₃
hypocenter	C₁₃	#10358
LCS	G ▷ C₂×C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #2844

subgroup

G/H

center

C₂³

#88207

commutator, cosocle

C₂×C₄²×C₁₃

C₂³

Frattini

C₂×C₄²

C₂²×D₁₃

Fitting

C₁₃×#2844

C₂

C₂³×C₁₃

C₂²

D₁₃

C₁₃

G ▷ C₂×C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #2844

N	H ≅ G/N	type
2. #13112717	C₂	semidirect
3. C₁₃×#2844	C₂	semidirect
19. #88128	Q₈	semidirect
46. #2844	D₁₃	semidirect
122. C₁₃	#10358	semidirect

H ≅ G/N

type

2. #13112717

C₂

semidirect

3. C₁₃×#2844

C₂

semidirect

19. #88128

Q₈

semidirect

46. #2844

D₁₃

semidirect

122. C₁₃

#10358

semidirect

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

subgroup	index
#13115659	2
#13112717	2
C₁₃×#2844	2
#10358	13

subgroup

index

#13115659

#13112717

C₁₃×#2844

#10358

Subgroups — #15765988

Conjugacy classes of subgroups

Complements

Maximal subgroups