Group #15852248 — subgroups

Subgroups — #15852248

Order 3328.

736 conjugacy classes of subgroups; only the 179 normal subgroups are shown.

normal subgroups only

This group has 736 conjugacy classes of subgroups — too many to draw. Only its 179 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	#13125581	C₂	1	3	yes	maximal, complemented
2	#13125211	C₂	1	3	yes	maximal
3	#13114496	C₂	1	6	yes	maximal, complemented
4	#13114448	C₂	1	2	yes	maximal, complemented
5	C₁₃×#1455	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#88128	C₂²	1	1	yes	characteristic
7	#11719175	C₂²	1	3	yes	—
8	C₄×C₁₃×M₁₆	C₂²	1	3	yes	—
9	#11718798	—	2	6	no	—
10	#11718792	—	2	6	no	—
11	#11718377	—	2	6	no	—
12	#11718374	—	2	6	no	—
13	#11718360	C₂²	1	3	yes	complemented
14	#11718355	C₂²	1	6	yes	—
15	#11718355	C₂²	1	6	yes	—
16	#11718342	C₂²	1	6	yes	complemented
17	#11718341	C₂²	1	3	yes	complemented
18	C₁₃×#322	C₂²	1	3	yes	complemented
19	C₈²×C₁₃	C₂²	1	1	yes	characteristic, complemented
20	C₂²×D₅₂	—	2	3	no	—
21	C₂×C₁₃×M₁₆	D₄	1	6	yes	—
22	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
23	#88180	—	2	3	no	—
24	#88178	—	2	6	no	—
25	C₂×Dic₅₂	—	2	6	no	—
26	#88128	—	2	3	no	—
27	#88128	C₂³	1	2	yes	—
28	#88124	—	2	3	no	—
29	#88123	C₂³	1	6	yes	—
30	C₄×Dic₂₆	—	2	6	no	—
31	C₁₃×#97	D₄	1	6	yes	—
32	C₄×C₈×C₁₃	C₂³	1	3	yes	—
33	C₄×C₈×C₁₃	C₂³	1	3	yes	—
34	#88061	—	2	12	no	—
35	#88061	—	2	12	no	—
36	#88057	—	2	6	no	—
37	#88056	—	2	12	no	—
38	#19753	—	13	1	no	maximal, Sylow
39	C₂³×D₁₃	—	4	1	no	—
40	C₂²×C₄×C₁₃	C₂×D₄	1	3	yes	—
41	C₂×D₅₂	—	2	6	no	—
42	C₂×D₅₂	—	4	12	no	—
43	C₂×D₅₂	—	4	3	no	—
44	C₂×Dic₂₆	—	2	6	no	—
45	C₁₃×M₁₆	—	4	6	no	—
46	C₂×C₈×C₁₃	C₂×D₄	1	6	yes	—
47	C₂×C₈×C₁₃	C₂×D₄	1	6	yes	—
48	C₄²×C₁₃	C₂×D₄	1	3	yes	—
49	C₄²×C₁₃	C₂⁴	1	1	yes	characteristic
50	C₂²⋊Dic₁₃	—	4	3	no	—
51	C₄⋊Dic₁₃	—	2	6	no	—
52	C₄⋊Dic₁₃	—	2	6	no	—
53	#6101	—	4	6	no	—
54	C₄×Dic₁₃	—	4	3	no	—
55	Dic₅₂	—	4	12	no	—
56	#3185	—	13	3	no	—
57	#3138	—	13	3	no	—
58	#1720	—	13	6	no	—
59	#1711	—	13	2	no	—
60	#1455	D₁₃	1	1	yes	characteristic, complemented
61	C₂³×C₁₃	#127	1	1	yes	characteristic
62	C₂²×D₁₃	—	4	2	no	—
63	C₂²×D₁₃	—	8	6	no	—
64	C₂×C₄×C₁₃	—	2	6	no	—
65	C₂×C₄×C₁₃	#127	1	3	yes	—
66	C₂×C₄×C₁₃	C₂²×D₄	1	3	yes	—
67	C₂×Dic₁₃	—	4	6	no	—
68	D₅₂	—	4	12	no	—
69	D₅₂	—	8	6	no	—
70	Dic₂₆	—	4	12	no	—
71	C₈×C₁₃	—	2	6	no	—
72	C₈×C₁₃	—	2	6	no	—
73	#533	—	13	3	no	—
74	C₂×#127	—	13	1	no	—
75	#498	—	13	3	no	—
76	#494	—	13	3	no	—
77	#490	—	26	6	no	—
78	#487	—	13	6	no	—
79	#486	—	13	6	no	—
80	#486	—	13	6	no	—
81	#470	—	26	6	no	—
82	#441	—	26	6	no	—
83	#418	—	26	6	no	—
84	C₄×M₁₆	C₂×D₁₃	1	3	yes	—
85	#322	C₂×D₁₃	1	3	yes	complemented
86	C₈²	C₂×D₁₃	1	1	yes	characteristic, complemented
87	C₂²×C₁₃	—	2	3	no	—
88	C₂²×C₁₃	C₂×#127	1	1	yes	characteristic
89	C₂×D₁₃	—	8	6	no	—
90	C₂×D₁₃	—	16	4	no	—
91	C₄×C₁₃	—	2	3	no	—
92	C₄×C₁₃	#575	1	6	yes	—
93	Dic₁₃	—	8	6	no	—
94	C₂²×D₄	—	26	3	no	—
95	C₂×Q₁₆	—	26	6	no	—
96	C₂×M₁₆	D₅₂	1	6	yes	—
97	C₄⋊Q₈	—	13	6	no	—
98	#127	—	13	2	no	—
99	#127	—	26	3	no	—
100	#125	—	26	3	no	—
101	C₂²⋊Q₈	—	26	6	no	—
102	C₄×Q₈	—	26	6	no	—
103	#117	—	26	3	no	—
104	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
105	#106	—	26	6	no	—
106	#103	—	26	12	no	—
107	#102	—	26	12	no	—
108	#102	—	26	12	no	—
109	#97	D₅₂	1	6	yes	—
110	C₄×C₈	C₂²×D₁₃	1	3	yes	—
111	C₄×C₈	C₂²×D₁₃	1	3	yes	—
112	C₂×C₁₃	—	4	1	no	—
113	C₂×C₁₃	#3153	1	3	yes	—
114	D₁₃	—	16	2	no	—
115	C₂⁴	—	52	1	no	—
116	C₂×Q₈	—	26	6	no	—
117	C₂×D₄	—	26	6	no	—
118	C₂×D₄	—	52	12	no	—
119	C₂×D₄	—	52	3	no	—
120	C₂²×C₄	C₂×D₅₂	1	3	yes	—
121	Q₁₆	—	52	12	no	—
122	M₁₆	—	4	6	no	—
123	C₂×C₈	C₂×D₅₂	1	6	yes	—
124	C₂×C₈	C₂×D₅₂	1	6	yes	—
125	C₄⋊C₄	—	26	6	no	—
126	C₄⋊C₄	—	26	6	no	—
127	C₄⋊C₄	—	52	6	no	—
128	C₂²⋊C₄	—	52	3	no	—
129	C₄²	—	52	3	no	—
130	C₄²	C₂×D₅₂	1	3	yes	—
131	C₄²	C₂³×D₁₃	1	1	yes	characteristic
132	C₁₃	#19753	1	1	yes	characteristic, complemented, Sylow
133	C₂³	—	52	2	no	—
134	C₂³	—	104	6	no	—
135	C₂³	#88128	1	1	yes	characteristic
136	Q₈	—	52	12	no	—
137	D₄	—	52	12	no	—
138	D₄	—	104	6	no	—
139	C₂×C₄	—	2	6	no	—
140	C₂×C₄	—	52	6	no	—
141	C₂×C₄	#88128	1	3	yes	—
142	C₂×C₄	C₂²×D₅₂	1	3	yes	—
143	C₈	—	2	6	no	—
144	C₈	—	2	6	no	—
145	C₂²	—	2	3	no	—
146	C₂²	—	104	6	no	—
147	C₂²	—	208	4	no	—
148	C₂²	C₂×#88128	1	1	yes	characteristic
149	C₄	—	2	3	no	—
150	C₄	—	104	6	no	—
151	C₄	#11719453	1	6	yes	—
152	C₂	—	4	1	no	—
153	C₂	—	208	2	no	—
154	C₂	#13125302	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#13125581

C₂

yes

maximal, complemented

#13125211

C₂

yes

maximal

#13114496

C₂

yes

maximal, complemented

#13114448

C₂

yes

maximal, complemented

C₁₃×#1455

C₂

yes

maximal, characteristic, complemented

C₂×#88128

C₂²

yes

characteristic

#11719175

C₂²

yes

—

C₄×C₁₃×M₁₆

C₂²

yes

—

—

—

—

—

C₂²

yes

complemented

#11718355

C₂²

yes

—

#11718355

C₂²

yes

—

#11718342

C₂²

yes

complemented

#11718341

C₂²

yes

complemented

C₁₃×#322

C₂²

yes

complemented

C₈²×C₁₃

C₂²

yes

characteristic, complemented

C₂²×D₅₂

—

C₂×C₁₃×M₁₆

D₄

yes

—

C₂×C₄²×C₁₃

C₂³

yes

characteristic

#88180

—

#88178

—

C₂×Dic₅₂

—

#88128

—

#88128

C₂³

yes

—

#88124

—

#88123

C₂³

yes

—

C₄×Dic₂₆

—

C₁₃×#97

D₄

yes

—

C₄×C₈×C₁₃

C₂³

yes

—

C₄×C₈×C₁₃

C₂³

yes

—

—

—

—

—

—

C₂³×D₁₃

—

C₂²×C₄×C₁₃

C₂×D₄

yes

—

C₂×D₅₂

—

C₂×D₅₂

—

C₂×D₅₂

—

C₂×Dic₂₆

—

C₁₃×M₁₆

—

C₂×C₈×C₁₃

C₂×D₄

yes

—

C₂×C₈×C₁₃

C₂×D₄

yes

—

C₄²×C₁₃

C₂×D₄

yes

—

C₄²×C₁₃

C₂⁴

yes

characteristic

C₂²⋊Dic₁₃

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

#6101

—

C₄×Dic₁₃

—

Dic₅₂

—

—

—

—

—

D₁₃

yes

characteristic, complemented

C₂³×C₁₃

#127

yes

characteristic

C₂²×D₁₃

—

C₂²×D₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

#127

yes

—

C₂×C₄×C₁₃

C₂²×D₄

yes

—

C₂×Dic₁₃

—

D₅₂

—

D₅₂

—

Dic₂₆

—

C₈×C₁₃

—

C₈×C₁₃

—

—

—

—

—

—

—

—

—

—

—

—

C₄×M₁₆

C₂×D₁₃

yes

—

#322

C₂×D₁₃

yes

complemented

C₈²

C₂×D₁₃

yes

characteristic, complemented

C₂²×C₁₃

—

C₂²×C₁₃

C₂×#127

yes

characteristic

C₂×D₁₃

—

C₂×D₁₃

—

C₄×C₁₃

—

C₄×C₁₃

#575

yes

—

Dic₁₃

—

C₂²×D₄

—

C₂×Q₁₆

—

C₂×M₁₆

D₅₂

yes

—

C₄⋊Q₈

—

#127

—

#127

—

100

#125

—

101

C₂²⋊Q₈

—

102

C₄×Q₈

—

103

#117

—

104

C₂×C₄²

C₂²×D₁₃

yes

characteristic

105

#106

—

106

#103

—

107

#102

—

108

#102

—

109

#97

D₅₂

yes

—

110

C₄×C₈

C₂²×D₁₃

yes

—

111

C₄×C₈

C₂²×D₁₃

yes

—

112

C₂×C₁₃

—

113

C₂×C₁₃

#3153

yes

—

114

D₁₃

—

115

C₂⁴

—

116

C₂×Q₈

—

117

C₂×D₄

—

118

C₂×D₄

—

119

C₂×D₄

—

120

C₂²×C₄

C₂×D₅₂

yes

—

121

Q₁₆

—

122

M₁₆

—

123

C₂×C₈

C₂×D₅₂

yes

—

124

C₂×C₈

C₂×D₅₂

yes

—

125

C₄⋊C₄

—

126

C₄⋊C₄

—

127

C₄⋊C₄

—

128

C₂²⋊C₄

—

129

C₄²

—

130

C₄²

C₂×D₅₂

yes

—

131

C₄²

C₂³×D₁₃

yes

characteristic

132

C₁₃

#19753

yes

characteristic, complemented, Sylow

133

C₂³

—

134

C₂³

—

104

—

135

C₂³

#88128

yes

characteristic

136

Q₈

—

137

D₄

—

138

D₄

—

104

—

139

C₂×C₄

—

140

C₂×C₄

—

141

C₂×C₄

#88128

yes

—

142

C₂×C₄

C₂²×D₅₂

yes

—

143

C₈

—

144

C₈

—

145

C₂²

—

146

C₂²

—

104

—

147

C₂²

—

208

—

148

C₂²

C₂×#88128

yes

characteristic

149

C₄

—

150

C₄

—

104

—

151

C₄

#11719453

yes

—

152

C₂

—

153

C₂

—

208

—

154

C₂

#13125302

yes

—

	subgroup	G/H
center	C₂²	C₂×#88128
commutator, cosocle	C₄²×C₁₃	C₂⁴
Frattini	C₄²	C₂³×D₁₃
Fitting	C₁₃×#1455	C₂
socle	C₂²×C₁₃	C₂×#127
prime radical	C₂×#88128	C₂²
hypercenter	#1455	D₁₃
hypocenter	C₁₃	#19753
LCS	G ▷ C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1455

subgroup

G/H

center

C₂²

C₂×#88128

commutator, cosocle

C₄²×C₁₃

C₂⁴

Frattini

C₄²

C₂³×D₁₃

Fitting

C₁₃×#1455

C₂

C₂²×C₁₃

C₂²

D₁₃

C₁₃

G ▷ C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1455

N	H ≅ G/N	type
1. #13125581	C₂	semidirect
3. #13114496	C₂	semidirect
4. #13114448	C₂	semidirect
5. C₁₃×#1455	C₂	semidirect
13. #11718360	C₂²	semidirect
16. #11718342	C₂²	semidirect
17. #11718341	C₂²	semidirect
18. C₁₃×#322	C₂²	semidirect
19. C₈²×C₁₃	C₂²	semidirect
60. #1455	D₁₃	semidirect
85. #322	C₂×D₁₃	semidirect
86. C₈²	C₂×D₁₃	semidirect
132. C₁₃	#19753	semidirect

H ≅ G/N

type

1. #13125581

C₂

semidirect

3. #13114496

C₂

semidirect

4. #13114448

C₂

semidirect

5. C₁₃×#1455

C₂

semidirect

13. #11718360

C₂²

semidirect

16. #11718342

C₂²

semidirect

17. #11718341

C₂²

semidirect

18. C₁₃×#322

C₂²

semidirect

19. C₈²×C₁₃

C₂²

semidirect

60. #1455

D₁₃

semidirect

85. #322

C₂×D₁₃

semidirect

86. C₈²

C₂×D₁₃

semidirect

132. C₁₃

#19753

semidirect

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

subgroup	index
#13125581	2
#13125211	2
#13114496	2
#13114448	2
C₁₃×#1455	2
#19753	13

subgroup

index

C₁₃×#1455

Subgroups — #15852248

Conjugacy classes of subgroups

Complements

Maximal subgroups