Group #15850724 — subgroups

Subgroups — #15850724

Order 3328.

690 conjugacy classes of subgroups; only the 227 normal subgroups are shown.

normal subgroups only

This group has 690 conjugacy classes of subgroups — too many to draw. Only its 227 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	#13125422	C₂	1	4	yes	maximal
2	#13125350	C₂	1	2	yes	maximal
3	#13114513	C₂	1	8	yes	maximal, complemented
4	C₂×C₁₃×#322	C₂	1	1	yes	maximal, characteristic
5	#11719175	C₂²	1	4	yes	—
6	C₂×C₄×C₈×C₁₃	C₂²	1	2	yes	—
7	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
8	#11718777	—	2	8	no	—
9	#11718767	—	2	4	no	—
10	#11718360	C₂²	1	4	yes	—
11	C₄×Dic₅₂	—	2	8	no	—
12	#11718344	—	2	4	no	—
13	#11718343	C₂²	1	4	yes	—
14	#11718343	C₂²	1	4	yes	—
15	#11718342	C₂²	1	8	yes	—
16	#11718341	C₂²	1	4	yes	—
17	C₁₃×#322	C₂²	1	4	yes	—
18	C₂²×C₈×C₁₃	D₄	1	4	yes	—
19	C₂²×C₈×C₁₃	D₄	1	2	yes	—
20	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
21	#88180	—	2	4	no	—
22	#88178	—	2	8	no	—
23	C₂×Dic₅₂	—	2	8	no	—
24	#88124	—	2	4	no	—
25	#88123	C₂³	1	8	yes	—
26	C₄×Dic₂₆	—	2	8	no	—
27	C₄×C₈×C₁₃	D₄	1	2	yes	—
28	C₄×C₈×C₁₃	D₄	1	4	yes	—
29	C₄×C₈×C₁₃	C₂³	1	2	yes	—
30	C₄×C₈×C₁₃	C₂³	1	4	yes	—
31	#88058	—	2	8	no	—
32	#88058	—	2	8	no	—
33	#88057	—	2	8	no	—
34	#88056	—	2	16	no	—
35	#19618	—	13	1	no	maximal, Sylow
36	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
37	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
38	C₂×Dic₂₆	—	2	8	no	—
39	C₂×C₈×C₁₃	—	4	4	no	—
40	C₂×C₈×C₁₃	Q₁₆	1	8	yes	complemented
41	C₂×C₈×C₁₃	C₂×D₄	1	8	yes	—
42	C₂×C₈×C₁₃	C₂×D₄	1	4	yes	—
43	C₄²×C₁₃	C₂×D₄	1	2	yes	—
44	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
45	C₄²×C₁₃	C₂⁴	1	1	yes	characteristic
46	C₂²⋊Dic₁₃	—	4	4	no	—
47	C₄⋊Dic₁₃	—	2	8	no	—
48	C₄⋊Dic₁₃	—	2	8	no	—
49	#6101	—	4	8	no	—
50	C₄×Dic₁₃	—	4	4	no	—
51	Dic₅₂	—	4	16	no	—
52	#3170	—	13	4	no	—
53	#3160	—	13	2	no	—
54	#1723	—	13	8	no	—
55	C₂×#322	D₁₃	1	1	yes	characteristic
56	C₂³×C₁₃	#127	1	1	yes	characteristic
57	C₂×C₄×C₁₃	—	2	4	no	—
58	C₂×C₄×C₁₃	#127	1	2	yes	—
59	C₂×C₄×C₁₃	#127	1	1	yes	characteristic
60	C₂×C₄×C₁₃	C₂×Q₁₆	1	4	yes	—
61	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
62	C₂×C₄×C₁₃	C₂²×D₄	1	2	yes	—
63	C₂×Dic₁₃	—	4	8	no	—
64	Dic₂₆	—	4	16	no	—
65	C₈×C₁₃	—	2	8	no	—
66	C₈×C₁₃	C₂×Q₁₆	1	8	yes	—
67	#533	—	13	4	no	—
68	#500	—	13	4	no	—
69	#500	—	13	4	no	—
70	#499	—	26	4	no	—
71	#498	—	13	4	no	—
72	#494	—	13	4	no	—
73	#487	—	13	8	no	—
74	#467	—	26	8	no	—
75	C₄×Q₁₆	—	26	8	no	—
76	#427	—	26	4	no	—
77	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
78	C₂×C₄×C₈	C₂×D₁₃	1	2	yes	—
79	#322	C₂×D₁₃	1	4	yes	—
80	C₂²×C₁₃	—	2	2	no	—
81	C₂²×C₁₃	#494	1	2	yes	—
82	C₂²×C₁₃	C₂×#127	1	1	yes	characteristic
83	C₄×C₁₃	—	2	2	no	—
84	C₄×C₁₃	#494	1	2	yes	—
85	C₄×C₁₃	C₂²×Q₁₆	1	2	yes	—
86	C₄×C₁₃	#575	1	4	yes	—
87	Dic₁₃	—	8	8	no	—
88	C₂×Q₁₆	—	26	8	no	—
89	C₂²×C₈	D₅₂	1	4	yes	—
90	C₂²×C₈	D₅₂	1	2	yes	—
91	C₄⋊Q₈	—	13	8	no	—
92	#125	—	26	4	no	—
93	C₂²⋊Q₈	—	26	8	no	—
94	C₄×Q₈	—	26	8	no	—
95	#117	—	26	4	no	—
96	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
97	#107	—	26	8	no	—
98	#107	—	26	8	no	—
99	#106	—	26	8	no	—
100	#103	—	26	16	no	—
101	C₄×C₈	D₅₂	1	4	yes	—
102	C₄×C₈	D₅₂	1	2	yes	—
103	C₄×C₈	C₂²×D₁₃	1	4	yes	—
104	C₄×C₈	C₂²×D₁₃	1	2	yes	—
105	C₂×C₁₃	—	2	2	no	—
106	C₂×C₁₃	C₂×#494	1	1	yes	characteristic
107	C₂×C₁₃	#3150	1	2	yes	—
108	C₂×Q₈	—	26	8	no	—
109	C₂²×C₄	C₂×D₅₂	1	2	yes	—
110	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
111	Q₁₆	—	52	16	no	—
112	C₂×C₈	—	4	4	no	—
113	C₂×C₈	Dic₅₂	1	8	yes	complemented
114	C₂×C₈	C₂×D₅₂	1	8	yes	—
115	C₂×C₈	C₂×D₅₂	1	4	yes	—
116	C₄⋊C₄	—	26	8	no	—
117	C₄⋊C₄	—	26	8	no	—
118	C₄⋊C₄	—	52	8	no	—
119	C₂²⋊C₄	—	52	4	no	—
120	C₄²	—	52	4	no	—
121	C₄²	C₂×D₅₂	1	1	yes	characteristic
122	C₄²	C₂×D₅₂	1	2	yes	—
123	C₄²	C₂³×D₁₃	1	1	yes	characteristic
124	C₁₃	#19618	1	1	yes	characteristic, complemented, Sylow
125	C₂³	#88128	1	1	yes	characteristic
126	Q₈	—	52	16	no	—
127	C₂×C₄	—	2	4	no	—
128	C₂×C₄	—	52	8	no	—
129	C₂×C₄	#88128	1	2	yes	—
130	C₂×C₄	#88128	1	1	yes	characteristic
131	C₂×C₄	C₂×Dic₅₂	1	4	yes	—
132	C₂×C₄	C₂²×D₅₂	1	1	yes	characteristic
133	C₂×C₄	C₂²×D₅₂	1	2	yes	—
134	C₈	—	2	8	no	—
135	C₈	C₂×Dic₅₂	1	8	yes	—
136	C₂²	—	2	2	no	—
137	C₂²	#11718360	1	2	yes	—
138	C₂²	C₂×#88128	1	1	yes	characteristic
139	C₄	—	2	2	no	—
140	C₄	—	104	8	no	—
141	C₄	#11718360	1	2	yes	—
142	C₄	C₂²×Dic₅₂	1	2	yes	—
143	C₄	#11719453	1	4	yes	—
144	C₂	—	2	2	no	—
145	C₂	C₂×#11718360	1	1	yes	characteristic
146	C₂	#13125282	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#13125422

C₂

yes

maximal

#13125350

C₂

yes

maximal

#13114513

C₂

yes

maximal, complemented

C₂×C₁₃×#322

C₂

yes

maximal, characteristic

#11719175

C₂²

yes

—

C₂×C₄×C₈×C₁₃

C₂²

yes

—

C₂×C₄×C₈×C₁₃

C₂²

yes

—

—

C₂²

yes

—

C₄×Dic₅₂

—

#11718344

—

#11718343

C₂²

yes

—

#11718343

C₂²

yes

—

#11718342

C₂²

yes

—

#11718341

C₂²

yes

—

C₁₃×#322

C₂²

yes

—

C₂²×C₈×C₁₃

D₄

yes

—

C₂²×C₈×C₁₃

D₄

yes

—

C₂×C₄²×C₁₃

C₂³

yes

characteristic

#88180

—

#88178

—

C₂×Dic₅₂

—

#88124

—

#88123

C₂³

yes

—

C₄×Dic₂₆

—

C₄×C₈×C₁₃

D₄

yes

—

C₄×C₈×C₁₃

D₄

yes

—

C₄×C₈×C₁₃

C₂³

yes

—

C₄×C₈×C₁₃

C₂³

yes

—

—

—

—

—

—

C₂²×C₄×C₁₃

C₂×D₄

yes

—

C₂²×C₄×C₁₃

C₂×D₄

yes

characteristic

C₂×Dic₂₆

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

Q₁₆

yes

complemented

C₂×C₈×C₁₃

C₂×D₄

yes

—

C₂×C₈×C₁₃

C₂×D₄

yes

—

C₄²×C₁₃

C₂×D₄

yes

—

C₄²×C₁₃

C₂×D₄

yes

characteristic

C₄²×C₁₃

C₂⁴

yes

characteristic

C₂²⋊Dic₁₃

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

#6101

—

C₄×Dic₁₃

—

Dic₅₂

—

—

—

—

D₁₃

yes

characteristic

C₂³×C₁₃

#127

yes

characteristic

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

#127

yes

—

C₂×C₄×C₁₃

#127

yes

characteristic

C₂×C₄×C₁₃

C₂×Q₁₆

yes

—

C₂×C₄×C₁₃

C₂²×D₄

yes

characteristic

C₂×C₄×C₁₃

C₂²×D₄

yes

—

C₂×Dic₁₃

—

Dic₂₆

—

C₈×C₁₃

—

C₈×C₁₃

C₂×Q₁₆

yes

—

—

—

—

—

—

—

—

—

C₄×Q₁₆

—

#427

—

C₂×C₄×C₈

C₂×D₁₃

yes

characteristic

C₂×C₄×C₈

C₂×D₁₃

yes

—

#322

C₂×D₁₃

yes

—

C₂²×C₁₃

—

C₂²×C₁₃

#494

yes

—

C₂²×C₁₃

C₂×#127

yes

characteristic

C₄×C₁₃

—

C₄×C₁₃

#494

yes

—

C₄×C₁₃

C₂²×Q₁₆

yes

—

C₄×C₁₃

#575

yes

—

Dic₁₃

—

C₂×Q₁₆

—

C₂²×C₈

D₅₂

yes

—

C₂²×C₈

D₅₂

yes

—

C₄⋊Q₈

—

#125

—

C₂²⋊Q₈

—

C₄×Q₈

—

#117

—

C₂×C₄²

C₂²×D₁₃

yes

—

—

—

100

#103

—

101

C₄×C₈

D₅₂

yes

—

102

C₄×C₈

D₅₂

yes

—

103

C₄×C₈

C₂²×D₁₃

yes

—

104

C₄×C₈

C₂²×D₁₃

yes

—

105

C₂×C₁₃

—

106

C₂×C₁₃

C₂×#494

yes

characteristic

107

C₂×C₁₃

#3150

yes

—

108

C₂×Q₈

—

109

C₂²×C₄

C₂×D₅₂

yes

—

110

C₂²×C₄

C₂×D₅₂

yes

characteristic

111

Q₁₆

—

112

C₂×C₈

—

113

C₂×C₈

Dic₅₂

yes

complemented

114

C₂×C₈

C₂×D₅₂

yes

—

115

C₂×C₈

C₂×D₅₂

yes

—

116

C₄⋊C₄

—

117

C₄⋊C₄

—

118

C₄⋊C₄

—

119

C₂²⋊C₄

—

120

C₄²

—

121

C₄²

C₂×D₅₂

yes

characteristic

122

C₄²

C₂×D₅₂

yes

—

123

C₄²

C₂³×D₁₃

yes

characteristic

124

C₁₃

#19618

yes

characteristic, complemented, Sylow

125

C₂³

#88128

yes

characteristic

126

Q₈

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

#88128

yes

—

130

C₂×C₄

#88128

yes

characteristic

131

C₂×C₄

C₂×Dic₅₂

yes

—

132

C₂×C₄

C₂²×D₅₂

yes

characteristic

133

C₂×C₄

C₂²×D₅₂

yes

—

134

C₈

—

135

C₈

C₂×Dic₅₂

yes

—

136

C₂²

—

137

C₂²

#11718360

yes

—

138

C₂²

C₂×#88128

yes

characteristic

139

C₄

—

140

C₄

—

104

—

141

C₄

#11718360

yes

—

142

C₄

C₂²×Dic₅₂

yes

—

143

C₄

#11719453

yes

—

144

C₂

—

145

C₂

C₂×#11718360

yes

characteristic

146

C₂

#13125282

yes

—

	subgroup	G/H
center	C₂×C₄	#88128
commutator, cosocle	C₄²×C₁₃	C₂⁴
Frattini	C₄²	C₂³×D₁₃
Fitting	C₂×C₁₃×#322	C₂
socle	C₂²×C₁₃	C₂×#127
prime radical	C₂³×C₁₃	#127
hypercenter	C₂×#322	D₁₃
hypocenter	C₁₃	#19618
LCS	G ▷ C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×#322

subgroup

G/H

center

C₂×C₄

#88128

commutator, cosocle

C₄²×C₁₃

C₂⁴

Frattini

C₄²

C₂³×D₁₃

Fitting

C₂×C₁₃×#322

C₂

C₂²×C₁₃

C₂³×C₁₃

D₁₃

C₁₃

G ▷ C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×#322

N	H ≅ G/N	type
3. #13114513	C₂	semidirect
40. C₂×C₈×C₁₃	Q₁₆	semidirect
113. C₂×C₈	Dic₅₂	semidirect
124. C₁₃	#19618	semidirect

H ≅ G/N

type

3. #13114513

C₂

semidirect

40. C₂×C₈×C₁₃

Q₁₆

semidirect

113. C₂×C₈

Dic₅₂

semidirect

124. C₁₃

#19618

semidirect

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

subgroup	index
#13125422	2
#13125350	2
#13114513	2
C₂×C₁₃×#322	2
#19618	13

subgroup

index

C₂×C₁₃×#322

Subgroups — #15850724

Conjugacy classes of subgroups

Complements

Maximal subgroups