Group #15777573 — subgroups

Subgroups — #15777573

Order 3328.

306 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 306 conjugacy classes of subgroups — too many to draw. Only its 87 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	#13117086	C₂	1	1	yes	maximal, characteristic
2	#13117061	C₂	1	2	yes	maximal, complemented
3	#13114429	C₂	1	2	yes	maximal, complemented
4	C₁₆×#1055	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₃×#2252	C₂	1	1	yes	maximal, characteristic, complemented
6	C₁₃×#500	C₂²	1	2	yes	—
7	#11718749	—	2	2	no	—
8	#11718746	—	2	2	no	—
9	C₂×D₂₀₈	—	2	2	no	—
10	#11718356	C₂²	1	2	yes	—
11	C₁₃×#366	—	2	2	no	—
12	C₄×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
13	#11718083	—	2	4	no	—
14	C₄×#6090	C₂²	1	1	yes	characteristic
15	C₈×#1055	C₂²	1	1	yes	characteristic
16	C₁₃×C₄⋊Q₈	—	2	2	no	—
17	C₂×#6106	—	4	4	no	—
18	C₂×D₁₀₄	—	2	2	no	—
19	C₂×D₁₀₄	—	2	2	no	—
20	#88128	—	2	2	no	—
21	C₂×C₁₃×C₁₆	D₄	1	2	yes	—
22	C₁₃×#107	—	2	2	no	—
23	C₁₃×#107	—	2	2	no	—
24	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
25	C₂×#6090	D₄	1	2	yes	—
26	#88049	—	4	4	no	—
27	C₄×#1055	D₄	1	1	yes	characteristic
28	C₄×#1055	D₄	1	1	yes	characteristic
29	D₂₀₈	—	4	4	no	—
30	#12503	—	13	1	no	maximal, Sylow
31	C₂×Q₈×C₁₃	—	4	2	no	—
32	C₂×D₅₂	—	4	2	no	—
33	C₂×D₅₂	—	4	2	no	—
34	C₂×D₅₂	—	4	2	no	—
35	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
36	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
37	C₁₃×C₄⋊C₄	—	4	2	no	—
38	C₁₃×C₄⋊C₄	—	4	2	no	—
39	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
40	#6106	—	8	8	no	—
41	C₂×#1055	D₈	1	2	yes	—
42	C₂×#1055	D₈	1	2	yes	—
43	D₁₀₄	—	4	4	no	—
44	D₁₀₄	—	4	4	no	—
45	C₁₃×C₁₆	SD₁₆	1	4	yes	complemented
46	#6090	—	2	2	no	—
47	#2252	D₁₃	1	1	yes	characteristic, complemented
48	#2246	—	13	1	no	—
49	#2240	—	13	2	no	—
50	#1706	—	13	2	no	—
51	C₈×C₁₆	—	13	1	no	—
52	C₂²×D₁₃	—	8	2	no	—
53	Q₈×C₁₃	—	8	4	no	—
54	C₂×C₄×C₁₃	—	8	2	no	—
55	C₂×C₄×C₁₃	#127	1	1	yes	characteristic
56	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
57	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
58	D₅₂	—	8	2	no	—
59	D₅₂	—	8	2	no	—
60	D₅₂	—	8	4	no	—
61	D₅₂	—	8	4	no	—
62	C₈×C₁₃	C₂×SD₁₆	1	2	yes	—
63	C₈×C₁₃	#135	1	2	yes	—
64	#1055	—	2	2	no	—
65	#1055	D₁₆	1	4	yes	complemented
66	C₂×D₁₆	—	26	2	no	—
67	#500	C₂×D₁₃	1	2	yes	—
68	#493	—	13	2	no	—
69	#492	—	26	2	no	—
70	#486	—	26	2	no	—
71	#366	—	2	2	no	—
72	#357	—	26	4	no	—
73	C₄×C₁₆	—	13	1	no	—
74	C₄×C₁₆	C₂×D₁₃	1	1	yes	characteristic
75	C₈²	—	13	1	no	—
76	C₂²×C₁₃	#493	1	1	yes	characteristic
77	C₂×D₁₃	—	16	2	no	—
78	C₂×D₁₃	—	16	2	no	—
79	C₂×D₁₃	—	16	2	no	—
80	C₄×C₁₃	—	16	2	no	—
81	C₄×C₁₃	#492	1	1	yes	characteristic
82	C₄×C₁₃	#495	1	1	yes	characteristic
83	C₄×C₁₃	C₂×D₁₆	1	2	yes	—
84	C₄×C₁₃	#508	1	2	yes	—
85	C₂×SD₁₆	—	52	4	no	—
86	C₂×D₈	—	26	2	no	—
87	C₂×D₈	—	26	2	no	—
88	C₄⋊Q₈	—	2	2	no	—
89	#127	—	26	2	no	—
90	D₁₆	—	52	4	no	—
91	C₂×C₁₆	—	13	2	no	—
92	C₂×C₁₆	C₂²⋊D₁₃	1	2	yes	—
93	#107	—	2	2	no	—
94	#107	—	2	2	no	—
95	#102	—	52	4	no	—
96	C₄×C₈	—	13	1	no	—
97	C₄×C₈	—	13	1	no	—
98	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
99	C₂×C₁₃	#1706	1	1	yes	characteristic
100	C₂×C₁₃	#2246	1	1	yes	characteristic
101	C₂×C₁₃	#2249	1	1	yes	characteristic
102	D₁₃	—	32	2	no	—
103	C₂×Q₈	—	4	2	no	—
104	C₂×D₄	—	52	2	no	—
105	C₂×D₄	—	52	2	no	—
106	C₂×D₄	—	52	2	no	—
107	SD₁₆	—	104	8	no	—
108	D₈	—	52	4	no	—
109	D₈	—	52	4	no	—
110	C₂×C₈	—	13	2	no	—
111	C₂×C₈	—	13	2	no	—
112	C₂×C₈	D₄×D₁₃	1	1	yes	characteristic
113	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
114	C₄⋊C₄	—	4	2	no	—
115	C₄⋊C₄	—	4	2	no	—
116	C₄²	D₄×D₁₃	1	1	yes	characteristic
117	C₁₆	—	26	2	no	—
118	C₁₆	#6106	1	4	yes	complemented
119	C₁₃	#12503	1	1	yes	characteristic, complemented, Sylow
120	C₂³	—	104	2	no	—
121	Q₈	—	8	4	no	—
122	D₄	—	104	2	no	—
123	D₄	—	104	4	no	—
124	D₄	—	104	4	no	—
125	D₄	—	104	2	no	—
126	C₂×C₄	—	8	2	no	—
127	C₂×C₄	D₈×D₁₃	1	1	yes	characteristic
128	C₂×C₄	D₈×D₁₃	1	1	yes	characteristic
129	C₂×C₄	#88194	1	1	yes	characteristic
130	C₈	—	13	4	no	—
131	C₈	—	26	2	no	—
132	C₈	#88173	1	2	yes	—
133	C₈	C₂×#6106	1	2	yes	—
134	C₂²	—	208	2	no	—
135	C₂²	—	208	2	no	—
136	C₂²	—	208	2	no	—
137	C₂²	#11718812	1	1	yes	characteristic
138	C₄	—	16	2	no	—
139	C₄	D₁₃×D₁₆	1	2	yes	—
140	C₄	#11718580	1	2	yes	—
141	C₄	#11718746	1	1	yes	characteristic
142	C₄	#11718852	1	1	yes	characteristic
143	C₂	—	416	2	no	—
144	C₂	#13114429	1	1	yes	characteristic
145	C₂	#13117084	1	1	yes	characteristic
146	C₂	#13117098	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13117086

C₂

yes

maximal, characteristic

#13117061

C₂

yes

maximal, complemented

#13114429

C₂

yes

maximal, complemented

C₁₆×#1055

C₂

yes

maximal, characteristic, complemented

C₁₃×#2252

C₂

yes

maximal, characteristic, complemented

C₁₃×#500

C₂²

yes

—

#11718749

—

#11718746

—

C₂×D₂₀₈

—

#11718356

C₂²

yes

—

C₁₃×#366

—

C₄×C₁₃×C₁₆

C₂²

yes

characteristic

#11718083

—

C₄×#6090

C₂²

yes

characteristic

C₈×#1055

C₂²

yes

characteristic

C₁₃×C₄⋊Q₈

—

C₂×#6106

—

C₂×D₁₀₄

—

C₂×D₁₀₄

—

#88128

—

C₂×C₁₃×C₁₆

D₄

yes

—

C₁₃×#107

—

C₁₃×#107

—

C₄×C₈×C₁₃

C₂³

yes

characteristic

C₂×#6090

D₄

yes

—

#88049

—

C₄×#1055

D₄

yes

characteristic

C₄×#1055

D₄

yes

characteristic

D₂₀₈

—

#12503

—

maximal, Sylow

C₂×Q₈×C₁₃

—

C₂×D₅₂

—

C₂×D₅₂

—

C₂×D₅₂

—

C₂×C₈×C₁₃

C₂×D₄

yes

characteristic

C₂×C₈×C₁₃

C₂×D₄

yes

characteristic

C₁₃×C₄⋊C₄

—

C₁₃×C₄⋊C₄

—

C₄²×C₁₃

C₂×D₄

yes

characteristic

#6106

—

C₂×#1055

D₈

yes

—

C₂×#1055

D₈

yes

—

D₁₀₄

—

D₁₀₄

—

C₁₃×C₁₆

SD₁₆

yes

complemented

#6090

—

#2252

D₁₃

yes

characteristic, complemented

#2246

—

#2240

—

#1706

—

C₈×C₁₆

—

C₂²×D₁₃

—

Q₈×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

#127

yes

characteristic

C₂×C₄×C₁₃

C₂×D₈

yes

characteristic

C₂×C₄×C₁₃

C₂×D₈

yes

characteristic

D₅₂

—

D₅₂

—

D₅₂

—

D₅₂

—

C₈×C₁₃

C₂×SD₁₆

yes

—

C₈×C₁₃

#135

yes

—

#1055

—

#1055

D₁₆

yes

complemented

C₂×D₁₆

—

#500

C₂×D₁₃

yes

—

—

—

—

—

—

C₄×C₁₆

—

C₄×C₁₆

C₂×D₁₃

yes

characteristic

C₈²

—

C₂²×C₁₃

#493

yes

characteristic

C₂×D₁₃

—

C₂×D₁₃

—

C₂×D₁₃

—

C₄×C₁₃

—

C₄×C₁₃

yes

C₄×C₁₃

yes

C₄×C₁₃

C₂×D₁₆

yes

—

C₄×C₁₃

#508

yes

—

C₂×SD₁₆

—

C₂×D₈

—

C₂×D₈

—

C₄⋊Q₈

—

#127

—

D₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₂²⋊D₁₃

yes

—

#107

—

#107

—

#102

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂²×D₁₃

yes

characteristic

C₂×C₁₃

#1706

yes

characteristic

100

C₂×C₁₃

#2246

yes

characteristic

101

C₂×C₁₃

#2249

yes

characteristic

102

D₁₃

—

103

C₂×Q₈

—

104

C₂×D₄

—

105

C₂×D₄

—

106

C₂×D₄

—

107

SD₁₆

—

104

—

108

D₈

—

109

D₈

—

110

C₂×C₈

—

111

C₂×C₈

—

112

C₂×C₈

D₄×D₁₃

yes

characteristic

113

C₂×C₈

C₂×C₂²⋊D₁₃

yes

characteristic

114

C₄⋊C₄

—

115

C₄⋊C₄

—

116

C₄²

D₄×D₁₃

yes

characteristic

117

C₁₆

—

118

C₁₆

#6106

yes

complemented

119

C₁₃

#12503

yes

characteristic, complemented, Sylow

120

C₂³

—

104

—

121

Q₈

—

122

D₄

—

104

—

123

D₄

—

104

—

124

D₄

—

104

—

125

D₄

—

104

—

126

C₂×C₄

—

127

C₂×C₄

D₈×D₁₃

yes

characteristic

128

C₂×C₄

D₈×D₁₃

yes

characteristic

129

C₂×C₄

#88194

yes

characteristic

130

C₈

—

131

C₈

—

132

C₈

#88173

yes

—

133

C₈

C₂×#6106

yes

—

134

C₂²

—

208

—

135

C₂²

—

208

—

136

C₂²

—

208

—

137

C₂²

#11718812

yes

characteristic

138

C₄

—

139

C₄

D₁₃×D₁₆

yes

—

140

C₄

#11718580

yes

—

141

C₄

#11718746

yes

characteristic

142

C₄

#11718852

yes

characteristic

143

C₂

—

416

—

144

C₂

#13114429

yes

characteristic

145

C₂

#13117084

yes

characteristic

146

C₂

#13117098

yes

characteristic

	subgroup	G/H
center	C₂²	#11718812
commutator, cosocle	C₄×C₈×C₁₃	C₂³
Frattini	C₄×C₈	C₂²×D₁₃
Fitting	C₁₃×#2252	C₂
socle	C₂²×C₁₃	#493
prime radical	#13117086	C₂
hypercenter	#2252	D₁₃
hypocenter	C₁₃	#12503
LCS	G ▷ C₄×C₈×C₁₃ ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2252

subgroup

G/H

center

C₂²

#11718812

commutator, cosocle

C₄×C₈×C₁₃

C₂³

Frattini

C₄×C₈

C₂²×D₁₃

Fitting

C₁₃×#2252

C₂

C₂²×C₁₃

C₂

D₁₃

C₁₃

G ▷ C₄×C₈×C₁₃ ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2252

N	H ≅ G/N	type
2. #13117061	C₂	semidirect
3. #13114429	C₂	semidirect
4. C₁₆×#1055	C₂	semidirect
5. C₁₃×#2252	C₂	semidirect
45. C₁₃×C₁₆	SD₁₆	semidirect
47. #2252	D₁₃	semidirect
65. #1055	D₁₆	semidirect
118. C₁₆	#6106	semidirect
119. C₁₃	#12503	semidirect

H ≅ G/N

type

2. #13117061

C₂

semidirect

3. #13114429

C₂

semidirect

4. C₁₆×#1055

C₂

semidirect

5. C₁₃×#2252

C₂

semidirect

45. C₁₃×C₁₆

SD₁₆

semidirect

47. #2252

D₁₃

semidirect

65. #1055

D₁₆

semidirect

118. C₁₆

#6106

semidirect

119. C₁₃

#12503

semidirect

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

subgroup	index
#13117086	2
#13117061	2
#13114429	2
C₁₆×#1055	2
C₁₃×#2252	2
#12503	13

subgroup

index

#13117086

#13117061

#13114429

C₁₆×#1055

C₁₃×#2252

#12503

Subgroups — #15777573

Conjugacy classes of subgroups

Complements

Maximal subgroups