Group #15777572 — subgroups

Subgroups — #15777572

Order 3328.

306 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 306 conjugacy classes of subgroups — too many to draw. Only its 99 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	#13117111	C₂	1	1	yes	maximal, characteristic, complemented
2	#13117086	C₂	1	1	yes	maximal, characteristic
3	#13117060	C₂	1	2	yes	maximal, complemented
4	#13114428	C₂	1	2	yes	maximal, complemented
5	C₈×C₁₃×C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₂₀₈	—	2	2	no	—
7	#11718356	C₂²	1	2	yes	—
8	#11718355	—	2	2	no	—
9	#11718354	—	2	2	no	—
10	#11718343	C₂²	1	2	yes	—
11	C₄×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
12	C₄×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
13	C₈²×C₁₃	C₂²	1	1	yes	characteristic
14	#11718101	—	2	4	no	—
15	#11718096	—	2	2	no	—
16	C₂×D₁₀₄	—	2	2	no	—
17	C₂×D₁₀₄	—	2	2	no	—
18	C₂×#6095	—	4	4	no	—
19	#88128	—	2	2	no	—
20	#88123	—	2	2	no	—
21	C₂×C₁₃×C₁₆	D₄	1	2	yes	—
22	C₂×C₁₃×C₁₆	D₄	1	2	yes	—
23	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
24	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
25	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
26	#88061	—	4	4	no	—
27	#88058	—	2	2	no	—
28	#88058	—	2	2	no	—
29	D₂₀₈	—	4	4	no	—
30	#12503	—	13	1	no	maximal, Sylow
31	C₂×D₅₂	—	4	2	no	—
32	C₂×D₅₂	—	4	2	no	—
33	C₂×D₅₂	—	4	2	no	—
34	C₂×Dic₂₆	—	4	2	no	—
35	C₂×C₈×C₁₃	D₈	1	2	yes	—
36	C₂×C₈×C₁₃	D₈	1	2	yes	—
37	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
38	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
39	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
40	C₄⋊Dic₁₃	—	4	2	no	—
41	C₄⋊Dic₁₃	—	4	2	no	—
42	D₁₀₄	—	4	4	no	—
43	D₁₀₄	—	4	4	no	—
44	#6095	—	8	8	no	—
45	C₁₃×C₁₆	—	2	2	no	—
46	C₁₃×C₁₆	SD₁₆	1	4	yes	complemented
47	#2252	—	13	1	no	—
48	#2246	—	13	1	no	—
49	#2240	—	13	2	no	—
50	#1706	—	13	2	no	—
51	C₈×C₁₆	D₁₃	1	1	yes	characteristic, complemented
52	C₂²×D₁₃	—	8	2	no	—
53	C₂×C₄×C₁₃	#127	1	1	yes	characteristic
54	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
55	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
56	C₂×Dic₁₃	—	8	2	no	—
57	D₅₂	—	8	2	no	—
58	D₅₂	—	8	2	no	—
59	D₅₂	—	8	4	no	—
60	D₅₂	—	8	4	no	—
61	Dic₂₆	—	8	4	no	—
62	C₈×C₁₃	—	2	2	no	—
63	C₈×C₁₃	D₁₆	1	4	yes	complemented
64	C₈×C₁₃	C₂×SD₁₆	1	2	yes	—
65	C₈×C₁₃	#135	1	2	yes	—
66	C₂×D₁₆	—	26	2	no	—
67	#500	—	13	2	no	—
68	#493	—	13	2	no	—
69	#492	—	26	2	no	—
70	#486	—	26	2	no	—
71	#366	—	26	2	no	—
72	#357	—	26	4	no	—
73	C₄×C₁₆	C₂×D₁₃	1	1	yes	characteristic
74	C₄×C₁₆	C₂×D₁₃	1	1	yes	characteristic
75	C₈²	C₂×D₁₃	1	1	yes	characteristic
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	#492	1	1	yes	characteristic
81	C₄×C₁₃	#495	1	1	yes	characteristic
82	C₄×C₁₃	C₂×D₁₆	1	2	yes	—
83	C₄×C₁₃	#508	1	2	yes	—
84	Dic₁₃	—	16	2	no	—
85	C₂×SD₁₆	—	52	4	no	—
86	C₂×D₈	—	26	2	no	—
87	C₂×D₈	—	26	2	no	—
88	C₄⋊Q₈	—	26	2	no	—
89	#127	—	26	2	no	—
90	D₁₆	—	52	4	no	—
91	C₂×C₁₆	D₅₂	1	2	yes	—
92	C₂×C₁₆	D₅₂	1	2	yes	—
93	#107	—	26	2	no	—
94	#107	—	26	2	no	—
95	#102	—	52	4	no	—
96	C₄×C₈	D₅₂	1	1	yes	characteristic
97	C₄×C₈	D₅₂	1	1	yes	characteristic
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	—	52	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	D₁₀₄	1	2	yes	—
111	C₂×C₈	D₁₀₄	1	2	yes	—
112	C₂×C₈	C₂×D₅₂	1	1	yes	characteristic
113	C₂×C₈	C₂×D₅₂	1	1	yes	characteristic
114	C₄⋊C₄	—	52	2	no	—
115	C₄⋊C₄	—	52	2	no	—
116	C₄²	C₂×D₅₂	1	1	yes	characteristic
117	C₁₆	—	2	2	no	—
118	C₁₆	#6095	1	4	yes	complemented
119	C₁₃	#12503	1	1	yes	characteristic, complemented, Sylow
120	C₂³	—	104	2	no	—
121	Q₈	—	104	4	no	—
122	D₄	—	104	4	no	—
123	D₄	—	104	4	no	—
124	D₄	—	104	2	no	—
125	D₄	—	104	2	no	—
126	C₂×C₄	—	104	2	no	—
127	C₂×C₄	#88128	1	1	yes	characteristic
128	C₂×C₄	C₂×D₁₀₄	1	1	yes	characteristic
129	C₂×C₄	C₂×D₁₀₄	1	1	yes	characteristic
130	C₈	—	2	2	no	—
131	C₈	D₂₀₈	1	4	yes	complemented
132	C₈	C₂×#6095	1	2	yes	—
133	C₈	#88158	1	2	yes	—
134	C₂²	—	208	2	no	—
135	C₂²	—	208	2	no	—
136	C₂²	—	208	2	no	—
137	C₂²	#11718356	1	1	yes	characteristic
138	C₄	—	208	2	no	—
139	C₄	#11718354	1	1	yes	characteristic
140	C₄	#11718357	1	1	yes	characteristic
141	C₄	C₂×D₂₀₈	1	2	yes	—
142	C₄	#11718565	1	2	yes	—
143	C₂	—	416	2	no	—
144	C₂	#13114428	1	1	yes	characteristic
145	C₂	#13117086	1	1	yes	characteristic
146	C₂	#13117097	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13117111

C₂

yes

maximal, characteristic, complemented

#13117086

C₂

yes

maximal, characteristic

#13117060

C₂

yes

maximal, complemented

#13114428

C₂

yes

maximal, complemented

C₈×C₁₃×C₁₆

C₂

yes

maximal, characteristic, complemented

C₂×D₂₀₈

—

#11718356

C₂²

yes

—

#11718355

—

#11718354

—

#11718343

C₂²

yes

—

C₄×C₁₃×C₁₆

C₂²

yes

characteristic

C₄×C₁₃×C₁₆

C₂²

yes

characteristic

C₈²×C₁₃

C₂²

yes

characteristic

#11718101

—

#11718096

—

C₂×D₁₀₄

—

C₂×D₁₀₄

—

C₂×#6095

—

#88128

—

#88123

—

C₂×C₁₃×C₁₆

D₄

yes

—

C₂×C₁₃×C₁₆

D₄

yes

—

C₄×C₈×C₁₃

D₄

yes

characteristic

C₄×C₈×C₁₃

D₄

yes

characteristic

C₄×C₈×C₁₃

C₂³

yes

—

—

—

D₂₀₈

—

#12503

—

maximal, Sylow

C₂×D₅₂

—

C₂×D₅₂

—

C₂×D₅₂

—

C₂×Dic₂₆

—

C₂×C₈×C₁₃

D₈

yes

—

C₂×C₈×C₁₃

D₈

yes

—

C₂×C₈×C₁₃

C₂×D₄

yes

characteristic

C₂×C₈×C₁₃

C₂×D₄

yes

characteristic

C₄²×C₁₃

C₂×D₄

yes

characteristic

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

D₁₀₄

—

D₁₀₄

—

#6095

—

C₁₃×C₁₆

—

C₁₃×C₁₆

SD₁₆

yes

—

—

—

—

C₈×C₁₆

D₁₃

yes

characteristic, complemented

C₂²×D₁₃

—

C₂×C₄×C₁₃

#127

yes

characteristic

C₂×C₄×C₁₃

C₂×D₈

yes

characteristic

C₂×C₄×C₁₃

C₂×D₈

yes

characteristic

C₂×Dic₁₃

—

D₅₂

—

D₅₂

—

D₅₂

—

D₅₂

—

Dic₂₆

—

C₈×C₁₃

—

C₈×C₁₃

D₁₆

yes

complemented

C₈×C₁₃

C₂×SD₁₆

yes

—

C₈×C₁₃

#135

yes

—

C₂×D₁₆

—

—

—

—

—

—

—

C₄×C₁₆

C₂×D₁₃

yes

characteristic

C₄×C₁₆

C₂×D₁₃

yes

characteristic

C₈²

C₂×D₁₃

yes

characteristic

C₂²×C₁₃

#493

yes

characteristic

C₂×D₁₃

—

C₂×D₁₃

—

C₂×D₁₃

—

C₄×C₁₃

yes

C₄×C₁₃

yes

C₄×C₁₃

C₂×D₁₆

yes

—

C₄×C₁₃

#508

yes

—

Dic₁₃

—

C₂×SD₁₆

—

C₂×D₈

—

C₂×D₈

—

C₄⋊Q₈

—

#127

—

D₁₆

—

C₂×C₁₆

D₅₂

yes

—

C₂×C₁₆

D₅₂

yes

—

#107

—

#107

—

#102

—

C₄×C₈

D₅₂

yes

characteristic

C₄×C₈

D₅₂

yes

characteristic

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

D₁₀₄

yes

—

111

C₂×C₈

D₁₀₄

yes

—

112

C₂×C₈

C₂×D₅₂

yes

characteristic

113

C₂×C₈

C₂×D₅₂

yes

characteristic

114

C₄⋊C₄

—

115

C₄⋊C₄

—

116

C₄²

C₂×D₅₂

yes

characteristic

117

C₁₆

—

118

C₁₆

#6095

yes

complemented

119

C₁₃

#12503

yes

characteristic, complemented, Sylow

120

C₂³

—

104

—

121

Q₈

—

104

—

122

D₄

—

104

—

123

D₄

—

104

—

124

D₄

—

104

—

125

D₄

—

104

—

126

C₂×C₄

—

104

—

127

C₂×C₄

#88128

yes

characteristic

128

C₂×C₄

C₂×D₁₀₄

yes

characteristic

129

C₂×C₄

C₂×D₁₀₄

yes

characteristic

130

C₈

—

131

C₈

D₂₀₈

yes

complemented

132

C₈

C₂×#6095

yes

—

133

C₈

#88158

yes

—

134

C₂²

—

208

—

135

C₂²

—

208

—

136

C₂²

—

208

—

137

C₂²

#11718356

yes

characteristic

138

C₄

—

208

—

139

C₄

#11718354

yes

characteristic

140

C₄

#11718357

yes

characteristic

141

C₄

C₂×D₂₀₈

yes

—

142

C₄

#11718565

yes

—

143

C₂

—

416

—

144

C₂

#13114428

yes

characteristic

145

C₂

#13117086

yes

characteristic

146

C₂

#13117097

yes

characteristic

	subgroup	G/H
center	C₂²	#11718356
commutator, cosocle	C₄×C₈×C₁₃	C₂³
Frattini	C₄×C₈	C₂²×D₁₃
Fitting	C₈×C₁₃×C₁₆	C₂
socle	C₂²×C₁₃	#493
prime radical	#13117086	C₂
hypercenter	C₈×C₁₆	D₁₃
hypocenter	C₁₃	#12503
LCS	G ▷ C₄×C₈×C₁₃ ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ C₈×C₁₆

subgroup

G/H

center

C₂²

#11718356

commutator, cosocle

C₄×C₈×C₁₃

C₂³

Frattini

C₄×C₈

C₂²×D₁₃

Fitting

C₈×C₁₃×C₁₆

C₂

C₂²×C₁₃

C₂

C₈×C₁₆

D₁₃

hypocenter

C₁₃

#12503

LCS

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ C₈×C₁₆

N	H ≅ G/N	type
1. #13117111	C₂	semidirect
3. #13117060	C₂	semidirect
4. #13114428	C₂	semidirect
5. C₈×C₁₃×C₁₆	C₂	semidirect
46. C₁₃×C₁₆	SD₁₆	semidirect
51. C₈×C₁₆	D₁₃	semidirect
63. C₈×C₁₃	D₁₆	semidirect
118. C₁₆	#6095	semidirect
119. C₁₃	#12503	semidirect
131. C₈	D₂₀₈	semidirect

H ≅ G/N

type

1. #13117111

C₂

semidirect

3. #13117060

C₂

semidirect

4. #13114428

C₂

semidirect

5. C₈×C₁₃×C₁₆

C₂

semidirect

46. C₁₃×C₁₆

SD₁₆

semidirect

51. C₈×C₁₆

D₁₃

semidirect

63. C₈×C₁₃

D₁₆

semidirect

118. C₁₆

#6095

semidirect

119. C₁₃

#12503

semidirect

131. C₈

D₂₀₈

semidirect

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

subgroup	index
#13117111	2
#13117086	2
#13117060	2
#13114428	2
C₈×C₁₃×C₁₆	2
#12503	13

subgroup

index

C₈×C₁₃×C₁₆

Subgroups — #15777572

Conjugacy classes of subgroups

Complements

Maximal subgroups