Group #15777278 — subgroups

Subgroups — #15777278

Order 3328.

204 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 204 conjugacy classes of subgroups — too many to draw. Only its 75 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
2	#13114230	C₂	1	2	yes	maximal
3	#13112908	C₂	1	1	yes	maximal, characteristic
4	#13112890	C₂	1	2	yes	maximal
5	Q₈×C₁₃×C₁₆	C₂	1	1	yes	maximal, characteristic
6	C₈×Q₈×C₁₃	C₂²	1	1	yes	characteristic
7	#11718692	—	2	2	no	—
8	#11718343	C₂²	1	2	yes	—
9	C₁₃×C₄⋊C₁₆	—	2	1	no	—
10	C₁₃×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
11	C₄×C₁₃×C₁₆	—	2	1	no	—
12	C₄×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
13	#11718096	—	2	1	no	—
14	#11718096	—	2	1	no	—
15	#11718078	—	2	2	no	—
16	#11718050	C₂²	1	2	yes	—
17	C₄×Q₈×C₁₃	D₄	1	1	yes	characteristic
18	C₂×#6107	—	4	2	no	—
19	#88123	—	2	2	no	—
20	C₂×C₁₃×C₁₆	—	2	1	no	—
21	C₂×C₁₃×C₁₆	D₄	1	1	yes	characteristic
22	C₂×C₁₃×C₁₆	D₄	1	1	yes	characteristic
23	C₁₃×C₄⋊C₈	—	2	1	no	—
24	C₁₃×C₄⋊C₈	D₄	1	1	yes	characteristic
25	C₄×C₈×C₁₃	—	2	1	no	—
26	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
27	#88058	—	2	2	no	—
28	#88058	—	2	2	no	—
29	#88050	—	4	2	no	—
30	C₄⋊#1055	—	2	2	no	—
31	#12457	—	13	1	no	maximal, Sylow
32	C₂×Q₈×C₁₃	D₈	1	1	yes	characteristic
33	C₂×Dic₂₆	—	4	2	no	—
34	C₂×C₈×C₁₃	—	2	1	no	—
35	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
36	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
37	C₁₃×C₄⋊C₄	—	2	1	no	—
38	C₁₃×C₄⋊C₄	D₈	1	1	yes	characteristic
39	C₄²×C₁₃	—	2	1	no	—
40	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
41	#6107	—	8	4	no	—
42	C₄⋊Dic₁₃	—	4	2	no	—
43	C₄⋊Dic₁₃	—	4	2	no	—
44	C₂×#1055	—	4	2	no	—
45	C₁₃×C₁₆	—	2	1	no	—
46	C₁₃×C₁₆	—	4	1	no	—
47	C₁₃×C₁₆	Q₁₆	1	2	yes	complemented
48	#2252	—	13	1	no	—
49	Q₈×C₁₆	D₁₃	1	1	yes	characteristic
50	#1674	—	13	2	no	—
51	#1371	—	13	1	no	—
52	#1364	—	13	2	no	—
53	Q₈×C₁₃	—	2	1	no	—
54	Q₈×C₁₃	D₁₆	1	2	yes	—
55	C₂×C₄×C₁₃	—	2	1	no	—
56	C₂×C₄×C₁₃	—	2	1	no	—
57	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
58	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
59	C₂×C₄×C₁₃	#135	1	1	yes	characteristic
60	C₂×Dic₁₃	—	8	2	no	—
61	Dic₂₆	—	8	2	no	—
62	Dic₂₆	—	8	2	no	—
63	C₈×C₁₃	—	2	1	no	—
64	C₈×C₁₃	—	4	1	no	—
65	C₈×C₁₃	C₂×Q₁₆	1	1	yes	characteristic
66	C₈×C₁₃	#137	1	1	yes	characteristic
67	#1055	—	8	2	no	—
68	#500	—	13	2	no	—
69	#462	—	26	2	no	—
70	C₈×Q₈	C₂×D₁₃	1	1	yes	characteristic
71	#366	—	26	1	no	—
72	#366	—	26	1	no	—
73	C₄⋊C₁₆	—	2	1	no	—
74	C₄⋊C₁₆	C₂×D₁₃	1	1	yes	characteristic
75	C₄×C₁₆	—	2	1	no	—
76	C₄×C₁₆	C₂×D₁₃	1	1	yes	characteristic
77	#335	—	13	2	no	—
78	#332	—	26	2	no	—
79	C₂²×C₁₃	#466	1	1	yes	characteristic
80	C₄×C₁₃	—	2	2	no	—
81	C₄×C₁₃	—	2	1	no	—
82	C₄×C₁₃	—	4	1	no	—
83	C₄×C₁₃	#462	1	1	yes	characteristic
84	C₄×C₁₃	#473	1	1	yes	characteristic
85	C₄×C₁₃	C₂×D₁₆	1	1	yes	characteristic
86	C₄×C₁₃	#508	1	1	yes	characteristic
87	Dic₁₃	—	16	2	no	—
88	C₂×Q₁₆	—	52	2	no	—
89	C₄⋊Q₈	—	26	2	no	—
90	C₄×Q₈	D₅₂	1	1	yes	characteristic
91	C₂×C₁₆	—	2	1	no	—
92	C₂×C₁₆	C₂²⋊D₁₃	1	1	yes	characteristic
93	C₂×C₁₆	C₂²⋊D₁₃	1	1	yes	characteristic
94	#107	—	26	2	no	—
95	#107	—	26	2	no	—
96	C₄⋊C₈	—	2	1	no	—
97	C₄⋊C₈	—	26	2	no	—
98	C₄⋊C₈	D₅₂	1	1	yes	characteristic
99	#103	—	52	2	no	—
100	C₄×C₈	—	2	1	no	—
101	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
102	C₂×C₁₃	#1674	1	1	yes	characteristic
103	C₂×C₁₃	#2215	1	1	yes	characteristic
104	C₂×C₁₃	#2222	1	1	yes	characteristic
105	C₂×Q₈	—	52	2	no	—
106	C₂×Q₈	D₁₀₄	1	1	yes	characteristic
107	Q₁₆	—	104	4	no	—
108	C₂×C₈	—	2	1	no	—
109	C₂×C₈	—	52	2	no	—
110	C₂×C₈	#6127	1	1	yes	characteristic
111	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
112	C₄⋊C₄	—	2	1	no	—
113	C₄⋊C₄	—	52	2	no	—
114	C₄⋊C₄	—	52	2	no	—
115	C₄⋊C₄	D₁₀₄	1	1	yes	characteristic
116	C₄²	—	2	1	no	—
117	C₄²	C₂×D₅₂	1	1	yes	characteristic
118	C₁₆	—	2	1	no	—
119	C₁₆	—	4	1	no	—
120	C₁₆	#6107	1	2	yes	complemented
121	C₁₃	#12457	1	1	yes	characteristic, complemented, Sylow
122	Q₈	—	2	1	no	—
123	Q₈	—	104	2	no	—
124	Q₈	—	104	2	no	—
125	Q₈	D₂₀₈	1	2	yes	—
126	C₂×C₄	—	2	1	no	—
127	C₂×C₄	—	2	1	no	—
128	C₂×C₄	—	104	2	no	—
129	C₂×C₄	C₂×D₁₀₄	1	1	yes	characteristic
130	C₂×C₄	#88158	1	1	yes	characteristic
131	C₂×C₄	#88184	1	1	yes	characteristic
132	C₈	—	2	1	no	—
133	C₈	—	4	1	no	—
134	C₈	—	104	2	no	—
135	C₈	C₂×#6107	1	1	yes	characteristic
136	C₈	#88205	1	1	yes	characteristic
137	C₂²	#11718776	1	1	yes	characteristic
138	C₄	—	2	1	no	—
139	C₄	—	2	2	no	—
140	C₄	—	4	1	no	—
141	C₄	—	208	2	no	—
142	C₄	C₂×D₂₀₈	1	1	yes	characteristic
143	C₄	#11718565	1	1	yes	characteristic
144	C₄	#11718692	1	1	yes	characteristic
145	C₄	#11718804	1	1	yes	characteristic
146	C₂	#13114230	1	1	yes	characteristic
147	C₂	#13116922	1	1	yes	characteristic
148	C₂	#13116961	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13117111

C₂

yes

maximal, characteristic

#13114230

C₂

yes

maximal

#13112908

C₂

yes

maximal, characteristic

#13112890

C₂

yes

maximal

Q₈×C₁₃×C₁₆

C₂

yes

maximal, characteristic

C₈×Q₈×C₁₃

C₂²

yes

characteristic

#11718692

—

#11718343

C₂²

yes

—

C₁₃×C₄⋊C₁₆

—

C₁₃×C₄⋊C₁₆

C₂²

yes

characteristic

C₄×C₁₃×C₁₆

—

C₄×C₁₃×C₁₆

C₂²

yes

—

—

—

C₂²

yes

—

C₄×Q₈×C₁₃

D₄

yes

characteristic

C₂×#6107

—

#88123

—

C₂×C₁₃×C₁₆

—

C₂×C₁₃×C₁₆

D₄

yes

characteristic

C₂×C₁₃×C₁₆

D₄

yes

characteristic

C₁₃×C₄⋊C₈

—

C₁₃×C₄⋊C₈

D₄

yes

characteristic

C₄×C₈×C₁₃

—

C₄×C₈×C₁₃

C₂³

yes

—

—

—

—

—

C₂×Q₈×C₁₃

D₈

yes

characteristic

C₂×Dic₂₆

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

C₂×D₄

yes

characteristic

C₂×C₈×C₁₃

#41

yes

characteristic

C₁₃×C₄⋊C₄

—

C₁₃×C₄⋊C₄

D₈

yes

characteristic

C₄²×C₁₃

—

C₄²×C₁₃

C₂×D₄

yes

characteristic

#6107

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

C₂×#1055

—

C₁₃×C₁₆

—

C₁₃×C₁₆

—

C₁₃×C₁₆

Q₁₆

yes

complemented

#2252

—

Q₈×C₁₆

D₁₃

yes

—

—

—

Q₈×C₁₃

—

Q₈×C₁₃

D₁₆

yes

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

#121

yes

characteristic

C₂×C₄×C₁₃

C₂×D₈

yes

characteristic

C₂×C₄×C₁₃

#135

yes

characteristic

C₂×Dic₁₃

—

Dic₂₆

—

Dic₂₆

—

C₈×C₁₃

—

C₈×C₁₃

—

C₈×C₁₃

C₂×Q₁₆

yes

C₈×C₁₃

yes

—

—

—

C₈×Q₈

C₂×D₁₃

yes

characteristic

#366

—

#366

—

C₄⋊C₁₆

—

C₄⋊C₁₆

C₂×D₁₃

yes

characteristic

C₄×C₁₆

—

C₄×C₁₆

C₂×D₁₃

yes

characteristic

#335

—

#332

—

C₂²×C₁₃

#466

yes

characteristic

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

yes

C₄×C₁₃

yes

C₄×C₁₃

C₂×D₁₆

yes

characteristic

C₄×C₁₃

#508

yes

characteristic

Dic₁₃

—

C₂×Q₁₆

—

C₄⋊Q₈

—

C₄×Q₈

D₅₂

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

C₂²⋊D₁₃

yes

characteristic

C₂×C₁₆

C₂²⋊D₁₃

yes

characteristic

#107

—

#107

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

D₅₂

yes

characteristic

#103

—

100

C₄×C₈

—

101

C₄×C₈

C₂²×D₁₃

yes

characteristic

102

C₂×C₁₃

#1674

yes

characteristic

103

C₂×C₁₃

#2215

yes

characteristic

104

C₂×C₁₃

#2222

yes

characteristic

105

C₂×Q₈

—

106

C₂×Q₈

D₁₀₄

yes

characteristic

107

Q₁₆

—

104

—

108

C₂×C₈

—

109

C₂×C₈

—

110

C₂×C₈

#6127

yes

characteristic

111

C₂×C₈

C₂×C₂²⋊D₁₃

yes

characteristic

112

C₄⋊C₄

—

113

C₄⋊C₄

—

114

C₄⋊C₄

—

115

C₄⋊C₄

D₁₀₄

yes

characteristic

116

C₄²

—

117

C₄²

C₂×D₅₂

yes

characteristic

118

C₁₆

—

119

C₁₆

—

120

C₁₆

#6107

yes

complemented

121

C₁₃

#12457

yes

characteristic, complemented, Sylow

122

Q₈

—

123

Q₈

—

104

—

124

Q₈

—

104

—

125

Q₈

D₂₀₈

yes

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

104

—

129

C₂×C₄

C₂×D₁₀₄

yes

characteristic

130

C₂×C₄

#88158

yes

characteristic

131

C₂×C₄

#88184

yes

characteristic

132

C₈

—

133

C₈

—

134

C₈

—

104

—

135

C₈

C₂×#6107

yes

characteristic

136

C₈

#88205

yes

characteristic

137

C₂²

#11718776

yes

characteristic

138

C₄

—

139

C₄

—

140

C₄

—

141

C₄

—

208

—

142

C₄

C₂×D₂₀₈

yes

characteristic

143

C₄

#11718565

yes

characteristic

144

C₄

#11718692

yes

characteristic

145

C₄

#11718804

yes

characteristic

146

C₂

#13114230

yes

characteristic

147

C₂

#13116922

yes

characteristic

148

C₂

#13116961

yes

characteristic

	subgroup	G/H
center	C₂²	#11718776
commutator, cosocle	C₄×C₈×C₁₃	C₂³
Frattini	C₄×C₈	C₂²×D₁₃
Fitting	Q₈×C₁₃×C₁₆	C₂
socle, prime radical	C₂²×C₁₃	#466
hypercenter	Q₈×C₁₆	D₁₃
hypocenter	C₁₃	#12457
LCS	G ▷ C₄×C₈×C₁₃ ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈×Q₈ ◁ Q₈×C₁₆

subgroup

G/H

center

C₂²

#11718776

commutator, cosocle

C₄×C₈×C₁₃

C₂³

Frattini

C₄×C₈

C₂²×D₁₃

Fitting

Q₈×C₁₃×C₁₆

C₂

socle, prime radical

C₂²×C₁₃

#466

hypercenter

Q₈×C₁₆

D₁₃

hypocenter

C₁₃

#12457

LCS

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈×Q₈ ◁ Q₈×C₁₆

N	H ≅ G/N	type
47. C₁₃×C₁₆	Q₁₆	semidirect
120. C₁₆	#6107	semidirect
121. C₁₃	#12457	semidirect

H ≅ G/N

type

47. C₁₃×C₁₆

Q₁₆

semidirect

120. C₁₆

#6107

semidirect

121. C₁₃

#12457

semidirect

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

subgroup	index
#13117111	2
#13114230	2
#13112908	2
#13112890	2
Q₈×C₁₃×C₁₆	2
#12457	13

subgroup

index

Q₈×C₁₃×C₁₆

Subgroups — #15777278

Conjugacy classes of subgroups

Complements

Maximal subgroups