Group #16777749 — subgroups

Subgroups — #16777749

Order 3328.

202 conjugacy classes of subgroups; only the 36 normal subgroups are shown.

normal subgroups only

This group has 202 conjugacy classes of subgroups — too many to draw. Only its 36 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	#13129782	C₂	1	2	yes	maximal, complemented
2	#13115668	C₂	1	1	yes	maximal, characteristic
3	#11719182	C₄	1	1	yes	characteristic
4	C₁₃×#376	C₄	1	1	yes	characteristic
5	C₂×#88045	—	2	2	no	—
6	#11718585	C₂²	1	1	yes	characteristic
7	#11718284	—	2	2	no	—
8	C₂×#6127	—	2	1	no	—
9	C₂×C₄²×C₁₃	—	2	1	no	—
10	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
11	#88184	—	4	1	no	—
12	C₂×#6099	D₄	1	2	yes	—
13	#88129	—	2	1	no	—
14	#88128	C₈	1	1	yes	characteristic
15	C₄×D₅₂	—	4	1	no	—
16	#88123	C₈	1	1	yes	characteristic
17	#88106	—	4	2	no	—
18	C₁₃×#95	—	4	1	no	—
19	#88045	—	4	4	no	—
20	C₄⋊#1055	—	4	1	no	—
21	#7356	—	13	1	no	maximal, Sylow
22	C₂²×C₄×C₁₃	—	2	1	no	—
23	C₂²×C₄×C₁₃	—	4	1	no	—
24	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
25	C₂×C₂²⋊D₁₃	—	4	1	no	—
26	#6127	—	4	2	no	—
27	#6127	—	4	2	no	—
28	C₂×D₅₂	—	2	1	no	—
29	C₂×D₅₂	—	4	1	no	—
30	C₂×C₄×D₁₃	—	4	1	no	—
31	C₂×Dic₂₆	—	2	1	no	—
32	C₄²×C₁₃	—	2	1	no	—
33	C₄²×C₁₃	—	2	2	no	—
34	C₄²×C₁₃	—	4	1	no	—
35	C₄²×C₁₃	C₂×C₈	1	1	yes	characteristic
36	C₄²×C₁₃	M₁₆	1	1	yes	characteristic
37	#6103	—	4	2	no	—
38	C₄⋊Dic₁₃	—	4	1	no	—
39	#6099	—	4	2	no	—
40	C₂×#1055	—	2	2	no	—
41	#6092	—	8	2	no	—
42	#1963	—	13	1	no	—
43	#1327	—	13	2	no	—
44	C₂³×C₁₃	#100	1	1	yes	characteristic
45	C₂²×D₁₃	—	4	1	no	—
46	C₂×C₄×C₁₃	—	2	1	no	—
47	C₂×C₄×C₁₃	—	2	1	no	—
48	C₂×C₄×C₁₃	—	2	1	no	—
49	C₂×C₄×C₁₃	—	2	1	no	—
50	C₂×C₄×C₁₃	—	4	2	no	—
51	C₂×C₄×C₁₃	—	8	1	no	—
52	C₂×C₄×C₁₃	—	8	2	no	—
53	C₂×C₄×C₁₃	C₂²⋊C₈	1	1	yes	characteristic
54	C₂×C₄×C₁₃	#99	1	1	yes	characteristic
55	C₂²⋊D₁₃	—	8	2	no	—
56	C₂×Dic₁₃	—	4	1	no	—
57	D₅₂	—	4	2	no	—
58	D₅₂	—	8	2	no	—
59	C₄×D₁₃	—	8	1	no	—
60	C₄×D₁₃	—	8	1	no	—
61	Dic₂₆	—	4	2	no	—
62	#1055	—	4	2	no	—
63	#532	—	13	1	no	—
64	#423	—	13	1	no	—
65	C₂×#104	—	26	2	no	—
66	#376	#261	1	1	yes	characteristic
67	#349	—	26	2	no	—
68	C₂²×C₁₃	—	2	1	no	—
69	C₂²×C₁₃	—	4	1	no	—
70	C₂²×C₁₃	#323	1	1	yes	characteristic
71	C₂×D₁₃	—	8	2	no	—
72	C₂×D₁₃	—	8	1	no	—
73	C₄×C₁₃	—	2	1	no	—
74	C₄×C₁₃	—	4	1	no	—
75	C₄×C₁₃	—	4	1	no	—
76	C₄×C₁₃	—	8	2	no	—
77	C₄×C₁₃	#353	1	1	yes	characteristic
78	C₄×C₁₃	#355	1	1	yes	characteristic
79	Dic₁₃	—	8	1	no	—
80	C₂×#41	—	26	1	no	—
81	C₂×M₁₆	—	13	2	no	—
82	C₄⋊Q₈	—	13	1	no	—
83	#127	—	13	1	no	—
84	#124	—	26	1	no	—
85	#121	—	52	1	no	—
86	C₄×D₄	—	52	1	no	—
87	C₂×C₄²	—	2	1	no	—
88	C₂×C₄²	C₂×#261	1	1	yes	characteristic
89	M₃₂	—	52	2	no	—
90	C₄⋊C₈	—	52	1	no	—
91	#104	—	52	4	no	—
92	#95	—	4	1	no	—
93	C₂×C₁₃	—	2	1	no	—
94	C₂×C₁₃	—	4	1	no	—
95	C₂×C₁₃	#1318	1	1	yes	characteristic
96	D₁₃	—	16	1	no	—
97	#41	—	52	2	no	—
98	#41	—	52	2	no	—
99	C₂×Q₈	—	26	1	no	—
100	C₂×D₄	—	26	1	no	—
101	C₂×D₄	—	52	1	no	—
102	C₂×D₄	—	52	1	no	—
103	C₂²×C₄	—	2	1	no	—
104	C₂²×C₄	—	4	1	no	—
105	C₂²×C₄	—	52	1	no	—
106	C₂²×C₄	C₂²⋊#261	1	1	yes	characteristic
107	M₁₆	—	52	2	no	—
108	C₂×C₈	—	26	2	no	—
109	C₄⋊C₄	—	52	1	no	—
110	C₂²⋊C₄	—	52	2	no	—
111	C₄²	—	2	1	no	—
112	C₄²	—	2	2	no	—
113	C₄²	—	4	1	no	—
114	C₄²	D₁₃⋊C₈	1	1	yes	characteristic
115	C₄²	#6118	1	1	yes	characteristic
116	C₁₆	—	104	2	no	—
117	C₁₃	#7356	1	1	yes	characteristic, complemented, Sylow
118	C₂³	—	52	1	no	—
119	C₂³	#88121	1	1	yes	characteristic
120	Q₈	—	52	2	no	—
121	D₄	—	52	2	no	—
122	D₄	—	104	2	no	—
123	D₄	—	104	2	no	—
124	C₂×C₄	—	2	1	no	—
125	C₂×C₄	—	2	1	no	—
126	C₂×C₄	—	2	1	no	—
127	C₂×C₄	—	2	1	no	—
128	C₂×C₄	—	4	2	no	—
129	C₂×C₄	—	8	1	no	—
130	C₂×C₄	—	8	2	no	—
131	C₂×C₄	—	52	1	no	—
132	C₂×C₄	—	104	1	no	—
133	C₂×C₄	—	104	1	no	—
134	C₂×C₄	#88107	1	1	yes	characteristic
135	C₂×C₄	#88111	1	1	yes	characteristic
136	C₈	—	52	2	no	—
137	C₂²	—	2	1	no	—
138	C₂²	—	4	1	no	—
139	C₂²	—	104	2	no	—
140	C₂²	—	104	1	no	—
141	C₂²	#11718287	1	1	yes	characteristic
142	C₄	—	2	1	no	—
143	C₄	—	4	1	no	—
144	C₄	—	4	1	no	—
145	C₄	—	8	2	no	—
146	C₄	—	104	1	no	—
147	C₄	#11718225	1	1	yes	characteristic
148	C₄	#11718228	1	1	yes	characteristic
149	C₂	—	2	1	no	—
150	C₂	—	4	1	no	—
151	C₂	—	208	1	no	—
152	C₂	#13129765	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13129782

C₂

yes

maximal, complemented

#13115668

C₂

yes

maximal, characteristic

#11719182

C₄

yes

characteristic

C₁₃×#376

C₄

yes

characteristic

C₂×#88045

—

#11718585

C₂²

yes

characteristic

#11718284

—

C₂×#6127

—

C₂×C₄²×C₁₃

—

C₂×C₄²×C₁₃

C₂×C₄

yes

characteristic

#88184

—

C₂×#6099

D₄

yes

—

#88129

—

#88128

C₈

yes

characteristic

C₄×D₅₂

—

#88123

C₈

yes

characteristic

#88106

—

C₁₃×#95

—

—

—

—

C₂²×C₄×C₁₃

—

C₂²×C₄×C₁₃

—

C₂²×C₄×C₁₃

C₂²⋊C₄

yes

characteristic

C₂×C₂²⋊D₁₃

—

#6127

—

#6127

—

C₂×D₅₂

—

C₂×D₅₂

—

C₂×C₄×D₁₃

—

C₂×Dic₂₆

—

C₄²×C₁₃

—

C₄²×C₁₃

—

C₄²×C₁₃

—

C₄²×C₁₃

C₂×C₈

yes

characteristic

C₄²×C₁₃

M₁₆

yes

characteristic

#6103

—

C₄⋊Dic₁₃

—

—

—

—

—

—

C₂³×C₁₃

#100

yes

characteristic

C₂²×D₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

C₂²⋊C₈

yes

characteristic

C₂×C₄×C₁₃

#99

yes

characteristic

C₂²⋊D₁₃

—

C₂×Dic₁₃

—

D₅₂

—

D₅₂

—

C₄×D₁₃

—

C₄×D₁₃

—

Dic₂₆

—

—

—

—

—

yes

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

#323

yes

characteristic

C₂×D₁₃

—

C₂×D₁₃

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

yes

C₄×C₁₃

yes

Dic₁₃

—

C₂×#41

—

C₂×M₁₆

—

C₄⋊Q₈

—

#127

—

#124

—

#121

—

C₄×D₄

—

C₂×C₄²

—

C₂×C₄²

C₂×#261

yes

characteristic

M₃₂

—

C₄⋊C₈

—

#104

—

#95

—

C₂×C₁₃

—

C₂×C₁₃

—

C₂×C₁₃

#1318

yes

characteristic

D₁₃

—

#41

—

#41

—

C₂×Q₈

—

100

C₂×D₄

—

101

C₂×D₄

—

102

C₂×D₄

—

103

C₂²×C₄

—

104

C₂²×C₄

—

105

C₂²×C₄

—

106

C₂²×C₄

C₂²⋊#261

yes

characteristic

107

M₁₆

—

108

C₂×C₈

—

109

C₄⋊C₄

—

110

C₂²⋊C₄

—

111

C₄²

—

112

C₄²

—

113

C₄²

—

114

C₄²

D₁₃⋊C₈

yes

characteristic

115

C₄²

#6118

yes

characteristic

116

C₁₆

—

104

—

117

C₁₃

#7356

yes

characteristic, complemented, Sylow

118

C₂³

—

119

C₂³

#88121

yes

characteristic

120

Q₈

—

121

D₄

—

122

D₄

—

104

—

123

D₄

—

104

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

104

—

133

C₂×C₄

—

104

—

134

C₂×C₄

#88107

yes

characteristic

135

C₂×C₄

#88111

yes

characteristic

136

C₈

—

137

C₂²

—

138

C₂²

—

139

C₂²

—

104

—

140

C₂²

—

104

—

141

C₂²

#11718287

yes

characteristic

142

C₄

—

143

C₄

—

144

C₄

—

145

C₄

—

146

C₄

—

104

—

147

C₄

#11718225

yes

characteristic

148

C₄

#11718228

yes

characteristic

149

C₂

—

150

C₂

—

151

C₂

—

208

—

152

C₂

#13129765

yes

characteristic

	subgroup	G/H
center	C₄	#11718225
commutator	C₄²×C₁₃	C₂×C₈
Frattini	C₂×C₄²	C₂×#261
Fitting	C₁₃×#376	C₄
socle	C₂×C₁₃	#1318
prime radical	#11719182	C₄
cosocle	#11718585	C₂²
hypercenter	#376	#261
hypocenter	C₁₃	#7356
LCS	G ▷ C₄²×C₁₃ ▷ C₂×C₄×C₁₃ ▷ C₂²×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ #376

subgroup

G/H

center

C₄

#11718225

commutator

C₄²×C₁₃

C₂×C₈

Frattini

C₂×C₄²

C₂×#261

Fitting

C₁₃×#376

C₄

C₂×C₁₃

C₄

C₂²

C₁₃

G ▷ C₄²×C₁₃ ▷ C₂×C₄×C₁₃ ▷ C₂²×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ #376

N	H ≅ G/N	type
1. #13129782	C₂	semidirect
117. C₁₃	#7356	semidirect

H ≅ G/N

type

1. #13129782

C₂

semidirect

117. C₁₃

#7356

semidirect

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

subgroup	index
#13129782	2
#13115668	2
#7356	13

subgroup

index

#13129782

#13115668

#7356

Subgroups — #16777749

Conjugacy classes of subgroups

Complements

Maximal subgroups