Finite Group Databasebeta

← #20276153

Subgroups — #20276153

Order 6784.

256 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 79 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#13133148C211yesmaximal, characteristic, complemented
2C2×#13133129C211yesmaximal, characteristic
3#16810480C211yesmaximal, characteristic
4#16810480C211yesmaximal, characteristic
5#16810386C211yesmaximal, characteristic, complemented
6#16810385C211yesmaximal, characteristic, complemented
7C53×#340C211yesmaximal, characteristic, complemented
8C22×C4×D53C2211yescharacteristic
9C22×#88322C2211yescharacteristic
10C2×C4⋊Dic53C2211yescharacteristic
11C2×C4⋊Dic53C2211yescharacteristic
12C2×#11719921—21no—
13C2×#11719921—21no—
14#13133148—22no—
15#13133129—22no—
16#13133124—21no—
17#13133124—21no—
18C22×C8×C53C2211yescharacteristic
19C2×C4⋊C4×C53C2211yescharacteristic
20C2×C4⋊C4×C53C2211yescharacteristic
21C23×D53D411yescharacteristic
22C22×C4×C53—21no—
23C22×C4×C53—21no—
24C22×C4×C53C2311yescharacteristic
25C22×Dic53—21no—
26C22×Dic53—21no—
27C22×Dic53D411yescharacteristic
28C2×C4×D53—21no—
29C2×C4×D53—21no—
30C2×C4×D53—21no—
31C2×C4×D53—21no—
32C2×C4×D53—21no—
33C2×C4×D53—21no—
34C2×C8×C53—21no—
35C2×C8×C53—21no—
36C2×C8×C53—21no—
37C4⋊C4×C53—21no—
38C4⋊C4×C53—21no—
39C4⋊C4×C53—21no—
40C4⋊C4×C53—21no—
41#11719921—42no—
42#11719921—42no—
43C4⋊Dic53—21no—
44C4⋊Dic53—21no—
45C4⋊Dic53—21no—
46C4⋊Dic53—21no—
47C2×#88322—21no—
48C2×#88322—21no—
49C2×#88322D412yes—
50C23×C53C2×D411yescharacteristic
51C22×D53—21no—
52C22×D53—21no—
53C22×D53—21no—
54C22×D53—21no—
55C22×D53—21no—
56C22×D53—21no—
57C22×D53SD1612yes—
58C2×C4×C53—41no—
59C2×C4×C53—41no—
60C2×C4×C53—41no—
61C2×C4×C53—41no—
62C2×C4×C53—41no—
63C2×C4×C53—41no—
64C2×C4×C53C2×D411yescharacteristic
65C2×C4×C53#4111yescharacteristic
66C2×C4×C53#4111yescharacteristic
67C2×C4×C53#4111yescharacteristic
68C2×C4×C53#4111yescharacteristic
69C2×C4×C53#4111yescharacteristic
70C2×Dic53—21no—
71C2×Dic53—21no—
72C2×Dic53—21no—
73C2×Dic53—41no—
74C2×Dic53—41no—
75C2×Dic53—41no—
76C2×Dic53—41no—
77C2×Dic53—41no—
78C2×Dic53—41no—
79C4×D53—41no—
80C4×D53—41no—
81C4×D53—41no—
82C4×D53—41no—
83C8×C53—41no—
84#88322—22no—
85C22×C53#12111yescharacteristic
86C22×C53#12311yescharacteristic
87C22×C53#12311yescharacteristic
88C22×C53C2×SD1611yescharacteristic
89C22×C53#13511yescharacteristic
90C22×C53AGL(1,ℤ/8)11yescharacteristic
91C22×C53#13711yescharacteristic
92C2×D53—22no—
93C2×D53—22no—
94C2×D53—22no—
95C2×D53—41no—
96C2×D53—41no—
97C2×D53—41no—
98C2×D53—41no—
99C4×C53—81no—
100C4×C53—81no—
101C4×C53#12311yescharacteristic
102C4×C53#12411yescharacteristic
103C4×C53#12611yescharacteristic
104C4×C53#12611yescharacteristic
105Dic53—41no—
106Dic53—81no—
107Dic53—81no—
108#2077—531nomaximal, Sylow
109C2×C53#39711yescharacteristic
110C2×C53#46511yescharacteristic
111C2×C53#46811yescharacteristic
112C2×C53#48111yescharacteristic
113C2×C53#48211yescharacteristic
114C2×C53#48311yescharacteristic
115C2×C53#48511yescharacteristic
116D53—42no—
117C2×#106—531no—
118C2×C22⋊C8—531no—
119#380—531no—
120#380—531no—
121#340—531no—
122#340—531no—
123#340D5311yescharacteristic, complemented
124C53#207711yescharacteristic, complemented, Sylow
125C23×C4—531no—
126C22×C8—531no—
127C22×C8C2×D5311yescharacteristic
128C2×C4⋊C4—531no—
129C2×C4⋊C4—531no—
130C2×C4⋊C4C2×D5311yescharacteristic
131C2×C4⋊C4C2×D5311yescharacteristic
132C2×C22⋊C4—1061no—
133C2×C22⋊C4—1061no—
134#106—1062no—
135C22⋊C8—1062no—
136#95—1061no—
137#95—1061no—
138C24—531no—
139C22×C4—21no—
140C22×C4—21no—
141C22×C4—531no—
142C22×C4—1061no—
143C22×C4—1061no—
144C22×C4—1061no—
145C22×C4—1061no—
146C22×C4—1061no—
147C22×C4—1061no—
148C22×C4—1061no—
149C22×C4—1061no—
150C22×C4C22×D5311yescharacteristic
151C2×C8—21no—
152C2×C8—21no—
153C2×C8—21no—
154C2×C8—532no—
155C2×C8—1061no—
156C2×C8—1061no—
157C4⋊C4—21no—
158C4⋊C4—21no—
159C4⋊C4—21no—
160C4⋊C4—21no—
161C4⋊C4—1061no—
162C4⋊C4—1061no—
163C4⋊C4—1061no—
164C4⋊C4—1061no—
165C22⋊C4—2122no—
166C22⋊C4—2122no—
167C23—532no—
168C23—1061no—
169C23—1061no—
170C23—1061no—
171C23—1061no—
172C23—1061no—
173C23—1061no—
174C23D4×D5311yescharacteristic
175C2×C4—41no—
176C2×C4—41no—
177C2×C4—41no—
178C2×C4—41no—
179C2×C4—41no—
180C2×C4—41no—
181C2×C4—1061no—
182C2×C4—1061no—
183C2×C4—1061no—
184C2×C4—2121no—
185C2×C4—2121no—
186C2×C4—2121no—
187C2×C4—2121no—
188C2×C4—2121no—
189C2×C4—2121no—
190C2×C4—2121no—
191C2×C4—2121no—
192C2×C4—2121no—
193C2×C4—2121no—
194C2×C4#1171994511yescharacteristic
195C2×C4#1171994511yescharacteristic
196C2×C4#1171994511yescharacteristic
197C2×C4D4×D5311yescharacteristic
198C2×C4#1171994711yescharacteristic
199C2×C4#1171994911yescharacteristic
200C8—41no—
201C8—1062no—
202C22—1062no—
203C22—1062no—
204C22—1062no—
205C22—2121no—
206C22—2121no—
207C22—2121no—
208C22—2121no—
209C22#1313319011yescharacteristic
210C22#1313319911yescharacteristic
211C22#1313320211yescharacteristic
212C22SD16×D5311yescharacteristic
213C22#1313323611yescharacteristic
214C22#1313324411yescharacteristic
215C22#1313324911yescharacteristic
216C4—81no—
217C4—81no—
218C4—4241no—
219C4—4241no—
220C4—2121no—
221C4#1313320211yescharacteristic
222C4#1313320311yescharacteristic
223C4#1313321111yescharacteristic
224C4#1313321211yescharacteristic
225C2—2122no—
226C2#1681055211yescharacteristic
227C2#1681085111yescharacteristic
228C2#1681087211yescharacteristic
229C2#1681094311yescharacteristic
230C2#1681094511yescharacteristic
231C2#1681095211yescharacteristic
232C2#1681096611yescharacteristic
subgroupG/H
centerC23D4×D53
commutatorC22×C4×C53C23
FrattiniC22×C4C22×D53
FittingC53×#340C2
socleC23×C53C2×D4
hypercenter#340D53
hypocenterC53#2077
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C22×C4 ◁ #340

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C2×#13133148C2semidirect
5. #16810386C2semidirect
6. #16810385C2semidirect
7. C53×#340C2semidirect
123. #340D53semidirect
124. C53#2077semidirect