Finite Group Databasebeta

← #20273285

Subgroups — #20273285

Order 6784.

288 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 288 conjugacy classes of subgroups — too many to draw. Only its 71 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
1C22⋊Q8×D53C211yesmaximal, characteristic
2C22⋊C8×D53C211yesmaximal, characteristic
3#16810482C211yesmaximal, characteristic
4#16810361C211yesmaximal, characteristic, complemented
5#16810360C211yesmaximal, characteristic, complemented
6#16810359C211yesmaximal, characteristic, complemented
7C53×#328C211yesmaximal, characteristic, complemented
8C2×Q8×D53C411yescharacteristic, complemented
9C22×C4×D53C2211yescharacteristic
10C2×C8×D53—21no—
11#13133197C411yescharacteristic, complemented
12#13133196C2211yescharacteristic
13#13133195—21no—
14#13133194—21no—
15#13133193C411yescharacteristic, complemented
16C2×#11719919—21no—
17C4⋊C4×D53—21no—
18C4⋊C4×D53C411yescharacteristic, complemented
19C2×#11719921—21no—
20C22⋊C4×D53—21no—
21C22⋊#88322C2211yescharacteristic
22#13133129—21no—
23#13133124C2211yescharacteristic
24C22⋊Q8×C53C2211yescharacteristic
25C22⋊C8×C53C2211yescharacteristic
26#95×C53C2211yescharacteristic
27C23×D53D411yescharacteristic
28C2×Q8×C53C2×C411yescharacteristic
29C22×C4×C53—21no—
30C22×C4×C53C2311yescharacteristic
31C22×Dic53—21no—
32C22×Dic53D411yescharacteristic
33Q8×D53—42no—
34C2×C4×D53—21no—
35C2×C4×D53—21no—
36C2×C4×D53—42no—
37C2×C4×D53C2×C411yescharacteristic
38C2×C4×D53C2×C411yescharacteristic
39C2×C4×D53D411yescharacteristic
40C2×C4×D53D411yescharacteristic
41C2×Dic106—21no—
42C2×Dic106C2×C411yescharacteristic
43C2×C8×C53—21no—
44C4⋊C4×C53—21no—
45C4⋊C4×C53C2×C411yescharacteristic
46C22⋊C4×C53—21no—
47C22⋊Dic53—21no—
48#11719921—21no—
49#11719921—21no—
50#11719921—42no—
51C4⋊Dic53—21no—
52C4⋊Dic53C2×C411yescharacteristic
53#11719919—21no—
54#11719919—21no—
55#11719919—21no—
56#11719919—42no—
57C2×#88322—21no—
58C8×D53—42no—
59C23×C53C2×D411yescharacteristic
60C22×D53—41no—
61C22×D53—41no—
62C22×D53C22⋊C411yescharacteristic
63C22×D53C22⋊C411yescharacteristic
64C22×D53SD1612yes—
65C22×D53Q1612yes—
66Q8×C53—41no—
67C2×C4×C53—21no—
68C2×C4×C53—21no—
69C2×C4×C53—41no—
70C2×C4×C53—41no—
71C2×C4×C53—41no—
72C2×C4×C53—41no—
73C2×C4×C53C22×C411yescharacteristic
74C2×C4×C53C2×D411yescharacteristic
75C2×Dic53—22no—
76C2×Dic53—21no—
77C2×Dic53—21no—
78C2×Dic53—41no—
79C2×Dic53—41no—
80C2×Dic53—41no—
81C2×Dic53C22⋊C411yescharacteristic
82C2×Dic53C22⋊C411yescharacteristic
83C4×D53—22no—
84C4×D53—22no—
85C4×D53—41no—
86C4×D53—41no—
87C4×D53—42no—
88C4×D53—42no—
89Dic106—42no—
90Dic106—41no—
91C8×C53—41no—
92#88322—41no—
93C22×C53—21no—
94C22×C53—21no—
95C22×C53C2×C22⋊C411yescharacteristic
96C22×C53C2×SD1611yescharacteristic
97C22×C53C2×Q1611yescharacteristic
98C2×D53—21no—
99C2×D53—21no—
100C2×D53—21no—
101C2×D53—21no—
102C2×D53—42no—
103C2×D53—42no—
104C2×D53#10312yes—
105C2×D53#10312yes—
106C4×C53—21no—
107C4×C53—21no—
108C4×C53—41no—
109C4×C53—41no—
110C4×C53—81no—
111Dic53—21no—
112Dic53—41no—
113Dic53—41no—
114Dic53—81no—
115Dic53#9912yes—
116#1520—531nomaximal, Sylow
117C2×C53—21no—
118C2×C53—21no—
119C2×C53C2×#9911yescharacteristic
120C2×C53C2×#10311yescharacteristic
121C2×C53#42111yescharacteristic
122D53—22no—
123D53—22no—
124C2×C22⋊Q8—531no—
125C2×C22⋊C8—531no—
126#380—531no—
127#328—531no—
128#328—531no—
129#328—531no—
130#328D5311yescharacteristic, complemented
131C53#152011yescharacteristic, complemented, Sylow
132C22×Q8—531no—
133C23×C4—531no—
134C22×C8—1061no—
135C22⋊Q8—531no—
136C22⋊Q8—531no—
137C22⋊Q8—531no—
138C22⋊Q8—1061no—
139C22⋊Q8—1061no—
140C22⋊Q8C2×D5311yescharacteristic
141C2×C4⋊C4—531no—
142C2×C4⋊C4—1061no—
143C2×C4⋊C4—1061no—
144C2×C22⋊C4—1061no—
145C2×C22⋊C4—1061no—
146C22⋊C8—531no—
147C22⋊C8—1061no—
148C22⋊C8C2×D5311yescharacteristic
149#95—531no—
150#95C2×D5311yescharacteristic
151C24—531no—
152C2×Q8—531no—
153C2×Q8—1061no—
154C2×Q8—2122no—
155C2×Q8C4×D5311yescharacteristic
156C22×C4—21no—
157C22×C4—531no—
158C22×C4—531no—
159C22×C4—531no—
160C22×C4—531no—
161C22×C4—531no—
162C22×C4—1061no—
163C22×C4—1061no—
164C22×C4—1061no—
165C22×C4—2122no—
166C22×C4C22×D5311yescharacteristic
167C2×C8—21no—
168C2×C8—1061no—
169C2×C8—2122no—
170C4⋊C4—21no—
171C4⋊C4—531no—
172C4⋊C4—1061no—
173C4⋊C4—1061no—
174C4⋊C4—1061no—
175C4⋊C4—1061no—
176C4⋊C4—2122no—
177C4⋊C4C4×D5311yescharacteristic
178C22⋊C4—21no—
179C22⋊C4—1061no—
180C22⋊C4—1061no—
181C22⋊C4—1061no—
182C22⋊C4—2122no—
183C23—531no—
184C23—531no—
185C23—532no—
186C23—532no—
187C23—2121no—
188C23—2121no—
189C23D4×D5311yescharacteristic
190Q8—41no—
191Q8—2121no—
192Q8—2122no—
193C2×C4—21no—
194C2×C4—21no—
195C2×C4—41no—
196C2×C4—41no—
197C2×C4—41no—
198C2×C4—41no—
199C2×C4—531no—
200C2×C4—531no—
201C2×C4—1062no—
202C2×C4—1062no—
203C2×C4—1062no—
204C2×C4—1061no—
205C2×C4—1061no—
206C2×C4—2121no—
207C2×C4—2121no—
208C2×C4—2122no—
209C2×C4—2122no—
210C2×C4—2121no—
211C2×C4—2121no—
212C2×C4—2121no—
213C2×C4C2×C4×D5311yescharacteristic
214C2×C4D4×D5311yescharacteristic
215C8—41no—
216C8—2121no—
217C22—21no—
218C22—21no—
219C22—532no—
220C22—532no—
221C22—1061no—
222C22—1061no—
223C22—1061no—
224C22—1061no—
225C22—2122no—
226C22—2122no—
227C22C22⋊C4×D5311yescharacteristic
228C22SD16×D5311yescharacteristic
229C22Q16×D5311yescharacteristic
230C4—21no—
231C4—21no—
232C4—41no—
233C4—41no—
234C4—81no—
235C4—532no—
236C4—1061no—
237C4—4241no—
238C4—2121no—
239C4—2121no—
240C2—21no—
241C2—21no—
242C2—1062no—
243C2—1062no—
244C2#99×D5311yescharacteristic
245C2#103×D5311yescharacteristic
246C2#1681063811yescharacteristic
subgroupG/H
centerC22C22⋊C4×D53
commutatorC2×C4×C53C22×C4
FrattiniC22×C4C22×D53
FittingC53×#328C2
socleC22×C53C2×C22⋊C4
hypercenter#328D53
hypocenterC53#1520
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ #328

Complements

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

NH ≅ G/Ntype
4. #16810361C2semidirect
5. #16810360C2semidirect
6. #16810359C2semidirect
7. C53×#328C2semidirect
8. C2×Q8×D53C4semidirect
11. #13133197C4semidirect
15. #13133193C4semidirect
18. C4⋊C4×D53C4semidirect
130. #328D53semidirect
131. C53#1520semidirect