Finite Group Databasebeta

← #20290150

Subgroups — #20290150

Order 6784.

340 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 340 conjugacy classes of subgroups — too many to draw. Only its 68 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×#13133274C211yesmaximal, characteristic, complemented
2#16811565C211yesmaximal, characteristic, complemented
3#16811475C212yesmaximal, complemented
4#16811474C212yesmaximal, complemented
5#121×D53C211yesmaximal, characteristic
6C22×#88324—21no—
7C2×C4⋊#6157—21no—
8C2×C22⋊#6157—21no—
9#13133274—21no—
10#13133274C2212yescomplemented
11(C2×#6157)⋊C4C2212yescomplemented
12C2×D4×D53—21no—
13C2×D4×D53C411yescharacteristic, complemented
14C22×C4×D53C2211yescharacteristic
15#13133192C2211yescharacteristic
16#13133191C411yescharacteristic, complemented
17#13133190—21no—
18#13133189—21no—
19#13133188C2211yescharacteristic
20C4⋊C4×D53C411yescharacteristic, complemented
21C22⋊C4×D53—21no—
22#121×C53C411yescharacteristic, complemented
23C23×D53—21no—
24C23×D53D411yescharacteristic
25C22×#6157—22no—
26C2×D4×C53—21no—
27C2×D4×C53C2×C411yescharacteristic
28C22×C4×C53C2×C411yescharacteristic
29C2×C22⋊D53—21no—
30C2×C22⋊D53—21no—
31C2×C22⋊D53—21no—
32C22×Dic53D411yescharacteristic
33D4×D53—42no—
34D4×D53—42no—
35D4×D53—42no—
36C2×D212—21no—
37C2×D212C2×C411yescharacteristic
38C2×C4×D53—21no—
39C2×C4×D53—42no—
40C2×C4×D53C2×C411yescharacteristic
41C2×C4×D53D411yescharacteristic
42C2×C4×D53D411yescharacteristic
43C2×C4×D53C2311yescharacteristic
44C22⋊#6157—42no—
45C2×#88324—22no—
46C2×#88324—42no—
47C4⋊#6157—42no—
48C4⋊C4×C53C2×C411yescharacteristic
49C22⋊C4×C53—21no—
50C22⋊Dic53—21no—
51#11719921—21no—
52#11719921—21no—
53C4⋊Dic53C2×C411yescharacteristic
54#11719919—21no—
55C23×C53—21no—
56C23×C53C22⋊C411yescharacteristic
57C22×D53—21no—
58C22×D53—41no—
59C22×D53—41no—
60C22×D53—42no—
61C22×D53—42no—
62C22×D53—42no—
63C22×D53C22⋊C411yescharacteristic
64C22×D53D812yes—
65C22×D53SD1612yes—
66C22×D53C2×D411yescharacteristic
67C2×#6157—42no—
68C2×#6157—42no—
69C2×#6157—42no—
70D4×C53—41no—
71D4×C53—42no—
72C2×C4×C53—21no—
73C2×C4×C53—41no—
74C2×C4×C53C22⋊C411yescharacteristic
75C2×C4×C53C22×C411yescharacteristic
76C22⋊D53—42no—
77C22⋊D53—41no—
78C22⋊D53—41no—
79C22⋊D53—42no—
80C2×Dic53—22no—
81C2×Dic53—21no—
82C2×Dic53C22⋊C411yescharacteristic
83C2×Dic53C2×D411yescharacteristic
84D212—41no—
85D212—41no—
86D212—41no—
87C4×D53—22no—
88C4×D53—22no—
89C4×D53—41no—
90C4×D53—41no—
91C4×D53—42no—
92#88324—42no—
93C22×C53—21no—
94C22×C53—21no—
95C22×C53—41no—
96C22×C53—41no—
97C22×C53—41no—
98C22×C53#10212yes—
99C22×C53C2×C22⋊C411yescharacteristic
100C2×D53—21no—
101C2×D53—21no—
102C2×D53—21no—
103C2×D53—21no—
104C2×D53—42no—
105C2×D53—42no—
106C2×D53—41no—
107C2×D53—41no—
108C2×D53—41no—
109C2×D53—84no—
110C2×D53#10212yes—
111C2×D53C2×D811yescharacteristic
112C2×D53C2×SD1611yescharacteristic
113#6157—82no—
114C4×C53—21no—
115C4×C53—21no—
116C4×C53—41no—
117Dic53—21no—
118Dic53—41no—
119Dic53#9912yes—
120#1519—531nomaximal, Sylow
121C2×C53—22no—
122C2×C53—81no—
123C2×C53C2×#9911yescharacteristic
124C2×C53C2×#10211yescharacteristic
125C2×C53#42111yescharacteristic
126D53—21no—
127D53—22no—
128D53—21no—
129D53—81no—
130C2×#121—531no—
131C2×C22⋊C8—531no—
132#380—531no—
133#327—532no—
134#327—532no—
135C53#151911yescharacteristic, complemented, Sylow
136C22×D4—531no—
137C22×D4—1061no—
138C23×C4—531no—
139C22×C8—1061no—
140#121—531no—
141#121—531no—
142#121—531no—
143#121—1061no—
144#121—1061no—
145#121#615711yescharacteristic, complemented
146C2×C4⋊C4—531no—
147C2×C4⋊C4—1061no—
148C2×C22⋊C4—1061no—
149C2×C22⋊C4—1061no—
150C22⋊C8—532no—
151C22⋊C8—1061no—
152#95—532no—
153C24—531no—
154C24—1061no—
155C2×D4—21no—
156C2×D4—531no—
157C2×D4—1061no—
158C2×D4—1061no—
159C2×D4—1061no—
160C2×D4—1061no—
161C2×D4—2122no—
162C2×D4—2122no—
163C2×D4—2122no—
164C2×D4C2×#615711yescharacteristic
165C22×C4—531no—
166C22×C4—531no—
167C22×C4—531no—
168C22×C4—531no—
169C22×C4—531no—
170C22×C4—1061no—
171C22×C4—1062no—
172C22×C4—2122no—
173C22×C4C2×#615711yescharacteristic
174C2×C8—1062no—
175C2×C8—2122no—
176C4⋊C4—531no—
177C4⋊C4—1061no—
178C4⋊C4—2122no—
179C4⋊C4C2×#615711yescharacteristic
180C22⋊C4—21no—
181C22⋊C4—1061no—
182C22⋊C4—1061no—
183C22⋊C4—1061no—
184C22⋊C4—2122no—
185C23—21no—
186C23—531no—
187C23—531no—
188C23—532no—
189C23—532no—
190C23—1061no—
191C23—2121no—
192C23—2121no—
193C23—2122no—
194C23—2122no—
195C23—2122no—
196C23C22⋊#615711yescharacteristic
197D4—41no—
198D4—42no—
199D4—2121no—
200D4—2121no—
201D4—2121no—
202D4—2122no—
203D4—2121no—
204D4—2121no—
205D4—2122no—
206C2×C4—21no—
207C2×C4—41no—
208C2×C4—531no—
209C2×C4—531no—
210C2×C4—1062no—
211C2×C4—1062no—
212C2×C4—1062no—
213C2×C4—1061no—
214C2×C4—2121no—
215C2×C4—2121no—
216C2×C4—2122no—
217C2×C4—2122no—
218C2×C4—2122no—
219C2×C4—2122no—
220C2×C4C22⋊#615711yescharacteristic
221C2×C4C22×#615711yescharacteristic
222C8—2122no—
223C22—21no—
224C22—21no—
225C22—41no—
226C22—41no—
227C22—41no—
228C22—531no—
229C22—532no—
230C22—531no—
231C22—1061no—
232C22—1061no—
233C22—1061no—
234C22—1061no—
235C22—4244no—
236C22—2122no—
237C22—2122no—
238C22—2121no—
239C22—2121no—
240C22—2121no—
241C22#1313327912yes—
242C22C2×C22⋊#615711yescharacteristic
243C4—21no—
244C4—21no—
245C4—41no—
246C4—532no—
247C4—1061no—
248C4—4242no—
249C4—2121no—
250C2—22no—
251C2—81no—
252C2—1061no—
253C2—1062no—
254C2—1061no—
255C2—4241no—
256C2D53⋊#9911yescharacteristic
257C2C2×#1313327911yescharacteristic
258C2#1681162011yescharacteristic
subgroupG/H
centerC22C2×C22⋊#6157
commutatorC2×C4×C53C22×C4
FrattiniC2×C4C22×#6157
Fitting#121×C53C4
socleC22×C53C2×C22⋊C4
hypercenter#121#6157
hypocenterC53#1519
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ #121

Complements

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

NH ≅ G/Ntype
1. C2×#13133274C2semidirect
2. #16811565C2semidirect
3. #16811475C2semidirect
4. #16811474C2semidirect
10. #13133274C22semidirect
11. (C2×#6157)⋊C4C22semidirect
13. C2×D4×D53C4semidirect
16. #13133191C4semidirect
20. C4⋊C4×D53C4semidirect
22. #121×C53C4semidirect
135. C53#1519semidirect
145. #121#6157semidirect