Finite Group Databasebeta

← #20275698

Subgroups — #20275698

Order 6784.

324 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 85 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×#13133194C211yesmaximal, characteristic
2#16810481C211yesmaximal, characteristic
3C22⋊C4×Dic53C211yesmaximal, characteristic
4#16810390C211yesmaximal, characteristic
5#16810389C211yesmaximal, characteristic
6#16810389C211yesmaximal, characteristic
7C53×#342C211yesmaximal, characteristic
8C22×Dic106C411yescharacteristic, complemented
9C23×Dic53C2211yescharacteristic
10#13133194—21no—
11#13133194—21no—
12#13133194—21no—
13#13133194—21no—
14C2×C4⋊Dic53C411yescharacteristic, complemented
15C2×#11719919—21no—
16C2×#11719919C411yescharacteristic, complemented
17C2×#11719919C411yescharacteristic, complemented
18C2×C22⋊Dic53C2211yescharacteristic
19C2×C22⋊Dic53C2211yescharacteristic
20C2×C22⋊Dic53C2211yescharacteristic
21C2×C4×Dic53—21no—
22#13133124—21no—
23#13133124—21no—
24C2×C22⋊C4×C53C2211yescharacteristic
25C2×C22⋊C4×C53C2211yescharacteristic
26C2×C22⋊C4×C53C2211yescharacteristic
27C24×C53C2311yescharacteristic
28C22×C4×C53—21no—
29C22×C4×C53—21no—
30C22×C4×C53C2×C411yescharacteristic
31C22×C4×C53C2×C411yescharacteristic
32C22×Dic53—21no—
33C22×Dic53—21no—
34C22×Dic53—42no—
35C22×Dic53C2×C411yescharacteristic
36C22×Dic53C2×C411yescharacteristic
37C22×Dic53C2×C411yescharacteristic
38C22×Dic53C2×C411yescharacteristic
39C22×Dic53D411yescharacteristic
40C22×Dic53D411yescharacteristic
41C22×Dic53D411yescharacteristic
42C22×Dic53D411yescharacteristic
43C2×Dic106—21no—
44C2×Dic106—21no—
45C2×Dic106—42no—
46C22⋊C4×C53—21no—
47C22⋊C4×C53—21no—
48C22⋊C4×C53—21no—
49C22⋊C4×C53—21no—
50C22⋊C4×C53—21no—
51C22⋊C4×C53Q812yescomplemented
52C22⋊Dic53—21no—
53C22⋊Dic53—21no—
54C22⋊Dic53—21no—
55C22⋊Dic53—21no—
56C22⋊Dic53—21no—
57C22⋊Dic53Q812yescomplemented
58C4⋊Dic53—21no—
59C4⋊Dic53—21no—
60#11719919—21no—
61#11719919—21no—
62#11719919—21no—
63#11719919—21no—
64#11719919—42no—
65C4×Dic53—42no—
66C23×C53—41no—
67C23×C53—41no—
68C23×C53C22×C411yescharacteristic
69C23×C53C2×D411yescharacteristic
70C23×C53C2×D411yescharacteristic
71C23×C53C2×Q811yescharacteristic
72C23×C53C2×Q811yescharacteristic
73C23×C53#4111yescharacteristic
74C23×C53#4111yescharacteristic
75C2×C4×C53—21no—
76C2×C4×C53—21no—
77C2×C4×C53—22no—
78C2×C4×C53—41no—
79C2×C4×C53—41no—
80C2×C4×C53—41no—
81C2×C4×C53—41no—
82C2×C4×C53—41no—
83C2×C4×C53—41no—
84C2×C4×C53—41no—
85C2×Dic53—22no—
86C2×Dic53—22no—
87C2×Dic53—22no—
88C2×Dic53—21no—
89C2×Dic53—21no—
90C2×Dic53—22no—
91C2×Dic53—41no—
92C2×Dic53—41no—
93C2×Dic53—41no—
94C2×Dic53—41no—
95C2×Dic53—41no—
96C2×Dic53—41no—
97C2×Dic53—41no—
98C2×Dic53—41no—
99C2×Dic53—41no—
100C2×Dic53—41no—
101C2×Dic53C22⋊C411yescharacteristic
102C2×Dic53C22⋊C411yescharacteristic
103C2×Dic53C22⋊C411yescharacteristic
104C2×Dic53C22⋊C411yescharacteristic
105Dic106—42no—
106Dic106—42no—
107C22×C53—21no—
108C22×C53—21no—
109C22×C53—21no—
110C22×C53—21no—
111C22×C53—21no—
112C22×C53—21no—
113C22×C53—41no—
114C22×C53—41no—
115C22×C53—41no—
116C22×C53—41no—
117C22×C53C2×C22⋊C411yescharacteristic
118C22×C53C4×Q811yescharacteristic
119C22×C53C4×Q811yescharacteristic
120C22×C53C22⋊Q811yescharacteristic
121C22×C53C22⋊Q811yescharacteristic
122C22×C53#12411yescharacteristic
123C22×C53C4⋊Q811yescharacteristic
124C4×C53—42no—
125C4×C53—41no—
126C4×C53—41no—
127C4×C53—81no—
128Dic53—21no—
129Dic53—21no—
130Dic53—21no—
131Dic53—42no—
132Dic53—41no—
133Dic53—41no—
134Dic53—81no—
135Dic53#9912yes—
136#1996—531nomaximal, Sylow
137C2×C53—21no—
138C2×C53—21no—
139C2×C53—21no—
140C2×C53—21no—
141C2×C53—21no—
142C2×C53—21no—
143C2×C53#39111yescharacteristic
144C2×C53C2×#9911yescharacteristic
145C2×C53#41011yescharacteristic
146C2×C22⋊Q8—531no—
147#380—531no—
148C4×C22⋊C4—531no—
149#342—531no—
150#342—531no—
151#342—531no—
152#342D5311yescharacteristic
153C53#199611yescharacteristic, complemented, Sylow
154C22×Q8—531no—
155C23×C4—531no—
156C22⋊Q8—1061no—
157C22⋊Q8—1061no—
158C22⋊Q8—1061no—
159C22⋊Q8—1061no—
160C2×C4⋊C4—531no—
161C2×C4⋊C4—531no—
162C2×C4⋊C4—531no—
163C2×C4⋊C4—1061no—
164C2×C22⋊C4—531no—
165C2×C22⋊C4—531no—
166C2×C22⋊C4—531no—
167C2×C22⋊C4C2×D5311yescharacteristic
168C2×C22⋊C4C2×D5311yescharacteristic
169C2×C22⋊C4C2×D5311yescharacteristic
170C2×C42—1061no—
171#95—1061no—
172#95—1061no—
173C24C22×D5311yescharacteristic
174C2×Q8—1061no—
175C2×Q8—1061no—
176C2×Q8—2122no—
177C22×C4—21no—
178C22×C4—21no—
179C22×C4—531no—
180C22×C4—531no—
181C22×C4—531no—
182C22×C4—531no—
183C22×C4—531no—
184C22×C4—531no—
185C22×C4—531no—
186C22×C4—531no—
187C22×C4—1061no—
188C22×C4—1061no—
189C22×C4—2122no—
190C22×C4C4×D5311yescharacteristic
191C22×C4C4×D5311yescharacteristic
192C4⋊C4—1061no—
193C4⋊C4—1061no—
194C4⋊C4—1061no—
195C4⋊C4—1061no—
196C4⋊C4—1061no—
197C4⋊C4—1061no—
198C4⋊C4—2122no—
199C22⋊C4—21no—
200C22⋊C4—21no—
201C22⋊C4—21no—
202C22⋊C4—21no—
203C22⋊C4—21no—
204C22⋊C4—532no—
205C22⋊C4—1061no—
206C22⋊C4—1061no—
207C22⋊C4—1061no—
208C22⋊C4—1061no—
209C22⋊C4—1061no—
210C22⋊C4Dic10612yescomplemented
211C42—2122no—
212C23—41no—
213C23—41no—
214C23C2×Dic10611yescharacteristic
215C23C2×C4×D5311yescharacteristic
216C23#1171994511yescharacteristic
217C23D4×D5311yescharacteristic
218C23D4×D5311yescharacteristic
219C23#1171994711yescharacteristic
220C23Q8×D5311yescharacteristic
221Q8—2122no—
222Q8—2122no—
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—22no—
226C2×C4—41no—
227C2×C4—41no—
228C2×C4—41no—
229C2×C4—41no—
230C2×C4—41no—
231C2×C4—41no—
232C2×C4—41no—
233C2×C4—531no—
234C2×C4—531no—
235C2×C4—531no—
236C2×C4—531no—
237C2×C4—1062no—
238C2×C4—1062no—
239C2×C4—1062no—
240C2×C4—1061no—
241C2×C4—1061no—
242C2×C4—1062no—
243C2×C4—2121no—
244C2×C4—2121no—
245C2×C4—2121no—
246C2×C4—2121no—
247C2×C4—2121no—
248C2×C4—2121no—
249C2×C4—2121no—
250C2×C4—2121no—
251C2×C4—2121no—
252C2×C4—2121no—
253C22—21no—
254C22—21no—
255C22—21no—
256C22—21no—
257C22—21no—
258C22—21no—
259C22—41no—
260C22—41no—
261C22—41no—
262C22—41no—
263C22C22⋊C4×D5311yescharacteristic
264C22#1313318211yescharacteristic
265C22C4×Dic10611yescharacteristic
266C22#1313319411yescharacteristic
267C22#1313319511yescharacteristic
268C22#1313320311yescharacteristic
269C22#1313321611yescharacteristic
270C4—42no—
271C4—41no—
272C4—41no—
273C4—81no—
274C4—532no—
275C4—1061no—
276C4—1061no—
277C4—1061no—
278C4—4241no—
279C4—2122no—
280C4—2121no—
281C4—2121no—
282C2—21no—
283C2—21no—
284C2—21no—
285C2—21no—
286C2—21no—
287C2—21no—
288C2#1681052711yescharacteristic
289C2#99×D5311yescharacteristic
290C2#1681059511yescharacteristic
subgroupG/H
centerC22C22⋊C4×D53
commutatorC23×C53C22×C4
FrattiniC24C22×D53
FittingC53×#342C2
socleC22×C53C2×C22⋊C4
hypercenter#342D53
hypocenterC53#1996
LCSG ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C24 ◁ #342

Complements

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

NH ≅ G/Ntype
8. C22×Dic106C4semidirect
14. C2×C4⋊Dic53C4semidirect
16. C2×#11719919C4semidirect
17. C2×#11719919C4semidirect
51. C22⋊C4×C53Q8semidirect
57. C22⋊Dic53Q8semidirect
153. C53#1996semidirect
210. C22⋊C4Dic106semidirect