Finite Group Databasebeta

← #20278047

Subgroups — #20278047

Order 6784.

864 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#11719947C211yesmaximal, characteristic
2C2×#13133200C211yesmaximal, characteristic, complemented
3C22×#11719919C211yesmaximal, characteristic, complemented
4#16810543C212yesmaximal, complemented
5#16810538C212yesmaximal, complemented
6#16810503C211yesmaximal, characteristic, complemented
7#16810500C211yesmaximal, characteristic, complemented
8#16810495C212yesmaximal, complemented
9#16810494C212yesmaximal, complemented
10#16810484C211yesmaximal, characteristic, complemented
11C2×#123×C53C211yesmaximal, characteristic, complemented
12C2×#11719947—24no—
13C2×#11719947—24no—
14C2×#11719947—24no—
15C22×Dic106C2211yescharacteristic
16C22×C22⋊D53C2211yescharacteristic
17C22×C22⋊D53C2211yescharacteristic
18C23×Dic53C2211yescharacteristic
19C23×Dic53C2211yescharacteristic
20C23×Dic53C2211yescharacteristic
21C22×C4×D53C2211yescharacteristic
22#13133200—24no—
23C2×#11719919—24no—
24C2×#11719919C2212yes—
25C2×#11719919C2212yes—
26C2×#11719919C2212yes—
27C2×C22⋊Dic53C2212yes—
28C2×C22⋊Dic53C2211yescharacteristic
29C2×#11719921C2212yes—
30C2×#11719921C2212yes—
31#13133124C2212yes—
32#13133124C2212yes—
33#13133124C2212yes—
34#13133124C2212yes—
35C22×D4×C53C2211yescharacteristic
36C23×C4×C53C2211yescharacteristic
37#123×C53—24no—
38C2×C4⋊C4×C53C2212yes—
39C2×C22⋊C4×C53C2211yescharacteristic
40C2×C22⋊C4×C53C2212yes—
41C24×C53C2311yescharacteristic
42C24×C53C2311yescharacteristic
43C23×D53C2311yescharacteristic
44C2×D4×C53—22no—
45C2×D4×C53—22no—
46C2×D4×C53D414yes—
47C22×C4×C53—22no—
48C22×C4×C53—22no—
49C22×C4×C53—22no—
50C22×C4×C53C2312yes—
51C22×C4×C53C2312yes—
52C22×C4×C53C2311yescharacteristic
53C2×C22⋊D53—22no—
54C2×C22⋊D53—22no—
55C2×C22⋊D53—22no—
56C2×C22⋊D53—22no—
57C2×C22⋊D53—22no—
58C2×C22⋊D53D414yes—
59C22×Dic53—22no—
60C22×Dic53—22no—
61C22×Dic53—22no—
62C22×Dic53—22no—
63C22×Dic53—22no—
64C22×Dic53—22no—
65C22×Dic53—24no—
66C22×Dic53D414yes—
67C22×Dic53C2312yes—
68C22×Dic53C2311yescharacteristic
69C22×Dic53C2311yescharacteristic
70C22×Dic53C2311yescharacteristic
71C22×Dic53C2312yes—
72#11719947—232no—
73C2×C4×D53—22no—
74C2×C4×D53—22no—
75C2×C4×D53—22no—
76C2×Dic106—22no—
77C2×Dic106—22no—
78C2×Dic106—22no—
79C4⋊C4×C53—24no—
80C22⋊C4×C53—24no—
81C22⋊C4×C53—22no—
82C22⋊Dic53—22no—
83C22⋊Dic53—24no—
84#11719921—24no—
85#11719921—24no—
86#11719919—24no—
87#11719919—24no—
88#11719919—24no—
89C23×C53—22no—
90C23×C53—22no—
91C23×C53—21no—
92C23×C53—21no—
93C23×C53—22no—
94C23×C53—22no—
95C23×C53C2×D412yes—
96C23×C53C2×D412yes—
97C23×C53C2×D412yes—
98C23×C53C2×D412yes—
99C23×C53C2411yescharacteristic
100C22×D53—21no—
101C22×D53—21no—
102C22×D53—22no—
103C22×D53—22no—
104C22×D53C2×D412yes—
105D4×C53—28no—
106C2×C4×C53—21no—
107C2×C4×C53—21no—
108C2×C4×C53—22no—
109C2×C4×C53—22no—
110C2×C4×C53—22no—
111C2×C4×C53—22no—
112C2×C4×C53—22no—
113C2×C4×C53—22no—
114C2×C4×C53—44no—
115C2×C4×C53C2×D412yes—
116C22⋊D53—28no—
117C22⋊D53—28no—
118C2×Dic53—28no—
119C2×Dic53—28no—
120C2×Dic53—28no—
121C2×Dic53—21no—
122C2×Dic53—21no—
123C2×Dic53—21no—
124C2×Dic53—21no—
125C2×Dic53—21no—
126C2×Dic53—21no—
127C2×Dic53—22no—
128C2×Dic53—22no—
129C2×Dic53—22no—
130C2×Dic53—22no—
131C2×Dic53—22no—
132C2×Dic53C2×D412yes—
133C2×Dic53C2×D412yes—
134C2×Dic53C2×D412yes—
135C2×Dic53#4114yescomplemented
136C4×D53—28no—
137Dic106—28no—
138C22×C53—22no—
139C22×C53—22no—
140C22×C53—24no—
141C22×C53—24no—
142C22×C53—22no—
143C22×C53—24no—
144C22×C53—41no—
145C22×C53—41no—
146C22×C53—42no—
147C22×C53#12014yes—
148C22×C53C22×D411yescharacteristic
149C22×C53C22×D411yescharacteristic
150C22×C53C22×D411yescharacteristic
151C22×C53C2×#4112yes—
152C22×C532−1+412yes—
153C2×D53—22no—
154C2×D53—24no—
155C2×D53—41no—
156C2×D53—41no—
157C2×D53—42no—
158C4×C53—22no—
159C4×C53—42no—
160C4×C53—42no—
161Dic53—22no—
162Dic53—22no—
163Dic53—22no—
164Dic53—24no—
165Dic53—42no—
166#2406—531nomaximal, Sylow
167C2×C53—24no—
168C2×C53—42no—
169C2×C53C2×#12011yescharacteristic
170C2×C53#52511yescharacteristic
171C2×C53#53611yescharacteristic
172C2×C53#54712yes—
173C2×C53#54712yes—
174D53—42no—
175C22×#41—531no—
176C2×#123—531no—
177C2×#123D5311yescharacteristic, complemented
178C22×C4⋊C4—531no—
179#395—532no—
180#394—532no—
181#386—531no—
182#386—531no—
183#385—532no—
184#385—532no—
185#381—531no—
186C53#240611yescharacteristic, complemented, Sylow
187C2×#41—1064no—
188C2×#41—1064no—
189C2×#41—1064no—
190C22×Q8—531no—
191C22×D4—531no—
192C22×D4—531no—
193C22×D4C2×D5311yescharacteristic
194C23×C4—531no—
195C23×C4—531no—
196C23×C4—531no—
197C23×C4—531no—
198C23×C4C2×D5311yescharacteristic
199#123—24no—
200#123—1064no—
201C2×C4⋊C4—532no—
202C2×C4⋊C4—532no—
203C2×C4⋊C4—532no—
204C2×C4⋊C4—1064no—
205C2×C4⋊C4C2×D5312yes—
206C2×C22⋊C4—531no—
207C2×C22⋊C4—532no—
208C2×C22⋊C4—532no—
209C2×C22⋊C4—532no—
210C2×C22⋊C4C2×D5311yescharacteristic
211C2×C22⋊C4C2×D5312yes—
212#95—532no—
213#95—532no—
214#95—532no—
215#95—532no—
216C24—531no—
217C24C22×D5311yescharacteristic
218C24C22×D5311yescharacteristic
219#41—10632no—
220C2×Q8—1062no—
221C2×Q8—1062no—
222C2×Q8—1062no—
223C2×D4—22no—
224C2×D4—22no—
225C2×D4—534no—
226C2×D4—1062no—
227C2×D4—1062no—
228C2×D4—1062no—
229C2×D4—1062no—
230C2×D4—1062no—
231C2×D4C22⋊D5314yes—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—534no—
236C22×C4—532no—
237C22×C4—531no—
238C22×C4—531no—
239C22×C4—531no—
240C22×C4—532no—
241C22×C4—1062no—
242C22×C4—1062no—
243C22×C4—1064no—
244C22×C4—1062no—
245C22×C4—1062no—
246C22×C4—1062no—
247C22×C4—1062no—
248C22×C4—1062no—
249C22×C4—1062no—
250C22×C4—1062no—
251C22×C4C22×D5312yes—
252C22×C4C22×D5312yes—
253C22×C4C22×D5311yescharacteristic
254C4⋊C4—24no—
255C4⋊C4—1064no—
256C4⋊C4—1064no—
257C4⋊C4—1064no—
258C22⋊C4—22no—
259C22⋊C4—24no—
260C22⋊C4—1064no—
261C22⋊C4—1064no—
262C22⋊C4—1064no—
263C22⋊C4—1062no—
264C23—22no—
265C23—22no—
266C23—21no—
267C23—21no—
268C23—22no—
269C23—22no—
270C23—532no—
271C23—1061no—
272C23—1061no—
273C23—1062no—
274C23—1062no—
275C23D4×D5312yes—
276C23D4×D5312yes—
277C23C2×C22⋊D5312yes—
278C23C2×C22⋊D5312yes—
279C23C23×D5311yescharacteristic
280Q8—1068no—
281D4—28no—
282D4—1068no—
283D4—1068no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—22no—
287C2×C4—22no—
288C2×C4—22no—
289C2×C4—22no—
290C2×C4—22no—
291C2×C4—22no—
292C2×C4—44no—
293C2×C4—532no—
294C2×C4—532no—
295C2×C4—532no—
296C2×C4—534no—
297C2×C4—1068no—
298C2×C4—1068no—
299C2×C4—1068no—
300C2×C4—1068no—
301C2×C4—1062no—
302C2×C4—1061no—
303C2×C4—1061no—
304C2×C4—1061no—
305C2×C4—1061no—
306C2×C4—1061no—
307C2×C4—1061no—
308C2×C4—1062no—
309C2×C4—1062no—
310C2×C4—1062no—
311C2×C4—1062no—
312C2×C4C2×C22⋊D5312yes—
313C22—24no—
314C22—24no—
315C22—22no—
316C22—22no—
317C22—24no—
318C22—22no—
319C22—42no—
320C22—41no—
321C22—41no—
322C22—1064no—
323C22—1062no—
324C22—2122no—
325C22—2121no—
326C22—2121no—
327C22#1313318514yes—
328C22C2×D4×D5311yescharacteristic
329C22C2×D4×D5311yescharacteristic
330C22C22×C22⋊D5311yescharacteristic
331C22#41×D5312yes—
332C22#1313326412yes—
333C4—22no—
334C4—42no—
335C4—42no—
336C4—1062no—
337C4—1062no—
338C4—1062no—
339C4—1064no—
340C4—2122no—
341C2—24no—
342C2—42no—
343C2—2122no—
344C2C2×#1313318511yescharacteristic
345C2#1681113111yescharacteristic
346C2#1681119511yescharacteristic
347C2#1681127112yes—
348C2#1681127212yes—
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC2×#123×C53C2
hypercenterC2×#123D53
hypocenterC53#2406
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×#123

Complements

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

NH ≅ G/Ntype
2. C2×#13133200C2semidirect
3. C22×#11719919C2semidirect
4. #16810543C2semidirect
5. #16810538C2semidirect
6. #16810503C2semidirect
7. #16810500C2semidirect
8. #16810495C2semidirect
9. #16810494C2semidirect
10. #16810484C2semidirect
11. C2×#123×C53C2semidirect
135. C2×Dic53#41semidirect
177. C2×#123D53semidirect
186. C53#2406semidirect