Finite Group Databasebeta

← #20278518

Subgroups — #20278518

Order 6784.

648 conjugacy classes of subgroups.

The subgroup lattice has 426 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#13133196C211yesmaximal, characteristic, complemented
2C2×#13133194C211yesmaximal, characteristic, complemented
3C22×#11719919C211yesmaximal, characteristic, complemented
4#16810559C211yesmaximal, characteristic, complemented
5#16810556C211yesmaximal, characteristic, complemented
6#16810546C211yesmaximal, characteristic, complemented
7#16810543C211yesmaximal, characteristic, complemented
8#16810521C211yesmaximal, characteristic, complemented
9#16810507C211yesmaximal, characteristic, complemented
10#16810503C211yesmaximal, characteristic
11#16810503C211yesmaximal, characteristic
12#16810498C211yesmaximal, characteristic, complemented
13#16810496C211yesmaximal, characteristic, complemented
14#16810496C211yesmaximal, characteristic, complemented
15C2×C4×D4×C53C211yesmaximal, characteristic
16C22×Dic106C2211yescharacteristic
17C23×Dic53C2211yescharacteristic
18C23×Dic53C2211yescharacteristic
19#13133196—24no—
20#13133194—24no—
21C2×C4⋊Dic53C2211yescharacteristic
22C2×C4⋊Dic53C2211yescharacteristic
23C2×C4⋊Dic53C2211yescharacteristic
24C2×#11719919—24no—
25C2×#11719919C2211yescharacteristic
26C2×#11719919C2211yescharacteristic
27C2×#11719919C2211yescharacteristic
28C2×#11719919C2211yescharacteristic
29C2×#11719919C2211yescharacteristic
30C2×#11719919C2211yescharacteristic
31C2×#11719919C2211yescharacteristic
32C2×#11719919C2211yescharacteristic
33C2×C22⋊Dic53C2211yescharacteristic
34C2×C22⋊Dic53C2211yescharacteristic
35C2×C22⋊Dic53C2211yescharacteristic
36C2×C22⋊Dic53C2211yescharacteristic
37C2×C22⋊Dic53C2211yescharacteristic
38C2×C22⋊Dic53C2211yescharacteristic
39#13133124C2211yescharacteristic
40#13133124C2211yescharacteristic
41#13133124C2211yescharacteristic
42#13133124C2211yescharacteristic
43#13133124C2211yescharacteristic
44#13133124C2211yescharacteristic
45#13133124C2211yescharacteristic
46#13133124C2211yescharacteristic
47C22×D4×C53C2211yescharacteristic
48C23×C4×C53C2211yescharacteristic
49C23×C4×C53C2211yescharacteristic
50C4×D4×C53—24no—
51C2×C4⋊C4×C53C2211yescharacteristic
52C2×C22⋊C4×C53C2211yescharacteristic
53C2×C22⋊C4×C53C2211yescharacteristic
54C2×C42×C53C2211yescharacteristic
55C24×C53C2311yescharacteristic
56C24×C53C2311yescharacteristic
57C2×D4×C53—22no—
58C2×D4×C53—22no—
59C2×D4×C53Q814yescomplemented
60C22×C4×C53—22no—
61C22×C4×C53—22no—
62C22×C4×C53—22no—
63C22×C4×C53—22no—
64C22×C4×C53—41no—
65C22×C4×C53—41no—
66C22×C4×C53C2311yescharacteristic
67C22×C4×C53C2311yescharacteristic
68C22×C4×C53C2311yescharacteristic
69C22×C4×C53C2311yescharacteristic
70C22×C4×C53C2311yescharacteristic
71C22×Dic53—22no—
72C22×Dic53—22no—
73C22×Dic53—22no—
74C22×Dic53—22no—
75C22×Dic53—22no—
76C22×Dic53—41no—
77C22×Dic53C2311yescharacteristic
78C22×Dic53C2311yescharacteristic
79C22×Dic53C2311yescharacteristic
80C22×Dic53C2311yescharacteristic
81C22×Dic53C2311yescharacteristic
82C22×Dic53C2311yescharacteristic
83C22×Dic53C2311yescharacteristic
84C22×Dic53C2311yescharacteristic
85C2×Dic106—22no—
86C2×Dic106—22no—
87C2×Dic106—41no—
88C4⋊C4×C53—22no—
89C22⋊C4×C53—22no—
90C22⋊C4×C53D414yes—
91C42×C53—22no—
92C22⋊Dic53—22no—
93C22⋊Dic53—22no—
94C22⋊Dic53—22no—
95C22⋊Dic53—22no—
96C22⋊Dic53—22no—
97C22⋊Dic53—41no—
98C4⋊Dic53—22no—
99C4⋊Dic53—22no—
100C4⋊Dic53—22no—
101#11719919—22no—
102#11719919—22no—
103#11719919—22no—
104#11719919—22no—
105#11719919—22no—
106#11719919—22no—
107#11719919—41no—
108#11719919D414yescomplemented
109C23×C53—21no—
110C23×C53—21no—
111C23×C53—21no—
112C23×C53—21no—
113C23×C53—21no—
114C23×C53—21no—
115C23×C53—21no—
116C23×C53—21no—
117C23×C53—21no—
118C23×C53—21no—
119C23×C53C2×D412yes—
120C23×C53C2×Q812yes—
121C23×C53C2×Q812yes—
122C23×C53#4112yes—
123C23×C53C2411yescharacteristic
124D4×C53—28no—
125C2×C4×C53—21no—
126C2×C4×C53—21no—
127C2×C4×C53—21no—
128C2×C4×C53—21no—
129C2×C4×C53—21no—
130C2×C4×C53—21no—
131C2×C4×C53—21no—
132C2×C4×C53—21no—
133C2×C4×C53—21no—
134C2×C4×C53—21no—
135C2×C4×C53—21no—
136C2×C4×C53—44no—
137C2×C4×C53—44no—
138C2×C4×C53C2×D412yes—
139C2×C4×C53C2×D412yes—
140C2×C4×C53C2×D412yes—
141C2×C4×C53C2×Q812yes—
142C2×Dic53—28no—
143C2×Dic53—21no—
144C2×Dic53—21no—
145C2×Dic53—21no—
146C2×Dic53—21no—
147C2×Dic53—21no—
148C2×Dic53—21no—
149C2×Dic53—21no—
150C2×Dic53—21no—
151C2×Dic53—21no—
152C2×Dic53—21no—
153C2×Dic53—21no—
154C2×Dic53—21no—
155C2×Dic53—21no—
156C2×Dic53—21no—
157C2×Dic53—21no—
158C2×Dic53—21no—
159C2×Dic53—21no—
160C2×Dic53—21no—
161C2×Dic53—21no—
162C2×Dic53—21no—
163C2×Dic53—21no—
164C2×Dic53—44no—
165C2×Dic53C2×D412yes—
166C2×Dic53C2×D412yes—
167C2×Dic53#4112yes—
168Dic106—44no—
169C22×C53—22no—
170C22×C53—22no—
171C22×C53—22no—
172C22×C53—22no—
173C22×C53—22no—
174C22×C53—22no—
175C22×C53—22no—
176C22×C53—22no—
177C22×C53—22no—
178C22×C53—41no—
179C22×C53—41no—
180C22×C53—41no—
181C22×C53—41no—
182C22×C53C22⋊Q814yescomplemented
183C22×C53C22×D411yescharacteristic
184C22×C53C22×D411yescharacteristic
185C22×C53C22×Q811yescharacteristic
186C22×C53C2×#4111yescharacteristic
187C22×C53C2×#4111yescharacteristic
188C22×C53H(5,𝔽2)11yescharacteristic
189C22×C532−1+411yescharacteristic
190C4×C53—22no—
191C4×C53—22no—
192C4×C53—22no—
193C4×C53—22no—
194C4×C53—41no—
195Dic53—22no—
196Dic53—22no—
197Dic53—22no—
198Dic53—41no—
199Dic53—41no—
200Dic53—41no—
201Dic53—41no—
202Dic53—41no—
203#2449—531nomaximal, Sylow
204C2×C53—24no—
205C2×C53—42no—
206C2×C53C2×C22⋊Q811yescharacteristic
207C2×C53#52511yescharacteristic
208C2×C53#53911yescharacteristic
209C2×C53D4211yescharacteristic
210C2×C53#54711yescharacteristic
211C2×C53D4×Q811yescharacteristic
212C2×C53#55411yescharacteristic
213C2×C22⋊Q8—531no—
214C2×C22⋊Q8—531no—
215C2×C4×D4D5311yescharacteristic
216C22×C4⋊C4—531no—
217#399—531no—
218#398—531no—
219#396—531no—
220#395—531no—
221#389—531no—
222#387—531no—
223#386—531no—
224#386—531no—
225#385—531no—
226#385—531no—
227#385—531no—
228C53#244911yescharacteristic, complemented, Sylow
229C22×Q8—531no—
230C22×D4C2×D5311yescharacteristic
231C23×C4—531no—
232C23×C4—531no—
233C23×C4C2×D5311yescharacteristic
234C23×C4C2×D5311yescharacteristic
235C22⋊Q8—1064no—
236C22⋊Q8—1064no—
237C4×D4—24no—
238C2×C4⋊C4—531no—
239C2×C4⋊C4—531no—
240C2×C4⋊C4—531no—
241C2×C4⋊C4—531no—
242C2×C4⋊C4—531no—
243C2×C4⋊C4—531no—
244C2×C4⋊C4—531no—
245C2×C4⋊C4—531no—
246C2×C4⋊C4—531no—
247C2×C4⋊C4—531no—
248C2×C4⋊C4—531no—
249C2×C4⋊C4—1064no—
250C2×C4⋊C4C2×D5311yescharacteristic
251C2×C22⋊C4—531no—
252C2×C22⋊C4—531no—
253C2×C22⋊C4—531no—
254C2×C22⋊C4—531no—
255C2×C22⋊C4—531no—
256C2×C22⋊C4—531no—
257C2×C22⋊C4C2×D5311yescharacteristic
258C2×C22⋊C4C2×D5311yescharacteristic
259C2×C42C2×D5311yescharacteristic
260#95—531no—
261#95—531no—
262#95—531no—
263#95—531no—
264#95—531no—
265#95—531no—
266#95—531no—
267#95—531no—
268C24C22×D5311yescharacteristic
269C24C22×D5311yescharacteristic
270C2×Q8—1062no—
271C2×Q8—1062no—
272C2×Q8—2121no—
273C2×D4—22no—
274C2×D4—22no—
275C2×D4Dic10614yescomplemented
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—41no—
281C22×C4—41no—
282C22×C4—531no—
283C22×C4—531no—
284C22×C4—531no—
285C22×C4—531no—
286C22×C4—531no—
287C22×C4—531no—
288C22×C4—531no—
289C22×C4—531no—
290C22×C4—1062no—
291C22×C4—1062no—
292C22×C4—1062no—
293C22×C4—1062no—
294C22×C4—1062no—
295C22×C4—2121no—
296C22×C4C22×D5311yescharacteristic
297C22×C4C22×D5311yescharacteristic
298C22×C4C22×D5311yescharacteristic
299C22×C4C22×D5311yescharacteristic
300C22×C4C22×D5311yescharacteristic
301C4⋊C4—22no—
302C4⋊C4—534no—
303C4⋊C4—1062no—
304C4⋊C4—1062no—
305C4⋊C4—1062no—
306C4⋊C4—1062no—
307C4⋊C4—1062no—
308C4⋊C4—1062no—
309C4⋊C4—1062no—
310C4⋊C4—1062no—
311C4⋊C4—1062no—
312C4⋊C4—2121no—
313C22⋊C4—22no—
314C22⋊C4—1062no—
315C22⋊C4—1062no—
316C22⋊C4—1062no—
317C22⋊C4—1062no—
318C22⋊C4—1062no—
319C22⋊C4—2121no—
320C22⋊C4C22⋊D5314yes—
321C42—22no—
322C23—21no—
323C23—21no—
324C23—21no—
325C23—21no—
326C23—21no—
327C23—21no—
328C23—21no—
329C23—21no—
330C23—21no—
331C23—21no—
332C23C2×Dic10612yes—
333C23C2×Dic10612yes—
334C23#1171994512yes—
335C23C2×C22⋊D5312yes—
336C23C23×D5311yescharacteristic
337Q8—2124no—
338D4—28no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—44no—
351C2×C4—44no—
352C2×C4—532no—
353C2×C4—532no—
354C2×C4—532no—
355C2×C4—1068no—
356C2×C4—1061no—
357C2×C4—1061no—
358C2×C4—1061no—
359C2×C4—1061no—
360C2×C4—1061no—
361C2×C4—1061no—
362C2×C4—1061no—
363C2×C4—1061no—
364C2×C4—1061no—
365C2×C4—1061no—
366C2×C4—1061no—
367C2×C4—1061no—
368C2×C4—1061no—
369C2×C4—1061no—
370C2×C4—1061no—
371C2×C4—1061no—
372C2×C4—1061no—
373C2×C4—1061no—
374C2×C4—1061no—
375C2×C4—1061no—
376C2×C4—1061no—
377C2×C4—2124no—
378C2×C4C2×Dic10612yes—
379C2×C4D4×D5312yes—
380C2×C4C2×C22⋊D5312yes—
381C2×C4C2×C22⋊D5312yes—
382C22—22no—
383C22—22no—
384C22—22no—
385C22—22no—
386C22—22no—
387C22—22no—
388C22—22no—
389C22—22no—
390C22—22no—
391C22—41no—
392C22—41no—
393C22—41no—
394C22—41no—
395C22#1313319614yescomplemented
396C22C2×D4×D5311yescharacteristic
397C22C22×C22⋊D5311yescharacteristic
398C22C22×Dic10611yescharacteristic
399C22#41×D5311yescharacteristic
400C22C2×#1171994511yescharacteristic
401C22#1313326211yescharacteristic
402C22#1313326411yescharacteristic
403C4—22no—
404C4—22no—
405C4—22no—
406C4—22no—
407C4—41no—
408C4—1062no—
409C4—1062no—
410C4—1062no—
411C4—2121no—
412C4—2121no—
413C4—2121no—
414C4—2121no—
415C4—2121no—
416C2—24no—
417C2—42no—
418C2C2×#1313319611yescharacteristic
419C2#1681113211yescharacteristic
420C2#1681121511yescharacteristic
421C2D4×C22⋊D5311yescharacteristic
422C2#1681127211yescharacteristic
423C2D4×Dic10611yescharacteristic
424C2#1681133211yescharacteristic
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC2×C4×D4×C53C2
hypercenterC2×C4×D4D53
hypocenterC53#2449
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C4×D4

Complements

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

NH ≅ G/Ntype
1. C2×#13133196C2semidirect
2. C2×#13133194C2semidirect
3. C22×#11719919C2semidirect
4. #16810559C2semidirect
5. #16810556C2semidirect
6. #16810546C2semidirect
7. #16810543C2semidirect
8. #16810521C2semidirect
9. #16810507C2semidirect
12. #16810498C2semidirect
13. #16810496C2semidirect
14. #16810496C2semidirect
59. C2×D4×C53Q8semidirect
108. #11719919D4semidirect
182. C22×C53C22⋊Q8semidirect
228. C53#2449semidirect
275. C2×D4Dic106semidirect
395. C22#13133196semidirect