Finite Group Databasebeta

← #20281209

Subgroups — #20281209

Order 6784.

536 conjugacy classes of subgroups.

The subgroup lattice has 412 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#13133197C211yesmaximal, characteristic, complemented
2C2×#13133195C211yesmaximal, characteristic, complemented
3#16810560C211yesmaximal, characteristic, complemented
4#16810555C211yesmaximal, characteristic, complemented
5#16810545C211yesmaximal, characteristic, complemented
6#16810544C211yesmaximal, characteristic, complemented
7#16810536C211yesmaximal, characteristic, complemented
8#16810536C211yesmaximal, characteristic, complemented
9#16810529C211yesmaximal, characteristic, complemented
10#16810525C211yesmaximal, characteristic
11#16810501C211yesmaximal, characteristic
12#16810501C211yesmaximal, characteristic
13#16810497C211yesmaximal, characteristic, complemented
14#16810480C211yesmaximal, characteristic, complemented
15C53×#387C211yesmaximal, characteristic, complemented
16C22×Dic106C2211yescharacteristic
17C22×Dic106C2211yescharacteristic
18C22×D212C2211yescharacteristic
19C22×C4×D53C2211yescharacteristic
20C22×C4×D53C2211yescharacteristic
21#13133197—24no—
22#13133195—24no—
23C2×C4⋊Dic53C2211yescharacteristic
24C2×C4⋊Dic53C2211yescharacteristic
25C2×C4⋊Dic53C2211yescharacteristic
26C2×C4⋊Dic53C2211yescharacteristic
27C2×C4⋊Dic53C2211yescharacteristic
28C2×#11719919C2211yescharacteristic
29C2×#11719919C2211yescharacteristic
30C2×#11719921C2211yescharacteristic
31C2×#11719921C2211yescharacteristic
32C2×#11719921C2211yescharacteristic
33C2×#11719921C2211yescharacteristic
34C2×#11719921C2211yescharacteristic
35C2×#11719921C2211yescharacteristic
36C2×#11719921C2211yescharacteristic
37C2×#11719921C2211yescharacteristic
38C2×#11719921C2211yescharacteristic
39C2×#11719921C2211yescharacteristic
40#13133124C2211yescharacteristic
41#13133124C2211yescharacteristic
42#13133124C2211yescharacteristic
43#13133124C2211yescharacteristic
44#13133124C2211yescharacteristic
45#13133124C2211yescharacteristic
46C2×C4⋊C4×C53C2211yescharacteristic
47C2×C4⋊C4×C53C2211yescharacteristic
48C2×C42×C53C2211yescharacteristic
49#95×C53C2211yescharacteristic
50#95×C53C2211yescharacteristic
51#95×C53C2211yescharacteristic
52#95×C53C2211yescharacteristic
53C23×D53C2311yescharacteristic
54C23×D53C2311yescharacteristic
55C22×C4×C53C2311yescharacteristic
56C22×C4×C53C2311yescharacteristic
57C22×C4×C53C2311yescharacteristic
58C22×C4×C53C2311yescharacteristic
59C22×C4×C53C2311yescharacteristic
60C22×C4×C53C2311yescharacteristic
61C22×C4×C53C2311yescharacteristic
62C22×Dic53C2311yescharacteristic
63C22×Dic53C2311yescharacteristic
64C22×Dic53C2311yescharacteristic
65C22×Dic53C2311yescharacteristic
66C22×Dic53C2311yescharacteristic
67C22×Dic53C2311yescharacteristic
68C2×D212—41no—
69C2×D212—41no—
70C2×D212Q814yescomplemented
71C2×C4×D53—22no—
72C2×C4×D53—22no—
73C2×C4×D53—22no—
74C2×C4×D53—41no—
75C2×C4×D53—41no—
76C2×C4×D53—41no—
77C2×Dic106—22no—
78C2×Dic106—22no—
79C2×Dic106—22no—
80C2×Dic106—41no—
81C2×Dic106—41no—
82C2×Dic106—41no—
83C4⋊C4×C53—41no—
84C4⋊C4×C53D414yescomplemented
85C42×C53—41no—
86#11719921—22no—
87#11719921—22no—
88#11719921—22no—
89#11719921—22no—
90#11719921—22no—
91#11719921—41no—
92#11719921—41no—
93#11719921—41no—
94#11719921—41no—
95#11719921—41no—
96C4⋊Dic53—22no—
97C4⋊Dic53—22no—
98C4⋊Dic53—41no—
99C4⋊Dic53—41no—
100C4⋊Dic53—41no—
101#11719919—22no—
102#11719919—22no—
103C23×C53C2411yescharacteristic
104C22×D53—21no—
105C22×D53—21no—
106C22×D53—21no—
107C22×D53—21no—
108C22×D53—21no—
109C22×D53—21no—
110C22×D53—21no—
111C22×D53—21no—
112C22×D53—21no—
113C22×D53—21no—
114C22×D53—21no—
115C22×D53—21no—
116C22×D53C2×Q812yes—
117C22×D53C2×Q812yes—
118C2×C4×C53—21no—
119C2×C4×C53—21no—
120C2×C4×C53—21no—
121C2×C4×C53—21no—
122C2×C4×C53—21no—
123C2×C4×C53—21no—
124C2×C4×C53—21no—
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×C53C2×D412yes—
136C2×C4×C53C2×D412yes—
137C2×C4×C53C2×D412yes—
138C2×C4×C53C2×Q812yes—
139C2×Dic53—21no—
140C2×Dic53—21no—
141C2×Dic53—21no—
142C2×Dic53—21no—
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#4112yes—
157D212—44no—
158C4×D53—44no—
159C4×D53—44no—
160Dic106—44no—
161Dic106—44no—
162C22×C53C22×D411yescharacteristic
163C22×C53C22×Q811yescharacteristic
164C22×C53C2×#4111yescharacteristic
165C22×C53H(5,𝔽2)11yescharacteristic
166C22×C53H(5,𝔽2)11yescharacteristic
167C22×C53H(5,𝔽2)11yescharacteristic
168C22×C532−1+411yescharacteristic
169C2×D53—22no—
170C2×D53—22no—
171C2×D53—22no—
172C2×D53—22no—
173C2×D53—22no—
174C2×D53—22no—
175C2×D53—41no—
176C2×D53—41no—
177C2×D53—41no—
178C2×D53—41no—
179C2×D53—41no—
180C2×D53—41no—
181C2×D53—41no—
182C2×D53—41no—
183C4×C53—22no—
184C4×C53—22no—
185C4×C53—22no—
186C4×C53—22no—
187C4×C53—41no—
188C4×C53—41no—
189C4×C53—41no—
190Dic53—22no—
191Dic53—41no—
192Dic53—41no—
193Dic53—41no—
194Dic53—41no—
195Dic53—41no—
196#2692—531nomaximal, Sylow
197C2×C53#53511yescharacteristic
198C2×C53#54211yescharacteristic
199C2×C53#54311yescharacteristic
200C2×C53#54611yescharacteristic
201C2×C53D4×Q811yescharacteristic
202C2×C53#55411yescharacteristic
203C2×C53#56211yescharacteristic
204D53—42no—
205D53—42no—
206C2×C22⋊Q8—531no—
207C2×C22⋊Q8—531no—
208#399—531no—
209#398—531no—
210#396—531no—
211#395—531no—
212#393—531no—
213#393—531no—
214#391—531no—
215#390—531no—
216#387D5311yescharacteristic, complemented
217#386—531no—
218#386—531no—
219#385—531no—
220#380—531no—
221C53#269211yescharacteristic, complemented, Sylow
222C22×Q8—531no—
223C22×Q8—531no—
224C22×D4—531no—
225C23×C4—531no—
226C23×C4—531no—
227C22⋊Q8—1064no—
228C22⋊Q8—1064no—
229C2×C4⋊C4—531no—
230C2×C4⋊C4—531no—
231C2×C4⋊C4—531no—
232C2×C4⋊C4—531no—
233C2×C4⋊C4—531no—
234C2×C4⋊C4—531no—
235C2×C4⋊C4—531no—
236C2×C4⋊C4C2×D5311yescharacteristic
237C2×C4⋊C4C2×D5311yescharacteristic
238C2×C22⋊C4—531no—
239C2×C22⋊C4—531no—
240C2×C22⋊C4—531no—
241C2×C22⋊C4—531no—
242C2×C22⋊C4—531no—
243C2×C22⋊C4—531no—
244C2×C22⋊C4—531no—
245C2×C22⋊C4—531no—
246C2×C22⋊C4—531no—
247C2×C22⋊C4—531no—
248C2×C42C2×D5311yescharacteristic
249#95—531no—
250#95—531no—
251#95—531no—
252#95—531no—
253#95—531no—
254#95—531no—
255#95C2×D5311yescharacteristic
256#95C2×D5311yescharacteristic
257#95C2×D5311yescharacteristic
258#95C2×D5311yescharacteristic
259C24—531no—
260C24—531no—
261C2×Q8—1062no—
262C2×Q8—1062no—
263C2×Q8—1062no—
264C2×Q8—2121no—
265C2×Q8—2121no—
266C2×Q8—2121no—
267C2×D4—534no—
268C2×D4—2121no—
269C2×D4—2121no—
270C22×C4—531no—
271C22×C4—531no—
272C22×C4—531no—
273C22×C4—531no—
274C22×C4—531no—
275C22×C4—531no—
276C22×C4—1062no—
277C22×C4—1062no—
278C22×C4—1062no—
279C22×C4—2121no—
280C22×C4—2121no—
281C22×C4—2121no—
282C22×C4C22×D5311yescharacteristic
283C22×C4C22×D5311yescharacteristic
284C22×C4C22×D5311yescharacteristic
285C22×C4C22×D5311yescharacteristic
286C22×C4C22×D5311yescharacteristic
287C22×C4C22×D5311yescharacteristic
288C22×C4C22×D5311yescharacteristic
289C4⋊C4—41no—
290C4⋊C4—1062no—
291C4⋊C4—1062no—
292C4⋊C4—1062no—
293C4⋊C4—1062no—
294C4⋊C4—2121no—
295C4⋊C4—2121no—
296C4⋊C4—2121no—
297C4⋊C4D21214yescomplemented
298C22⋊C4—1062no—
299C22⋊C4—1062no—
300C22⋊C4—1062no—
301C22⋊C4—1062no—
302C22⋊C4—1062no—
303C22⋊C4—2121no—
304C22⋊C4—2121no—
305C22⋊C4—2121no—
306C22⋊C4—2121no—
307C22⋊C4—2121no—
308C42—41no—
309C23—532no—
310C23—532no—
311C23—1061no—
312C23—1061no—
313C23—1061no—
314C23—1061no—
315C23—1061no—
316C23—1061no—
317C23—1061no—
318C23—1061no—
319C23—1061no—
320C23—1061no—
321C23—1061no—
322C23—1061no—
323C23C23×D5311yescharacteristic
324Q8—2124no—
325Q8—2124no—
326D4—2124no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—532no—
345C2×C4—1061no—
346C2×C4—1061no—
347C2×C4—1061no—
348C2×C4—1061no—
349C2×C4—1061no—
350C2×C4—1061no—
351C2×C4—1061no—
352C2×C4—1061no—
353C2×C4—1061no—
354C2×C4—1061no—
355C2×C4—1061no—
356C2×C4—1061no—
357C2×C4—1061no—
358C2×C4—1061no—
359C2×C4—1061no—
360C2×C4—1061no—
361C2×C4—1061no—
362C2×C4—2124no—
363C2×C4—2124no—
364C2×C4C2×D21212yes—
365C2×C4C2×D21212yes—
366C2×C4C2×D21212yes—
367C2×C4Q8×D5312yes—
368C22—1062no—
369C22—1062no—
370C22—1062no—
371C22—1062no—
372C22—1062no—
373C22—1062no—
374C22—2121no—
375C22—2121no—
376C22—2121no—
377C22—2121no—
378C22—2121no—
379C22—2121no—
380C22—2121no—
381C22—2121no—
382C22C22×D21211yescharacteristic
383C22C2×Q8×D5311yescharacteristic
384C22#41×D5311yescharacteristic
385C22#1313326211yescharacteristic
386C22#1313326311yescharacteristic
387C22#1313326311yescharacteristic
388C22#1313326411yescharacteristic
389C4—22no—
390C4—22no—
391C4—22no—
392C4—22no—
393C4—41no—
394C4—41no—
395C4—41no—
396C4—1062no—
397C4—2121no—
398C4—2121no—
399C4—2121no—
400C4—2121no—
401C4—2121no—
402C2—2122no—
403C2—2122no—
404C2#1681119211yescharacteristic
405C2#1681123911yescharacteristic
406C2#1681124811yescharacteristic
407C2#1681126811yescharacteristic
408C2Q8×D21211yescharacteristic
409C2#1681132911yescharacteristic
410C2#1681137011yescharacteristic
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#387C2
hypercenter#387D53
hypocenterC53#2692
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #387

Complements

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

NH ≅ G/Ntype
1. C2×#13133197C2semidirect
2. C2×#13133195C2semidirect
3. #16810560C2semidirect
4. #16810555C2semidirect
5. #16810545C2semidirect
6. #16810544C2semidirect
7. #16810536C2semidirect
8. #16810536C2semidirect
9. #16810529C2semidirect
13. #16810497C2semidirect
14. #16810480C2semidirect
15. C53×#387C2semidirect
70. C2×D212Q8semidirect
84. C4⋊C4×C53D4semidirect
216. #387D53semidirect
221. C53#2692semidirect
297. C4⋊C4D212semidirect