Finite Group Databasebeta

← #20281084

Subgroups — #20281084

Order 6784.

592 conjugacy classes of subgroups.

The subgroup lattice has 424 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#13133197C211yesmaximal, characteristic, complemented
2C2×#13133195C211yesmaximal, characteristic, complemented
3C2×#13133194C211yesmaximal, characteristic, complemented
4C2×#13133194C211yesmaximal, characteristic, complemented
5#16810547C211yesmaximal, characteristic, complemented
6#16810546C211yesmaximal, characteristic, complemented
7#16810545C211yesmaximal, characteristic, complemented
8#16810544C211yesmaximal, characteristic, complemented
9#16810543C211yesmaximal, characteristic, complemented
10#16810523C211yesmaximal, characteristic
11#16810519C211yesmaximal, characteristic, complemented
12#16810518C211yesmaximal, characteristic, complemented
13#16810500C211yesmaximal, characteristic
14#16810499C211yesmaximal, characteristic
15C53×#396C211yesmaximal, characteristic, complemented
16C22×Dic106C2211yescharacteristic
17C22×Dic106C2211yescharacteristic
18C22×C22⋊D53C2211yescharacteristic
19C23×Dic53C2211yescharacteristic
20C22×C4×D53C2211yescharacteristic
21#13133197—24no—
22#13133195—24no—
23#13133194—24no—
24#13133194—24no—
25C2×C4⋊Dic53C2211yescharacteristic
26C2×C4⋊Dic53C2211yescharacteristic
27C2×C4⋊Dic53C2211yescharacteristic
28C2×#11719919C2211yescharacteristic
29C2×#11719919C2211yescharacteristic
30C2×#11719919C2211yescharacteristic
31C2×#11719919C2211yescharacteristic
32C2×#11719919C2211yescharacteristic
33C2×#11719919C2211yescharacteristic
34C2×#11719919C2211yescharacteristic
35C2×C22⋊Dic53C2211yescharacteristic
36C2×C22⋊Dic53C2211yescharacteristic
37C2×#11719921C2211yescharacteristic
38C2×#11719921C2211yescharacteristic
39C2×#11719921C2211yescharacteristic
40C2×#11719921C2211yescharacteristic
41C2×#11719921C2211yescharacteristic
42C2×C4×Dic53C2211yescharacteristic
43#13133124C2211yescharacteristic
44#13133124C2211yescharacteristic
45#13133124C2211yescharacteristic
46#13133124C2211yescharacteristic
47#13133124C2211yescharacteristic
48C2×C4⋊C4×C53C2211yescharacteristic
49C2×C4⋊C4×C53C2211yescharacteristic
50C2×C4⋊C4×C53C2211yescharacteristic
51C2×C22⋊C4×C53C2211yescharacteristic
52C2×C22⋊C4×C53C2211yescharacteristic
53C2×C22⋊C4×C53C2211yescharacteristic
54#95×C53C2211yescharacteristic
55C24×C53C2311yescharacteristic
56C23×D53C2311yescharacteristic
57C22×C4×C53C2311yescharacteristic
58C22×C4×C53C2311yescharacteristic
59C22×C4×C53C2311yescharacteristic
60C22×C4×C53C2311yescharacteristic
61C22×C4×C53C2311yescharacteristic
62C22×C4×C53C2311yescharacteristic
63C2×C22⋊D53—41no—
64C2×C22⋊D53—41no—
65C2×C22⋊D53Q814yescomplemented
66C22×Dic53—22no—
67C22×Dic53—22no—
68C22×Dic53—41no—
69C22×Dic53C2311yescharacteristic
70C22×Dic53C2311yescharacteristic
71C22×Dic53C2311yescharacteristic
72C22×Dic53C2311yescharacteristic
73C22×Dic53C2311yescharacteristic
74C22×Dic53C2311yescharacteristic
75C22×Dic53C2311yescharacteristic
76C2×C4×D53—22no—
77C2×C4×D53—22no—
78C2×C4×D53—41no—
79C2×Dic106—22no—
80C2×Dic106—22no—
81C2×Dic106—22no—
82C2×Dic106—22no—
83C2×Dic106—41no—
84C2×Dic106—41no—
85C4⋊C4×C53—22no—
86C4⋊C4×C53—22no—
87C4⋊C4×C53—41no—
88C22⋊C4×C53—22no—
89C22⋊C4×C53—22no—
90C22⋊C4×C53—41no—
91C22⋊Dic53—22no—
92C22⋊Dic53—22no—
93#11719921—22no—
94#11719921—22no—
95#11719921—22no—
96#11719921—22no—
97#11719921—41no—
98C4⋊Dic53—22no—
99C4⋊Dic53—22no—
100C4⋊Dic53D414yescomplemented
101#11719919—22no—
102#11719919—22no—
103#11719919—22no—
104#11719919—22no—
105#11719919—41no—
106#11719919—41no—
107#11719919D414yescomplemented
108C4×Dic53—41no—
109C23×C53—21no—
110C23×C53—21no—
111C23×C53—21no—
112C23×C53—21no—
113C23×C53—21no—
114C23×C53—21no—
115C23×C53C2×Q812yes—
116C23×C53C2411yescharacteristic
117C22×D53—21no—
118C22×D53—21no—
119C22×D53—21no—
120C22×D53—21no—
121C22×D53—21no—
122C22×D53—21no—
123C22×D53C2×Q812yes—
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×C53—21no—
136C2×C4×C53—21no—
137C2×C4×C53—21no—
138C2×C4×C53—21no—
139C2×C4×C53—21no—
140C2×C4×C53C2×D412yes—
141C2×C4×C53C2×D412yes—
142C22⋊D53—44no—
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—44no—
160C2×Dic53C2×D412yes—
161C2×Dic53C2×D412yes—
162C2×Dic53C2×D412yes—
163C2×Dic53C2×D412yes—
164C2×Dic53C2×Q812yes—
165C4×D53—44no—
166Dic106—44no—
167Dic106—44no—
168C22×C53—22no—
169C22×C53—22no—
170C22×C53—22no—
171C22×C53—41no—
172C22×C53—41no—
173C22×C53—41no—
174C22×C53—41no—
175C22×C53C22×D411yescharacteristic
176C22×C53C22×D411yescharacteristic
177C22×C53C22×Q811yescharacteristic
178C22×C53H(5,𝔽2)11yescharacteristic
179C22×C53H(5,𝔽2)11yescharacteristic
180C22×C532−1+411yescharacteristic
181C22×C532−1+411yescharacteristic
182C2×D53—22no—
183C2×D53—22no—
184C2×D53—22no—
185C2×D53—41no—
186C2×D53—41no—
187C2×D53—41no—
188C2×D53—41no—
189C4×C53—22no—
190C4×C53—22no—
191C4×C53—41no—
192C4×C53—41no—
193C4×C53—41no—
194C4×C53—41no—
195Dic53—22no—
196Dic53—22no—
197Dic53—22no—
198Dic53—22no—
199Dic53—22no—
200Dic53—41no—
201Dic53—41no—
202#2683—531nomaximal, Sylow
203C2×C53—42no—
204C2×C53#53711yescharacteristic
205C2×C53#53711yescharacteristic
206C2×C53#54311yescharacteristic
207C2×C53D4211yescharacteristic
208C2×C53D4×Q811yescharacteristic
209C2×C53D4×Q811yescharacteristic
210C2×C53#56311yescharacteristic
211D53—42no—
212C2×C22⋊Q8—531no—
213C2×C22⋊Q8—531no—
214C2×C22⋊Q8—531no—
215C2×C22⋊Q8—531no—
216#396—531no—
217#396—531no—
218#396—531no—
219#396D5311yescharacteristic, complemented
220#395—531no—
221#395—531no—
222#390—531no—
223#389—531no—
224#389—531no—
225#386—531no—
226#386—531no—
227C53#268311yescharacteristic, complemented, Sylow
228C22×Q8—531no—
229C22×Q8—531no—
230C22×D4—531no—
231C23×C4—531no—
232C23×C4—531no—
233C22⋊Q8—1064no—
234C22⋊Q8—1064no—
235C22⋊Q8—1064no—
236C22⋊Q8—1064no—
237C2×C4⋊C4—531no—
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⋊C4C2×D5311yescharacteristic
248C2×C4⋊C4C2×D5311yescharacteristic
249C2×C4⋊C4C2×D5311yescharacteristic
250C2×C22⋊C4—531no—
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×C22⋊C4C2×D5311yescharacteristic
260C2×C42—531no—
261#95—531no—
262#95—531no—
263#95—531no—
264#95—531no—
265#95—531no—
266#95C2×D5311yescharacteristic
267C24—531no—
268C24C22×D5311yescharacteristic
269C2×Q8—1062no—
270C2×Q8—1062no—
271C2×Q8—1062no—
272C2×Q8—1062no—
273C2×Q8—2121no—
274C2×Q8—2121no—
275C2×D4—534no—
276C2×D4—2121no—
277C2×D4—2121no—
278C22×C4—531no—
279C22×C4—531no—
280C22×C4—531no—
281C22×C4—531no—
282C22×C4—531no—
283C22×C4—531no—
284C22×C4—531no—
285C22×C4—1062no—
286C22×C4—1062no—
287C22×C4—1062no—
288C22×C4—1062no—
289C22×C4—2121no—
290C22×C4—2121no—
291C22×C4C22×D5311yescharacteristic
292C22×C4C22×D5311yescharacteristic
293C22×C4C22×D5311yescharacteristic
294C22×C4C22×D5311yescharacteristic
295C22×C4C22×D5311yescharacteristic
296C22×C4C22×D5311yescharacteristic
297C4⋊C4—22no—
298C4⋊C4—22no—
299C4⋊C4—41no—
300C4⋊C4—534no—
301C4⋊C4—534no—
302C4⋊C4—1062no—
303C4⋊C4—1062no—
304C4⋊C4—1062no—
305C4⋊C4—1062no—
306C4⋊C4—1062no—
307C4⋊C4—1062no—
308C4⋊C4—2121no—
309C4⋊C4—2121no—
310C22⋊C4—22no—
311C22⋊C4—22no—
312C22⋊C4—41no—
313C22⋊C4—1062no—
314C22⋊C4—1062no—
315C22⋊C4—1062no—
316C22⋊C4—1062no—
317C22⋊C4—1062no—
318C22⋊C4—1062no—
319C22⋊C4—2121no—
320C42—2121no—
321C23—21no—
322C23—21no—
323C23—21no—
324C23—21no—
325C23—21no—
326C23—21no—
327C23—532no—
328C23—1061no—
329C23—1061no—
330C23—1061no—
331C23—1061no—
332C23—1061no—
333C23—1061no—
334C23Q8×D5312yes—
335C23C23×D5311yescharacteristic
336Q8—2124no—
337Q8—2124no—
338D4—2124no—
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—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—532no—
356C2×C4—532no—
357C2×C4—532no—
358C2×C4—532no—
359C2×C4—532no—
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—2124no—
377C2×C4—2124no—
378C2×C4D4×D5312yes—
379C2×C4D4×D5312yes—
380C22—22no—
381C22—22no—
382C22—22no—
383C22—41no—
384C22—41no—
385C22—41no—
386C22—41no—
387C22—1062no—
388C22—1062no—
389C22—1062no—
390C22—2121no—
391C22—2121no—
392C22—2121no—
393C22—2121no—
394C22C2×D4×D5311yescharacteristic
395C22C2×D4×D5311yescharacteristic
396C22C2×Q8×D5311yescharacteristic
397C22#1313326211yescharacteristic
398C22#1313326311yescharacteristic
399C22#1313326411yescharacteristic
400C22#1313326411yescharacteristic
401C4—22no—
402C4—22no—
403C4—41no—
404C4—41no—
405C4—41no—
406C4—41no—
407C4—1062no—
408C4—1062no—
409C4—1062no—
410C4—1062no—
411C4—1062no—
412C4—2121no—
413C4—2121no—
414C2—42no—
415C2—2122no—
416C2#1681120011yescharacteristic
417C2#1681120211yescharacteristic
418C2#1681124811yescharacteristic
419C2#1681125711yescharacteristic
420C2#1681128611yescharacteristic
421C2#1681128611yescharacteristic
422C2#1681137511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#396C2
hypercenter#396D53
hypocenterC53#2683
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #396

Complements

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

NH ≅ G/Ntype
1. C2×#13133197C2semidirect
2. C2×#13133195C2semidirect
3. C2×#13133194C2semidirect
4. C2×#13133194C2semidirect
5. #16810547C2semidirect
6. #16810546C2semidirect
7. #16810545C2semidirect
8. #16810544C2semidirect
9. #16810543C2semidirect
11. #16810519C2semidirect
12. #16810518C2semidirect
15. C53×#396C2semidirect
65. C2×C22⋊D53Q8semidirect
100. C4⋊Dic53D4semidirect
107. #11719919D4semidirect
219. #396D53semidirect
227. C53#2683semidirect