Finite Group Databasebeta

← #20277613

Subgroups — #20277613

Order 6784.

776 conjugacy classes of subgroups.

The subgroup lattice has 438 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×C22⋊D53C211yesmaximal, characteristic
2C2×#13133177C211yesmaximal, characteristic
3C2×#13133177C211yesmaximal, characteristic
4C22×#11719919C211yesmaximal, characteristic, complemented
5C2×C4⋊C4×D53C211yesmaximal, characteristic, complemented
6#16810520C211yesmaximal, characteristic, complemented
7#16810518C211yesmaximal, characteristic, complemented
8#16810498C211yesmaximal, characteristic, complemented
9#16810497C211yesmaximal, characteristic, complemented
10#16810494C211yesmaximal, characteristic, complemented
11#16810492C211yesmaximal, characteristic, complemented
12#16810482C211yesmaximal, characteristic, complemented
13#16810481C211yesmaximal, characteristic, complemented
14#16810474C211yesmaximal, characteristic, complemented
15C53×#385C211yesmaximal, characteristic, complemented
16C22×C22⋊D53C2211yescharacteristic
17C23×Dic53C2211yescharacteristic
18C23×Dic53C2211yescharacteristic
19C22×C4×D53C2211yescharacteristic
20C22×C4×D53C2211yescharacteristic
21C22×C4×D53C2211yescharacteristic
22C4×C22⋊D53—24no—
23#13133177—24no—
24#13133177C418yescomplemented
25C2×C4⋊Dic53C2211yescharacteristic
26C2×C4⋊Dic53C2211yescharacteristic
27C2×#11719919—24no—
28C2×#11719919C2211yescharacteristic
29C2×#11719919C2211yescharacteristic
30C2×#11719919C2211yescharacteristic
31C2×#11719919C2211yescharacteristic
32C2×#11719919C2211yescharacteristic
33C2×#11719919C2211yescharacteristic
34C2×#11719919C2211yescharacteristic
35C2×#11719919C2211yescharacteristic
36C2×#11719919C2211yescharacteristic
37C4⋊C4×D53—24no—
38C2×C22⋊Dic53C2211yescharacteristic
39C2×#11719921C2211yescharacteristic
40C2×#11719921C2211yescharacteristic
41C2×#11719921C2211yescharacteristic
42C2×C4×Dic53C2211yescharacteristic
43C2×C4×Dic53C2211yescharacteristic
44C2×C4×Dic53C2211yescharacteristic
45#13133124C2211yescharacteristic
46#13133124C2211yescharacteristic
47#13133124C2211yescharacteristic
48#13133124C2211yescharacteristic
49C23×C4×C53C2211yescharacteristic
50C2×C4⋊C4×C53C2211yescharacteristic
51C2×C4⋊C4×C53C2211yescharacteristic
52C2×C22⋊C4×C53C2211yescharacteristic
53C2×C22⋊C4×C53C2211yescharacteristic
54#95×C53C2211yescharacteristic
55#95×C53C2211yescharacteristic
56C24×C53C2311yescharacteristic
57C23×D53C2311yescharacteristic
58C22×C4×C53—22no—
59C22×C4×C53—22no—
60C22×C4×C53—41no—
61C22×C4×C53C2311yescharacteristic
62C22×C4×C53C2311yescharacteristic
63C22×C4×C53C2311yescharacteristic
64C22×C4×C53C2311yescharacteristic
65C22×C4×C53C2311yescharacteristic
66C22×C4×C53C2311yescharacteristic
67C2×C22⋊D53C2×C414yes—
68C2×C22⋊D53D414yes—
69C2×C22⋊D53Q814yes—
70C22×Dic53—22no—
71C22×Dic53—22no—
72C22×Dic53—22no—
73C22×Dic53—22no—
74C22×Dic53—41no—
75C22×Dic53C2×C414yescomplemented
76C22×Dic53C2311yescharacteristic
77C22×Dic53C2311yescharacteristic
78C22×Dic53C2311yescharacteristic
79C22×Dic53C2311yescharacteristic
80C22×Dic53C2311yescharacteristic
81C22×Dic53C2311yescharacteristic
82C22×Dic53C2311yescharacteristic
83C2×C4×D53—22no—
84C2×C4×D53—22no—
85C2×C4×D53—22no—
86C2×C4×D53—22no—
87C2×C4×D53—22no—
88C2×C4×D53—22no—
89C2×C4×D53—41no—
90C2×C4×D53—41no—
91C2×C4×D53C2×C414yescomplemented
92C4⋊C4×C53—22no—
93C4⋊C4×C53—41no—
94C22⋊C4×C53—22no—
95C22⋊C4×C53C2×C414yescomplemented
96C22⋊Dic53—22no—
97#11719921—22no—
98#11719921—22no—
99#11719921C2×C414yescomplemented
100C4⋊Dic53—22no—
101C4⋊Dic53—41no—
102#11719919—22no—
103#11719919—22no—
104#11719919—22no—
105#11719919—22no—
106#11719919—22no—
107#11719919—22no—
108#11719919—22no—
109#11719919C2×C414yescomplemented
110#11719919D414yescomplemented
111C4×Dic53—22no—
112C4×Dic53—22no—
113C4×Dic53C2×C414yescomplemented
114C23×C53—21no—
115C23×C53—21no—
116C23×C53—21no—
117C23×C53—21no—
118C23×C53C22×C412yes—
119C23×C53C2×D412yes—
120C23×C53C2×Q812yes—
121C23×C53C2411yescharacteristic
122C22×D53—21no—
123C22×D53—21no—
124C22×D53—21no—
125C22×D53—21no—
126C22×D53C22×C412yes—
127C22×D53C2×D412yes—
128C22×D53C2×Q812yes—
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×C53—21no—
141C2×C4×C53—21no—
142C2×C4×C53—21no—
143C2×C4×C53—21no—
144C2×C4×C53—44no—
145C2×C4×C53C22×C412yes—
146C2×C4×C53C22×C412yes—
147C2×C4×C53C2×D412yes—
148C22⋊D53C4⋊C4116yescomplemented
149C2×Dic53—21no—
150C2×Dic53—21no—
151C2×Dic53—21no—
152C2×Dic53—21no—
153C2×Dic53—28no—
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—44no—
164C2×Dic53C22×C412yes—
165C2×Dic53C22×C412yes—
166C2×Dic53C22×C412yes—
167C2×Dic53C2×D412yes—
168C2×Dic53C2×D412yes—
169C2×Dic53C2×D412yes—
170C2×Dic53C2×Q812yes—
171C2×Dic53#4112yes—
172C4×D53—28no—
173C4×D53—44no—
174C4×D53—44no—
175C22×C53—22no—
176C22×C53—22no—
177C22×C53—22no—
178C22×C53—22no—
179C22×C53—22no—
180C22×C53—22no—
181C22×C53C2×C4⋊C414yescomplemented
182C22×C53C23×C411yescharacteristic
183C22×C53C22×D411yescharacteristic
184C22×C53C22×D411yescharacteristic
185C22×C53C22×Q811yescharacteristic
186C22×C53C2×#4111yescharacteristic
187C22×C53H(5,𝔽2)11yescharacteristic
188C22×C532−1+411yescharacteristic
189C2×D53—22no—
190C2×D53—22no—
191C2×D53—22no—
192C2×D53—22no—
193C2×D53—22no—
194C2×D53—22no—
195C2×D53C2×C4⋊C414yescomplemented
196C4×C53—22no—
197C4×C53—22no—
198C4×C53—22no—
199C4×C53—41no—
200C4×C53—41no—
201C4×C53—41no—
202Dic53—22no—
203Dic53—22no—
204Dic53—41no—
205Dic53—41no—
206Dic53—41no—
207Dic53C2×C4⋊C414yescomplemented
208Dic53C4×D414yescomplemented
209D4×C4⋊C4—531nomaximal, Sylow
210C2×C53—24no—
211C2×C53C22×C4⋊C411yescharacteristic
212C2×C53C2×C4×D411yescharacteristic
213C2×C53#52011yescharacteristic
214C2×C53D4211yescharacteristic
215C2×C53#54711yescharacteristic
216C2×C53D4×Q811yescharacteristic
217C2×C53#55411yescharacteristic
218D53—24no—
219C2×C4×D4—531no—
220C2×C4×D4—531no—
221C2×C4×D4—531no—
222C22×C4⋊C4—531no—
223C22×C4⋊C4—531no—
224#389—531no—
225#389—531no—
226#385—531no—
227#385—531no—
228#385—531no—
229#385D5311yescharacteristic, complemented
230#384—531no—
231#380—531no—
232#380—531no—
233C4×C4⋊C4—531no—
234C53D4×C4⋊C411yescharacteristic, complemented, Sylow
235C22×D4—531no—
236C23×C4—531no—
237C23×C4—531no—
238C23×C4—531no—
239C23×C4—531no—
240C23×C4—531no—
241C23×C4C2×D5311yescharacteristic
242C4×D4—538no—
243C4×D4—1064no—
244C4×D4—1064no—
245C2×C4⋊C4—531no—
246C2×C4⋊C4—531no—
247C2×C4⋊C4—531no—
248C2×C4⋊C4—531no—
249C2×C4⋊C4—531no—
250C2×C4⋊C4—531no—
251C2×C4⋊C4—531no—
252C2×C4⋊C4—531no—
253C2×C4⋊C4—531no—
254C2×C4⋊C4—531no—
255C2×C4⋊C4—531no—
256C2×C4⋊C4—1064no—
257C2×C4⋊C4—1064no—
258C2×C4⋊C4C2×D5311yescharacteristic
259C2×C4⋊C4C2×D5311yescharacteristic
260C2×C22⋊C4—531no—
261C2×C22⋊C4—531no—
262C2×C22⋊C4—531no—
263C2×C22⋊C4—531no—
264C2×C22⋊C4C2×D5311yescharacteristic
265C2×C22⋊C4C2×D5311yescharacteristic
266C2×C42—531no—
267C2×C42—531no—
268C2×C42—531no—
269#95—531no—
270#95—531no—
271#95—531no—
272#95—531no—
273#95C2×D5311yescharacteristic
274#95C2×D5311yescharacteristic
275C24—531no—
276C24C22×D5311yescharacteristic
277C2×D4—534no—
278C2×D4—534no—
279C2×D4—534no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—41no—
283C22×C4—531no—
284C22×C4—531no—
285C22×C4—534no—
286C22×C4—531no—
287C22×C4—531no—
288C22×C4—534no—
289C22×C4—531no—
290C22×C4—531no—
291C22×C4—531no—
292C22×C4—1062no—
293C22×C4—1062no—
294C22×C4—1062no—
295C22×C4—1062no—
296C22×C4—1062no—
297C22×C4—1062no—
298C22×C4—1062no—
299C22×C4—1062no—
300C22×C4—1062no—
301C22×C4—1062no—
302C22×C4—2121no—
303C22×C4—2121no—
304C22×C4—2121no—
305C22×C4C22×D5311yescharacteristic
306C22×C4C22×D5311yescharacteristic
307C22×C4C22×D5311yescharacteristic
308C22×C4C22×D5311yescharacteristic
309C22×C4C22×D5311yescharacteristic
310C22×C4C22×D5311yescharacteristic
311C4⋊C4—22no—
312C4⋊C4—41no—
313C4⋊C4—534no—
314C4⋊C4—534no—
315C4⋊C4—1062no—
316C4⋊C4—1062no—
317C4⋊C4—1062no—
318C4⋊C4—1062no—
319C4⋊C4—1062no—
320C4⋊C4—1062no—
321C4⋊C4—1062no—
322C4⋊C4—1062no—
323C4⋊C4—2121no—
324C22⋊C4—22no—
325C22⋊C4—534no—
326C22⋊C4—1062no—
327C22⋊C4—1062no—
328C22⋊C4—1062no—
329C22⋊C4C4×D5314yescomplemented
330C42—534no—
331C42—1062no—
332C42—1062no—
333C23—21no—
334C23—21no—
335C23—21no—
336C23—21no—
337C23—532no—
338C23—532no—
339C23—532no—
340C23—1061no—
341C23—1061no—
342C23—1061no—
343C23—1061no—
344C23C2×C4×D5312yes—
345C23D4×D5312yes—
346C23Q8×D5312yes—
347C23C23×D5311yescharacteristic
348D4—5316no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—44no—
365C2×C4—532no—
366C2×C4—532no—
367C2×C4—532no—
368C2×C4—532no—
369C2×C4—532no—
370C2×C4—532no—
371C2×C4—532no—
372C2×C4—532no—
373C2×C4—1068no—
374C2×C4—1068no—
375C2×C4—1061no—
376C2×C4—1061no—
377C2×C4—1061no—
378C2×C4—1061no—
379C2×C4—1061no—
380C2×C4—1061no—
381C2×C4—1061no—
382C2×C4—1061no—
383C2×C4—1061no—
384C2×C4—1061no—
385C2×C4—1061no—
386C2×C4—1061no—
387C2×C4—1061no—
388C2×C4—2124no—
389C2×C4—2124no—
390C2×C4—2124no—
391C2×C4C2×C4×D5312yes—
392C2×C4C2×C4×D5312yes—
393C2×C4D4×D5312yes—
394C22—22no—
395C22—22no—
396C22—22no—
397C22—22no—
398C22—22no—
399C22—22no—
400C22—534no—
401C22—1062no—
402C22—1062no—
403C22—1062no—
404C22—1062no—
405C22—1062no—
406C22—1062no—
407C22C4⋊C4×D5314yescomplemented
408C22C22×C4×D5311yescharacteristic
409C22C2×D4×D5311yescharacteristic
410C22C2×D4×D5311yescharacteristic
411C22C2×Q8×D5311yescharacteristic
412C22#41×D5311yescharacteristic
413C22#1313326211yescharacteristic
414C22#1313326411yescharacteristic
415C4—22no—
416C4—22no—
417C4—22no—
418C4—41no—
419C4—41no—
420C4—41no—
421C4—534no—
422C4—534no—
423C4—1062no—
424C4—1062no—
425C4—2121no—
426C4—2121no—
427C4—2121no—
428C2—24no—
429C2—1064no—
430C2C2×C4⋊C4×D5311yescharacteristic
431C2C4×D4×D5311yescharacteristic
432C2#1681110211yescharacteristic
433C2#1681125511yescharacteristic
434C2#1681127111yescharacteristic
435C2#1681128611yescharacteristic
436C2#1681132711yescharacteristic
subgroupG/H
center, FrattiniC23C23×D53
commutatorC22×C53C23×C4
FittingC53×#385C2
socleC23×C53C24
hypercenter#385D53
hypocenterC53D4×C4⋊C4
LCSG ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ #385

Complements

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

NH ≅ G/Ntype
4. C22×#11719919C2semidirect
5. C2×C4⋊C4×D53C2semidirect
6. #16810520C2semidirect
7. #16810518C2semidirect
8. #16810498C2semidirect
9. #16810497C2semidirect
10. #16810494C2semidirect
11. #16810492C2semidirect
12. #16810482C2semidirect
13. #16810481C2semidirect
14. #16810474C2semidirect
15. C53×#385C2semidirect
24. #13133177C4semidirect
75. C22×Dic53C2×C4semidirect
91. C2×C4×D53C2×C4semidirect
95. C22⋊C4×C53C2×C4semidirect
99. #11719921C2×C4semidirect
109. #11719919C2×C4semidirect
110. #11719919D4semidirect
113. C4×Dic53C2×C4semidirect
148. C22⋊D53C4⋊C4semidirect
181. C22×C53C2×C4⋊C4semidirect
195. C2×D53C2×C4⋊C4semidirect
207. Dic53C2×C4⋊C4semidirect
208. Dic53C4×D4semidirect
229. #385D53semidirect
234. C53D4×C4⋊C4semidirect
329. C22⋊C4C4×D53semidirect
407. C22C4⋊C4×D53semidirect