Finite Group Databasebeta

← #27355881

Subgroups — #27355881

Order 13568.

600 conjugacy classes of subgroups.

The subgroup lattice has 394 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20285285C211yesmaximal, characteristic
2#20283719C212yesmaximal, complemented
3#20273613C211yesmaximal, characteristic, complemented
4#20273601C211yesmaximal, characteristic, complemented
5#20273560C212yesmaximal, complemented
6#20273553C212yesmaximal, complemented
7C8×Dic212C211yesmaximal, characteristic, complemented
8C8×D424C211yesmaximal, characteristic, complemented
9#20273463C211yesmaximal, characteristic, complemented
10#20273124C212yesmaximal, complemented
11C8×M16×C53C211yesmaximal, characteristic, complemented
12#16811157C2212yes—
13#16811025C411yescharacteristic
14#16811007C411yescharacteristic
15#16810978C412yes—
16#16810864C411yescharacteristic
17#16810859C411yescharacteristic
18#16810852C412yes—
19#16810753C2212yes—
20C8×Dic106C2212yes—
21C4×Dic212C2211yescharacteristic
22C4×#11719913C2212yes—
23C4×D424C2211yescharacteristic
24#16810702C2212yes—
25#16810699—22no—
26#16810692—22no—
27C8×D212C2212yes—
28#16810685—22no—
29#16810676C2212yes—
30#16810580—22no—
31#16810376C2211yescharacteristic
32#16810376C2211yescharacteristic
33#16810374C2212yes—
34#16810355C2212yes—
35#16810355C2212yes—
36#16810352C2212yes—
37#16810352C2212yes—
38C53×#431C2211yescharacteristic
39C4×M16×C53C2211yescharacteristic
40C2×C4×C8×C53C2211yescharacteristic
41C53×#322C2211yescharacteristic
42C53×#322C2211yescharacteristic
43C82×C53C2211yescharacteristic
44C82×C53C2211yescharacteristic
45C2×Dic212C2×C411yescharacteristic
46C2×#11719913C2×C412yes—
47C2×D424C2×C411yescharacteristic
48C2×#11719912—22no—
49C2×C8×D53—22no—
50#13133211—22no—
51#13133209C2×C412yes—
52#13133205C2×C412yes—
53#13133200—22no—
54#13133196C2×C412yes—
55#13133191C2×C412yes—
56C4×Dic106C2312yes—
57C4×D212C2312yes—
58C4×C22⋊D53—22no—
59#13133174—22no—
60#13133151C2×C411yescharacteristic
61#13133151C2×C411yescharacteristic
62#13133149C2×C412yes—
63C4⋊#88322C2312yes—
64C4⋊#88322C2312yes—
65#13133146—22no—
66#13133146—22no—
67#13133142C2×C412yes—
68#13133142C2×C412yes—
69#13133139C2×C412yes—
70#13133139C2×C412yes—
71C22⋊#88322—22no—
72#13133129—22no—
73#13133129—22no—
74#13133127—22no—
75C4×#88322—22no—
76C8×Dic53—22no—
77C2×M16×C53D411yescharacteristic
78C2×M16×C53D411yescharacteristic
79C22×C8×C53C2×C411yescharacteristic
80C22×C8×C53C2×C411yescharacteristic
81C2×C42×C53C2311yescharacteristic
82C4⋊C8×C53—21no—
83C22⋊C8×C53—21no—
84#97×C53D411yescharacteristic
85#97×C53D411yescharacteristic
86C4×C8×C53C2×C411yescharacteristic
87C4×C8×C53C2×C411yescharacteristic
88C4×C8×C53C2311yescharacteristic
89C4×C8×C53C2311yescharacteristic
90C4×C8×C53C2311yescharacteristic
91C4×C8×C53C2311yescharacteristic
92C4×C8×C53C2311yescharacteristic
93C4×C8×C53C2311yescharacteristic
94C22×C4×C53C22×C411yescharacteristic
95C22×C4×C53C2×D411yescharacteristic
96C22×C4×C53#4111yescharacteristic
97C2×C22⋊D53—22no—
98C2×D212C22×C412yes—
99C2×C4×D53—22no—
100C2×Dic106C22×C412yes—
101M16×C53—22no—
102M16×C53—22no—
103C2×C8×C53—22no—
104C2×C8×C53—21no—
105C2×C8×C53—21no—
106C2×C8×C53—41no—
107C2×C8×C53C22×C411yescharacteristic
108C2×C8×C53C22×C411yescharacteristic
109C2×C8×C53C22×C411yescharacteristic
110C2×C8×C53C22×C411yescharacteristic
111C2×C8×C53C2×D411yescharacteristic
112C2×C8×C53C2×D411yescharacteristic
113C2×C8×C53C2×D411yescharacteristic
114C2×C8×C53C2×D411yescharacteristic
115C42×C53C22×C411yescharacteristic
116C42×C53C2×D411yescharacteristic
117C42×C53#4111yescharacteristic
118C42×C53C2411yescharacteristic
119C22⋊Dic53—22no—
120#11719921—22no—
121#11719921—22no—
122#11719921—22no—
123#11719921—22no—
124C4⋊Dic53C22×C412yes—
125C4⋊Dic53C22×C412yes—
126#11719919—22no—
127#11719919—22no—
128#11719919—22no—
129#11719919—22no—
130C4×Dic53—22no—
131C2×#88322—22no—
132C2×#88322—22no—
133Dic212—22no—
134D424—22no—
135#11719913—42no—
136#11719912—44no—
137C8×D53—44no—
138C23×C53C4×D411yescharacteristic
139C22×D53—22no—
140C2×C4×C53—21no—
141C2×C4×C53—21no—
142C2×C4×C53—21no—
143C2×C4×C53—21no—
144C2×C4×C53—41no—
145C2×C4×C53C4×D411yescharacteristic
146C2×C4×C53C4×D411yescharacteristic
147C2×C4×C53C4×D411yescharacteristic
148C2×C4×C53C23×C411yescharacteristic
149C2×C4×C53C22×D411yescharacteristic
150C2×C4×C53C2×#4111yescharacteristic
151C22⋊D53—82no—
152C2×Dic53—22no—
153C2×Dic53—22no—
154C2×Dic53—22no—
155D212—22no—
156D212—22no—
157C4×D53—42no—
158C4×D53—42no—
159Dic106—22no—
160Dic106—22no—
161C8×C53—21no—
162C8×C53—21no—
163C8×C53—21no—
164C8×C53—21no—
165C8×C53—21no—
166C8×C53—41no—
167C8×C53—41no—
168C8×C53#13112yes—
169C8×C53#13512yes—
170C8×C53#13512yes—
171#88322—42no—
172#88322—42no—
173#18069—531nomaximal, Sylow
174C22×C53—21no—
175C22×C53—21no—
176C22×C53—21no—
177C22×C53C2×C4×D411yescharacteristic
178C2×D53—42no—
179C2×D53—42no—
180C2×D53—42no—
181C4×C53—21no—
182C4×C53—21no—
183C4×C53—21no—
184C4×C53—21no—
185C4×C53C2×#13111yescharacteristic
186C4×C53#56811yescharacteristic
187C4×C53C2×#13511yescharacteristic
188C4×C53C2×#13511yescharacteristic
189Dic53—42no—
190Dic53—42no—
191Dic53—42no—
192#3168—531no—
193#2969—532no—
194#1591—531no—
195#1589—531no—
196#1583—532no—
197#1582—532no—
198C8×Q16—531no—
199C8×D8—531no—
200#1564—531no—
201#1482—532no—
202C8×M16D5311yescharacteristic, complemented
203C2×C53—41no—
204C2×C53#293511yescharacteristic
205C2×C53C4×#13511yescharacteristic
206C2×C53#295711yescharacteristic
207D53—82no—
208#529—532no—
209#499—531no—
210#495—531no—
211#488—532no—
212#467—531no—
213#466—531no—
214#465—532no—
215#446—532no—
216C8×Q8—532no—
217C4×Q16—531no—
218C4×SD16—532no—
219C4×D8—531no—
220#436—532no—
221#436—1062no—
222#435—1062no—
223C8×D4—532no—
224#433—1062no—
225#431—532no—
226#431C2×D5311yescharacteristic
227#405—1062no—
228C4×M16C2×D5311yescharacteristic
229C2×C4×C8C2×D5311yescharacteristic
230#335—531no—
231#335—531no—
232#334—532no—
233Q8⋊C8—532no—
234Q8⋊C8—532no—
235D4⋊C8—532no—
236D4⋊C8—532no—
237#322C2×D5311yescharacteristic
238#322C2×D5311yescharacteristic
239C82C2×D5311yescharacteristic
240C82C2×D5311yescharacteristic
241C53#1806911yescharacteristic, complemented, Sylow
242C2×Q16—531no—
243C2×SD16—532no—
244C2×D8—531no—
245C2×M16—1062no—
246C2×M16D21211yescharacteristic
247C2×M16D21211yescharacteristic
248C22×C8—1062no—
249C22×C8C4×D5311yescharacteristic
250C22×C8C4×D5311yescharacteristic
251#126—1062no—
252#125—532no—
253#124—532no—
254#123—1062no—
255C22⋊Q8—532no—
256#121—532no—
257C4×Q8—532no—
258C4×D4—532no—
259C4×D4—1062no—
260#117—1062no—
261C2×C42C22×D5311yescharacteristic
262#107—531no—
263#107—531no—
264#106—532no—
265C4⋊C8—21no—
266C4⋊C8—532no—
267C4⋊C8—532no—
268C4⋊C8—1062no—
269C4⋊C8—1062no—
270#103—532no—
271#103—532no—
272#102—532no—
273#102—532no—
274C22⋊C8—21no—
275C22⋊C8—1062no—
276C22⋊C8—1062no—
277C22⋊C8—1062no—
278#97—1062no—
279#97D21211yescharacteristic
280#97D21211yescharacteristic
281C4×C8—1062no—
282C4×C8—1062no—
283C4×C8C4×D5311yescharacteristic
284C4×C8C4×D5311yescharacteristic
285C4×C8C22×D5311yescharacteristic
286C4×C8C22×D5311yescharacteristic
287C4×C8C22×D5311yescharacteristic
288C4×C8C22×D5311yescharacteristic
289C4×C8C22×D5311yescharacteristic
290C4×C8C22×D5311yescharacteristic
291C2×Q8—532no—
292C2×D4—532no—
293C2×D4—1062no—
294C22×C4—1062no—
295C22×C4C2×C4×D5311yescharacteristic
296C22×C4C2×D21211yescharacteristic
297C22×C4#1171994511yescharacteristic
298Q16—1062no—
299SD16—2122no—
300D8—1062no—
301M16—22no—
302M16—22no—
303M16—2124no—
304C2×C8—22no—
305C2×C8—21no—
306C2×C8—21no—
307C2×C8—41no—
308C2×C8—1062no—
309C2×C8—1062no—
310C2×C8—2124no—
311C2×C8C2×C4×D5311yescharacteristic
312C2×C8C2×C4×D5311yescharacteristic
313C2×C8C2×C4×D5311yescharacteristic
314C2×C8C2×C4×D5311yescharacteristic
315C2×C8C2×D21211yescharacteristic
316C2×C8C2×D21211yescharacteristic
317C2×C8C2×D21211yescharacteristic
318C2×C8C2×D21211yescharacteristic
319C4⋊C4—532no—
320C4⋊C4—532no—
321C4⋊C4—1062no—
322C4⋊C4—1062no—
323C4⋊C4—1062no—
324C4⋊C4—1062no—
325C22⋊C4—1062no—
326C22⋊C4—1062no—
327C22⋊C4—1062no—
328C22⋊C4—1062no—
329C22⋊C4—1062no—
330C42—1062no—
331C42C2×C4×D5311yescharacteristic
332C42C2×D21211yescharacteristic
333C42#1171994511yescharacteristic
334C42C23×D5311yescharacteristic
335C23—1062no—
336C23C4×D21211yescharacteristic
337Q8—1062no—
338Q8—1062no—
339D4—1062no—
340D4—1062no—
341D4—4242no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—41no—
347C2×C4—1062no—
348C2×C4—1062no—
349C2×C4—1062no—
350C2×C4—2122no—
351C2×C4—2122no—
352C2×C4C4×D21211yescharacteristic
353C2×C4C4×D21211yescharacteristic
354C2×C4C4×D21211yescharacteristic
355C2×C4C22×C4×D5311yescharacteristic
356C2×C4C22×D21211yescharacteristic
357C2×C4C2×#1171994511yescharacteristic
358C8—21no—
359C8—21no—
360C8—21no—
361C8—21no—
362C8—21no—
363C8—41no—
364C8—41no—
365C8—2122no—
366C8—2122no—
367C8#1313322512yes—
368C8#1313323912yes—
369C8#1313323912yes—
370C22—21no—
371C22—21no—
372C22—21no—
373C22—2122no—
374C22—2122no—
375C22—2122no—
376C22C2×C4×D21211yescharacteristic
377C4—21no—
378C4—21no—
379C4—21no—
380C4—21no—
381C4—2122no—
382C4—2122no—
383C4—2122no—
384C4C2×#1313322511yescharacteristic
385C4#1681138911yescharacteristic
386C4C2×#1313323911yescharacteristic
387C4C2×#1313323911yescharacteristic
388C2—41no—
389C2—4242no—
390C2#2028346111yescharacteristic
391C2C4×#1313323911yescharacteristic
392C2#2028363311yescharacteristic
subgroupG/H
centerC2×C4C22×D212
commutatorC2×C4×C53C23×C4
FrattiniC42C23×D53
FittingC8×M16×C53C2
socleC22×C53C2×C4×D4
prime radical#16810859C4
cosocleC42×C53C24
hypercenterC8×M16D53
hypocenterC53#18069
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C4×M16 ◁ C8×M16

Complements

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

NH ≅ G/Ntype
2. #20283719C2semidirect
3. #20273613C2semidirect
4. #20273601C2semidirect
5. #20273560C2semidirect
6. #20273553C2semidirect
7. C8×Dic212C2semidirect
8. C8×D424C2semidirect
9. #20273463C2semidirect
10. #20273124C2semidirect
11. C8×M16×C53C2semidirect
202. C8×M16D53semidirect
241. C53#18069semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#202852852
#202837192
#202736132
#202736012
#202735602
#202735532
C8×Dic2122
C8×D4242
#202734632
#202731242
C8×M16×C532
#1806953