Finite Group Databasebeta

← #27539437

Subgroups — #27539437

Order 13568.

570 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20285254C211yesmaximal, characteristic, complemented
2#20285209C211yesmaximal, characteristic
3#20276961C212yesmaximal, complemented
4#20276952C212yesmaximal, complemented
5#20276947C212yesmaximal, complemented
6#20276943C212yesmaximal, complemented
7#20276436C211yesmaximal, characteristic, complemented
8#20276428C211yesmaximal, characteristic
9#20276423C211yesmaximal, characteristic
10C2×#16810428C211yesmaximal, characteristic, complemented
11C53×#2108C211yesmaximal, characteristic, complemented
12C2×#13133250—21no—
13C2×#13133239C2211yescharacteristic
14C22×Dic212C2211yescharacteristic
15#16811251—21no—
16#16811199—21no—
17#16811032—22no—
18#16811028C2212yes—
19#16811025C2212yes—
20#16811017—22no—
21#16811007C2212yes—
22#16811003C2212yes—
23#16810665—21no—
24#16810661C2211yescharacteristic
25C2×#13133151C2211yescharacteristic
26#16810428C2212yes—
27#16810428C2212yes—
28#16810428C2211yescharacteristic
29#16810428C2211yescharacteristic
30#16810428C2212yes—
31#16810428C2212yes—
32#16810428C2212yes—
33#16810425C2211yescharacteristic
34#16810425C2211yescharacteristic
35#16810425C2212yes—
36C2×M32×C53C2211yescharacteristic
37C22×C16×C53C2211yescharacteristic
38C53×#405C2211yescharacteristic
39C53×#346C2212yes—
40C4×C16×C53C2212yes—
41C2×#11719945—21no—
42C22×Dic106—21no—
43#13133250—44no—
44#13133239—22no—
45#13133239—22no—
46C2×Dic212—21no—
47C2×Dic212—22no—
48C2×Dic212—22no—
49C2×Dic212—22no—
50C2×Dic212C2312yes—
51C2×Dic212C2311yescharacteristic
52C2×#11719913—21no—
53C2×#11719913—22no—
54C2×D424C2311yescharacteristic
55#13133217—22no—
56#13133217—22no—
57#13133216—41no—
58#13133209—22no—
59#13133208—41no—
60#13133205—22no—
61#13133201—42no—
62#13133197—41no—
63#13133197—41no—
64#13133194—42no—
65#13133176—21no—
66C2×C4⋊Dic53—21no—
67#13133151—22no—
68#13133151—21no—
69#13133151C2311yescharacteristic
70#13133151C2312yes—
71#13133151C2311yescharacteristic
72#13133149—22no—
73#13133149—21no—
74C2×M16×C53D411yescharacteristic
75C22×C8×C53C2311yescharacteristic
76#117×C53D411yescharacteristic
77M32×C53—22no—
78C2×C16×C53—22no—
79C2×C16×C53C2311yescharacteristic
80C2×C16×C53C2311yescharacteristic
81C2×C16×C53C2312yes—
82#97×C53D412yes—
83C4×C8×C53C2312yes—
84C22×C4×C53C2×D411yescharacteristic
85C2×C22⋊D53—41no—
86C22×Dic53—41no—
87C22×Dic53—41no—
88#11719945—42no—
89#11719945—42no—
90C2×D212—21no—
91C2×C4×D53—41no—
92C2×Dic106—22no—
93C2×Dic106—21no—
94C2×Dic106—42no—
95C2×Dic106—42no—
96C2×Dic106—41no—
97C2×Dic106—42no—
98M16×C53SD1614yes—
99C2×C8×C53C2×D411yescharacteristic
100C2×C8×C53C2×D412yes—
101C2×C8×C53#4112yes—
102C2×C8×C53#4112yes—
103C2×C8×C53C2411yescharacteristic
104C4⋊C4×C53D811yescharacteristic
105C4⋊C4×C53D811yescharacteristic
106C22⋊C4×C53D812yes—
107C42×C53C2×D412yes—
108C22⋊Dic53—41no—
109#11719921—42no—
110#11719921—42no—
111C4⋊Dic53—21no—
112C4⋊Dic53—22no—
113C4⋊Dic53—21no—
114C4⋊Dic53—41no—
115C4⋊Dic53—42no—
116C4⋊Dic53—42no—
117C4⋊Dic53—42no—
118#11719919—42no—
119#11719919—42no—
120C4×Dic53—41no—
121Dic212—22no—
122Dic212—22no—
123Dic212—21no—
124Dic212—21no—
125Dic212—42no—
126Dic212—44no—
127D424—21no—
128D424—21no—
129#11719913—42no—
130#11719913—44no—
131C16×C53—21no—
132C16×C53—21no—
133C16×C53—22no—
134C23×C53C2×D811yescharacteristic
135C22×D53—41no—
136C2×C4×C53#12412yes—
137C2×C4×C53C2×D812yes—
138C2×C4×C53C2×D812yes—
139C2×C4×C53C2×D811yescharacteristic
140C2×C4×C53C2×SD1612yes—
141C2×C4×C53C22×D411yescharacteristic
142C22⋊D53—82no—
143C2×Dic53—42no—
144C2×Dic53—41no—
145C2×Dic53—41no—
146C2×Dic53—42no—
147C2×Dic53—41no—
148C2×Dic53—82no—
149C2×Dic53—161no—
150D212—41no—
151D212—41no—
152C4×D53—82no—
153Dic106—42no—
154Dic106—41no—
155Dic106—42no—
156Dic106—41no—
157Dic106—82no—
158Dic106—162no—
159C8×C53#12412yes—
160C8×C53C2×SD1614yes—
161C8×C53C2×#4111yescharacteristic
162C8×C53C2×#4111yescharacteristic
163#33703—531nomaximal, Sylow
164C22×C53—21no—
165C22×C53—21no—
166C22×C53#48612yes—
167C22×C53C22×D811yescharacteristic
168C2×D53—81no—
169C2×D53—81no—
170C2×D53—81no—
171C4×C53—22no—
172C4×C53—22no—
173C4×C53#48612yes—
174C4×C53C2×#12411yescharacteristic
175C4×C53C22×SD1611yescharacteristic
176Dic53—82no—
177Dic53—81no—
178Dic53—81no—
179Dic53—82no—
180Dic53—81no—
181#3164—531no—
182#3156—531no—
183#2245—532no—
184#2243—532no—
185#2242—532no—
186#2241—532no—
187#2141—531no—
188#2139—531no—
189#2138—531no—
190C2×#358—531no—
191#2108D5311yescharacteristic, complemented
192C2×C53—22no—
193C2×C53C2×#48611yescharacteristic
194C2×C53#341611yescharacteristic
195C2×C53#341711yescharacteristic
196D53—161no—
197C2×#137—1061no—
198C2×#135—531no—
199C22×Q16—531no—
200#544—1061no—
201#536—1061no—
202C2×M32C2×D5311yescharacteristic
203C22×C16C2×D5311yescharacteristic
204#501—1062no—
205#500—532no—
206#499—532no—
207#497—1062no—
208#495—532no—
209#494—532no—
210#428—1061no—
211#427—531no—
212C2×#107—531no—
213#405C2×D5311yescharacteristic
214#358—532no—
215#358—532no—
216#358—531no—
217#358—531no—
218#358—532no—
219#358—532no—
220#358—532no—
221#357—531no—
222#357—531no—
223#357—532no—
224#346C2×D5312yes—
225C4×C16C2×D5312yes—
226C53#3370311yescharacteristic, complemented, Sylow
227C2×#41—1061no—
228C22×Q8—1061no—
229#137—2124no—
230#135—1062no—
231#135—1062no—
232C2×Q16—532no—
233C2×Q16—531no—
234C2×Q16—1061no—
235C2×Q16—1062no—
236C2×Q16—1062no—
237C2×Q16—1062no—
238C2×SD16—1061no—
239C2×SD16—1062no—
240C2×D8—531no—
241C2×M16D21211yescharacteristic
242C22×C8C22×D5311yescharacteristic
243C4⋊Q8—1062no—
244C4⋊Q8—1062no—
245C4⋊Q8—2121no—
246#125—1062no—
247#125—2121no—
248#124—1062no—
249#123—2122no—
250C22⋊Q8—2121no—
251C22⋊Q8—2121no—
252C22⋊Q8—2122no—
253#117—1061no—
254#117D21211yescharacteristic
255C2×C4⋊C4—1061no—
256M32—22no—
257C2×C16—22no—
258C2×C16C22×D5311yescharacteristic
259C2×C16C22×D5312yes—
260C2×C16C22×D5311yescharacteristic
261#107—532no—
262#107—531no—
263#107—531no—
264#107—1061no—
265#107—1062no—
266#106—1061no—
267#106—1062no—
268#97D21212yes—
269C4×C8C22×D5312yes—
270#41—2122no—
271#41—2122no—
272C2×Q8—1062no—
273C2×Q8—1061no—
274C2×Q8—2122no—
275C2×Q8—2122no—
276C2×Q8—2122no—
277C2×Q8—2121no—
278C2×D4—1061no—
279C2×D4—2121no—
280C22×C4—2121no—
281C22×C4—2121no—
282C22×C4—2121no—
283C22×C4C2×D21211yescharacteristic
284Q16—1062no—
285Q16—1062no—
286Q16—1061no—
287Q16—1061no—
288Q16—2122no—
289Q16—2124no—
290SD16—2122no—
291SD16—2124no—
292D8—1061no—
293D8—1061no—
294M16#1171991314yes—
295C2×C8C2×D21211yescharacteristic
296C2×C8C2×D21212yes—
297C2×C8#1171994512yes—
298C2×C8#1171994512yes—
299C2×C8C23×D5311yescharacteristic
300C4⋊C4—1061no—
301C4⋊C4—1062no—
302C4⋊C4—1061no—
303C4⋊C4—2121no—
304C4⋊C4—2122no—
305C4⋊C4—2122no—
306C4⋊C4—2122no—
307C4⋊C4—2122no—
308C4⋊C4—2122no—
309C4⋊C4D42411yescharacteristic
310C4⋊C4D42411yescharacteristic
311C22⋊C4—2121no—
312C22⋊C4—2122no—
313C22⋊C4—2122no—
314C22⋊C4D42412yes—
315C42—2121no—
316C42C2×D21212yes—
317C16—21no—
318C16—21no—
319C16—22no—
320C23—2121no—
321C23C2×D42411yescharacteristic
322Q8—8482no—
323Q8—4242no—
324Q8—2121no—
325Q8—2122no—
326Q8—2122no—
327Q8—2121no—
328D4—4242no—
329D4—2121no—
330D4—2121no—
331C2×C4—8481no—
332C2×C4—4242no—
333C2×C4—4242no—
334C2×C4—2121no—
335C2×C4—2121no—
336C2×C4—2122no—
337C2×C4—2122no—
338C2×C4—2121no—
339C2×C4#1313320512yes—
340C2×C4C2×D42412yes—
341C2×C4C2×D42412yes—
342C2×C4C2×D42411yescharacteristic
343C2×C4C2×#1171991312yes—
344C2×C4C22×D21211yescharacteristic
345C8#1313320512yes—
346C8C2×#1171991314yes—
347C8C2×#1171994511yescharacteristic
348C8C2×#1171994511yescharacteristic
349C22—21no—
350C22—21no—
351C22—4241no—
352C22—4241no—
353C22—4241no—
354C22#1681097012yes—
355C22C22×D42411yescharacteristic
356C4—22no—
357C4—22no—
358C4—4241no—
359C4—4242no—
360C4—4242no—
361C4—4241no—
362C4—4241no—
363C4#1681097012yes—
364C4C2×#1313320511yescharacteristic
365C4C22×#1171991311yescharacteristic
366C2—22no—
367C2—8481no—
368C2C2×#1681097011yescharacteristic
369C2#2028786311yescharacteristic
370C2#2028786811yescharacteristic
subgroupG/H
centerC22C22×D424
commutator, cosocleC2×C8×C53C24
FrattiniC2×C8C23×D53
FittingC53×#2108C2
socleC22×C53C22×D8
prime radicalC2×#13133239C22
hypercenter#2108D53
hypocenterC53#33703
LCSG ▷ C2×C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ #117 ◁ #405 ◁ #2108

Complements

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

NH ≅ G/Ntype
1. #20285254C2semidirect
3. #20276961C2semidirect
4. #20276952C2semidirect
5. #20276947C2semidirect
6. #20276943C2semidirect
7. #20276436C2semidirect
10. C2×#16810428C2semidirect
11. C53×#2108C2semidirect
191. #2108D53semidirect
226. C53#33703semidirect

Maximal subgroups

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

subgroupindex
#202852542
#202852092
#202769612
#202769522
#202769472
#202769432
#202764362
#202764282
#202764232
C2×#168104282
C53×#21082
#3370353