Finite Group Databasebeta

← #27539440

Subgroups — #27539440

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#20285209C211yesmaximal, characteristic
2#20276962C212yesmaximal, complemented
3#20276954C212yesmaximal, complemented
4#20276949C212yesmaximal, complemented
5#20276944C212yesmaximal, complemented
6#20276434C211yesmaximal, characteristic, complemented
7#20276425C211yesmaximal, characteristic
8#20276420C211yesmaximal, characteristic
9C2×#16810426C211yesmaximal, characteristic, complemented
10#20276312C211yesmaximal, characteristic, complemented
11C53×#3164C211yesmaximal, characteristic, complemented
12C2×#13133250—21no—
13C2×#13133239C2211yescharacteristic
14C22×Dic212C2211yescharacteristic
15#16811199—21no—
16C2×#13133157C2211yescharacteristic
17C22×#11719908C2211yescharacteristic
18#16811017—22no—
19#16811007C2212yes—
20#16811003C2212yes—
21#16810426C2212yes—
22#16810426C2212yes—
23#16810426C2211yescharacteristic
24#16810426C2211yescharacteristic
25#16810426C2212yes—
26#16810426C2212yes—
27#16810426C2212yes—
28#16810423C2211yescharacteristic
29#16810423C2211yescharacteristic
30#16810423C2212yes—
31#16810399C2212yes—
32C4×#11719908C2212yes—
33C53×#544—21no—
34C53×#501—22no—
35C53×#500C2212yes—
36C53×#499C2212yes—
37C53×#428—21no—
38C53×#427C2211yescharacteristic
39C2×#107×C53C2211yescharacteristic
40C53×#405C2211yescharacteristic
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#13133205—22no—
57#13133201—42no—
58#13133197—41no—
59#13133197—41no—
60#13133157—22no—
61C2×#11719908—22no—
62C2×#11719908C2311yescharacteristic
63C2×#11719908C2311yescharacteristic
64C2×#11719908C2312yes—
65C2×M16×C53D411yescharacteristic
66C22×C8×C53C2311yescharacteristic
67C4⋊Q8×C53—22no—
68C4⋊Q8×C53—41no—
69#125×C53—22no—
70#125×C53—41no—
71C22⋊Q8×C53—42no—
72#117×C53—21no—
73#117×C53D411yescharacteristic
74C2×C4⋊C4×C53—21no—
75#107×C53—22no—
76#107×C53—21no—
77#107×C53C2311yescharacteristic
78#107×C53C2312yes—
79#107×C53C2311yescharacteristic
80#106×C53—22no—
81#106×C53—21no—
82#97×C53D412yes—
83C4×C8×C53C2312yes—
84C2×Q8×C53—42no—
85C22×C4×C53—41no—
86C22×C4×C53C2×D411yescharacteristic
87C2×C22⋊D53—41no—
88C22×Dic53—41no—
89#11719945—42no—
90#11719945—42no—
91C2×D212—21no—
92C2×C4×D53—41no—
93C2×Dic106—22no—
94C2×Dic106—21no—
95C2×Dic106—42no—
96C2×Dic106—42no—
97C2×Dic106—41no—
98M16×C53SD1614yes—
99C2×C8×C53C2×D411yescharacteristic
100C2×C8×C53C2×D412yes—
101C2×C8×C53#4112yes—
102C2×C8×C53#4112yes—
103C2×C8×C53C2411yescharacteristic
104C4⋊C4×C53—21no—
105C4⋊C4×C53—22no—
106C4⋊C4×C53—21no—
107C4⋊C4×C53—41no—
108C4⋊C4×C53—42no—
109C4⋊C4×C53—42no—
110C4⋊C4×C53—42no—
111C4⋊C4×C53D811yescharacteristic
112C4⋊C4×C53D811yescharacteristic
113C22⋊C4×C53—41no—
114C22⋊C4×C53D812yes—
115C42×C53—41no—
116C42×C53C2×D412yes—
117#11719921—42no—
118#11719921—42no—
119C4⋊Dic53—42no—
120C4⋊Dic53—42no—
121Dic212—22no—
122Dic212—22no—
123Dic212—21no—
124Dic212—21no—
125Dic212—42no—
126Dic212—44no—
127D424—21no—
128D424—21no—
129#11719913—42no—
130#11719913—44no—
131#11719908—21no—
132#11719908—21no—
133#11719908—22no—
134C23×C53C2×D811yescharacteristic
135C22×D53—41no—
136Q8×C53—162no—
137C2×C4×C53—42no—
138C2×C4×C53—41no—
139C2×C4×C53—41no—
140C2×C4×C53—161no—
141C2×C4×C53#12412yes—
142C2×C4×C53C2×D812yes—
143C2×C4×C53C2×D812yes—
144C2×C4×C53C2×D811yescharacteristic
145C2×C4×C53C2×SD1612yes—
146C2×C4×C53C22×D411yescharacteristic
147C22⋊D53—82no—
148C2×Dic53—42no—
149C2×Dic53—41no—
150C2×Dic53—82no—
151D212—41no—
152D212—41no—
153C4×D53—82no—
154Dic106—42no—
155Dic106—41no—
156Dic106—42no—
157Dic106—41no—
158Dic106—82no—
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—82no—
174C4×C53—81no—
175C4×C53—81no—
176C4×C53#48612yes—
177C4×C53C2×#12411yescharacteristic
178C4×C53C22×SD1611yescharacteristic
179Dic53—82no—
180Dic53—81no—
181#3164D5311yescharacteristic, complemented
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#2108—531no—
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—21no—
201#536—1061no—
202C2×M32—531no—
203C22×C16—531no—
204#501—22no—
205#500C2×D5312yes—
206#499C2×D5312yes—
207#497—1062no—
208#495—532no—
209#494—532no—
210#428—21no—
211#427C2×D5311yescharacteristic
212C2×#107C2×D5311yescharacteristic
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#346—532no—
225C4×C16—532no—
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—1062no—
235C2×Q16—1062no—
236C2×Q16—1061no—
237C2×Q16—1062no—
238C2×SD16—1061no—
239C2×SD16—1062no—
240C2×D8—531no—
241C2×M16C22⋊D5311yescharacteristic
242C22×C8C22×D5311yescharacteristic
243C4⋊Q8—22no—
244C4⋊Q8—41no—
245C4⋊Q8—1062no—
246#125—22no—
247#125—41no—
248#124—1062no—
249#123—2122no—
250C22⋊Q8—42no—
251C22⋊Q8—2121no—
252C22⋊Q8—2121no—
253#117—21no—
254#117C22⋊D5311yescharacteristic
255C2×C4⋊C4—21no—
256M32—1062no—
257C2×C16—531no—
258C2×C16—532no—
259C2×C16—531no—
260C2×C16—1062no—
261#107—21no—
262#107—22no—
263#107C22×D5312yes—
264#107C22×D5311yescharacteristic
265#107C22×D5311yescharacteristic
266#106—21no—
267#106—22no—
268#97C22⋊D5312yes—
269C4×C8C22×D5312yes—
270#41—2122no—
271#41—2122no—
272C2×Q8—42no—
273C2×Q8—1062no—
274C2×Q8—1061no—
275C2×Q8—2122no—
276C2×Q8—2122no—
277C2×Q8—2121no—
278C2×D4—1061no—
279C2×D4—2121no—
280C22×C4—41no—
281C22×C4—2121no—
282C22×C4—2121no—
283C22×C4C2×C22⋊D5311yescharacteristic
284Q16—1062no—
285Q16—1062no—
286Q16—1061no—
287Q16—1061no—
288Q16—2122no—
289Q16—2124no—
290SD16—2122no—
291SD16—2124no—
292D8—1061no—
293D8—1061no—
294M16#1171992414yes—
295C2×C8#1171994912yes—
296C2×C8#1171994912yes—
297C2×C8C2×C22⋊D5311yescharacteristic
298C2×C8C2×C22⋊D5312yes—
299C2×C8C23×D5311yescharacteristic
300C4⋊C4—21no—
301C4⋊C4—22no—
302C4⋊C4—21no—
303C4⋊C4—41no—
304C4⋊C4—42no—
305C4⋊C4—42no—
306C4⋊C4—42no—
307C4⋊C4—2122no—
308C4⋊C4—2122no—
309C4⋊C4#1171992211yescharacteristic
310C4⋊C4#1171992211yescharacteristic
311C22⋊C4—41no—
312C22⋊C4—2122no—
313C22⋊C4—2122no—
314C22⋊C4#1171992212yes—
315C42—41no—
316C42C2×C22⋊D5312yes—
317C16—1061no—
318C16—1061no—
319C16—1062no—
320C23—2121no—
321C23C2×#1171992211yescharacteristic
322Q8—162no—
323Q8—4242no—
324Q8—2122no—
325Q8—2121no—
326Q8—2121no—
327Q8—2122no—
328D4—4242no—
329D4—2121no—
330D4—2121no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—42no—
334C2×C4—161no—
335C2×C4—4242no—
336C2×C4—4242no—
337C2×C4—2122no—
338C2×C4—2121no—
339C2×C4#1313320612yes—
340C2×C4C2×#1171992212yes—
341C2×C4C2×#1171992212yes—
342C2×C4C2×#1171992211yescharacteristic
343C2×C4C2×#1171992412yes—
344C2×C4C22×C22⋊D5311yescharacteristic
345C8#1313320612yes—
346C8C2×#1171992414yes—
347C8C2×#1171994911yescharacteristic
348C8C2×#1171994911yescharacteristic
349C22—21no—
350C22—21no—
351C22—4241no—
352C22—4241no—
353C22—4241no—
354C22#1681097112yes—
355C22C22×#1171992211yescharacteristic
356C4—22no—
357C4—22no—
358C4—82no—
359C4—81no—
360C4—81no—
361C4—4241no—
362C4—4242no—
363C4#1681097112yes—
364C4C2×#1313320611yescharacteristic
365C4C22×#1171992411yescharacteristic
366C2—22no—
367C2—8481no—
368C2C2×#1681097111yescharacteristic
369C2#2028786211yescharacteristic
370C2#2028786611yescharacteristic
subgroupG/H
centerC22C22×#11719922
commutator, cosocleC2×C8×C53C24
FrattiniC2×C8C23×D53
FittingC53×#3164C2
socleC22×C53C22×D8
prime radicalC2×#13133239C22
hypercenter#3164D53
hypocenterC53#33703
LCSG ▷ C2×C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ #117 ◁ #405 ◁ #3164

Complements

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

NH ≅ G/Ntype
2. #20276962C2semidirect
3. #20276954C2semidirect
4. #20276949C2semidirect
5. #20276944C2semidirect
6. #20276434C2semidirect
9. C2×#16810426C2semidirect
10. #20276312C2semidirect
11. C53×#3164C2semidirect
181. #3164D53semidirect
226. C53#33703semidirect

Maximal subgroups

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

subgroupindex
#202852092
#202769622
#202769542
#202769492
#202769442
#202764342
#202764252
#202764202
C2×#168104262
#202763122
C53×#31642
#3370353