Finite Group Databasebeta

← #27369444

Subgroups — #27369444

Order 13568.

676 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20285838C211yesmaximal, characteristic
2#20284187C211yesmaximal, characteristic
3C2×#16810833C211yesmaximal, characteristic, complemented
4#20274159C214yesmaximal, complemented
5#20274109C214yesmaximal, complemented
6#20273490C211yesmaximal, characteristic, complemented
7#20273386C211yesmaximal, characteristic, complemented
8#20273362C211yesmaximal, characteristic
9C53×#2982C211yesmaximal, characteristic, complemented
10C2×#13133248—21no—
11C22×#11719925—21no—
12#16811173C2211yescharacteristic
13C2×#13133217C2211yescharacteristic
14#16811028C2212yes—
15#16810970C2212yes—
16#16810937—22no—
17#16810904—22no—
18#16810845—22no—
19#16810833—22no—
20#16810833C2212yes—
21#16810833C2214yes—
22#16810823C2212yes—
23#16810647C2211yescharacteristic
24C2×C4⋊#88322C2211yescharacteristic
25#16810622—21no—
26C2×#13133140—21no—
27#16810376C2214yes—
28#16810368C2212yes—
29#16810368C2214yes—
30#16810366C2212yes—
31C2×C4×Q8×C53C2211yescharacteristic
32C8×Q8×C53—22no—
33C8×Q8×C53C2214yes—
34C53×#431—22no—
35C53×#431C2211yescharacteristic
36C53×#431C2211yescharacteristic
37C2×#11719945—21no—
38C22×Dic106—21no—
39#13133248—44no—
40C2×#11719925—24no—
41C2×#11719925—22no—
42C2×#11719925—44no—
43C2×#11719924—22no—
44C2×#11719917—21no—
45C22×#88322—21no—
46#13133217—21no—
47#13133217—21no—
48#13133217—21no—
49#13133217C2311yescharacteristic
50#13133217C2312yes—
51#13133214C2311yescharacteristic
52#13133205—21no—
53#13133191—21no—
54#13133191—21no—
55C4×D212—21no—
56C4×D212—21no—
57C2×C4⋊Dic53—21no—
58C2×C4⋊Dic53—21no—
59#13133151—24no—
60C4⋊#88322—21no—
61C4⋊#88322—21no—
62C4⋊#88322C2312yes—
63C4⋊#88322C2312yes—
64#13133140—24no—
65#13133140—22no—
66#13133139—24no—
67#13133137—22no—
68C22×Q8×C53D411yescharacteristic
69C4×Q8×C53—21no—
70C4×Q8×C53—21no—
71C4×Q8×C53D412yes—
72C4×Q8×C53C2312yes—
73C2×C4⋊C4×C53—21no—
74C2×C4⋊C4×C53D411yescharacteristic
75C2×C42×C53—21no—
76C2×C42×C53C2311yescharacteristic
77C4⋊C8×C53—24no—
78C4⋊C8×C53D412yes—
79C4⋊C8×C53C2312yes—
80C22⋊C8×C53—21no—
81C22⋊C8×C53D412yes—
82C4×C8×C53—24no—
83C4×C8×C53—21no—
84C4×C8×C53C2312yes—
85C2×Q8×C53—21no—
86C2×Q8×C53—22no—
87C2×Q8×C53D814yes—
88C2×Q8×C53C2×D412yes—
89C22×C4×C53—21no—
90C22×C4×C53—21no—
91C22×C4×C53C2×D411yescharacteristic
92C22×C4×C53C2×D411yescharacteristic
93C22×C4×C53#4111yescharacteristic
94C2×C22⋊D53—41no—
95C22×Dic53—41no—
96#11719945—42no—
97#11719945—42no—
98C2×D212—21no—
99C2×D212—21no—
100C2×D212—21no—
101C2×C4×D53—41no—
102C2×Dic106—21no—
103C2×Dic106—22no—
104C2×Dic106—42no—
105C2×Dic106—42no—
106C2×Dic106—41no—
107C2×C8×C53—22no—
108C2×C8×C53C2×D414yes—
109C4⋊C4×C53—22no—
110C4⋊C4×C53—21no—
111C4⋊C4×C53—22no—
112C4⋊C4×C53C2×D412yes—
113C42×C53—22no—
114C42×C53—22no—
115C42×C53C2×D411yescharacteristic
116C42×C53C2×D411yescharacteristic
117C42×C53#4111yescharacteristic
118C42×C53C2411yescharacteristic
119#11719925—48no—
120#11719925—44no—
121#11719924—44no—
122#11719921—41no—
123#11719921—41no—
124C4⋊Dic53—22no—
125C4⋊Dic53—21no—
126C4⋊Dic53—22no—
127C4⋊Dic53—21no—
128C4⋊Dic53—41no—
129C4⋊Dic53—41no—
130#11719917—42no—
131C2×#88322—22no—
132C2×#88322—22no—
133C2×#88322—42no—
134C23×C53#12111yescharacteristic
135C22×D53—41no—
136Q8×C53—22no—
137Q8×C53—24no—
138Q8×C53C2×D814yes—
139C2×C4×C53—22no—
140C2×C4×C53—24no—
141C2×C4×C53—22no—
142C2×C4×C53—22no—
143C2×C4×C53—41no—
144C2×C4×C53—81no—
145C2×C4×C53#12111yescharacteristic
146C2×C4×C53#12111yescharacteristic
147C2×C4×C53#12111yescharacteristic
148C2×C4×C53C2×D812yes—
149C2×C4×C53#13512yes—
150C2×C4×C53C22×D411yescharacteristic
151C2×C4×C53C22×D411yescharacteristic
152C2×C4×C53C2×#4111yescharacteristic
153C22⋊D53—82no—
154C2×Dic53—42no—
155C2×Dic53—41no—
156C2×Dic53—82no—
157D212—41no—
158D212—41no—
159D212—42no—
160D212—81no—
161C4×D53—82no—
162Dic106—42no—
163Dic106—41no—
164Dic106—42no—
165Dic106—41no—
166Dic106—82no—
167C8×C53—24no—
168C8×C53—42no—
169#88322—42no—
170#88322—42no—
171#19404—531nomaximal, Sylow
172C22×C53—21no—
173C22×C53—21no—
174C22×C53#46612yes—
175C22×C53C2×#12111yescharacteristic
176C2×D53—81no—
177C2×D53—81no—
178C2×D53—81no—
179C4×C53—21no—
180C4×C53—24no—
181C4×C53—21no—
182C4×C53—21no—
183C4×C53—42no—
184C4×C53#46612yes—
185C4×C53C22×D811yescharacteristic
186C4×C53C2×#13511yescharacteristic
187C4×C53#57812yes—
188Dic53—82no—
189Dic53—81no—
190#3226—531no—
191#3037—531no—
192C2×#462—531no—
193#2982D5311yescharacteristic, complemented
194#1682—534no—
195#1674—534no—
196#1571—531no—
197#1546—531no—
198#1540—531no—
199C2×C53—22no—
200C2×C53#304411yescharacteristic
201C2×C53C2×#46611yescharacteristic
202C2×C53#306911yescharacteristic
203D53—161no—
204C2×#137—1061no—
205C22×Q16—1061no—
206#532—531no—
207C2×C4⋊Q8—531no—
208C2×C4×Q8C2×D5311yescharacteristic
209#500—532no—
210#486—532no—
211#480—1062no—
212#474—1062no—
213#464—1062no—
214#462—532no—
215#462—534no—
216#462—1062no—
217#460—532no—
218C8×Q8—22no—
219C8×Q8C2×D5314yes—
220#431—22no—
221#431C2×D5311yescharacteristic
222#431C2×D5311yescharacteristic
223#423—531no—
224C2×C4⋊C8—531no—
225#417—1061no—
226C2×#103—1061no—
227#335—534no—
228#332—532no—
229#332—534no—
230#331—532no—
231C53#1940411yescharacteristic, complemented, Sylow
232C2×#41—1061no—
233C22×Q8—1061no—
234C22×Q8D21211yescharacteristic
235#137—2124no—
236C2×Q16—1064no—
237C2×Q16—1062no—
238C2×Q16—2124no—
239C2×SD16—1062no—
240C2×M16—1061no—
241C22×C8—1061no—
242C4⋊Q8—532no—
243C4⋊Q8—531no—
244C4⋊Q8—1061no—
245C4⋊Q8—1061no—
246C4⋊Q8—1061no—
247#127—531no—
248#124—1061no—
249#121—1061no—
250#121—1061no—
251C4×Q8—21no—
252C4×Q8—21no—
253C4×Q8D21212yes—
254C4×Q8C22×D5312yes—
255C4×D4—1061no—
256C4×D4—1061no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—1061no—
259C2×C4⋊C4—1061no—
260C2×C4⋊C4D21211yescharacteristic
261C2×C42—21no—
262C2×C42C22×D5311yescharacteristic
263#107—1064no—
264C4⋊C8—24no—
265C4⋊C8—532no—
266C4⋊C8—532no—
267C4⋊C8—1061no—
268C4⋊C8—1061no—
269C4⋊C8C22⋊D5312yes—
270C4⋊C8C22×D5312yes—
271#103—1064no—
272#103—1062no—
273#102—1064no—
274#102—1062no—
275C22⋊C8—21no—
276C22⋊C8C22⋊D5312yes—
277C4×C8—24no—
278C4×C8—21no—
279C4×C8C22×D5312yes—
280#41—2122no—
281#41—2122no—
282C2×Q8—21no—
283C2×Q8—22no—
284C2×Q8—1061no—
285C2×Q8—1062no—
286C2×Q8—2122no—
287C2×Q8—2122no—
288C2×Q8—2121no—
289C2×Q8D42414yes—
290C2×Q8C2×D21212yes—
291C2×D4—1061no—
292C2×D4—1061no—
293C2×D4—1061no—
294C2×D4—2121no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—2121no—
298C22×C4—2121no—
299C22×C4C2×D21211yescharacteristic
300C22×C4#1171994511yescharacteristic
301C22×C4C2×C22⋊D5311yescharacteristic
302Q16—2128no—
303Q16—2124no—
304SD16—2124no—
305M16—2122no—
306C2×C8—22no—
307C2×C8—1062no—
308C2×C8—1062no—
309C2×C8—2122no—
310C2×C8C2×C22⋊D5314yes—
311C4⋊C4—22no—
312C4⋊C4—21no—
313C4⋊C4—22no—
314C4⋊C4—1062no—
315C4⋊C4—1061no—
316C4⋊C4—1062no—
317C4⋊C4—1061no—
318C4⋊C4—2121no—
319C4⋊C4—2121no—
320C4⋊C4C2×D21212yes—
321C22⋊C4—2121no—
322C22⋊C4—2121no—
323C42—22no—
324C42—22no—
325C42C2×D21211yescharacteristic
326C42#1171994511yescharacteristic
327C42C2×C22⋊D5311yescharacteristic
328C42C23×D5311yescharacteristic
329C23—2121no—
330C23#1313319111yescharacteristic
331Q8—24no—
332Q8—22no—
333Q8—4242no—
334Q8—2122no—
335Q8—2121no—
336Q8—2122no—
337Q8—2121no—
338Q8C2×D42414yes—
339D4—4241no—
340D4—4242no—
341D4—2121no—
342D4—2121no—
343D4—2122no—
344C2×C4—24no—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—41no—
349C2×C4—81no—
350C2×C4—4242no—
351C2×C4—4242no—
352C2×C4—2121no—
353C2×C4—2122no—
354C2×C4#1313319111yescharacteristic
355C2×C4#1313319111yescharacteristic
356C2×C4#1313319111yescharacteristic
357C2×C4C2×D42412yes—
358C2×C4#1313323912yes—
359C2×C4C22×C22⋊D5311yescharacteristic
360C2×C4C22×D21211yescharacteristic
361C2×C4C2×#1171994511yescharacteristic
362C8—24no—
363C8—42no—
364C8—2122no—
365C8—2122no—
366C22—21no—
367C22—21no—
368C22—4241no—
369C22—4241no—
370C22—4241no—
371C22#1681085912yes—
372C22C2×#1313319111yescharacteristic
373C4—21no—
374C4—21no—
375C4—21no—
376C4—24no—
377C4—42no—
378C4—4241no—
379C4—4242no—
380C4#1681085912yes—
381C4C22×D42411yescharacteristic
382C4C2×#1313323911yescharacteristic
383C4#1681143812yes—
384C2—22no—
385C2—8481no—
386C2#2028424511yescharacteristic
387C2C2×#1681085911yescharacteristic
388C2#2028445211yescharacteristic
subgroupG/H
centerC22C2×#13133191
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingC53×#2982C2
socleC22×C53C2×#121
prime radical#16811173C22
hypercenter#2982D53
hypocenterC53#19404
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2982

Complements

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

NH ≅ G/Ntype
3. C2×#16810833C2semidirect
4. #20274159C2semidirect
5. #20274109C2semidirect
6. #20273490C2semidirect
7. #20273386C2semidirect
9. C53×#2982C2semidirect
193. #2982D53semidirect
231. C53#19404semidirect

Maximal subgroups

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

subgroupindex
#202858382
#202841872
C2×#168108332
#202741592
#202741092
#202734902
#202733862
#202733622
C53×#29822
#1940453