Finite Group Databasebeta

← #27294434

Subgroups — #27294434

Order 13568.

486 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 486 conjugacy classes of subgroups — too many to draw. Only its 83 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20280599C211yesmaximal, characteristic
2#20273036C211yesmaximal, characteristic, complemented
3#20272738C212yesmaximal, complemented
4#20272711C212yesmaximal, complemented
5C4×#99×C53C211yesmaximal, characteristic, complemented
6C2×#13133191C411yescharacteristic, complemented
7#16810676—21no—
8#16810562C411yescharacteristic, complemented
9#16810559C2211yescharacteristic
10#16810540C411yescharacteristic, complemented
11#16810538—21no—
12#16810532C2211yescharacteristic
13#16810531—21no—
14#16810514—21no—
15#16810502—21no—
16#16810486C411yescharacteristic, complemented
17#16810347—21no—
18#16810347C2212yes—
19C2×C4×D4×C53C2211yescharacteristic
20C2×#99×C53—21no—
21C4×C22⋊C4×C53—21no—
22C53×#342C2212yes—
23C22×D212C2×C411yescharacteristic
24C22×C22⋊D53—21no—
25C22×C22⋊D53—21no—
26#13133191—41no—
27#13133191—41no—
28C2×C4⋊Dic53C2×C411yescharacteristic
29C2×#11719919—21no—
30C2×C22⋊Dic53—21no—
31C2×C22⋊Dic53—21no—
32C2×#11719921—21no—
33C2×#11719921—21no—
34C2×#11719921—21no—
35C2×#11719921—21no—
36C4⋊#88322—41no—
37C22⋊#88322—22no—
38C4×#88322—41no—
39#13133124—21no—
40#13133124—21no—
41#13133124—21no—
42#13133124C2×C411yescharacteristic
43#13133124C2×C411yescharacteristic
44C22×D4×C53D411yescharacteristic
45C23×C4×C53C2×C411yescharacteristic
46C23×C4×C53D411yescharacteristic
47C4×D4×C53—41no—
48C4×D4×C53—41no—
49C2×C4⋊C4×C53C2×C411yescharacteristic
50C2×C22⋊C4×C53—22no—
51C2×C22⋊C4×C53D411yescharacteristic
52C2×C22⋊C4×C53C2311yescharacteristic
53C2×C42×C53—41no—
54C2×C42×C53D411yescharacteristic
55#99×C53—24no—
56#95×C53—41no—
57C24×C53C22⋊C411yescharacteristic
58C24×C53C2×D411yescharacteristic
59C23×D53—21no—
60C2×D4×C53—22no—
61C2×D4×C53—22no—
62C2×D4×C53—41no—
63C22×C4×C53—21no—
64C22×C4×C53—21no—
65C22×C4×C53—21no—
66C22×C4×C53—21no—
67C22×C4×C53—42no—
68C22×C4×C53—81no—
69C22×C4×C53—81no—
70C22×C4×C53C22⋊C411yescharacteristic
71C22×C4×C53C22⋊C411yescharacteristic
72C22×C4×C53C22⋊C411yescharacteristic
73C22×C4×C53C22×C411yescharacteristic
74C22×C4×C53C2×D411yescharacteristic
75C2×C22⋊D53—42no—
76C2×C22⋊D53—42no—
77C2×C22⋊D53—81no—
78C2×C22⋊D53—81no—
79C2×C22⋊D53—81no—
80C2×C22⋊D53—81no—
81C22×Dic53—21no—
82C22×Dic53—21no—
83C22×Dic53—21no—
84C2×D212—41no—
85C2×D212—81no—
86C4⋊C4×C53—41no—
87C22⋊C4×C53—24no—
88C22⋊C4×C53—41no—
89C22⋊C4×C53—41no—
90C22⋊C4×C53—42no—
91C42×C53—21no—
92C42×C53—21no—
93C42×C53—82no—
94C22⋊Dic53—81no—
95C22⋊Dic53—81no—
96#11719921—41no—
97#11719921—41no—
98#11719921—42no—
99#11719921—81no—
100#11719921—81no—
101C4⋊Dic53—41no—
102#11719919—81no—
103C2×#88322—42no—
104C23×C53—21no—
105C23×C53—21no—
106C23×C53—21no—
107C23×C53—21no—
108C23×C53—21no—
109C23×C53—41no—
110C23×C53—41no—
111C23×C53—41no—
112C23×C53—41no—
113C23×C53C2×C22⋊C411yescharacteristic
114C23×C53#13512yes—
115C22×D53—41no—
116C22×D53—41no—
117C22×D53—41no—
118C22×D53—41no—
119C22×D53—41no—
120C22×D53—41no—
121C22×D53—41no—
122D4×C53—44no—
123C2×C4×C53—21no—
124C2×C4×C53—21no—
125C2×C4×C53—21no—
126C2×C4×C53—21no—
127C2×C4×C53—21no—
128C2×C4×C53—41no—
129C2×C4×C53—42no—
130C2×C4×C53—41no—
131C2×C4×C53—41no—
132C2×C4×C53—41no—
133C2×C4×C53—44no—
134C2×C4×C53—81no—
135C2×C4×C53—81no—
136C2×C4×C53—81no—
137C2×C4×C53—82no—
138C2×C4×C53—82no—
139C2×C4×C53#9911yescharacteristic
140C2×C4×C53#9911yescharacteristic
141C2×C4×C53#10412yes—
142C22⋊D53—84no—
143C22⋊D53—84no—
144C2×Dic53—41no—
145C2×Dic53—41no—
146C2×Dic53—41no—
147C2×Dic53—41no—
148C2×Dic53—41no—
149C2×Dic53—41no—
150C2×Dic53—41no—
151C2×Dic53—41no—
152C2×Dic53—41no—
153D212—82no—
154#88322—82no—
155#11729—531nomaximal, Sylow
156C22×C53—21no—
157C22×C53—21no—
158C22×C53—22no—
159C22×C53—42no—
160C22×C53—41no—
161C22×C53—41no—
162C22×C53—41no—
163C22×C53—41no—
164C22×C53—41no—
165C22×C53—81no—
166C22×C53—81no—
167C22×C53—81no—
168C22×C53C2×#9911yescharacteristic
169C22×C53#41611yescharacteristic
170C22×C53C2×#10411yescharacteristic
171C2×D53—81no—
172C2×D53—81no—
173C2×D53—81no—
174C2×D53—81no—
175C2×D53—81no—
176C2×D53—81no—
177C2×D53—81no—
178C4×C53—21no—
179C4×C53—21no—
180C4×C53—22no—
181C4×C53—41no—
182C4×C53—41no—
183C4×C53—41no—
184C4×C53—84no—
185Dic53—81no—
186Dic53—81no—
187Dic53—81no—
188#2635—531no—
189C4×#99D5311yescharacteristic, complemented
190#1460—531no—
191#1352—532no—
192#1343—532no—
193C2×C53—21no—
194C2×C53—21no—
195C2×C53—42no—
196C2×C53—81no—
197C2×C53#150111yescharacteristic
198C2×C53#212111yescharacteristic
199C2×C53#212811yescharacteristic
200D53—161no—
201C2×#121—531no—
202C2×C4×D4C2×D5311yescharacteristic
203#431—1061no—
204C2×#99—21no—
205#400—531no—
206#399—531no—
207#394—531no—
208#394—1061no—
209#392—531no—
210#392—1061no—
211#388—1061no—
212#386—1061no—
213#381—531no—
214C4×C22⋊C4—21no—
215#342C2×D5312yes—
216#323—532no—
217#323—1061no—
218C53#1172911yescharacteristic, complemented, Sylow
219C22×D4—531no—
220C22×D4—1061no—
221C22×D4—1061no—
222C22×D4C22⋊D5311yescharacteristic
223C23×C4C4×D5311yescharacteristic
224C23×C4C22⋊D5311yescharacteristic
225#121—2121no—
226#121—2121no—
227C4×D4—41no—
228C4×D4—41no—
229C2×C4⋊C4—531no—
230C2×C4⋊C4—1061no—
231C2×C4⋊C4C4×D5311yescharacteristic
232C2×C22⋊C4—22no—
233C2×C22⋊C4—1061no—
234C2×C22⋊C4—1061no—
235C2×C22⋊C4—1061no—
236C2×C22⋊C4—1061no—
237C2×C22⋊C4—1061no—
238C2×C22⋊C4—1061no—
239C2×C22⋊C4D21211yescharacteristic
240C2×C22⋊C4C22×D5311yescharacteristic
241C2×C42—41no—
242C2×C42D21211yescharacteristic
243C4⋊C8—2121no—
244#99—24no—
245C22⋊C8—1062no—
246C4×C8—2121no—
247#95—41no—
248#95—531no—
249#95—531no—
250#95—1061no—
251#95—1061no—
252#95—1061no—
253C24—1061no—
254C24#1171992111yescharacteristic
255C24C2×C22⋊D5311yescharacteristic
256C2×D4—22no—
257C2×D4—22no—
258C2×D4—41no—
259C2×D4—4241no—
260C2×D4—4241no—
261C2×D4—4241no—
262C2×D4—4241no—
263C2×D4—4241no—
264C2×D4—2121no—
265C2×D4—2122no—
266C2×D4—2122no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—42no—
272C22×C4—81no—
273C22×C4—81no—
274C22×C4—1061no—
275C22×C4—1061no—
276C22×C4—1061no—
277C22×C4#1171992111yescharacteristic
278C22×C4#1171992111yescharacteristic
279C22×C4#1171992111yescharacteristic
280C22×C4C2×C4×D5311yescharacteristic
281C22×C4C2×D21211yescharacteristic
282C2×C8—2122no—
283C4⋊C4—41no—
284C4⋊C4—4241no—
285C4⋊C4—2121no—
286C22⋊C4—24no—
287C22⋊C4—41no—
288C22⋊C4—41no—
289C22⋊C4—42no—
290C22⋊C4—4241no—
291C22⋊C4—4241no—
292C22⋊C4—4241no—
293C22⋊C4—4241no—
294C22⋊C4—2121no—
295C22⋊C4—2121no—
296C22⋊C4—2122no—
297C42—21no—
298C42—21no—
299C42—82no—
300C23—21no—
301C23—21no—
302C23—21no—
303C23—21no—
304C23—21no—
305C23—41no—
306C23—41no—
307C23—41no—
308C23—41no—
309C23—2121no—
310C23—2121no—
311C23—2121no—
312C23—2121no—
313C23—2121no—
314C23—2121no—
315C23—2121no—
316C23C2×#1171992111yescharacteristic
317C23#1313323712yes—
318D4—44no—
319D4—4242no—
320D4—4244no—
321D4—4244no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—41no—
328C2×C4—42no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—44no—
333C2×C4—81no—
334C2×C4—81no—
335C2×C4—81no—
336C2×C4—82no—
337C2×C4—82no—
338C2×C4—2121no—
339C2×C4—2121no—
340C2×C4—2121no—
341C2×C4—2121no—
342C2×C4—2121no—
343C2×C4—2121no—
344C2×C4—2121no—
345C2×C4—2121no—
346C2×C4—2121no—
347C2×C4#1313313111yescharacteristic
348C2×C4#1313313111yescharacteristic
349C2×C4#1313314512yes—
350C8—4242no—
351C22—21no—
352C22—21no—
353C22—22no—
354C22—42no—
355C22—41no—
356C22—41no—
357C22—41no—
358C22—41no—
359C22—41no—
360C22—81no—
361C22—81no—
362C22—81no—
363C22—4241no—
364C22—4241no—
365C22—4241no—
366C22—4241no—
367C22—4241no—
368C22—4241no—
369C22—4241no—
370C22C2×#1313313111yescharacteristic
371C22#1681061711yescharacteristic
372C22C2×#1313314511yescharacteristic
373C4—22no—
374C4—21no—
375C4—21no—
376C4—41no—
377C4—41no—
378C4—41no—
379C4—84no—
380C4—4241no—
381C4—4241no—
382C4—4241no—
383C2—21no—
384C2—21no—
385C2—42no—
386C2—81no—
387C2—8481no—
388C2#2027320411yescharacteristic
389C2#2027635911yescharacteristic
390C2#2027638611yescharacteristic
subgroupG/H
centerC22C2×#13133131
commutatorC22×C4×C53C22×C4
FrattiniC2×C22⋊C4C22×D53
FittingC4×#99×C53C2
socleC22×C53C2×#99
prime radical#20280599C2
cosocleC2×C22⋊C4×C53C23
hypercenterC4×#99D53
hypocenterC53#11729
LCSG ▷ C22×C4×C53 ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ C4×#99

Complements

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

NH ≅ G/Ntype
2. #20273036C2semidirect
3. #20272738C2semidirect
4. #20272711C2semidirect
5. C4×#99×C53C2semidirect
6. C2×#13133191C4semidirect
8. #16810562C4semidirect
10. #16810540C4semidirect
16. #16810486C4semidirect
189. C4×#99D53semidirect
218. C53#11729semidirect

Maximal subgroups

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

subgroupindex
#202805992
#202730362
#202727382
#202727112
C4×#99×C532
#1172953