Finite Group Databasebeta

← #27294439

Subgroups — #27294439

Order 13568.

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

normal subgroups only

This group has 500 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#20280567C211yesmaximal, characteristic
2#20273033C211yesmaximal, characteristic, complemented
3#20272735C212yesmaximal, complemented
4#20272711C212yesmaximal, complemented
5C4×#99×C53C211yesmaximal, characteristic, complemented
6C2×#13133205C411yescharacteristic, complemented
7C2×C22⋊#88322—21no—
8#16810549—21no—
9#16810535C411yescharacteristic, complemented
10#16810535C2211yescharacteristic
11#16810532C411yescharacteristic, complemented
12#16810532C2211yescharacteristic
13#16810531—21no—
14#16810502—21no—
15#16810502—21no—
16#16810491C411yescharacteristic, complemented
17#16810349—21no—
18#16810347C2212yes—
19C2×C4×D4×C53C2211yescharacteristic
20C2×#99×C53—21no—
21C4×C22⋊C4×C53—21no—
22C53×#342C2212yes—
23C22×Dic106C2×C411yescharacteristic
24C22×D212C2×C411yescharacteristic
25C22×C22⋊D53—21no—
26C22×C22⋊D53—21no—
27C22×#88322—41no—
28#13133205—41no—
29#13133205—41no—
30C2×C22⋊Dic53—21no—
31C2×C22⋊Dic53—21no—
32C2×#11719921—21no—
33C2×#11719921—21no—
34C2×#11719921—21no—
35C2×#11719921—21no—
36C22⋊#88322—22no—
37C22⋊#88322—41no—
38#13133124—21no—
39#13133124—21no—
40#13133124—21no—
41#13133124—21no—
42#13133124C2×C411yescharacteristic
43#13133124C2×C411yescharacteristic
44C22×D4×C53D411yescharacteristic
45C23×C4×C53D411yescharacteristic
46C23×C4×C53D411yescharacteristic
47C4×D4×C53—41no—
48C4×D4×C53—41no—
49C2×C4⋊C4×C53D411yescharacteristic
50C2×C22⋊C4×C53—22no—
51C2×C22⋊C4×C53C2×C411yescharacteristic
52C2×C22⋊C4×C53C2311yescharacteristic
53C2×C42×C53—41no—
54C2×C42×C53C2×C411yescharacteristic
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—
86C2×Dic106—41no—
87C2×Dic106—81no—
88C4⋊C4×C53—41no—
89C22⋊C4×C53—24no—
90C22⋊C4×C53—41no—
91C22⋊C4×C53—41no—
92C22⋊C4×C53—42no—
93C42×C53—21no—
94C42×C53—21no—
95C42×C53—82no—
96C22⋊Dic53—81no—
97C22⋊Dic53—81no—
98#11719921—42no—
99#11719921—41no—
100#11719921—41no—
101#11719921—81no—
102#11719921—81no—
103C2×#88322—42no—
104C2×#88322—82no—
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×C53#10412yes—
114C23×C53C2×C22⋊C411yescharacteristic
115C23×C53#13512yes—
116C22×D53—41no—
117C22×D53—41no—
118C22×D53—41no—
119C22×D53—41no—
120C22×D53—41no—
121C22×D53—41no—
122C22×D53—41no—
123D4×C53—44no—
124C2×C4×C53—21no—
125C2×C4×C53—21no—
126C2×C4×C53—21no—
127C2×C4×C53—21no—
128C2×C4×C53—21no—
129C2×C4×C53—21no—
130C2×C4×C53—41no—
131C2×C4×C53—42no—
132C2×C4×C53—42no—
133C2×C4×C53—41no—
134C2×C4×C53—41no—
135C2×C4×C53—41no—
136C2×C4×C53—44no—
137C2×C4×C53—81no—
138C2×C4×C53—81no—
139C2×C4×C53—82no—
140C2×C4×C53—82no—
141C2×C4×C53#9911yescharacteristic
142C2×C4×C53#9911yescharacteristic
143C22⋊D53—84no—
144C22⋊D53—84no—
145C2×Dic53—41no—
146C2×Dic53—41no—
147C2×Dic53—41no—
148C2×Dic53—41no—
149C2×Dic53—41no—
150C2×Dic53—41no—
151C2×Dic53—41no—
152C2×Dic53—41no—
153C2×Dic53—41no—
154D212—82no—
155Dic106—82no—
156#88322—82no—
157#11730—531nomaximal, Sylow
158C22×C53—21no—
159C22×C53—21no—
160C22×C53—22no—
161C22×C53—22no—
162C22×C53—42no—
163C22×C53—42no—
164C22×C53—41no—
165C22×C53—41no—
166C22×C53—41no—
167C22×C53—41no—
168C22×C53—81no—
169C22×C53—81no—
170C22×C53C2×#9911yescharacteristic
171C22×C53#41611yescharacteristic
172C22×C53C2×#10411yescharacteristic
173C2×D53—81no—
174C2×D53—81no—
175C2×D53—81no—
176C2×D53—81no—
177C2×D53—81no—
178C2×D53—81no—
179C2×D53—81no—
180C4×C53—21no—
181C4×C53—21no—
182C4×C53—41no—
183C4×C53—41no—
184C4×C53—41no—
185C4×C53—41no—
186C4×C53—84no—
187Dic53—81no—
188Dic53—81no—
189Dic53—81no—
190#2631—531no—
191C4×#99D5311yescharacteristic, complemented
192#1459—531no—
193#1351—532no—
194#1343—532no—
195C2×C53—21no—
196C2×C53—21no—
197C2×C53—42no—
198C2×C53—42no—
199C2×C53#150111yescharacteristic
200C2×C53#212211yescharacteristic
201C2×C53#212711yescharacteristic
202D53—161no—
203C2×#124—531no—
204C2×C4×D4C2×D5311yescharacteristic
205C2×#99—21no—
206C2×C22⋊C8—1061no—
207#397—1061no—
208#393—531no—
209#393—531no—
210#392—531no—
211#392—531no—
212#392—1061no—
213#386—1061no—
214#386—1061no—
215#383—531no—
216C4×C22⋊C4—21no—
217#342C2×D5312yes—
218#324—1061no—
219#323—532no—
220C53#1173011yescharacteristic, complemented, Sylow
221C22×Q8—531no—
222C22×D4—531no—
223C22×D4—1061no—
224C22×D4—1061no—
225C22×D4C22⋊D5311yescharacteristic
226C23×C4D21211yescharacteristic
227C23×C4C22⋊D5311yescharacteristic
228C22×C8—2121no—
229#124—2121no—
230#124—2121no—
231C4×D4—41no—
232C4×D4—41no—
233C2×C4⋊C4D21211yescharacteristic
234C2×C22⋊C4—22no—
235C2×C22⋊C4—1061no—
236C2×C22⋊C4—1061no—
237C2×C22⋊C4—1061no—
238C2×C22⋊C4—1061no—
239C2×C22⋊C4—1061no—
240C2×C22⋊C4—1061no—
241C2×C22⋊C4C4×D5311yescharacteristic
242C2×C22⋊C4C22×D5311yescharacteristic
243C2×C42—41no—
244C2×C42C4×D5311yescharacteristic
245#99—24no—
246C22⋊C8—1062no—
247C22⋊C8—2121no—
248#95—41no—
249#95—531no—
250#95—531no—
251#95—1061no—
252#95—1061no—
253#95—1061no—
254#95—1061no—
255C24—1061no—
256C24#1171992111yescharacteristic
257C24C2×C22⋊D5311yescharacteristic
258C2×Q8—4241no—
259C2×Q8—2121no—
260C2×D4—22no—
261C2×D4—22no—
262C2×D4—41no—
263C2×D4—4241no—
264C2×D4—4241no—
265C2×D4—4241no—
266C2×D4—4241no—
267C2×D4—4241no—
268C2×D4—2121no—
269C2×D4—2122no—
270C2×D4—2122no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—42no—
276C22×C4—81no—
277C22×C4—81no—
278C22×C4—1061no—
279C22×C4—1061no—
280C22×C4—1061no—
281C22×C4#1171992111yescharacteristic
282C22×C4#1171992111yescharacteristic
283C22×C4#1171992111yescharacteristic
284C22×C4C2×C4×D5311yescharacteristic
285C22×C4C2×D21211yescharacteristic
286C2×C8—4242no—
287C2×C8—2122no—
288C4⋊C4—41no—
289C22⋊C4—24no—
290C22⋊C4—41no—
291C22⋊C4—41no—
292C22⋊C4—42no—
293C22⋊C4—4241no—
294C22⋊C4—4241no—
295C22⋊C4—4241no—
296C22⋊C4—4241no—
297C22⋊C4—2122no—
298C22⋊C4—2121no—
299C22⋊C4—2121no—
300C42—21no—
301C42—21no—
302C42—82no—
303C23—21no—
304C23—21no—
305C23—21no—
306C23—21no—
307C23—41no—
308C23—41no—
309C23—41no—
310C23—41no—
311C23—2121no—
312C23—2121no—
313C23—2121no—
314C23—2121no—
315C23—2121no—
316C23—2121no—
317C23—2121no—
318C23#1313314512yes—
319C23C2×#1171992111yescharacteristic
320C23#1313323712yes—
321Q8—4242no—
322D4—44no—
323D4—4242no—
324D4—4244no—
325D4—4244no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—41no—
333C2×C4—42no—
334C2×C4—42no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—44no—
339C2×C4—81no—
340C2×C4—81no—
341C2×C4—82no—
342C2×C4—82no—
343C2×C4—2121no—
344C2×C4—2121no—
345C2×C4—2121no—
346C2×C4—2121no—
347C2×C4—2121no—
348C2×C4—2121no—
349C2×C4—2121no—
350C2×C4—2121no—
351C2×C4—2121no—
352C2×C4#1313313111yescharacteristic
353C2×C4#1313313111yescharacteristic
354C8—4242no—
355C22—21no—
356C22—21no—
357C22—22no—
358C22—22no—
359C22—42no—
360C22—42no—
361C22—41no—
362C22—41no—
363C22—41no—
364C22—41no—
365C22—81no—
366C22—81no—
367C22—4241no—
368C22—4241no—
369C22—4241no—
370C22—4241no—
371C22—4241no—
372C22—4241no—
373C22—4241no—
374C22C2×#1313313111yescharacteristic
375C22#1681061711yescharacteristic
376C22C2×#1313314511yescharacteristic
377C4—21no—
378C4—21no—
379C4—41no—
380C4—41no—
381C4—41no—
382C4—41no—
383C4—84no—
384C4—4241no—
385C4—4241no—
386C4—4241no—
387C2—21no—
388C2—21no—
389C2—42no—
390C2—42no—
391C2—8481no—
392C2#2027320411yescharacteristic
393C2#2027636411yescharacteristic
394C2#2027638211yescharacteristic
subgroupG/H
centerC22C2×#13133131
commutatorC22×C4×C53C22×C4
FrattiniC2×C22⋊C4C22×D53
FittingC4×#99×C53C2
socleC22×C53C2×#99
prime radical#20280567C2
cosocleC2×C22⋊C4×C53C23
hypercenterC4×#99D53
hypocenterC53#11730
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. #20273033C2semidirect
3. #20272735C2semidirect
4. #20272711C2semidirect
5. C4×#99×C53C2semidirect
6. C2×#13133205C4semidirect
9. #16810535C4semidirect
11. #16810532C4semidirect
16. #16810491C4semidirect
191. C4×#99D53semidirect
220. C53#11730semidirect

Maximal subgroups

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

subgroupindex
#202805672
#202730332
#202727352
#202727112
C4×#99×C532
#1173053