Finite Group Databasebeta

← #27294571

Subgroups — #27294571

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#20280402C211yesmaximal, characteristic
2#20273049C211yesmaximal, characteristic, complemented
3#20272735C212yesmaximal, complemented
4#20272711C212yesmaximal
5C53×#1740C211yesmaximal, characteristic, complemented
6C2×#13133205C411yescharacteristic, complemented
7#16810680—21no—
8#16810543C411yescharacteristic, complemented
9#16810540C411yescharacteristic, complemented
10#16810535C2211yescharacteristic
11#16810532C2211yescharacteristic
12#16810523—21no—
13#16810514—21no—
14#16810513—21no—
15#16810502—21no—
16#16810469C411yescharacteristic, complemented
17#16810349—21no—
18#16810347C2212yes—
19C2×#117×C53—21no—
20C2×#117×C53C2211yescharacteristic
21C53×#342—21no—
22C53×#342C2212yes—
23C22×Dic106C2×C411yescharacteristic
24C22×D212C2×C411yescharacteristic
25C22×C22⋊D53—21no—
26#13133205—41no—
27#13133205—41no—
28C2×#11719919—21no—
29C2×C22⋊Dic53—21no—
30C2×#11719921—21no—
31C2×#11719921—21no—
32C2×#11719921—21no—
33C2×#11719921—21no—
34C2×#11719921—21no—
35C2×C4×Dic53—21no—
36C4⋊#88322—41no—
37C22⋊#88322—22no—
38#13133128—41no—
39#13133124—21no—
40#13133124—21no—
41#13133124—21no—
42#13133124C2×C411yescharacteristic
43#13133124C2×C411yescharacteristic
44C23×C4×C53D411yescharacteristic
45#117×C53—24no—
46#117×C53—41no—
47#117×C53—41no—
48#117×C53—42no—
49C2×C4⋊C4×C53—41no—
50C2×C4⋊C4×C53C2×C411yescharacteristic
51C2×C4⋊C4×C53D411yescharacteristic
52C2×C22⋊C4×C53—22no—
53C2×C22⋊C4×C53D411yescharacteristic
54C2×C22⋊C4×C53C2311yescharacteristic
55C2×C42×C53—41no—
56C2×C42×C53C2×C411yescharacteristic
57C2×C42×C53D411yescharacteristic
58C24×C53C2×D411yescharacteristic
59C23×D53—21no—
60C22×C4×C53—21no—
61C22×C4×C53—21no—
62C22×C4×C53—21no—
63C22×C4×C53—21no—
64C22×C4×C53—22no—
65C22×C4×C53—42no—
66C22×C4×C53C22⋊C411yescharacteristic
67C22×C4×C53C22⋊C411yescharacteristic
68C22×C4×C53C22⋊C411yescharacteristic
69C22×C4×C53C22⋊C411yescharacteristic
70C22×C4×C53C22×C411yescharacteristic
71C22×C4×C53C2×D411yescharacteristic
72C2×C22⋊D53—42no—
73C2×C22⋊D53—41no—
74C2×C22⋊D53—41no—
75C2×C22⋊D53—81no—
76C22×Dic53—21no—
77C22×Dic53—21no—
78C22×Dic53—21no—
79C2×D212—41no—
80C2×D212—81no—
81C2×Dic106—41no—
82C2×Dic106—81no—
83C4⋊C4×C53—41no—
84C4⋊C4×C53—41no—
85C4⋊C4×C53—44no—
86C22⋊C4×C53—24no—
87C22⋊C4×C53—41no—
88C22⋊C4×C53—41no—
89C22⋊C4×C53—42no—
90C42×C53—21no—
91C42×C53—21no—
92C42×C53—21no—
93C42×C53—21no—
94C42×C53—44no—
95C22⋊Dic53—81no—
96#11719921—42no—
97#11719921—42no—
98#11719921—41no—
99#11719921—41no—
100#11719921—42no—
101#11719921—81no—
102#11719919—81no—
103C4×Dic53—42no—
104C2×#88322—42no—
105C23×C53—21no—
106C23×C53—21no—
107C23×C53—41no—
108C23×C53—41no—
109C23×C53C2×C22⋊C411yescharacteristic
110C23×C53AGL(1,ℤ/8)11yescharacteristic
111C23×C53#13711yescharacteristic
112C22×D53—41no—
113C22×D53—41no—
114C22×D53—41no—
115C22×D53—41no—
116C22×D53—41no—
117C22×D53—41no—
118C22×D53—41no—
119C2×C4×C53—22no—
120C2×C4×C53—22no—
121C2×C4×C53—22no—
122C2×C4×C53—22no—
123C2×C4×C53—21no—
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—21no—
131C2×C4×C53—41no—
132C2×C4×C53—41no—
133C2×C4×C53—41no—
134C2×C4×C53—41no—
135C2×C4×C53—44no—
136C2×C4×C53—82no—
137C2×C4×C53—82no—
138C2×C4×C53#9911yescharacteristic
139C2×C4×C53#9911yescharacteristic
140C2×C4×C53#10412yes—
141C22⋊D53—82no—
142C22⋊D53—82no—
143C2×Dic53—22no—
144C2×Dic53—41no—
145C2×Dic53—41no—
146C2×Dic53—41no—
147C2×Dic53—41no—
148C2×Dic53—41no—
149C2×Dic53—41no—
150C2×Dic53—41no—
151C2×Dic53—41no—
152D212—82no—
153Dic106—82no—
154#88322—82no—
155#11758—531nomaximal, Sylow
156C22×C53—21no—
157C22×C53—21no—
158C22×C53—21no—
159C22×C53—21no—
160C22×C53—41no—
161C22×C53—41no—
162C22×C53—41no—
163C22×C53—41no—
164C22×C53—81no—
165C22×C53C2×#9911yescharacteristic
166C22×C53#41711yescharacteristic
167C22×C53C2×#10411yescharacteristic
168C2×D53—81no—
169C2×D53—81no—
170C2×D53—81no—
171C2×D53—81no—
172C2×D53—81no—
173C2×D53—81no—
174C2×D53—81no—
175C4×C53—21no—
176C4×C53—21no—
177C4×C53—22no—
178C4×C53—41no—
179C4×C53—41no—
180C4×C53—41no—
181C4×C53—41no—
182C4×C53—84no—
183Dic53—42no—
184Dic53—81no—
185Dic53—81no—
186#2610—531no—
187#1740D5311yescharacteristic, complemented
188#1464—531no—
189#1351—532no—
190#1343—532no—
191C2×C53—21no—
192C2×C53—21no—
193C2×C53—41no—
194C2×C53—41no—
195C2×C53#150411yescharacteristic
196C2×C53#212211yescharacteristic
197C2×C53#212611yescharacteristic
198D53—161no—
199C2×#124—531no—
200C2×#117—21no—
201C2×#117C2×D5311yescharacteristic
202#432—1061no—
203#395—531no—
204#394—531no—
205#393—531no—
206#392—531no—
207#390—1061no—
208#388—1061no—
209#388—1061no—
210#386—1061no—
211#376—531no—
212#342—21no—
213#342C2×D5312yes—
214#324—1061no—
215#323—532no—
216C53#1175811yescharacteristic, complemented, Sylow
217C22×Q8—531no—
218C22×D4—531no—
219C22×D4—1061no—
220C23×C4C22⋊D5311yescharacteristic
221#124—2121no—
222#124—2121no—
223#117—24no—
224#117—41no—
225#117—41no—
226#117—42no—
227C2×C4⋊C4—41no—
228C2×C4⋊C4—1061no—
229C2×C4⋊C4C4×D5311yescharacteristic
230C2×C4⋊C4D21211yescharacteristic
231C2×C22⋊C4—22no—
232C2×C22⋊C4—1061no—
233C2×C22⋊C4—1061no—
234C2×C22⋊C4—1061no—
235C2×C22⋊C4—1061no—
236C2×C22⋊C4—1061no—
237C2×C22⋊C4—1061no—
238C2×C22⋊C4C22⋊D5311yescharacteristic
239C2×C22⋊C4C22×D5311yescharacteristic
240C2×C42—41no—
241C2×C42—1061no—
242C2×C42C4×D5311yescharacteristic
243C2×C42D21211yescharacteristic
244C4⋊C8—2121no—
245C22⋊C8—1062no—
246#97—2121no—
247#95—531no—
248#95—531no—
249#95—1061no—
250#95—1061no—
251#95—1061no—
252C24—1061no—
253C24C2×C22⋊D5311yescharacteristic
254C2×Q8—4241no—
255C2×Q8—2121no—
256C2×D4—4241no—
257C2×D4—4241no—
258C2×D4—2121no—
259C2×D4—2122no—
260C2×D4—2121no—
261C2×D4—2121no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—22no—
267C22×C4—42no—
268C22×C4—1061no—
269C22×C4—1061no—
270C22×C4—1061no—
271C22×C4#1171992111yescharacteristic
272C22×C4#1171992111yescharacteristic
273C22×C4#1171992111yescharacteristic
274C22×C4#1171992111yescharacteristic
275C22×C4C2×C4×D5311yescharacteristic
276C22×C4C2×D21211yescharacteristic
277C2×C8—2122no—
278C4⋊C4—41no—
279C4⋊C4—41no—
280C4⋊C4—44no—
281C4⋊C4—4241no—
282C22⋊C4—24no—
283C22⋊C4—41no—
284C22⋊C4—41no—
285C22⋊C4—42no—
286C22⋊C4—4241no—
287C22⋊C4—4241no—
288C22⋊C4—2122no—
289C22⋊C4—2122no—
290C22⋊C4—2121no—
291C22⋊C4—2121no—
292C22⋊C4—2122no—
293C42—21no—
294C42—21no—
295C42—21no—
296C42—21no—
297C42—44no—
298C42—2122no—
299C23—21no—
300C23—21no—
301C23—41no—
302C23—41no—
303C23—2121no—
304C23—2121no—
305C23—2121no—
306C23—2121no—
307C23—2121no—
308C23—2121no—
309C23—2121no—
310C23C2×#1171992111yescharacteristic
311C23#1313324311yescharacteristic
312C23#1313324811yescharacteristic
313Q8—4242no—
314D4—4242no—
315D4—4242no—
316D4—4242no—
317C2×C4—22no—
318C2×C4—22no—
319C2×C4—22no—
320C2×C4—22no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—44no—
334C2×C4—82no—
335C2×C4—82no—
336C2×C4—1062no—
337C2×C4—2121no—
338C2×C4—2121no—
339C2×C4—2121no—
340C2×C4—2121no—
341C2×C4—2121no—
342C2×C4—2121no—
343C2×C4—2121no—
344C2×C4—2121no—
345C2×C4#1313313111yescharacteristic
346C2×C4#1313313111yescharacteristic
347C2×C4#1313314512yes—
348C8—4242no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—21no—
353C22—41no—
354C22—41no—
355C22—41no—
356C22—41no—
357C22—81no—
358C22—4241no—
359C22—4241no—
360C22—4241no—
361C22—4241no—
362C22—4241no—
363C22—4241no—
364C22—4241no—
365C22C2×#1313313111yescharacteristic
366C22#1681062211yescharacteristic
367C22C2×#1313314511yescharacteristic
368C4—21no—
369C4—21no—
370C4—22no—
371C4—41no—
372C4—41no—
373C4—41no—
374C4—41no—
375C4—84no—
376C4—4241no—
377C4—4241no—
378C4—2122no—
379C2—21no—
380C2—21no—
381C2—41no—
382C2—41no—
383C2—8481no—
384C2#2027321711yescharacteristic
385C2#2027636411yescharacteristic
386C2#2027637811yescharacteristic
subgroupG/H
centerC22C2×#13133131
commutatorC22×C4×C53C22×C4
FrattiniC2×C22⋊C4C22×D53
FittingC53×#1740C2
socleC22×C53C2×#99
prime radical#16810532C22
cosocleC2×C22⋊C4×C53C23
hypercenter#1740D53
hypocenterC53#11758
LCSG ▷ C22×C4×C53 ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #1740

Complements

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

NH ≅ G/Ntype
2. #20273049C2semidirect
3. #20272735C2semidirect
5. C53×#1740C2semidirect
6. C2×#13133205C4semidirect
8. #16810543C4semidirect
9. #16810540C4semidirect
16. #16810469C4semidirect
187. #1740D53semidirect
216. C53#11758semidirect

Maximal subgroups

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

subgroupindex
#202804022
#202730492
#202727352
#202727112
C53×#17402
#1175853