Finite Group Databasebeta

← #27294961

Subgroups — #27294961

Order 13568.

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

normal subgroups only

This group has 506 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#20278989C211yesmaximal, characteristic
2#20274789C211yesmaximal, characteristic
3#20272740C212yesmaximal, complemented
4#20272716C212yesmaximal, complemented
5C2×C53×#323C211yesmaximal, characteristic
6C2×#13133176C411yescharacteristic, complemented
7#16810559C411yescharacteristic, complemented
8#16810559C2211yescharacteristic
9#16810548C411yescharacteristic, complemented
10#16810548C2211yescharacteristic
11#16810511—21no—
12#16810511—21no—
13#16810496—21no—
14#16810486C411yescharacteristic, complemented
15#16810479—21no—
16#16810479—21no—
17#16810390—21no—
18#16810390C2212yes—
19C22×C22⋊C4×C53C2211yescharacteristic
20C2×C22⋊C8×C53—21no—
21C53×#323—21no—
22C53×#323C2212yes—
23#13133176—41no—
24#13133176—41no—
25C2×C4⋊Dic53C2×C411yescharacteristic
26C2×C4⋊Dic53C2×C411yescharacteristic
27C2×#11719919—21no—
28C2×#11719919—21no—
29C2×C22⋊Dic53—21no—
30C2×C22⋊Dic53—21no—
31C2×C22⋊Dic53—21no—
32C2×C22⋊Dic53—21no—
33C2×C22⋊Dic53—21no—
34C2×C22⋊Dic53—21no—
35C2×C22⋊Dic53—22no—
36C2×C22⋊Dic53—41no—
37C2×C22⋊Dic53—41no—
38C2×C4×Dic53—21no—
39#13133124—21no—
40#13133124—21no—
41#13133124—21no—
42#13133124C2×C411yescharacteristic
43#13133124C2×C411yescharacteristic
44C25×C53D411yescharacteristic
45C23×C4×C53C2×C411yescharacteristic
46C23×C4×C53D411yescharacteristic
47C22×C8×C53—41no—
48C2×C22⋊C4×C53—41no—
49C2×C22⋊C4×C53—41no—
50C2×C22⋊C4×C53C2×C411yescharacteristic
51C2×C22⋊C4×C53D411yescharacteristic
52C2×C22⋊C4×C53D411yescharacteristic
53C2×C22⋊C4×C53C2311yescharacteristic
54C22⋊C8×C53—22no—
55C22⋊C8×C53—41no—
56C24×C53—21no—
57C24×C53—21no—
58C24×C53—22no—
59C24×C53—41no—
60C24×C53—42no—
61C24×C53—81no—
62C24×C53C22⋊C411yescharacteristic
63C24×C53C22⋊C411yescharacteristic
64C24×C53C2×D411yescharacteristic
65C22×C4×C53—21no—
66C22×C4×C53—21no—
67C22×C4×C53—21no—
68C22×C4×C53—21no—
69C22×C4×C53—42no—
70C22×C4×C53—81no—
71C22×C4×C53C22⋊C411yescharacteristic
72C22×C4×C53C22⋊C411yescharacteristic
73C22×C4×C53C22×C411yescharacteristic
74C22×C4×C53C2×D411yescharacteristic
75C22×Dic53—21no—
76C22×Dic53—21no—
77C22×Dic53—21no—
78C22×Dic53—21no—
79C22×Dic53—42no—
80C2×C8×C53—42no—
81C2×C8×C53—82no—
82C22⋊C4×C53—41no—
83C22⋊C4×C53—41no—
84C22⋊C4×C53—41no—
85C22⋊C4×C53—41no—
86C22⋊Dic53—42no—
87C22⋊Dic53—42no—
88C22⋊Dic53—41no—
89C22⋊Dic53—41no—
90C22⋊Dic53—42no—
91C22⋊Dic53—42no—
92C22⋊Dic53—42no—
93C22⋊Dic53—81no—
94C22⋊Dic53—81no—
95C22⋊Dic53—82no—
96C22⋊Dic53—82no—
97C4⋊Dic53—41no—
98C4⋊Dic53—41no—
99#11719919—81no—
100#11719919—81no—
101C4×Dic53—42no—
102C23×C53—22no—
103C23×C53—21no—
104C23×C53—21no—
105C23×C53—21no—
106C23×C53—21no—
107C23×C53—21no—
108C23×C53—21no—
109C23×C53—21no—
110C23×C53—41no—
111C23×C53—42no—
112C23×C53—42no—
113C23×C53—41no—
114C23×C53—41no—
115C23×C53—41no—
116C23×C53—41no—
117C23×C53—42no—
118C23×C53—42no—
119C23×C53—81no—
120C23×C53—81no—
121C23×C53—84no—
122C23×C53—84no—
123C23×C53#9911yescharacteristic
124C23×C53#9911yescharacteristic
125C23×C53C2×C22⋊C411yescharacteristic
126C23×C53#13512yes—
127C2×C4×C53—21no—
128C2×C4×C53—21no—
129C2×C4×C53—21no—
130C2×C4×C53—21no—
131C2×C4×C53—42no—
132C2×C4×C53—42no—
133C2×C4×C53—42no—
134C2×C4×C53—41no—
135C2×C4×C53—41no—
136C2×C4×C53—41no—
137C2×C4×C53—41no—
138C2×C4×C53—81no—
139C2×Dic53—41no—
140C2×Dic53—41no—
141C2×Dic53—41no—
142C2×Dic53—41no—
143C2×Dic53—41no—
144C2×Dic53—41no—
145C2×Dic53—41no—
146C2×Dic53—41no—
147C2×Dic53—41no—
148C2×Dic53—41no—
149C2×Dic53—41no—
150C2×Dic53—41no—
151C2×Dic53—82no—
152C2×Dic53—82no—
153C2×Dic53—82no—
154C8×C53—82no—
155#11841—531nomaximal, Sylow
156C22×C53—21no—
157C22×C53—21no—
158C22×C53—21no—
159C22×C53—21no—
160C22×C53—21no—
161C22×C53—21no—
162C22×C53—22no—
163C22×C53—21no—
164C22×C53—42no—
165C22×C53—42no—
166C22×C53—42no—
167C22×C53—42no—
168C22×C53—41no—
169C22×C53—41no—
170C22×C53—41no—
171C22×C53—41no—
172C22×C53—41no—
173C22×C53—81no—
174C22×C53—84no—
175C22×C53—81no—
176C22×C53—84no—
177C22×C53#35212yes—
178C22×C53C2×#9911yescharacteristic
179C22×C53#41611yescharacteristic
180C22×C53#42111yescharacteristic
181C4×C53—41no—
182C4×C53—41no—
183C4×C53—41no—
184C4×C53—41no—
185Dic53—81no—
186Dic53—81no—
187Dic53—81no—
188Dic53—81no—
189Dic53—162no—
190#2482—531no—
191#1819—531no—
192C2×#323D5311yescharacteristic
193#1352—532no—
194#1344—532no—
195C2×C53—21no—
196C2×C53—21no—
197C2×C53—22no—
198C2×C53—42no—
199C2×C53—41no—
200C2×C53—81no—
201C2×C53#151611yescharacteristic
202C2×C53C2×#35211yescharacteristic
203C2×C53#212711yescharacteristic
204C2×#117—531no—
205C22×C22⋊C4C2×D5311yescharacteristic
206C2×C22⋊C8—21no—
207#399—531no—
208#399—531no—
209#397—531no—
210#397—531no—
211#388—1061no—
212#388—1061no—
213#385—1061no—
214#381—531no—
215#379—1061no—
216#379—1061no—
217#342—532no—
218#342—1061no—
219#323—21no—
220#323C2×D5312yes—
221C53#1184111yescharacteristic, complemented, Sylow
222C25D21211yescharacteristic
223C23×C4C4×D5311yescharacteristic
224C23×C4C22⋊D5311yescharacteristic
225C22×C8—41no—
226#117—2121no—
227#117—2121no—
228C2×C4⋊C4—531no—
229C2×C4⋊C4—531no—
230C2×C4⋊C4—1061no—
231C2×C4⋊C4—1061no—
232C2×C22⋊C4—41no—
233C2×C22⋊C4—41no—
234C2×C22⋊C4—1061no—
235C2×C22⋊C4—1061no—
236C2×C22⋊C4—1061no—
237C2×C22⋊C4—1061no—
238C2×C22⋊C4—1061no—
239C2×C22⋊C4—1061no—
240C2×C22⋊C4—1062no—
241C2×C22⋊C4—2121no—
242C2×C22⋊C4—2121no—
243C2×C22⋊C4C4×D5311yescharacteristic
244C2×C22⋊C4D21211yescharacteristic
245C2×C22⋊C4C22⋊D5311yescharacteristic
246C2×C22⋊C4C22×D5311yescharacteristic
247C2×C42—1061no—
248C22⋊C8—22no—
249C22⋊C8—41no—
250#95—531no—
251#95—531no—
252#95—1061no—
253#95—1061no—
254#95—1061no—
255C24—21no—
256C24—21no—
257C24—22no—
258C24—41no—
259C24—42no—
260C24—81no—
261C24#1171992111yescharacteristic
262C24#1171992111yescharacteristic
263C24C2×D21211yescharacteristic
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—42no—
269C22×C4—81no—
270C22×C4—1061no—
271C22×C4—1061no—
272C22×C4—1061no—
273C22×C4—1061no—
274C22×C4—2122no—
275C22×C4#1171992111yescharacteristic
276C22×C4#1171992111yescharacteristic
277C22×C4C2×C4×D5311yescharacteristic
278C22×C4C2×C22⋊D5311yescharacteristic
279C2×C8—42no—
280C2×C8—82no—
281C4⋊C4—4241no—
282C4⋊C4—4241no—
283C4⋊C4—2121no—
284C4⋊C4—2121no—
285C22⋊C4—41no—
286C22⋊C4—41no—
287C22⋊C4—41no—
288C22⋊C4—41no—
289C22⋊C4—4242no—
290C22⋊C4—4242no—
291C22⋊C4—4241no—
292C22⋊C4—4241no—
293C22⋊C4—2122no—
294C22⋊C4—2122no—
295C22⋊C4—2122no—
296C22⋊C4—2122no—
297C22⋊C4—2121no—
298C22⋊C4—2121no—
299C22⋊C4—2122no—
300C42—2122no—
301C23—21no—
302C23—21no—
303C23—21no—
304C23—22no—
305C23—21no—
306C23—21no—
307C23—21no—
308C23—21no—
309C23—41no—
310C23—42no—
311C23—42no—
312C23—41no—
313C23—41no—
314C23—41no—
315C23—41no—
316C23—42no—
317C23—42no—
318C23—81no—
319C23—81no—
320C23—84no—
321C23—84no—
322C23#1313313111yescharacteristic
323C23#1313313111yescharacteristic
324C23C2×#1171992111yescharacteristic
325C23#1313323912yes—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—42no—
331C2×C4—42no—
332C2×C4—42no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—81no—
338C2×C4—4242no—
339C2×C4—4242no—
340C2×C4—4242no—
341C2×C4—2121no—
342C2×C4—2121no—
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—2121no—
353C8—82no—
354C22—21no—
355C22—21no—
356C22—21no—
357C22—21no—
358C22—22no—
359C22—21no—
360C22—21no—
361C22—21no—
362C22—42no—
363C22—42no—
364C22—42no—
365C22—42no—
366C22—41no—
367C22—41no—
368C22—41no—
369C22—41no—
370C22—41no—
371C22—81no—
372C22—84no—
373C22—81no—
374C22—84no—
375C22#1681041412yes—
376C22C2×#1313313111yescharacteristic
377C22#1681062011yescharacteristic
378C22#1681063911yescharacteristic
379C4—41no—
380C4—41no—
381C4—41no—
382C4—41no—
383C4—8482no—
384C4—4241no—
385C4—4241no—
386C4—4241no—
387C4—4241no—
388C2—21no—
389C2—21no—
390C2—22no—
391C2—42no—
392C2—41no—
393C2—81no—
394C2#2027327211yescharacteristic
395C2C2×#1681041411yescharacteristic
396C2#2027638211yescharacteristic
subgroupG/H
centerC22C2×#13133131
commutatorC22×C4×C53C22×C4
FrattiniC2×C22⋊C4C22×D53
FittingC2×C53×#323C2
socleC22×C53C2×#99
prime radicalC25×C53D4
cosocleC2×C22⋊C4×C53C23
hypercenterC2×#323D53
hypocenterC53#11841
LCSG ▷ C22×C4×C53 ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C24 ◁ C22×C22⋊C4 ◁ C2×#323

Complements

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

NH ≅ G/Ntype
3. #20272740C2semidirect
4. #20272716C2semidirect
6. C2×#13133176C4semidirect
7. #16810559C4semidirect
9. #16810548C4semidirect
14. #16810486C4semidirect
221. C53#11841semidirect

Maximal subgroups

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

subgroupindex
#202789892
#202747892
#202727402
#202727162
C2×C53×#3232
#1184153