Finite Group Databasebeta

← #27294815

Subgroups — #27294815

Order 13568.

496 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 496 conjugacy classes of subgroups — too many to draw. Only its 79 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#20280520C211yesmaximal, characteristic
2#20274789C211yesmaximal, characteristic
3#20272740C212yesmaximal, complemented
4#20272737C212yesmaximal, complemented
5C53×#1462C211yesmaximal, characteristic
6C2×#13133196C411yescharacteristic, complemented
7#16810559C411yescharacteristic, complemented
8#16810559C2211yescharacteristic
9#16810546—21no—
10#16810546—21no—
11#16810535C411yescharacteristic, complemented
12#16810535C2211yescharacteristic
13#16810496—21no—
14#16810483C411yescharacteristic, complemented
15#16810479—21no—
16#16810479—21no—
17#16810390—21no—
18#16810390C2212yes—
19C22×C22⋊C4×C53C2211yescharacteristic
20C53×#407—21no—
21C53×#323—21no—
22C53×#323C2212yes—
23C22×Dic106C2×C411yescharacteristic
24#13133196—41no—
25#13133196—41no—
26C2×C4⋊Dic53—21no—
27C2×C4⋊Dic53C2×C411yescharacteristic
28C2×#11719919—21no—
29C2×#11719919—21no—
30C2×#11719919—21no—
31C2×C22⋊Dic53—21no—
32C2×C22⋊Dic53—21no—
33C2×C22⋊Dic53—21no—
34C2×C22⋊Dic53—21no—
35C2×C22⋊Dic53—21no—
36C2×C22⋊Dic53—21no—
37C2×C22⋊Dic53—22no—
38C2×C22⋊Dic53—41no—
39C2×C22⋊Dic53—41no—
40#13133124—21no—
41#13133124—21no—
42#13133124C2×C411yescharacteristic
43#13133124C2×C411yescharacteristic
44C25×C53D411yescharacteristic
45C23×C4×C53C2×C411yescharacteristic
46C23×C4×C53D411yescharacteristic
47C2×M16×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—81no—
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—81no—
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×Dic106—41no—
81C2×Dic106—81no—
82M16×C53—82no—
83C2×C8×C53—42no—
84C22⋊C4×C53—41no—
85C22⋊C4×C53—41no—
86C22⋊C4×C53—41no—
87C22⋊C4×C53—41no—
88C22⋊Dic53—42no—
89C22⋊Dic53—41no—
90C22⋊Dic53—41no—
91C22⋊Dic53—42no—
92C22⋊Dic53—42no—
93C22⋊Dic53—81no—
94C22⋊Dic53—81no—
95C22⋊Dic53—81no—
96C22⋊Dic53—81no—
97C22⋊Dic53—82no—
98C22⋊Dic53—82no—
99C4⋊Dic53—41no—
100C4⋊Dic53—81no—
101#11719919—42no—
102#11719919—81no—
103#11719919—81no—
104C23×C53—21no—
105C23×C53—21no—
106C23×C53—21no—
107C23×C53—21no—
108C23×C53—21no—
109C23×C53—21no—
110C23×C53—21no—
111C23×C53—41no—
112C23×C53—41no—
113C23×C53—42no—
114C23×C53—42no—
115C23×C53—41no—
116C23×C53—41no—
117C23×C53—41no—
118C23×C53—41no—
119C23×C53—42no—
120C23×C53—42no—
121C23×C53—81no—
122C23×C53—81no—
123C23×C53—84no—
124C23×C53—84no—
125C23×C53#9911yescharacteristic
126C23×C53#9911yescharacteristic
127C23×C53C2×C22⋊C411yescharacteristic
128C23×C53#13712yes—
129C2×C4×C53—21no—
130C2×C4×C53—21no—
131C2×C4×C53—21no—
132C2×C4×C53—21no—
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×C4×C53—81no—
140C2×C4×C53—81no—
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—41no—
152C2×Dic53—41no—
153C2×Dic53—82no—
154C2×Dic53—82no—
155C2×Dic53—82no—
156Dic106—82no—
157C8×C53—82no—
158#11811—531nomaximal, Sylow
159C22×C53—21no—
160C22×C53—21no—
161C22×C53—21no—
162C22×C53—21no—
163C22×C53—21no—
164C22×C53—21no—
165C22×C53—22no—
166C22×C53—21no—
167C22×C53—21no—
168C22×C53—42no—
169C22×C53—42no—
170C22×C53—42no—
171C22×C53—42no—
172C22×C53—41no—
173C22×C53—41no—
174C22×C53—41no—
175C22×C53—41no—
176C22×C53—41no—
177C22×C53—81no—
178C22×C53—81no—
179C22×C53—84no—
180C22×C53—84no—
181C22×C53C2×#9911yescharacteristic
182C22×C53#41911yescharacteristic
183C22×C53#42111yescharacteristic
184C4×C53—41no—
185C4×C53—41no—
186C4×C53—41no—
187C4×C53—41no—
188Dic53—81no—
189Dic53—81no—
190Dic53—81no—
191Dic53—81no—
192Dic53—162no—
193#2626—531no—
194#1819—531no—
195#1462D5311yescharacteristic
196#1352—532no—
197#1351—532no—
198C2×C53—21no—
199C2×C53—21no—
200C2×C53—42no—
201C2×C53—41no—
202C2×C53—41no—
203C2×C53—81no—
204C2×C53#151111yescharacteristic
205C2×C53#212111yescharacteristic
206C2×C53#212711yescharacteristic
207C2×C22⋊Q8—531no—
208C22×C22⋊C4C2×D5311yescharacteristic
209#407—21no—
210#399—531no—
211#399—531no—
212#396—1061no—
213#396—1061no—
214#393—531no—
215#393—531no—
216#385—1061no—
217#380—531no—
218#379—1061no—
219#379—1061no—
220#342—532no—
221#342—1061no—
222#323—21no—
223#323C2×D5312yes—
224C53#1181111yescharacteristic, complemented, Sylow
225C25D21211yescharacteristic
226C22×Q8—531no—
227C23×C4C4×D5311yescharacteristic
228C23×C4C22⋊D5311yescharacteristic
229C2×M16—41no—
230C22⋊Q8—2121no—
231C22⋊Q8—2121no—
232C2×C4⋊C4—531no—
233C2×C4⋊C4—1061no—
234C2×C4⋊C4—1061no—
235C2×C4⋊C4—1061no—
236C2×C4⋊C4—1061no—
237C2×C22⋊C4—41no—
238C2×C22⋊C4—41no—
239C2×C22⋊C4—1061no—
240C2×C22⋊C4—1061no—
241C2×C22⋊C4—1061no—
242C2×C22⋊C4—1061no—
243C2×C22⋊C4—1061no—
244C2×C22⋊C4—1061no—
245C2×C22⋊C4—1062no—
246C2×C22⋊C4—2121no—
247C2×C22⋊C4—2121no—
248C2×C22⋊C4C4×D5311yescharacteristic
249C2×C22⋊C4D21211yescharacteristic
250C2×C22⋊C4C22⋊D5311yescharacteristic
251C2×C22⋊C4C22×D5311yescharacteristic
252C22⋊C8—22no—
253C22⋊C8—41no—
254#95—531no—
255#95—531no—
256#95—1061no—
257#95—1061no—
258C24—21no—
259C24—21no—
260C24—22no—
261C24—41no—
262C24—81no—
263C24—81no—
264C24#1171992111yescharacteristic
265C24#1171992111yescharacteristic
266C24C2×D21211yescharacteristic
267C2×Q8—4241no—
268C2×Q8—2121no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—81no—
274C22×C4—81no—
275C22×C4—1061no—
276C22×C4—1061no—
277C22×C4—1061no—
278C22×C4—1061no—
279C22×C4—2122no—
280C22×C4#1171992111yescharacteristic
281C22×C4#1171992111yescharacteristic
282C22×C4C2×C4×D5311yescharacteristic
283C22×C4C2×C22⋊D5311yescharacteristic
284M16—82no—
285C2×C8—42no—
286C4⋊C4—4241no—
287C4⋊C4—4241no—
288C4⋊C4—4241no—
289C4⋊C4—2121no—
290C4⋊C4—2122no—
291C22⋊C4—41no—
292C22⋊C4—41no—
293C22⋊C4—41no—
294C22⋊C4—41no—
295C22⋊C4—4242no—
296C22⋊C4—4242no—
297C22⋊C4—4241no—
298C22⋊C4—4241no—
299C22⋊C4—4241no—
300C22⋊C4—4241no—
301C22⋊C4—2122no—
302C22⋊C4—2122no—
303C22⋊C4—2122no—
304C22⋊C4—2121no—
305C22⋊C4—2121no—
306C23—21no—
307C23—21no—
308C23—21no—
309C23—21no—
310C23—21no—
311C23—21no—
312C23—21no—
313C23—41no—
314C23—41no—
315C23—42no—
316C23—42no—
317C23—41no—
318C23—41no—
319C23—41no—
320C23—41no—
321C23—42no—
322C23—42no—
323C23—81no—
324C23—81no—
325C23—84no—
326C23—84no—
327C23#1313313111yescharacteristic
328C23#1313313111yescharacteristic
329C23C2×#1171992111yescharacteristic
330C23#1313325012yes—
331Q8—4242no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—42no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—81no—
342C2×C4—81no—
343C2×C4—81no—
344C2×C4—4242no—
345C2×C4—4242no—
346C2×C4—4242no—
347C2×C4—2121no—
348C2×C4—2121no—
349C2×C4—2121no—
350C2×C4—2121no—
351C2×C4—2121no—
352C2×C4—2121no—
353C2×C4—2121no—
354C2×C4—2121no—
355C2×C4—2121no—
356C2×C4—2121no—
357C2×C4—2121no—
358C2×C4—2121no—
359C8—82no—
360C22—21no—
361C22—21no—
362C22—22no—
363C22—21no—
364C22—21no—
365C22—21no—
366C22—21no—
367C22—21no—
368C22—21no—
369C22—42no—
370C22—42no—
371C22—42no—
372C22—42no—
373C22—41no—
374C22—41no—
375C22—41no—
376C22—41no—
377C22—41no—
378C22—81no—
379C22—81no—
380C22—84no—
381C22—84no—
382C22C2×#1313313111yescharacteristic
383C22#1681063311yescharacteristic
384C22#1681063911yescharacteristic
385C4—41no—
386C4—41no—
387C4—41no—
388C4—41no—
389C4—8482no—
390C4—4241no—
391C4—4241no—
392C4—4241no—
393C4—4241no—
394C2—21no—
395C2—21no—
396C2—42no—
397C2—41no—
398C2—41no—
399C2—81no—
400C2#2027324911yescharacteristic
401C2#2027636211yescharacteristic
402C2#2027638211yescharacteristic
subgroupG/H
centerC22C2×#13133131
commutatorC22×C4×C53C22×C4
FrattiniC2×C22⋊C4C22×D53
FittingC53×#1462C2
socleC22×C53C2×#99
prime radicalC25×C53D4
cosocleC2×C22⋊C4×C53C23
hypercenter#1462D53
hypocenterC53#11811
LCSG ▷ C22×C4×C53 ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C24 ◁ C22×C22⋊C4 ◁ #1462

Complements

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

NH ≅ G/Ntype
3. #20272740C2semidirect
4. #20272737C2semidirect
6. C2×#13133196C4semidirect
7. #16810559C4semidirect
11. #16810535C4semidirect
14. #16810483C4semidirect
224. C53#11811semidirect

Maximal subgroups

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

subgroupindex
#202805202
#202747892
#202727402
#202727372
C53×#14622
#1181153