Finite Group Databasebeta

← #27293468

Subgroups — #27293468

Order 13568.

814 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 814 conjugacy classes of subgroups — too many to draw. Only its 95 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
1C2×#16810524C211yesmaximal, characteristic
2#20273045C211yesmaximal, characteristic, complemented
3#20272646C214yesmaximal, complemented
4C53×#1463C211yesmaximal, characteristic, complemented
5C23×D212C411yescharacteristic
6C22×#11719921—21no—
7C22×#11719921—21no—
8C22×C4×Dic53C411yescharacteristic
9#16810587—21no—
10#16810524—21no—
11#16810524—21no—
12#16810524C412yes—
13#16810524C2212yes—
14#16810349—21no—
15#16810349C2212yescomplemented
16C22×C4⋊C4×C53C2211yescharacteristic
17C53×#407—21no—
18C53×#324—21no—
19C53×#324C2212yescomplemented
20C24×D53—21no—
21C22×D212—21no—
22C22×D212—24no—
23C22×D212—41no—
24C22×D212—41no—
25C22×D212—42no—
26C22×D212C2×C412yes—
27C23×Dic53—21no—
28C2×#11719917—41no—
29C2×#11719921—22no—
30C2×#11719921—22no—
31C2×#11719921—22no—
32C2×#11719921—22no—
33C2×#11719921—44no—
34C2×#11719921—44no—
35C2×C4×Dic53—21no—
36C2×C4×Dic53—42no—
37C2×C4×Dic53C2×C412yes—
38C22⋊#88322—22no—
39C22⋊#88322—41no—
40C23×C4×C53C2×C411yescharacteristic
41C23×C4×C53D411yescharacteristic
42C23×C4×C53D411yescharacteristic
43C2×M16×C53—41no—
44C2×C4⋊C4×C53—41no—
45C2×C4⋊C4×C53—41no—
46C2×C4⋊C4×C53C2×C411yescharacteristic
47C2×C4⋊C4×C53D411yescharacteristic
48C2×C4⋊C4×C53D411yescharacteristic
49C2×C4⋊C4×C53C2311yescharacteristic
50C22⋊C8×C53—22no—
51C22⋊C8×C53—41no—
52C24×C53C22⋊C411yescharacteristic
53C23×D53—24no—
54C23×D53—22no—
55C23×D53—42no—
56C23×D53—42no—
57C23×D53—42no—
58C23×D53—42no—
59C23×D53—42no—
60C23×D53—41no—
61C23×D53—41no—
62C22×C4×C53—21no—
63C22×C4×C53—21no—
64C22×C4×C53—22no—
65C22×C4×C53—22no—
66C22×C4×C53—21no—
67C22×C4×C53—21no—
68C22×C4×C53—21no—
69C22×C4×C53—21no—
70C22×C4×C53—42no—
71C22×C4×C53—42no—
72C22×C4×C53C22⋊C411yescharacteristic
73C22×C4×C53C22⋊C411yescharacteristic
74C22×C4×C53C22⋊C411yescharacteristic
75C22×C4×C53C22×C411yescharacteristic
76C22×C4×C53C2×D411yescharacteristic
77C22×C4×C53C2×D411yescharacteristic
78C22×Dic53—24no—
79C22×Dic53—22no—
80C22×Dic53—42no—
81C22×Dic53—42no—
82C2×D212—24no—
83C2×D212—22no—
84C2×D212—48no—
85C2×D212—48no—
86C2×D212—42no—
87C2×D212—42no—
88C2×D212—42no—
89C2×D212—42no—
90C2×D212M1614yescomplemented
91M16×C53—82no—
92C2×C8×C53—42no—
93C4⋊C4×C53—41no—
94C4⋊C4×C53—41no—
95C4⋊C4×C53—41no—
96C4⋊C4×C53—41no—
97#11719921—44no—
98#11719921—44no—
99#11719921—44no—
100#11719921—44no—
101C4×Dic53—42no—
102C4×Dic53—42no—
103#11719917—82no—
104C2×#88322—42no—
105C23×C53—22no—
106C23×C53—22no—
107C23×C53—21no—
108C23×C53—21no—
109C23×C53#9911yescharacteristic
110C23×C53#9911yescharacteristic
111C23×C53C2×C22⋊C411yescharacteristic
112C22×D53—24no—
113C22×D53—44no—
114C22×D53—44no—
115C22×D53—48no—
116C22×D53—48no—
117C22×D53—42no—
118C22×D53—42no—
119C22×D53—42no—
120C22×D53—42no—
121C22×D53—42no—
122C22×D53—42no—
123C22×D53—84no—
124C22×D53—84no—
125C22×D53—82no—
126C22×D53—82no—
127C22×D53—82no—
128C22×D53—82no—
129C2×C4×C53—24no—
130C2×C4×C53—24no—
131C2×C4×C53—21no—
132C2×C4×C53—21no—
133C2×C4×C53—21no—
134C2×C4×C53—21no—
135C2×C4×C53—21no—
136C2×C4×C53—21no—
137C2×C4×C53—21no—
138C2×C4×C53—21no—
139C2×C4×C53—42no—
140C2×C4×C53—42no—
141C2×C4×C53—41no—
142C2×C4×C53—41no—
143C2×C4×C53—41no—
144C2×C4×C53—41no—
145C2×C4×C53—81no—
146C2×C4×C53—81no—
147C2×C4×C53#10012yes—
148C2×C4×C53C2×M1612yes—
149C2×Dic53—24no—
150C2×Dic53—48no—
151C2×Dic53—42no—
152C2×Dic53—42no—
153D212—44no—
154D212—48no—
155D212—44no—
156C8×C53—82no—
157#88322—82no—
158#11505—531nomaximal, Sylow
159C22×C53—24no—
160C22×C53—24no—
161C22×C53—21no—
162C22×C53—21no—
163C22×C53—21no—
164C22×C53—21no—
165C22×C53—21no—
166C22×C53—21no—
167C22×C53#35312yes—
168C22×C53#35512yes—
169C22×C53#40711yescharacteristic
170C22×C53C2×#9911yescharacteristic
171C22×C53C2×#10011yescharacteristic
172C2×D53—44no—
173C2×D53—44no—
174C2×D53—44no—
175C2×D53—82no—
176C2×D53—82no—
177C2×D53—88no—
178C2×D53—88no—
179C2×D53—82no—
180C2×D53—82no—
181C4×C53—22no—
182C4×C53—22no—
183C4×C53—41no—
184C4×C53—41no—
185C4×C53—41no—
186C4×C53—41no—
187Dic53—44no—
188C2×#390—531no—
189#1463—531no—
190#1463D5311yescharacteristic, complemented
191#1318—534no—
192C2×C53—21no—
193C2×C53—21no—
194C2×C53—22no—
195C2×C53—22no—
196C2×C53#146211yescharacteristic
197C2×C53C2×#35311yescharacteristic
198C2×C53C2×#35511yescharacteristic
199D53—84no—
200C23×D4—531no—
201C22×C4⋊C4C2×D5311yescharacteristic
202C22×C22⋊C4—1061no—
203C22×C22⋊C4—1061no—
204C22×C42—531no—
205#407—21no—
206#407—1061no—
207#390—532no—
208#390—532no—
209#390—1061no—
210#390—1061no—
211#324—21no—
212#324—532no—
213#324—1061no—
214#324C2×D5312yescomplemented
215C53#1150511yescharacteristic, complemented, Sylow
216C25—1061no—
217C22×D4—532no—
218C22×D4—1064no—
219C22×D4—1061no—
220C22×D4—2121no—
221C22×D4—2121no—
222C22×D4—2122no—
223C23×C4—1061no—
224C23×C4C4×D5311yescharacteristic
225C23×C4D21211yescharacteristic
226C23×C4C22⋊D5311yescharacteristic
227C2×M16—41no—
228C2×M16—2121no—
229C2×C4⋊C4—41no—
230C2×C4⋊C4—41no—
231C2×C4⋊C4C4×D5311yescharacteristic
232C2×C4⋊C4D21211yescharacteristic
233C2×C4⋊C4C22⋊D5311yescharacteristic
234C2×C4⋊C4C22×D5311yescharacteristic
235C2×C22⋊C4—1062no—
236C2×C22⋊C4—1062no—
237C2×C22⋊C4—1062no—
238C2×C22⋊C4—1062no—
239C2×C22⋊C4—2124no—
240C2×C22⋊C4—2124no—
241C2×C42—532no—
242C2×C42—1061no—
243C2×C42—2122no—
244C22⋊C8—22no—
245C22⋊C8—41no—
246C22⋊C8—1062no—
247C22⋊C8—2121no—
248C24—1064no—
249C24—1062no—
250C24—2122no—
251C24—2122no—
252C24—2122no—
253C24—2122no—
254C24—2122no—
255C24—2121no—
256C24—2121no—
257C24#1171992111yescharacteristic
258C2×D4—534no—
259C2×D4—1064no—
260C2×D4—1062no—
261C2×D4—2122no—
262C2×D4—2122no—
263C2×D4—2128no—
264C2×D4—2128no—
265C2×D4—2122no—
266C2×D4—2122no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—42no—
276C22×C4—42no—
277C22×C4—1064no—
278C22×C4—1062no—
279C22×C4—2122no—
280C22×C4—2122no—
281C22×C4#1171992111yescharacteristic
282C22×C4#1171992111yescharacteristic
283C22×C4#1171992111yescharacteristic
284C22×C4C2×C4×D5311yescharacteristic
285C22×C4C2×D21211yescharacteristic
286C22×C4C2×C22⋊D5311yescharacteristic
287M16—82no—
288M16—4242no—
289C2×C8—42no—
290C2×C8—2122no—
291C4⋊C4—41no—
292C4⋊C4—41no—
293C4⋊C4—41no—
294C4⋊C4—41no—
295C22⋊C4—2124no—
296C22⋊C4—2124no—
297C22⋊C4—2124no—
298C22⋊C4—2124no—
299C42—2122no—
300C42—2122no—
301C23—21no—
302C23—21no—
303C23—22no—
304C23—22no—
305C23—1064no—
306C23—4244no—
307C23—4244no—
308C23—4242no—
309C23—4242no—
310C23—4242no—
311C23—4242no—
312C23—2122no—
313C23—2122no—
314C23—2124no—
315C23—2124no—
316C23—2128no—
317C23—2128no—
318C23—2122no—
319C23—2122no—
320C23—2122no—
321C23—2122no—
322C23#1313313111yescharacteristic
323C23#1313313111yescharacteristic
324C23C2×#1171992111yescharacteristic
325D4—2124no—
326D4—2128no—
327D4—2124no—
328C2×C4—24no—
329C2×C4—24no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—42no—
339C2×C4—42no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—81no—
345C2×C4—81no—
346C2×C4—1064no—
347C2×C4—2128no—
348C2×C4—2122no—
349C2×C4—2122no—
350C2×C4#1313313312yes—
351C2×C4M16×D5312yes—
352C8—82no—
353C8—4242no—
354C22—24no—
355C22—24no—
356C22—21no—
357C22—21no—
358C22—21no—
359C22—21no—
360C22—21no—
361C22—21no—
362C22—4242no—
363C22—4242no—
364C22—4248no—
365C22—4248no—
366C22—4242no—
367C22—4242no—
368C22—2124no—
369C22—2124no—
370C22—2124no—
371C22#1681041512yes—
372C22#1681041912yes—
373C22#1681058611yescharacteristic
374C22C2×#1313313111yescharacteristic
375C22C2×#1313313311yescharacteristic
376C4—22no—
377C4—22no—
378C4—41no—
379C4—41no—
380C4—41no—
381C4—41no—
382C4—2124no—
383C2—21no—
384C2—22no—
385C2—22no—
386C2—21no—
387C2—4244no—
388C2#2027304111yescharacteristic
389C2C2×#1681041511yescharacteristic
390C2C2×#1681041911yescharacteristic
subgroupG/H
centerC22C2×#13133131
commutatorC22×C4×C53C22×C4
FrattiniC2×C4⋊C4C22×D53
FittingC53×#1463C2
socleC22×C53C2×#99
prime radicalC23×D212C4
cosocleC2×C4⋊C4×C53C23
hypercenter#1463D53
hypocenterC53#11505
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C24 ◁ C22×C4⋊C4 ◁ #1463

Complements

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

NH ≅ G/Ntype
2. #20273045C2semidirect
3. #20272646C2semidirect
4. C53×#1463C2semidirect
15. #16810349C22semidirect
19. C53×#324C22semidirect
90. C2×D212M16semidirect
190. #1463D53semidirect
214. #324C2×D53semidirect
215. C53#11505semidirect

Maximal subgroups

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

subgroupindex
C2×#168105242
#202730452
#202726462
C53×#14632
#1150553