Finite Group Databasebeta

← #27294554

Subgroups — #27294554

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#20280281C211yesmaximal, characteristic
2#20274448C211yesmaximal, characteristic, complemented
3#20272740C212yesmaximal, complemented
4#20272713C212yesmaximal
5C53×#1464C211yesmaximal, characteristic, complemented
6C2×C4×D212C411yescharacteristic, complemented
7C2×#13133176—21no—
8#16810562C411yescharacteristic, complemented
9#16810559C2211yescharacteristic
10#16810540C411yescharacteristic, complemented
11#16810532C2211yescharacteristic
12#16810514—21no—
13#16810502—21no—
14#16810499—21no—
15#16810494—21no—
16#16810491C411yescharacteristic, complemented
17#16810390—21no—
18#16810390C2212yes—
19C2×#117×C53C2211yescharacteristic
20C53×#432—21no—
21C53×#324—21no—
22C53×#323C2212yes—
23C22×D212C2×C411yescharacteristic
24C22×C22⋊D53—21no—
25C22×C4×D53—21no—
26C4×D212—41no—
27C4×D212—41no—
28#13133176—24no—
29#13133176—42no—
30C2×C4⋊Dic53—41no—
31C2×C4⋊Dic53C2×C411yescharacteristic
32C2×#11719919—21no—
33C2×C22⋊Dic53—21no—
34C2×C22⋊Dic53—22no—
35C2×#11719921—21no—
36C2×#11719921—21no—
37C2×#11719921—21no—
38C2×#11719921—21no—
39C2×C4×Dic53—41no—
40#13133124—21no—
41#13133124—21no—
42#13133124—21no—
43#13133124—21no—
44#13133124C2×C411yescharacteristic
45#13133124C2×C411yescharacteristic
46C23×C4×C53D411yescharacteristic
47#117×C53—41no—
48#117×C53—41no—
49C2×C4⋊C4×C53C2×C411yescharacteristic
50C2×C4⋊C4×C53D411yescharacteristic
51C2×C22⋊C4×C53D411yescharacteristic
52C2×C22⋊C4×C53C2311yescharacteristic
53C2×C42×C53C2×C411yescharacteristic
54C2×C42×C53D411yescharacteristic
55C4⋊C8×C53—41no—
56C22⋊C8×C53—22no—
57#97×C53—41no—
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×C53C22⋊C411yescharacteristic
66C22×C4×C53C22⋊C411yescharacteristic
67C22×C4×C53C22⋊C411yescharacteristic
68C22×C4×C53C22⋊C411yescharacteristic
69C22×C4×C53C22×C411yescharacteristic
70C22×C4×C53C2×D411yescharacteristic
71C2×C22⋊D53—42no—
72C2×C22⋊D53—41no—
73C2×C22⋊D53—41no—
74C2×C22⋊D53—81no—
75C22×Dic53—21no—
76C22×Dic53—21no—
77C22×Dic53—21no—
78C22×Dic53—42no—
79C2×D212—41no—
80C2×D212—42no—
81C2×C4×D53—42no—
82C2×C4×D53—81no—
83C2×C4×D53—81no—
84C2×C8×C53—42no—
85C4⋊C4×C53—41no—
86C4⋊C4×C53—41no—
87C22⋊C4×C53—41no—
88C22⋊C4×C53—41no—
89C42×C53—21no—
90C42×C53—21no—
91C42×C53—21no—
92C42×C53—21no—
93C22⋊Dic53—24no—
94C22⋊Dic53—42no—
95C22⋊Dic53—81no—
96#11719921—42no—
97#11719921—41no—
98#11719921—41no—
99#11719921—42no—
100#11719921—81no—
101C4⋊Dic53—44no—
102C4⋊Dic53—41no—
103#11719919—81no—
104C4×Dic53—44no—
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—22no—
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#9911yescharacteristic
136C2×C4×C53#9911yescharacteristic
137C2×C4×C53#10412yes—
138C22⋊D53—82no—
139C22⋊D53—82no—
140C2×Dic53—41no—
141C2×Dic53—41no—
142C2×Dic53—41no—
143C2×Dic53—41no—
144C2×Dic53—41no—
145C2×Dic53—41no—
146C2×Dic53—44no—
147C2×Dic53—41no—
148C2×Dic53—41no—
149C2×Dic53—41no—
150C2×Dic53—82no—
151C2×Dic53—82no—
152D212—42no—
153D212—42no—
154C4×D53—84no—
155C8×C53—82no—
156#11754—531nomaximal, Sylow
157C22×C53—21no—
158C22×C53—21no—
159C22×C53—21no—
160C22×C53—21no—
161C22×C53—41no—
162C22×C53—41no—
163C22×C53—41no—
164C22×C53—41no—
165C22×C53—81no—
166C22×C53C2×#9911yescharacteristic
167C22×C53#41711yescharacteristic
168C22×C53C2×#10411yescharacteristic
169C2×D53—42no—
170C2×D53—42no—
171C2×D53—42no—
172C2×D53—81no—
173C2×D53—81no—
174C2×D53—81no—
175C2×D53—81no—
176C4×C53—21no—
177C4×C53—21no—
178C4×C53—22no—
179C4×C53—41no—
180C4×C53—41no—
181C4×C53—41no—
182C4×C53—41no—
183Dic53—81no—
184Dic53—81no—
185Dic53—84no—
186Dic53—81no—
187#2598—531no—
188#1740—531no—
189#1464D5311yescharacteristic, complemented
190#1352—532no—
191#1343—532no—
192C2×C53—21no—
193C2×C53—21no—
194C2×C53—41no—
195C2×C53—41no—
196C2×C53#150411yescharacteristic
197C2×C53#212011yescharacteristic
198C2×C53#212811yescharacteristic
199D53—82no—
200C2×C4×D4—531no—
201C2×#117—1061no—
202C2×#117C2×D5311yescharacteristic
203#432—21no—
204#400—531no—
205#399—531no—
206#394—531no—
207#392—531no—
208#388—1061no—
209#386—1061no—
210#386—1061no—
211#385—1061no—
212#383—531no—
213#342—532no—
214#342—1061no—
215#324—21no—
216#323C2×D5312yes—
217C53#1175411yescharacteristic, complemented, Sylow
218C22×D4—531no—
219C22×D4—1061no—
220C23×C4—1061no—
221C23×C4D21211yescharacteristic
222C4×D4—2121no—
223C4×D4—2121no—
224#117—41no—
225#117—41no—
226#117—1064no—
227#117—2122no—
228C2×C4⋊C4—531no—
229C2×C4⋊C4—1061no—
230C2×C4⋊C4—2121no—
231C2×C4⋊C4C4×D5311yescharacteristic
232C2×C4⋊C4C22⋊D5311yescharacteristic
233C2×C22⋊C4—1062no—
234C2×C22⋊C4—1061no—
235C2×C22⋊C4—1061no—
236C2×C22⋊C4—1061no—
237C2×C22⋊C4—1061no—
238C2×C22⋊C4—1061no—
239C2×C22⋊C4D21211yescharacteristic
240C2×C22⋊C4C22×D5311yescharacteristic
241C2×C42—2121no—
242C2×C42C4×D5311yescharacteristic
243C2×C42C22⋊D5311yescharacteristic
244C4⋊C8—41no—
245C22⋊C8—22no—
246#97—41no—
247#95—531no—
248#95—531no—
249#95—1061no—
250#95—1061no—
251#95—1061no—
252#95—1061no—
253C24—1061no—
254C24C2×D21211yescharacteristic
255C2×D4—4241no—
256C2×D4—2121no—
257C2×D4—2122no—
258C2×D4—2122no—
259C2×D4—2121no—
260C2×D4—2121no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—22no—
266C22×C4—1061no—
267C22×C4—1061no—
268C22×C4—1061no—
269C22×C4—4241no—
270C22×C4—4241no—
271C22×C4—2122no—
272C22×C4—2122no—
273C22×C4#1171992111yescharacteristic
274C22×C4#1171992111yescharacteristic
275C22×C4#1171992111yescharacteristic
276C22×C4#1171992111yescharacteristic
277C22×C4C2×C4×D5311yescharacteristic
278C22×C4C2×C22⋊D5311yescharacteristic
279C2×C8—42no—
280C4⋊C4—41no—
281C4⋊C4—41no—
282C4⋊C4—4241no—
283C4⋊C4—2121no—
284C4⋊C4—2124no—
285C22⋊C4—41no—
286C22⋊C4—41no—
287C22⋊C4—1064no—
288C22⋊C4—4241no—
289C22⋊C4—4241no—
290C22⋊C4—2122no—
291C22⋊C4—2122no—
292C22⋊C4—2121no—
293C22⋊C4—2121no—
294C22⋊C4—2122no—
295C42—21no—
296C42—21no—
297C42—21no—
298C42—21no—
299C42—2124no—
300C23—21no—
301C23—21no—
302C23—41no—
303C23—41no—
304C23—1062no—
305C23—2121no—
306C23—2121no—
307C23—2121no—
308C23—2121no—
309C23—2121no—
310C23—2121no—
311C23C2×#1171992111yescharacteristic
312C23#1313324511yescharacteristic
313C23#1313325011yescharacteristic
314D4—4242no—
315D4—4242no—
316D4—2122no—
317D4—2122no—
318C2×C4—22no—
319C2×C4—22no—
320C2×C4—22no—
321C2×C4—22no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—4244no—
335C2×C4—4242no—
336C2×C4—4242no—
337C2×C4—2124no—
338C2×C4—2121no—
339C2×C4—2121no—
340C2×C4—2121no—
341C2×C4—2121no—
342C2×C4—2121no—
343C2×C4—2121no—
344C2×C4—2121no—
345C2×C4—2121no—
346C2×C4—2121no—
347C2×C4#1313313111yescharacteristic
348C2×C4#1313313111yescharacteristic
349C2×C4#1313314312yes—
350C8—82no—
351C22—21no—
352C22—21no—
353C22—21no—
354C22—21no—
355C22—41no—
356C22—41no—
357C22—41no—
358C22—41no—
359C22—81no—
360C22—4241no—
361C22—4241no—
362C22—4241no—
363C22—4241no—
364C22—2122no—
365C22—2122no—
366C22—2122no—
367C22C2×#1313313111yescharacteristic
368C22#1681062511yescharacteristic
369C22C2×#1313314311yescharacteristic
370C4—21no—
371C4—21no—
372C4—22no—
373C4—41no—
374C4—41no—
375C4—41no—
376C4—41no—
377C4—4241no—
378C4—4241no—
379C4—4244no—
380C4—4241no—
381C2—21no—
382C2—21no—
383C2—41no—
384C2—41no—
385C2—4242no—
386C2#2027322011yescharacteristic
387C2#2027635711yescharacteristic
388C2#2027638611yescharacteristic
subgroupG/H
centerC22C2×#13133131
commutatorC22×C4×C53C22×C4
FrattiniC2×C22⋊C4C22×D53
FittingC53×#1464C2
socleC22×C53C2×#99
prime radical#16810532C22
cosocleC2×C22⋊C4×C53C23
hypercenter#1464D53
hypocenterC53#11754
LCSG ▷ C22×C4×C53 ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #1464

Complements

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

NH ≅ G/Ntype
2. #20274448C2semidirect
3. #20272740C2semidirect
5. C53×#1464C2semidirect
6. C2×C4×D212C4semidirect
8. #16810562C4semidirect
10. #16810540C4semidirect
16. #16810491C4semidirect
189. #1464D53semidirect
217. C53#11754semidirect

Maximal subgroups

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

subgroupindex
#202802812
#202744482
#202727402
#202727132
C53×#14642
#1175453