Finite Group Databasebeta

← #27281801

Subgroups — #27281801

Order 13568.

690 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 690 conjugacy classes of subgroups — too many to draw. Only its 107 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#20279661C211yesmaximal, characteristic
2#20273428C212yesmaximal, complemented
3#20273056C211yesmaximal, characteristic, complemented
4#20272572C212yesmaximal, complemented
5C53×#2973C211yesmaximal, characteristic, complemented
6C2×#13133214C2211yescharacteristic
7C2×#13133213C412yes—
8C2×#13133191—21no—
9C2×#13133191—21no—
10#16810680C2212yes—
11#16810532—21no—
12#16810532—21no—
13#16810524C412yes—
14C4×C4⋊Dic53C2211yescharacteristic
15#16810366—24no—
16#16810347—21no—
17#16810347—21no—
18C2×C4×D4×C53C2211yescharacteristic
19C53×#435—24no—
20C53×#432C2212yes—
21C53×#407—21no—
22C2×C22⋊C8×C53—21no—
23C22×D212—21no—
24C22×D212—21no—
25C22×D212C2×C412yes—
26C22×C22⋊D53—22no—
27C22×C22⋊D53—22no—
28#13133214—22no—
29#13133214—22no—
30#13133213—44no—
31#13133191—44no—
32#13133191—44no—
33C2×C4⋊Dic53—21no—
34C2×#11719921—22no—
35C2×#11719921—22no—
36C2×C4×Dic53C2×C412yes—
37C4⋊#88322—22no—
38C22⋊#88322—22no—
39#13133128—22no—
40#13133124—21no—
41C22×D4×C53C2×C411yescharacteristic
42C23×C4×C53D411yescharacteristic
43C23×C4×C53D411yescharacteristic
44C2×M16×C53—22no—
45C22×C8×C53—22no—
46C4×D4×C53—22no—
47C4×D4×C53—22no—
48C2×C4⋊C4×C53C2×C411yescharacteristic
49C2×C22⋊C4×C53D411yescharacteristic
50C2×C22⋊C4×C53D411yescharacteristic
51C2×C42×C53C2311yescharacteristic
52C4⋊C8×C53D414yescomplemented
53C22⋊C8×C53—22no—
54C22⋊C8×C53—22no—
55C22⋊C8×C53—22no—
56#97×C53—22no—
57C24×C53C22⋊C411yescharacteristic
58C24×C53C22⋊C411yescharacteristic
59C23×D53—22no—
60C2×D4×C53—22no—
61C2×D4×C53—22no—
62C2×D4×C53—41no—
63C22×C4×C53—21no—
64C22×C4×C53—21no—
65C22×C4×C53—21no—
66C22×C4×C53—21no—
67C22×C4×C53—42no—
68C22×C4×C53—42no—
69C22×C4×C53C22⋊C411yescharacteristic
70C22×C4×C53C22⋊C411yescharacteristic
71C22×C4×C53C22×C411yescharacteristic
72C22×C4×C53C2×D411yescharacteristic
73C22×C4×C53C2×D411yescharacteristic
74C2×C22⋊D53—44no—
75C2×C22⋊D53—44no—
76C2×C22⋊D53—44no—
77C2×C22⋊D53—44no—
78C2×C22⋊D53—82no—
79C2×C22⋊D53—82no—
80C22×Dic53—22no—
81C2×D212—24no—
82C2×D212—42no—
83C2×D212—42no—
84C2×D212—42no—
85C2×D212—42no—
86C2×D212—41no—
87C2×D212—42no—
88C2×D212—41no—
89C2×D212—82no—
90M16×C53—82no—
91C2×C8×C53—24no—
92C2×C8×C53—82no—
93C4⋊C4×C53—22no—
94C22⋊C4×C53—22no—
95C22⋊C4×C53—22no—
96C42×C53C2×D412yes—
97C42×C53#4112yes—
98#11719921—44no—
99#11719921—44no—
100C4⋊Dic53—24no—
101C4×Dic53—42no—
102C2×#88322—42no—
103C23×C53—21no—
104C23×C53—21no—
105C23×C53—21no—
106C23×C53—21no—
107C23×C53—21no—
108C23×C53—41no—
109C23×C53—41no—
110C23×C53—41no—
111C23×C53—41no—
112C23×C53#10012yes—
113C23×C53C2×C22⋊C411yescharacteristic
114C22×D53—42no—
115C22×D53—42no—
116C22×D53—42no—
117C22×D53—42no—
118C22×D53—42no—
119C22×D53—42no—
120C22×D53—42no—
121D4×C53—44no—
122C2×C4×C53—21no—
123C2×C4×C53—21no—
124C2×C4×C53—22no—
125C2×C4×C53—21no—
126C2×C4×C53—21no—
127C2×C4×C53—41no—
128C2×C4×C53—42no—
129C2×C4×C53—42no—
130C2×C4×C53—41no—
131C2×C4×C53—41no—
132C2×C4×C53—81no—
133C2×C4×C53—81no—
134C2×C4×C53C4×D412yes—
135C2×C4×C53#12011yescharacteristic
136C2×C4×C53#12112yes—
137C2×C4×C53#12311yescharacteristic
138C22⋊D53—88no—
139C22⋊D53—88no—
140C2×Dic53—24no—
141C2×Dic53—42no—
142C2×Dic53—42no—
143D212—84no—
144D212—82no—
145D212—84no—
146D212—82no—
147C8×C53—44no—
148#88322—82no—
149#9457—531nomaximal, Sylow
150C22×C53—21no—
151C22×C53—21no—
152C22×C53—22no—
153C22×C53—42no—
154C22×C53—42no—
155C22×C53—41no—
156C22×C53—41no—
157C22×C53—41no—
158C22×C53—41no—
159C22×C53—41no—
160C22×C53—81no—
161C22×C53—81no—
162C22×C53#38611yescharacteristic
163C22×C53C2×#10011yescharacteristic
164C22×C53#41311yescharacteristic
165C2×D53—82no—
166C2×D53—82no—
167C2×D53—82no—
168C2×D53—82no—
169C2×D53—82no—
170C2×D53—82no—
171C2×D53—82no—
172C4×C53—21no—
173C4×C53—22no—
174C4×C53—22no—
175C4×C53—41no—
176C4×C53—41no—
177C4×C53#45312yes—
178Dic53—44no—
179#2973D5311yescharacteristic, complemented
180#2541—531no—
181#1555—532no—
182#1466—531no—
183#1285—532no—
184C2×C53—21no—
185C2×C53—21no—
186C2×C53—42no—
187C2×C53—81no—
188C2×C53#188711yescharacteristic
189C2×C53#190011yescharacteristic
190C2×C53#191011yescharacteristic
191D53—162no—
192C2×#127—532no—
193C2×#127—531no—
194C2×#121—1061no—
195C2×#121—1061no—
196C2×C4×D4C2×D5311yescharacteristic
197#435—24no—
198#432—532no—
199#432C2×D5312yes—
200#407—21no—
201C2×C22⋊C8—21no—
202#392—1061no—
203#392—1061no—
204#390—532no—
205C4×C4⋊C4—531no—
206#331—1064no—
207#323—1061no—
208#323—1061no—
209C53#945711yescharacteristic, complemented, Sylow
210C22×D4—532no—
211C22×D4—1062no—
212C22×D4—1062no—
213C22×D4—1061no—
214C22×D4—1061no—
215C22×D4C4×D5311yescharacteristic
216C23×C4C22⋊D5311yescharacteristic
217C23×C4C22⋊D5311yescharacteristic
218C2×M16—22no—
219C22×C8—22no—
220#127—1062no—
221#127—1062no—
222#127—2124no—
223#121—2124no—
224#121—2124no—
225C4×D4—22no—
226C4×D4—22no—
227C2×C4⋊C4—1061no—
228C2×C4⋊C4C4×D5311yescharacteristic
229C2×C22⋊C4—1062no—
230C2×C22⋊C4—1062no—
231C2×C22⋊C4D21211yescharacteristic
232C2×C22⋊C4D21211yescharacteristic
233C2×C42—532no—
234C2×C42C22×D5311yescharacteristic
235C4⋊C8—1062no—
236C4⋊C8C22⋊D5314yescomplemented
237C22⋊C8—22no—
238C22⋊C8—22no—
239C22⋊C8—22no—
240C22⋊C8—1062no—
241#97—22no—
242#97—1062no—
243#95—1061no—
244C24—1062no—
245C24#1171992111yescharacteristic
246C24#1171992111yescharacteristic
247C2×D4—22no—
248C2×D4—22no—
249C2×D4—41no—
250C2×D4—1064no—
251C2×D4—4242no—
252C2×D4—4242no—
253C2×D4—4242no—
254C2×D4—2122no—
255C2×D4—2122no—
256C2×D4—2122no—
257C2×D4—2122no—
258C2×D4—2124no—
259C2×D4—2124no—
260C2×D4—2124no—
261C2×D4—2124no—
262C2×D4—2121no—
263C2×D4—2122no—
264C2×D4—2121no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—42no—
270C22×C4—42no—
271C22×C4—1062no—
272C22×C4#1171992111yescharacteristic
273C22×C4#1171992111yescharacteristic
274C22×C4C2×C4×D5311yescharacteristic
275C22×C4C2×D21211yescharacteristic
276C22×C4C2×C22⋊D5311yescharacteristic
277M16—82no—
278C2×C8—24no—
279C2×C8—82no—
280C2×C8—2122no—
281C4⋊C4—22no—
282C4⋊C4—1064no—
283C22⋊C4—22no—
284C22⋊C4—22no—
285C22⋊C4—2124no—
286C22⋊C4—2124no—
287C42—2122no—
288C42#1171994512yes—
289C42C2×C22⋊D5312yes—
290C23—21no—
291C23—21no—
292C23—21no—
293C23—21no—
294C23—21no—
295C23—41no—
296C23—41no—
297C23—41no—
298C23—41no—
299C23—2122no—
300C23—2122no—
301C23—2122no—
302C23—2122no—
303C23—2122no—
304C23—2122no—
305C23—2122no—
306C23#1313313312yes—
307C23C2×#1171992111yescharacteristic
308D4—44no—
309D4—4244no—
310D4—4244no—
311D4—4242no—
312D4—4248no—
313D4—4248no—
314D4—4242no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—22no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—41no—
321C2×C4—42no—
322C2×C4—42no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—81no—
326C2×C4—81no—
327C2×C4—1064no—
328C2×C4—2122no—
329C2×C4—2122no—
330C2×C4C4×C22⋊D5312yes—
331C2×C4#1313318611yescharacteristic
332C2×C4#1313319112yes—
333C2×C4#1313320011yescharacteristic
334C8—44no—
335C8—4242no—
336C22—21no—
337C22—21no—
338C22—22no—
339C22—42no—
340C22—42no—
341C22—41no—
342C22—41no—
343C22—41no—
344C22—41no—
345C22—41no—
346C22—81no—
347C22—81no—
348C22—4242no—
349C22—4242no—
350C22—4242no—
351C22—4242no—
352C22—4242no—
353C22—4242no—
354C22—4242no—
355C22#1681050211yescharacteristic
356C22C2×#1313313311yescharacteristic
357C22#1681060511yescharacteristic
358C4—22no—
359C4—21no—
360C4—22no—
361C4—41no—
362C4—41no—
363C4—2124no—
364C4#1681079112yes—
365C2—21no—
366C2—21no—
367C2—42no—
368C2—81no—
369C2—8482no—
370C2#2027511311yescharacteristic
371C2#2027517811yescharacteristic
372C2#2027524211yescharacteristic
subgroupG/H
centerC22#16810502
commutatorC22×C4×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#2973C2
socleC22×C53#386
prime radical#20279661C2
cosocleC2×C42×C53C23
hypercenter#2973D53
hypocenterC53#9457
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2973

Complements

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

NH ≅ G/Ntype
2. #20273428C2semidirect
3. #20273056C2semidirect
4. #20272572C2semidirect
5. C53×#2973C2semidirect
52. C4⋊C8×C53D4semidirect
179. #2973D53semidirect
209. C53#9457semidirect
236. C4⋊C8C22⋊D53semidirect

Maximal subgroups

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

subgroupindex
#202796612
#202734282
#202730562
#202725722
C53×#29732
#945753