Finite Group Databasebeta

← #27281754

Subgroups — #27281754

Order 13568.

486 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 486 conjugacy classes of subgroups — too many to draw. Only its 99 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#20283800C211yesmaximal, characteristic, complemented
2#20279691C211yesmaximal, characteristic
3#20273338C212yesmaximal
4#20272572C212yesmaximal, complemented
5C53×#1467C211yesmaximal, characteristic, complemented
6C2×#13133215C411yescharacteristic
7C2×#13133206C411yescharacteristic
8C2×#13133205C2211yescharacteristic
9#16810754—24no—
10#16810680—21no—
11#16810680C2212yes—
12#16810676—21no—
13#16810545—21no—
14#16810545—21no—
15#16810528C411yescharacteristic
16#16810524C411yescharacteristic
17#16810514—21no—
18#16810514—21no—
19C4×C4⋊Dic53C2211yescharacteristic
20#16810362—24no—
21C2×C4×Q8×C53C2211yescharacteristic
22C53×#432C2212yes—
23C53×#324—21no—
24C53×#324—21no—
25C22×Dic106C2×C411yescharacteristic
26C22×D212C2×C411yescharacteristic
27#13133215—42no—
28#13133206—42no—
29#13133205—22no—
30#13133205—22no—
31C2×C4⋊Dic53—21no—
32C2×#11719919—21no—
33C2×#11719919—21no—
34C2×#11719921—21no—
35C2×#11719921—21no—
36C2×#11719921—21no—
37C2×#11719921—21no—
38C2×#11719921—21no—
39C2×#11719921—21no—
40C2×C4×Dic53C2×C411yescharacteristic
41C2×C4×Dic53C2×C411yescharacteristic
42C4⋊#88322—22no—
43C4⋊#88322—22no—
44C4⋊#88322—22no—
45C22⋊#88322—22no—
46#13133128—22no—
47#13133128D414yes—
48C4×#88322—22no—
49#13133124—21no—
50#13133124—21no—
51#13133124—21no—
52C22×Q8×C53C2×C411yescharacteristic
53C4×Q8×C53—22no—
54C4×Q8×C53—22no—
55C2×C4⋊C4×C53C2×C411yescharacteristic
56C2×C4⋊C4×C53D411yescharacteristic
57C2×C4⋊C4×C53D411yescharacteristic
58C2×C42×C53D411yescharacteristic
59C2×C42×C53D411yescharacteristic
60C2×C42×C53C2311yescharacteristic
61C4⋊C8×C53—22no—
62C22⋊C8×C53—22no—
63#97×C53—22no—
64C23×D53—21no—
65C2×Q8×C53—22no—
66C2×Q8×C53—22no—
67C2×Q8×C53—41no—
68C22×C4×C53C22⋊C411yescharacteristic
69C22×C4×C53C22⋊C411yescharacteristic
70C22×C4×C53C22⋊C411yescharacteristic
71C22×C4×C53C22⋊C411yescharacteristic
72C22×C4×C53C22×C411yescharacteristic
73C22×C4×C53C2×D411yescharacteristic
74C22×C4×C53C2×D411yescharacteristic
75C22×Dic53—21no—
76C22×Dic53—21no—
77C22×Dic53—21no—
78C2×D212—22no—
79C2×D212—41no—
80C2×D212—41no—
81C2×Dic106—22no—
82C2×Dic106—41no—
83C2×Dic106—41no—
84C2×C8×C53—42no—
85C4⋊C4×C53—22no—
86C4⋊C4×C53—22no—
87C4⋊C4×C53—22no—
88C42×C53—21no—
89C42×C53—21no—
90C42×C53—21no—
91C42×C53—21no—
92C42×C53C2×D412yes—
93C42×C53#4112yes—
94#11719921—42no—
95#11719921—42no—
96#11719921—42no—
97#11719921—42no—
98#11719921—41no—
99#11719921—42no—
100#11719921—41no—
101C4⋊Dic53—42no—
102#11719919—42no—
103#11719919—42no—
104C4×Dic53—41no—
105C4×Dic53—41no—
106C2×#88322—24no—
107C23×C53C2×C22⋊C411yescharacteristic
108C22×D53—41no—
109C22×D53—41no—
110C22×D53—41no—
111C22×D53—41no—
112C22×D53—41no—
113C22×D53—41no—
114C22×D53—41no—
115Q8×C53—44no—
116C2×C4×C53—21no—
117C2×C4×C53—21no—
118C2×C4×C53—21no—
119C2×C4×C53—21no—
120C2×C4×C53—21no—
121C2×C4×C53—21no—
122C2×C4×C53—21no—
123C2×C4×C53—21no—
124C2×C4×C53—21no—
125C2×C4×C53—41no—
126C2×C4×C53—41no—
127C2×C4×C53—41no—
128C2×C4×C53—41no—
129C2×C4×C53#10012yes—
130C2×C4×C53C4×D412yes—
131C2×C4×C53#12011yescharacteristic
132C2×C4×C53#12112yes—
133C2×C4×C53#12311yescharacteristic
134C2×Dic53—22no—
135C2×Dic53—41no—
136C2×Dic53—41no—
137C2×Dic53—41no—
138C2×Dic53—41no—
139C2×Dic53—41no—
140C2×Dic53—41no—
141C2×Dic53—41no—
142C2×Dic53—41no—
143D212—82no—
144Dic106—82no—
145C8×C53—82no—
146#88322—44no—
147#9448—531nomaximal, Sylow
148C22×C53—21no—
149C22×C53—21no—
150C22×C53#38611yescharacteristic
151C22×C53C2×#10011yescharacteristic
152C22×C53#41311yescharacteristic
153C2×D53—81no—
154C2×D53—81no—
155C2×D53—81no—
156C2×D53—81no—
157C2×D53—81no—
158C2×D53—81no—
159C2×D53—81no—
160C4×C53—22no—
161C4×C53—22no—
162C4×C53—22no—
163C4×C53—21no—
164C4×C53—21no—
165C4×C53—41no—
166C4×C53—41no—
167C4×C53—41no—
168Dic53—42no—
169Dic53—81no—
170Dic53—81no—
171#2983—531no—
172#2545—531no—
173#1533—532no—
174#1467D5311yescharacteristic, complemented
175#1285—532no—
176C2×C53—21no—
177C2×C53—21no—
178C2×C53#188411yescharacteristic
179C2×C53#190011yescharacteristic
180C2×C53#191211yescharacteristic
181D53—161no—
182C2×C4⋊Q8—531no—
183C2×#124—531no—
184C2×#124—531no—
185C2×C4×Q8C2×D5311yescharacteristic
186#446—1064no—
187#432—532no—
188#432—1061no—
189#432C2×D5312yes—
190#431—1061no—
191#396—1061no—
192#396—1061no—
193#391—531no—
194#390—531no—
195#388—1061no—
196#388—1061no—
197C4×C4⋊C4—531no—
198#329—1064no—
199#324—21no—
200#324—21no—
201C53#944811yescharacteristic, complemented, Sylow
202C22×Q8—531no—
203C22×Q8C4×D5311yescharacteristic
204C22×D4—531no—
205C4⋊Q8—2122no—
206#124—1062no—
207#124—1062no—
208#124—2122no—
209C4×Q8—22no—
210C4×Q8—22no—
211C2×C4⋊C4—1061no—
212C2×C4⋊C4—1061no—
213C2×C4⋊C4—1061no—
214C2×C4⋊C4C4×D5311yescharacteristic
215C2×C4⋊C4C22⋊D5311yescharacteristic
216C2×C4⋊C4C22⋊D5311yescharacteristic
217C2×C22⋊C4—1061no—
218C2×C22⋊C4—1061no—
219C2×C22⋊C4—1061no—
220C2×C22⋊C4—1061no—
221C2×C22⋊C4—1061no—
222C2×C22⋊C4—1061no—
223C2×C42—531no—
224C2×C42—531no—
225C2×C42D21211yescharacteristic
226C2×C42D21211yescharacteristic
227C2×C42C22×D5311yescharacteristic
228C4⋊C8—22no—
229C4⋊C8—1062no—
230C4⋊C8—1062no—
231C4⋊C8—1062no—
232C22⋊C8—22no—
233C22⋊C8—1062no—
234#97—22no—
235#97—534no—
236#97—1062no—
237C4×C8—1062no—
238#95—1061no—
239#95—1061no—
240#95—1061no—
241C24—1061no—
242C2×Q8—22no—
243C2×Q8—22no—
244C2×Q8—41no—
245C2×Q8—1062no—
246C2×Q8—2121no—
247C2×Q8—2121no—
248C2×D4—1062no—
249C2×D4—2121no—
250C2×D4—2121no—
251C22×C4—1061no—
252C22×C4—1061no—
253C22×C4—1061no—
254C22×C4#1171992111yescharacteristic
255C22×C4#1171992111yescharacteristic
256C22×C4#1171992111yescharacteristic
257C22×C4#1171992111yescharacteristic
258C22×C4C2×C4×D5311yescharacteristic
259C22×C4C2×D21211yescharacteristic
260C22×C4C2×C22⋊D5311yescharacteristic
261C2×C8—42no—
262C2×C8—1064no—
263C4⋊C4—22no—
264C4⋊C4—22no—
265C4⋊C4—22no—
266C4⋊C4—2122no—
267C4⋊C4—2122no—
268C4⋊C4—2122no—
269C22⋊C4—2122no—
270C22⋊C4—2122no—
271C22⋊C4—2122no—
272C22⋊C4—2122no—
273C22⋊C4—2121no—
274C22⋊C4—2122no—
275C22⋊C4—2121no—
276C42—21no—
277C42—21no—
278C42—21no—
279C42—21no—
280C42—2121no—
281C42—2121no—
282C42D4×D5312yes—
283C42#1171994712yes—
284C23—2121no—
285C23—2121no—
286C23—2121no—
287C23—2121no—
288C23—2121no—
289C23—2121no—
290C23—2121no—
291C23C2×#1171992111yescharacteristic
292Q8—44no—
293Q8—4242no—
294D4—4242no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—1062no—
309C2×C4—2121no—
310C2×C4—2121no—
311C2×C4—2121no—
312C2×C4—2121no—
313C2×C4—2121no—
314C2×C4—2121no—
315C2×C4—2121no—
316C2×C4—2121no—
317C2×C4#1313313312yes—
318C2×C4#1313317712yes—
319C2×C4#1313318711yescharacteristic
320C2×C4#1313318912yes—
321C2×C4#1313320111yescharacteristic
322C8—82no—
323C8—2124no—
324C22—21no—
325C22—21no—
326C22—4241no—
327C22—4241no—
328C22—4241no—
329C22—4241no—
330C22—4241no—
331C22—4241no—
332C22—4241no—
333C22#1681050011yescharacteristic
334C22C2×#1313313311yescharacteristic
335C22#1681060511yescharacteristic
336C4—21no—
337C4—21no—
338C4—22no—
339C4—22no—
340C4—22no—
341C4—41no—
342C4—41no—
343C4—41no—
344C4—4241no—
345C4—4241no—
346C4—2122no—
347C2—21no—
348C2—21no—
349C2—8481no—
350C2#2027509611yescharacteristic
351C2#2027517611yescharacteristic
352C2#2027525011yescharacteristic
subgroupG/H
centerC22#16810500
commutatorC22×C4×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#1467C2
socleC22×C53#386
prime radicalC22×D212C2×C4
cosocleC2×C42×C53C23
hypercenter#1467D53
hypocenterC53#9448
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1467

Complements

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

NH ≅ G/Ntype
1. #20283800C2semidirect
4. #20272572C2semidirect
5. C53×#1467C2semidirect
174. #1467D53semidirect
201. C53#9448semidirect

Maximal subgroups

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

subgroupindex
#202838002
#202796912
#202733382
#202725722
C53×#14672
#944853