Finite Group Databasebeta

← #27281734

Subgroups — #27281734

Order 13568.

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

normal subgroups only

This group has 474 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
1#20283353C211yesmaximal, characteristic
2#20279164C211yesmaximal, characteristic
3#20273247C212yesmaximal
4#20272567C212yesmaximal
5C53×#1820C211yesmaximal, characteristic
6C2×#13133196C2211yescharacteristic
7#16810684—24no—
8#16810647—21no—
9#16810587C2212yes—
10C4×#11719917—21no—
11#16810551—21no—
12#16810551—21no—
13#16810528C411yescharacteristic, complemented
14#16810528C411yescharacteristic, complemented
15#16810507—21no—
16#16810507—21no—
17#16810492C411yescharacteristic, complemented
18#16810488C411yescharacteristic, complemented
19C4×C22⋊Dic53C2211yescharacteristic
20#16810359—24no—
21C2×#117×C53C2211yescharacteristic
22C53×#381C2212yes—
23C53×#342—21no—
24C53×#342—21no—
25C22×Dic106C2×C411yescharacteristic
26C2×#11719917—22no—
27#13133196—22no—
28#13133196—22no—
29C2×C4⋊Dic53C2×C411yescharacteristic
30C2×#11719919—21no—
31C2×#11719919—21no—
32C2×#11719919—21no—
33C2×C22⋊Dic53—21no—
34C2×C22⋊Dic53—21no—
35C2×C4×Dic53C2×C411yescharacteristic
36C2×C4×Dic53C2×C411yescharacteristic
37C4⋊#88322—24no—
38C22⋊#88322—22no—
39C22⋊#88322D414yes—
40#13133128—22no—
41C4×#88322—22no—
42#13133124—21no—
43#13133124—21no—
44#13133124—21no—
45#13133124—21no—
46#13133124—21no—
47#13133124—21no—
48#13133124—21no—
49C23×C4×C53C2311yescharacteristic
50#117×C53—22no—
51#117×C53—22no—
52C2×C4⋊C4×C53C2×C411yescharacteristic
53C2×C4⋊C4×C53C2×C411yescharacteristic
54C2×C22⋊C4×C53—22no—
55C2×C22⋊C4×C53D411yescharacteristic
56C2×C22⋊C4×C53D411yescharacteristic
57C2×C42×C53D411yescharacteristic
58C2×C42×C53D411yescharacteristic
59#95×C53—22no—
60#95×C53—22no—
61C24×C53C2×D411yescharacteristic
62C22×C4×C53—41no—
63C22×C4×C53—41no—
64C22×C4×C53—42no—
65C22×C4×C53C22⋊C411yescharacteristic
66C22×C4×C53C22⋊C411yescharacteristic
67C22×C4×C53C22⋊C411yescharacteristic
68C22×C4×C53C22⋊C411yescharacteristic
69C22×C4×C53C22×C411yescharacteristic
70C22×C4×C53C2×D411yescharacteristic
71C22×C4×C53C2×D412yes—
72C22×C4×C53#4112yes—
73C22×Dic53—21no—
74C22×Dic53—21no—
75C22×Dic53—21no—
76C22×Dic53—21no—
77C2×Dic106—22no—
78C2×Dic106—41no—
79C2×Dic106—41no—
80C4⋊C4×C53—22no—
81C4⋊C4×C53—22no—
82C22⋊C4×C53—22no—
83C22⋊C4×C53—22no—
84C22⋊C4×C53—42no—
85C22⋊C4×C53—42no—
86C42×C53—21no—
87C42×C53—21no—
88C42×C53—21no—
89C42×C53—21no—
90C22⋊Dic53—42no—
91C22⋊Dic53—42no—
92C4⋊Dic53—22no—
93#11719919—42no—
94#11719919—42no—
95#11719919—41no—
96#11719919—41no—
97C4×Dic53—41no—
98C4×Dic53—41no—
99#11719917—42no—
100#11719917—42no—
101C2×#88322—24no—
102C23×C53—21no—
103C23×C53—21no—
104C23×C53—41no—
105C23×C53—41no—
106C23×C53C2×C22⋊C411yescharacteristic
107C23×C53#12112yes—
108C2×C4×C53—21no—
109C2×C4×C53—21no—
110C2×C4×C53—21no—
111C2×C4×C53—21no—
112C2×C4×C53—21no—
113C2×C4×C53—21no—
114C2×C4×C53—21no—
115C2×C4×C53—21no—
116C2×C4×C53—42no—
117C2×C4×C53—42no—
118C2×C4×C53—41no—
119C2×C4×C53—41no—
120C2×C4×C53—41no—
121C2×C4×C53—41no—
122C2×C4×C53—82no—
123C2×C4×C53—82no—
124C2×C4×C53—82no—
125C2×C4×C53C4×D412yes—
126C2×C4×C53#12011yescharacteristic
127C2×C4×C53#12311yescharacteristic
128C2×Dic53—22no—
129C2×Dic53—41no—
130C2×Dic53—41no—
131C2×Dic53—41no—
132C2×Dic53—41no—
133C2×Dic53—41no—
134C2×Dic53—41no—
135C2×Dic53—41no—
136C2×Dic53—41no—
137C2×Dic53—41no—
138C2×Dic53—41no—
139C2×Dic53—41no—
140Dic106—82no—
141#88322—44no—
142#9444—531nomaximal, Sylow
143C22×C53—21no—
144C22×C53—21no—
145C22×C53—22no—
146C22×C53—22no—
147C22×C53—22no—
148C22×C53—81no—
149C22×C53—81no—
150C22×C53#38611yescharacteristic
151C22×C53#41011yescharacteristic
152C22×C53#41011yescharacteristic
153C4×C53—21no—
154C4×C53—22no—
155C4×C53—21no—
156C4×C53—41no—
157C4×C53—41no—
158C4×C53—41no—
159C4×C53—41no—
160C4×C53—162no—
161Dic53—42no—
162Dic53—81no—
163Dic53—81no—
164Dic53—81no—
165#2921—531no—
166#2499—531no—
167#1820D5311yescharacteristic
168#1511—532no—
169#1284—532no—
170C2×C53—21no—
171C2×C53—21no—
172C2×C53—42no—
173C2×C53#188411yescharacteristic
174C2×C53#189911yescharacteristic
175C2×C53#190911yescharacteristic
176C2×C22⋊Q8—531no—
177C2×#117C2×D5311yescharacteristic
178#433—1064no—
179#423—1061no—
180#407—532no—
181C4×M16—1061no—
182#397—1061no—
183#397—1061no—
184#391—531no—
185#391—531no—
186#387—1061no—
187#387—1061no—
188#384—531no—
189#382—531no—
190#381C2×D5312yes—
191C4×C22⋊C4—531no—
192#342—21no—
193#342—21no—
194#328—1064no—
195C53#944411yescharacteristic, complemented, Sylow
196C22×Q8—531no—
197C23×C4C22×D5311yescharacteristic
198C2×M16—1062no—
199C22⋊Q8—1062no—
200C22⋊Q8—1062no—
201#117—22no—
202#117—22no—
203C2×C4⋊C4—531no—
204C2×C4⋊C4—1061no—
205C2×C4⋊C4—1061no—
206C2×C4⋊C4—1061no—
207C2×C4⋊C4C4×D5311yescharacteristic
208C2×C4⋊C4C4×D5311yescharacteristic
209C2×C22⋊C4—22no—
210C2×C22⋊C4—1061no—
211C2×C22⋊C4—1061no—
212C2×C22⋊C4C22⋊D5311yescharacteristic
213C2×C22⋊C4C22⋊D5311yescharacteristic
214C2×C42—531no—
215C2×C42—531no—
216C2×C42D21211yescharacteristic
217C2×C42D21211yescharacteristic
218C4⋊C8—1064no—
219C22⋊C8—534no—
220C22⋊C8—1062no—
221#97—1062no—
222C4×C8—1062no—
223#95—22no—
224#95—22no—
225#95—1061no—
226#95—1061no—
227#95—1061no—
228#95—1061no—
229#95—1061no—
230#95—1061no—
231#95—1061no—
232C24C2×C22⋊D5311yescharacteristic
233C2×Q8—1062no—
234C2×Q8—2121no—
235C2×Q8—2121no—
236C22×C4—41no—
237C22×C4—41no—
238C22×C4—42no—
239C22×C4—1061no—
240C22×C4—1061no—
241C22×C4—1061no—
242C22×C4—1061no—
243C22×C4#1171992111yescharacteristic
244C22×C4#1171992111yescharacteristic
245C22×C4#1171992111yescharacteristic
246C22×C4#1171992111yescharacteristic
247C22×C4C2×C4×D5311yescharacteristic
248C22×C4C2×D21211yescharacteristic
249C22×C4D4×D5312yes—
250C22×C4#1171994712yes—
251M16—2122no—
252M16—2122no—
253C2×C8—1064no—
254C4⋊C4—22no—
255C4⋊C4—22no—
256C4⋊C4—1062no—
257C4⋊C4—2122no—
258C4⋊C4—2122no—
259C4⋊C4—2121no—
260C4⋊C4—2121no—
261C22⋊C4—22no—
262C22⋊C4—22no—
263C22⋊C4—42no—
264C22⋊C4—42no—
265C22⋊C4—2122no—
266C22⋊C4—2122no—
267C42—21no—
268C42—21no—
269C42—21no—
270C42—21no—
271C42—2121no—
272C42—2121no—
273C23—21no—
274C23—21no—
275C23—41no—
276C23—41no—
277C23C2×#1171992111yescharacteristic
278C23#1313318912yes—
279Q8—4242no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—42no—
289C2×C4—42no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—82no—
295C2×C4—82no—
296C2×C4—82no—
297C2×C4—1062no—
298C2×C4—2121no—
299C2×C4—2121no—
300C2×C4—2121no—
301C2×C4—2121no—
302C2×C4—2121no—
303C2×C4—2121no—
304C2×C4—2121no—
305C2×C4—2121no—
306C2×C4—2121no—
307C2×C4—2121no—
308C2×C4—2121no—
309C2×C4#1313317712yes—
310C2×C4#1313318711yescharacteristic
311C2×C4#1313320111yescharacteristic
312C8—2124no—
313C22—21no—
314C22—21no—
315C22—22no—
316C22—22no—
317C22—22no—
318C22—81no—
319C22—81no—
320C22#1681050011yescharacteristic
321C22#1681059611yescharacteristic
322C22#1681059611yescharacteristic
323C4—21no—
324C4—22no—
325C4—21no—
326C4—41no—
327C4—41no—
328C4—41no—
329C4—41no—
330C4—162no—
331C4—4241no—
332C4—4241no—
333C4—4241no—
334C4—2122no—
335C2—21no—
336C2—21no—
337C2—42no—
338C2#2027509611yescharacteristic
339C2#2027517111yescharacteristic
340C2#2027523511yescharacteristic
subgroupG/H
centerC22#16810500
commutatorC22×C4×C53C22×C4
FrattiniC23×C4C22×D53
FittingC53×#1820C2
socleC22×C53#386
prime radicalC24×C53C2×D4
cosocleC23×C4×C53C23
hypercenter#1820D53
hypocenterC53#9444
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1820

Complements

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

NH ≅ G/Ntype
13. #16810528C4semidirect
14. #16810528C4semidirect
17. #16810492C4semidirect
18. #16810488C4semidirect
195. C53#9444semidirect

Maximal subgroups

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

subgroupindex
#202833532
#202791642
#202732472
#202725672
C53×#18202
#944453