Finite Group Databasebeta

← #28312499

Subgroups — #28312499

Order 13568.

334 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 60 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×#16811532C211yesmaximal, characteristic
2C2×#16811530C211yesmaximal, characteristic
3#20289861C212yesmaximal, complemented
4#20289860C212yesmaximal, complemented
5#20285104C211yesmaximal, characteristic
6C2×#13133278—21no—
7C2×#13133276—21no—
8#16811532—21no—
9#16811532C2212yes—
10#16811530—21no—
11#16811530C2212yes—
12C2×#13133205C2211yescharacteristic
13#16810985—21no—
14#16810985—21no—
15#16810980C411yescharacteristic, complemented
16#16810980C411yescharacteristic, complemented
17#16810980C2211yescharacteristic
18#16810980C2211yescharacteristic
19#16810631—21no—
20#16810627—21no—
21C4×#11719917C411yescharacteristic, complemented
22C53×#533C411yescharacteristic, complemented
23C2×#11719940—41no—
24C2×C22⋊#6157—41no—
25#13133278—22no—
26#13133278—41no—
27#13133276—22no—
28#13133276—41no—
29C22×Dic106D411yescharacteristic
30C22×D212D411yescharacteristic
31C2×#11719917—21no—
32#13133205—41no—
33#13133205C2×C411yescharacteristic
34#13133205D411yescharacteristic
35#13133205D411yescharacteristic
36#13133205C2311yescharacteristic
37C2×#11719921—41no—
38#13133140—21no—
39#13133140—21no—
40#13133140—21no—
41#13133140—21no—
42#13133137—21no—
43#13133137—21no—
44#13133137—21no—
45#13133137—21no—
46#13133128—21no—
47C4×#88322C2×C411yescharacteristic, complemented
48C4×#88322C2×C411yescharacteristic, complemented
49C4⋊Q8×C53—21no—
50#125×C53C2×C411yescharacteristic, complemented
51#125×C53C2×C411yescharacteristic, complemented
52C22⋊Q8×C53—41no—
53C4×Q8×C53—41no—
54#117×C53—21no—
55C2×C42×C53C2×C411yescharacteristic
56C23×D53—21no—
57C22×#6157—81no—
58C2×Q8×C53—41no—
59C22×C4×C53—21no—
60C22×C4×C53C22⋊C411yescharacteristic
61C22×Dic53—21no—
62C2×D212—21no—
63C2×D212—41no—
64C2×D212—41no—
65C2×D212C22⋊C411yescharacteristic
66C2×D212C2×D411yescharacteristic
67C2×Dic106—21no—
68C2×Dic106—41no—
69C2×Dic106—41no—
70C2×Dic106C22⋊C411yescharacteristic
71C2×Dic106C2×D411yescharacteristic
72C22⋊#6157—42no—
73C22⋊#6157—81no—
74#11719940—42no—
75#11719940—81no—
76C2×#88324—81no—
77C4⋊C4×C53—21no—
78C4⋊C4×C53—21no—
79C4⋊C4×C53—41no—
80C4⋊C4×C53—41no—
81C4⋊C4×C53—41no—
82C22⋊C4×C53—41no—
83C42×C53—21no—
84C42×C53—41no—
85C42×C53C22⋊C411yescharacteristic
86C42×C53C22×C411yescharacteristic
87#11719921—41no—
88#11719921—41no—
89#11719921—41no—
90#11719921—41no—
91#11719917—41no—
92#11719917—41no—
93C2×#88322—21no—
94C2×#88322—21no—
95C23×C53#9911yescharacteristic
96C22×D53—21no—
97C22×D53—21no—
98C22×D53—41no—
99C22×D53—41no—
100C22×D53—82no—
101C2×#6157—82no—
102C2×#6157—162no—
103Q8×C53—82no—
104C2×C4×C53—21no—
105C2×C4×C53—21no—
106C2×C4×C53—21no—
107C2×C4×C53—21no—
108C2×C4×C53—41no—
109C2×C4×C53—41no—
110C2×C4×C53—81no—
111C2×C4×C53#9911yescharacteristic
112C2×C4×C53C2×C22⋊C411yescharacteristic
113C2×Dic53—21no—
114C2×Dic53—21no—
115C2×Dic53—41no—
116C2×Dic53—41no—
117D212—41no—
118D212—41no—
119D212—81no—
120Dic106—41no—
121Dic106—41no—
122Dic106—81no—
123#88324—82no—
124#88322—41no—
125#88322—41no—
126#13603—531nomaximal, Sylow
127C22×C53—41no—
128C22×C53#35411yescharacteristic
129C22×C53#35411yescharacteristic
130C22×C53C2×#9911yescharacteristic
131C2×D53—41no—
132C2×D53—41no—
133C2×D53—82no—
134C2×D53—81no—
135C2×D53—81no—
136C2×D53—161no—
137#6157—162no—
138C4×C53—21no—
139C4×C53—21no—
140C4×C53—41no—
141C4×C53—41no—
142C4×C53—81no—
143C4×C53—81no—
144Dic53—41no—
145Dic53—41no—
146#3140—531no—
147C2×#355—531no—
148C2×#354—531no—
149#1414—532no—
150#1414—532no—
151C2×C53—21no—
152C2×C53—21no—
153C2×C53—21no—
154C2×C53C2×#35411yescharacteristic
155D53—81no—
156D53—81no—
157#533#615711yescharacteristic, complemented
158C2×#124—531no—
159#489—1061no—
160#489—1061no—
161#488—531no—
162#488—531no—
163#488—531no—
164#488—531no—
165#419—1061no—
166#418—1061no—
167C2×#101—1061no—
168C2×#99—1061no—
169C4×M16—531no—
170#355—532no—
171#355—1061no—
172#354—532no—
173#354—1061no—
174C53#1360311yescharacteristic, complemented, Sylow
175C22×Q8—531no—
176C22×D4—531no—
177C2×M16—1061no—
178C2×M16—2121no—
179C4⋊Q8—21no—
180#125C2×#615711yescharacteristic, complemented
181#125C2×#615711yescharacteristic, complemented
182#124—531no—
183#124—531no—
184#124—531no—
185#124—531no—
186#124—2121no—
187C22⋊Q8—41no—
188C4×Q8—41no—
189#117—21no—
190C2×C22⋊C4—2121no—
191C2×C22⋊C4—2121no—
192C2×C42C2×#615711yescharacteristic
193#103—1061no—
194#103—1061no—
195#103—1061no—
196#103—1061no—
197#102—1061no—
198#102—1061no—
199#102—1061no—
200#102—1061no—
201#101—1062no—
202#101—2121no—
203#99—1062no—
204#99—2121no—
205#97—1061no—
206C4×C8—531no—
207C4×C8—531no—
208C24—1061no—
209C2×Q8—41no—
210C2×Q8—531no—
211C2×Q8—531no—
212C2×Q8—1061no—
213C2×Q8—2121no—
214C2×Q8—2121no—
215C2×D4—531no—
216C2×D4—531no—
217C2×D4—1061no—
218C2×D4—2121no—
219C2×D4—2121no—
220C22×C4—21no—
221C22×C4—1061no—
222C22×C4—4241no—
223C22×C4C22⋊#615711yescharacteristic
224M16—4241no—
225M16—2122no—
226M16—2121no—
227M16—2121no—
228C2×C8—1061no—
229C2×C8—1061no—
230C2×C8—4241no—
231C4⋊C4—21no—
232C4⋊C4—21no—
233C4⋊C4—41no—
234C4⋊C4—41no—
235C4⋊C4—41no—
236C22⋊C4—41no—
237C22⋊C4—4241no—
238C22⋊C4—2121no—
239C22⋊C4—2121no—
240C22⋊C4—2121no—
241C22⋊C4—2121no—
242C22⋊C4—2122no—
243C42—21no—
244C42—41no—
245C42C22⋊#615711yescharacteristic
246C42C22×#615711yescharacteristic
247C23—1061no—
248C23—1061no—
249C23—4242no—
250C23—2121no—
251C23—2121no—
252C23#1313327511yescharacteristic
253Q8—82no—
254Q8—4241no—
255Q8—2121no—
256Q8—2121no—
257D4—4241no—
258D4—2121no—
259D4—2121no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—41no—
265C2×C4—41no—
266C2×C4—81no—
267C2×C4—8482no—
268C2×C4—1061no—
269C2×C4—1061no—
270C2×C4—4242no—
271C2×C4—2121no—
272C2×C4—2121no—
273C2×C4#1313327511yescharacteristic
274C2×C4C2×C22⋊#615711yescharacteristic
275C8—4242no—
276C8—2121no—
277C8—2121no—
278C22—41no—
279C22—8481no—
280C22—4241no—
281C22—4241no—
282C22—4242no—
283C22—2121no—
284C22—2121no—
285C22#1681152911yescharacteristic
286C22#1681152911yescharacteristic
287C22C2×#1313327511yescharacteristic
288C4—21no—
289C4—21no—
290C4—41no—
291C4—41no—
292C4—81no—
293C4—81no—
294C4—8482no—
295C4—2121no—
296C4—2121no—
297C2—21no—
298C2—21no—
299C2—21no—
300C2—4241no—
301C2—4241no—
302C2C2×#1681152911yescharacteristic
subgroupG/H
centerC2C2×#16811529
commutatorC42×C53C22×C4
FrattiniC42C22×#6157
FittingC53×#533C4
socleC2×C53C2×#354
prime radicalC22×D212D4
cosocle#13133205C23
hypercenter#533#6157
hypocenterC53#13603
LCSG ▷ C42×C53 ▷ C2×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C23 ◁ C22×C4 ◁ C2×C42 ◁ #533

Complements

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

NH ≅ G/Ntype
3. #20289861C2semidirect
4. #20289860C2semidirect
15. #16810980C4semidirect
16. #16810980C4semidirect
21. C4×#11719917C4semidirect
22. C53×#533C4semidirect
47. C4×#88322C2×C4semidirect
48. C4×#88322C2×C4semidirect
50. #125×C53C2×C4semidirect
51. #125×C53C2×C4semidirect
157. #533#6157semidirect
174. C53#13603semidirect
180. #125C2×#6157semidirect
181. #125C2×#6157semidirect

Maximal subgroups

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

subgroupindex
C2×#168115322
C2×#168115302
#202898612
#202898602
#202851042
#1360353