Finite Group Databasebeta

← #27287273

Subgroups — #27287273

Order 13568.

360 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 75 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#20280364C211yesmaximal, characteristic
2#20275660C211yesmaximal, characteristic
3#20273227C211yesmaximal, characteristic
4#20272607C211yesmaximal, characteristic
5#20272579C211yesmaximal, characteristic
6#20272570C211yesmaximal, characteristic
7C53×#1515C211yesmaximal, characteristic
8#16810680C2211yescharacteristic
9C4×#11719917C2211yescharacteristic
10#16810565—21no—
11#16810544—21no—
12#16810527—21no—
13#16810527—21no—
14#16810510—21no—
15#16810506—21no—
16C4×C4⋊Dic53C2211yescharacteristic
17#16810469C2211yescharacteristic
18#16810382—21no—
19#16810382—21no—
20#16810360—21no—
21#16810360—21no—
22C53×#533C2211yescharacteristic
23C53×#432C2211yescharacteristic
24C53×#383C2211yescharacteristic
25C53×#328—21no—
26C53×#328—21no—
27C22×Dic106—21no—
28C2×#11719917—21no—
29C2×C4⋊Dic53—21no—
30C2×C4⋊Dic53—21no—
31C2×#11719919—21no—
32C2×#11719919—21no—
33C2×C4×Dic53—21no—
34C2×C4×Dic53D411yescharacteristic
35C2×C4×Dic53D411yescharacteristic
36C4⋊#88322—21no—
37C22⋊#88322—21no—
38#13133128—21no—
39#13133128—21no—
40C4×#88322—21no—
41#13133124—21no—
42#13133124—21no—
43#13133124—21no—
44#13133124—21no—
45#13133124—21no—
46#13133124—21no—
47#13133124D411yescharacteristic
48#13133124D411yescharacteristic
49C4⋊Q8×C53Q811yescharacteristic, complemented
50C4⋊Q8×C53Q811yescharacteristic, complemented
51#125×C53—21no—
52C22⋊Q8×C53—21no—
53C22⋊Q8×C53—21no—
54C4×Q8×C53—21no—
55C4×Q8×C53—21no—
56#117×C53—21no—
57C2×C42×C53C2311yescharacteristic
58C4⋊C8×C53—21no—
59C22⋊C8×C53—21no—
60#97×C53—21no—
61#95×C53—21no—
62#95×C53—21no—
63#95×C53D411yescharacteristic
64#95×C53D411yescharacteristic
65C2×Q8×C53—21no—
66C2×Q8×C53—21no—
67C22×C4×C53—21no—
68C22×C4×C53—21no—
69C22×C4×C53C2×D411yescharacteristic
70C22×C4×C53C2×D411yescharacteristic
71C22×C4×C53C2×D411yescharacteristic
72C22×Dic53—21no—
73C22×Dic53—21no—
74C22×Dic53—21no—
75C22×Dic53—21no—
76C2×Dic106—42no—
77C2×Dic106—42no—
78C2×Dic106—81no—
79C2×C8×C53—41no—
80C4⋊C4×C53—21no—
81C4⋊C4×C53—21no—
82C4⋊C4×C53—21no—
83C4⋊C4×C53—21no—
84C4⋊C4×C53—41no—
85C4⋊C4×C53—41no—
86C22⋊C4×C53—41no—
87C42×C53—41no—
88C42×C53C2×Q811yescharacteristic
89C42×C53#4111yescharacteristic
90C42×C53#4111yescharacteristic
91C42×C53#4111yescharacteristic
92C4⋊Dic53—42no—
93C4⋊Dic53—42no—
94#11719919—42no—
95#11719919—81no—
96C4×Dic53—42no—
97C4×Dic53—41no—
98C4×Dic53—41no—
99#11719917—22no—
100#11719917—41no—
101C2×#88322—41no—
102C2×#88322—41no—
103C23×C53#12011yescharacteristic
104Q8×C53—81no—
105Q8×C53—81no—
106C2×C4×C53—21no—
107C2×C4×C53—21no—
108C2×C4×C53—41no—
109C2×C4×C53—41no—
110C2×C4×C53—41no—
111C2×C4×C53—41no—
112C2×C4×C53—41no—
113C2×C4×C53—41no—
114C2×C4×C53—41no—
115C2×C4×C53—41no—
116C2×C4×C53—41no—
117C2×C4×C53—41no—
118C2×C4×C53C22⋊Q811yescharacteristic
119C2×C4×C53C22⋊Q811yescharacteristic
120C2×C4×C53C22⋊Q811yescharacteristic
121C2×C4×C53#12411yescharacteristic
122C2×C4×C53#12411yescharacteristic
123C2×C4×C53#12411yescharacteristic
124C2×Dic53—22no—
125C2×Dic53—22no—
126C2×Dic53—41no—
127C2×Dic53—41no—
128C2×Dic53—41no—
129C2×Dic53—41no—
130C2×Dic53—41no—
131C2×Dic53—41no—
132C2×Dic53—41no—
133C2×Dic53—41no—
134C2×Dic53—41no—
135C2×Dic53—41no—
136Dic106—84no—
137C8×C53—81no—
138#88322—42no—
139#88322—81no—
140#10379—531nomaximal, Sylow
141C22×C53—21no—
142C22×C53—21no—
143C22×C53#39311yescharacteristic
144C22×C53#45511yescharacteristic
145C22×C53#45611yescharacteristic
146C4×C53—21no—
147C4×C53—21no—
148C4×C53—21no—
149C4×C53—21no—
150C4×C53—21no—
151C4×C53—21no—
152C4×C53—81no—
153C4×C53—81no—
154C4×C53—81no—
155C4×C53—81no—
156Dic53—42no—
157Dic53—42no—
158Dic53—81no—
159Dic53—81no—
160#2605—531no—
161#1986—531no—
162#1515D5311yescharacteristic
163#1506—531no—
164#1300—531no—
165#1288—531no—
166#1285—531no—
167C2×C53—21no—
168C2×C53—21no—
169C2×C53#202811yescharacteristic
170C2×C53#203111yescharacteristic
171C2×C53#203111yescharacteristic
172#533C2×D5311yescharacteristic
173#432—531no—
174#432C2×D5311yescharacteristic
175C4×M16—531no—
176#400—1061no—
177#395—1061no—
178#391—1061no—
179#391—1061no—
180#387—1061no—
181#387—1061no—
182#383C2×D5311yescharacteristic
183C4×C4⋊C4—531no—
184#376—531no—
185#339—1061no—
186#339—1061no—
187#328—21no—
188#328—21no—
189#328—1061no—
190#328—1061no—
191C53#1037911yescharacteristic, complemented, Sylow
192C22×Q8—1061no—
193C2×M16—1061no—
194C4⋊Q8Dic10611yescharacteristic, complemented
195C4⋊Q8Dic10611yescharacteristic, complemented
196#125—21no—
197C22⋊Q8—21no—
198C22⋊Q8—21no—
199C4×Q8—21no—
200C4×Q8—21no—
201#117—21no—
202C2×C4⋊C4—1061no—
203C2×C4⋊C4—1061no—
204C2×C4⋊C4—1061no—
205C2×C4⋊C4—1061no—
206C2×C42—531no—
207C2×C42—531no—
208C2×C42—1061no—
209C2×C42C22×D5311yescharacteristic
210C4⋊C8—21no—
211C4⋊C8—1061no—
212C22⋊C8—21no—
213C22⋊C8—1061no—
214#97—21no—
215#97—1061no—
216#97—1061no—
217C4×C8—1061no—
218#95—21no—
219#95—21no—
220#95—531no—
221#95—531no—
222#95—1061no—
223#95—1061no—
224#95—1061no—
225#95—1061no—
226#95—1061no—
227#95—1061no—
228#95C22⋊D5311yescharacteristic
229#95C22⋊D5311yescharacteristic
230C2×Q8—21no—
231C2×Q8—21no—
232C2×Q8—4241no—
233C2×Q8—2122no—
234C2×Q8—2122no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—1061no—
238C22×C4—1061no—
239C22×C4—1061no—
240C22×C4—1061no—
241C22×C4D4×D5311yescharacteristic
242C22×C4D4×D5311yescharacteristic
243C22×C4C2×C22⋊D5311yescharacteristic
244M16—1062no—
245M16—2121no—
246C2×C8—41no—
247C2×C8—2121no—
248C2×C8—2121no—
249C4⋊C4—21no—
250C4⋊C4—21no—
251C4⋊C4—21no—
252C4⋊C4—21no—
253C4⋊C4—41no—
254C4⋊C4—41no—
255C4⋊C4—4241no—
256C4⋊C4—2122no—
257C4⋊C4—2122no—
258C4⋊C4—2122no—
259C22⋊C4—41no—
260C42—41no—
261C42—2121no—
262C42—2121no—
263C42—2122no—
264C42C2×Dic10611yescharacteristic
265C42#1171994511yescharacteristic
266C42#1171994711yescharacteristic
267C42#1171994711yescharacteristic
268C23#1313318511yescharacteristic
269Q8—81no—
270Q8—81no—
271Q8—4244no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—1062no—
285C2×C4—1062no—
286C2×C4—2121no—
287C2×C4—2121no—
288C2×C4—2121no—
289C2×C4—2121no—
290C2×C4—2121no—
291C2×C4—2121no—
292C2×C4—2121no—
293C2×C4—2121no—
294C2×C4—2121no—
295C2×C4—2121no—
296C2×C4#1313319411yescharacteristic
297C2×C4#1313319411yescharacteristic
298C2×C4#1313319611yescharacteristic
299C2×C4#1313320311yescharacteristic
300C2×C4#1313320311yescharacteristic
301C2×C4#1313320411yescharacteristic
302C8—81no—
303C8—4241no—
304C8—2122no—
305C22—21no—
306C22—21no—
307C22#1681053511yescharacteristic
308C22#1681079911yescharacteristic
309C22#1681080411yescharacteristic
310C4—21no—
311C4—21no—
312C4—21no—
313C4—21no—
314C4—21no—
315C4—21no—
316C4—81no—
317C4—81no—
318C4—81no—
319C4—81no—
320C4—4241no—
321C4—4241no—
322C4—2122no—
323C4—2122no—
324C2—21no—
325C2—21no—
326C2#2027587411yescharacteristic
327C2#2027588511yescharacteristic
328C2#2027588911yescharacteristic
subgroupG/H
centerC22#16810535
commutator, cosocleC2×C42×C53C23
FrattiniC2×C42C22×D53
FittingC53×#1515C2
socleC22×C53#393
prime radicalC23×C53#120
hypercenter#1515D53
hypocenterC53#10379
LCSG ▷ C2×C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1515

Complements

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

NH ≅ G/Ntype
49. C4⋊Q8×C53Q8semidirect
50. C4⋊Q8×C53Q8semidirect
191. C53#10379semidirect
194. C4⋊Q8Dic106semidirect
195. C4⋊Q8Dic106semidirect

Maximal subgroups

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

subgroupindex
#202803642
#202756602
#202732272
#202726072
#202725792
#202725702
C53×#15152
#1037953