Finite Group Databasebeta

← #27302753

Subgroups — #27302753

Order 13568.

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

normal subgroups only

This group has 616 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#20278080C211yesmaximal, characteristic
2#20274707C211yesmaximal, characteristic
3#20272832C214yesmaximal
4C4×#99×C53C211yesmaximal, characteristic
5C23×Dic106C411yescharacteristic, complemented
6C2×#13133196—21no—
7#16810647—21no—
8C2×#13133134C2212yes—
9#16810543—21no—
10#16810535—21no—
11#16810529—21no—
12#16810503C2212yes—
13#16810495C412yescomplemented
14#16810483C411yescharacteristic, complemented
15#16810392—21no—
16C2×C4×D4×C53C2211yescharacteristic
17C2×#99×C53C2212yes—
18C4×C22⋊C4×C53—21no—
19C53×#342—21no—
20C22×Dic106—21no—
21C22×Dic106—21no—
22C22×Dic106—44no—
23C22×Dic106—41no—
24C22×Dic106—41no—
25C23×Dic53C2×C412yes—
26C2×#11719917—22no—
27#13133196—44no—
28C2×C4⋊Dic53—21no—
29C2×#11719919—22no—
30C2×C22⋊Dic53—22no—
31C2×C22⋊Dic53C2×C412yes—
32C4⋊#88322—41no—
33C4⋊#88322—41no—
34#13133134—24no—
35#13133124—22no—
36#13133124—21no—
37#13133124—22no—
38C22×D4×C53C2311yescharacteristic
39C23×C4×C53C2×C411yescharacteristic
40C23×C4×C53D411yescharacteristic
41C4×D4×C53—41no—
42C4×D4×C53—41no—
43C2×C4⋊C4×C53D411yescharacteristic
44C2×C22⋊C4×C53—22no—
45C2×C22⋊C4×C53C2×C411yescharacteristic
46C2×C22⋊C4×C53D411yescharacteristic
47C2×C42×C53—41no—
48C2×C42×C53D411yescharacteristic
49#99×C53—22no—
50#99×C53Q814yescomplemented
51#95×C53—41no—
52C24×C53C22×C411yescharacteristic
53C24×C53C2×D411yescharacteristic
54C2×D4×C53—41no—
55C2×D4×C53—41no—
56C2×D4×C53C2×Q812yes—
57C2×D4×C53#4112yes—
58C22×C4×C53—21no—
59C22×C4×C53—21no—
60C22×C4×C53—21no—
61C22×C4×C53—21no—
62C22×C4×C53—42no—
63C22×C4×C53—42no—
64C22×C4×C53—42no—
65C22×C4×C53C22⋊C411yescharacteristic
66C22×C4×C53C22⋊C411yescharacteristic
67C22×C4×C53C22⋊C411yescharacteristic
68C22×C4×C53C22⋊C411yescharacteristic
69C22×C4×C53C2×D411yescharacteristic
70C22×Dic53—22no—
71C22×Dic53—22no—
72C22×Dic53—44no—
73C22×Dic53—42no—
74C2×Dic106—24no—
75C2×Dic106—44no—
76C2×Dic106—42no—
77C2×Dic106—42no—
78C2×Dic106—48no—
79C2×Dic106—44no—
80C2×Dic106—41no—
81C2×Dic106—41no—
82C2×Dic106—44no—
83C4⋊C4×C53—41no—
84C22⋊C4×C53—24no—
85C22⋊C4×C53—41no—
86C22⋊C4×C53—41no—
87C22⋊C4×C53—42no—
88C42×C53—21no—
89C42×C53—21no—
90C42×C53—82no—
91C22⋊Dic53—44no—
92C22⋊Dic53—42no—
93C4⋊Dic53—42no—
94#11719919—44no—
95#11719917—42no—
96#11719917—42no—
97C2×#88322—42no—
98C23×C53—21no—
99C23×C53—21no—
100C23×C53—21no—
101C23×C53—21no—
102C23×C53—41no—
103C23×C53—41no—
104C23×C53—41no—
105C23×C53—41no—
106C23×C53C2×C22⋊C411yescharacteristic
107C23×C53C4×Q812yes—
108C23×C53C22⋊Q812yes—
109D4×C53—42no—
110D4×C53—42no—
111C2×C4×C53—21no—
112C2×C4×C53—21no—
113C2×C4×C53—22no—
114C2×C4×C53—21no—
115C2×C4×C53—21no—
116C2×C4×C53—21no—
117C2×C4×C53—42no—
118C2×C4×C53—42no—
119C2×C4×C53—42no—
120C2×C4×C53—41no—
121C2×C4×C53—41no—
122C2×C4×C53—41no—
123C2×C4×C53—44no—
124C2×C4×C53—81no—
125C2×C4×C53—82no—
126C2×C4×C53—82no—
127C2×C4×C53#9911yescharacteristic
128C2×C4×C53#9911yescharacteristic
129C2×C4×C53#12411yescharacteristic
130C2×C4×C53C4⋊Q811yescharacteristic
131C2×Dic53—24no—
132C2×Dic53—42no—
133C2×Dic53—44no—
134C2×Dic53—42no—
135C2×Dic53—44no—
136C2×Dic53—42no—
137C2×Dic53—42no—
138C2×Dic53—42no—
139Dic106—44no—
140Dic106—48no—
141Dic106—44no—
142#88322—82no—
143#13296—531nomaximal, Sylow
144C22×C53—21no—
145C22×C53—21no—
146C22×C53—22no—
147C22×C53—22no—
148C22×C53—42no—
149C22×C53—42no—
150C22×C53—42no—
151C22×C53—41no—
152C22×C53—41no—
153C22×C53—41no—
154C22×C53—41no—
155C22×C53—81no—
156C22×C53#39111yescharacteristic
157C22×C53C2×#9911yescharacteristic
158C22×C53#41011yescharacteristic
159C4×C53—21no—
160C4×C53—21no—
161C4×C53—21no—
162C4×C53—41no—
163C4×C53—41no—
164C4×C53—41no—
165C4×C53—84no—
166C4×C53AΣL(1,𝔽16)12yes—
167Dic53—44no—
168Dic53—82no—
169#2409—531no—
170#1800—531no—
171C4×#99D5311yescharacteristic
172#1390—534no—
173C2×C53—21no—
174C2×C53—21no—
175C2×C53—42no—
176C2×C53—42no—
177C2×C53#199611yescharacteristic
178C2×C53C2×AΣL(1,𝔽16)11yescharacteristic
179C2×C53#212311yescharacteristic
180C23×Q8—531no—
181C2×C22⋊Q8—1061no—
182C2×C4×D4C2×D5311yescharacteristic
183#423—1061no—
184C2×#100—532no—
185C2×#99C2×D5312yes—
186#395—1061no—
187#393—1061no—
188#391—1061no—
189#386—532no—
190#385—532no—
191#380—531no—
192C4×C22⋊C4—21no—
193#343—1061no—
194#342—21no—
195C53#1329611yescharacteristic, complemented, Sylow
196C22×Q8—1061no—
197C22×Q8—1061no—
198C22×Q8—2121no—
199C22×Q8—2121no—
200C22×Q8—2124no—
201C22×D4C22×D5311yescharacteristic
202C23×C4—532no—
203C23×C4C4×D5311yescharacteristic
204C23×C4C22⋊D5311yescharacteristic
205C2×M16—1062no—
206C22⋊Q8—2124no—
207C4×D4—41no—
208C4×D4—41no—
209C2×C4⋊C4—1062no—
210C2×C4⋊C4—1061no—
211C2×C4⋊C4D21211yescharacteristic
212C2×C22⋊C4—22no—
213C2×C22⋊C4—532no—
214C2×C22⋊C4—1062no—
215C2×C22⋊C4C4×D5311yescharacteristic
216C2×C22⋊C4C22⋊D5311yescharacteristic
217C2×C42—41no—
218C2×C42D21211yescharacteristic
219C4⋊C8—2121no—
220C4⋊C8—2121no—
221#100—1064no—
222#99—22no—
223#99Dic10614yescomplemented
224#95—41no—
225#95—1062no—
226#95—1061no—
227#95—1062no—
228C24C2×C4×D5311yescharacteristic
229C24C2×C22⋊D5311yescharacteristic
230C2×Q8—1064no—
231C2×Q8—2122no—
232C2×Q8—2122no—
233C2×Q8—2124no—
234C2×Q8—2124no—
235C2×Q8—2128no—
236C2×Q8—2124no—
237C2×Q8—2121no—
238C2×Q8—2121no—
239C2×D4—41no—
240C2×D4—41no—
241C2×D4C2×Dic10612yes—
242C2×D4#1171994512yes—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—42no—
248C22×C4—42no—
249C22×C4—42no—
250C22×C4—1062no—
251C22×C4—1062no—
252C22×C4—2124no—
253C22×C4—2122no—
254C22×C4#1171992111yescharacteristic
255C22×C4#1171992111yescharacteristic
256C22×C4#1171992111yescharacteristic
257C22×C4#1171992111yescharacteristic
258C22×C4C2×D21211yescharacteristic
259M16—2122no—
260M16—2122no—
261C2×C8—2122no—
262C4⋊C4—41no—
263C4⋊C4—2122no—
264C4⋊C4—2124no—
265C22⋊C4—24no—
266C22⋊C4—41no—
267C22⋊C4—41no—
268C22⋊C4—42no—
269C22⋊C4—2124no—
270C22⋊C4—2122no—
271C42—21no—
272C42—21no—
273C42—82no—
274C23—21no—
275C23—21no—
276C23—21no—
277C23—21no—
278C23—41no—
279C23—41no—
280C23—41no—
281C23—41no—
282C23C2×#1171992111yescharacteristic
283C23C4×Dic10612yes—
284C23#1313319612yes—
285Q8—2124no—
286Q8—2128no—
287Q8—2124no—
288D4—42no—
289D4—42no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—22no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—42no—
297C2×C4—42no—
298C2×C4—42no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—44no—
303C2×C4—81no—
304C2×C4—82no—
305C2×C4—82no—
306C2×C4—1064no—
307C2×C4—2122no—
308C2×C4—2122no—
309C2×C4—2122no—
310C2×C4—2122no—
311C2×C4—2124no—
312C2×C4—2122no—
313C2×C4—2124no—
314C2×C4#1313313111yescharacteristic
315C2×C4#1313313111yescharacteristic
316C2×C4#1313320511yescharacteristic
317C2×C4#1313321711yescharacteristic
318C8—4242no—
319C22—21no—
320C22—21no—
321C22—22no—
322C22—22no—
323C22—42no—
324C22—42no—
325C22—42no—
326C22—41no—
327C22—41no—
328C22—41no—
329C22—41no—
330C22—81no—
331C22#1681052911yescharacteristic
332C22C2×#1313313111yescharacteristic
333C22#1681059611yescharacteristic
334C4—21no—
335C4—21no—
336C4—21no—
337C4—41no—
338C4—41no—
339C4—41no—
340C4—84no—
341C4—4242no—
342C4—2124no—
343C4#1681040912yes—
344C2—21no—
345C2—21no—
346C2—42no—
347C2—42no—
348C2#2027570011yescharacteristic
349C2C2×#1681040911yescharacteristic
350C2#2027636811yescharacteristic
subgroupG/H
centerC22C2×#13133131
commutatorC24×C53C22×C4
FrattiniC22×D4C22×D53
FittingC4×#99×C53C2
socleC22×C53C2×#99
prime radical, cosocleC22×D4×C53C23
hypercenterC4×#99D53
hypocenterC53#13296
LCSG ▷ C24×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ C4×#99

Complements

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

NH ≅ G/Ntype
5. C23×Dic106C4semidirect
13. #16810495C4semidirect
14. #16810483C4semidirect
50. #99×C53Q8semidirect
195. C53#13296semidirect
223. #99Dic106semidirect

Maximal subgroups

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

subgroupindex
#202780802
#202747072
#202728322
C4×#99×C532
#1329653