Finite Group Databasebeta

← #27302739

Subgroups — #27302739

Order 13568.

612 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 612 conjugacy classes of subgroups — too many to draw. Only its 103 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
2C4×#13133134C211yesmaximal, characteristic
3#20272832C214yesmaximal
4C53×#1799C211yesmaximal, characteristic
5C23×Dic106C411yescharacteristic, complemented
6C2×#13133196—21no—
7C2×#13133134C2212yes—
8C4×#11719917—21no—
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—
18C53×#380—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—
32#13133134—22no—
33#13133134Q814yescomplemented
34#13133128—41no—
35C4×#88322—41no—
36#13133124—22no—
37#13133124—21no—
38#13133124—22no—
39C22×D4×C53C2311yescharacteristic
40C23×C4×C53C2×C411yescharacteristic
41C23×C4×C53D411yescharacteristic
42C4×D4×C53—41no—
43C4×D4×C53—41no—
44C2×C4⋊C4×C53—41no—
45C2×C4⋊C4×C53D411yescharacteristic
46C2×C22⋊C4×C53—22no—
47C2×C22⋊C4×C53C2×C411yescharacteristic
48C2×C22⋊C4×C53D411yescharacteristic
49C2×C42×C53D411yescharacteristic
50#99×C53—24no—
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—
84C4⋊C4×C53—82no—
85C22⋊C4×C53—41no—
86C22⋊C4×C53—41no—
87C22⋊C4×C53—42no—
88C22⋊C4×C53—42no—
89C42×C53—21no—
90C42×C53—21no—
91C22⋊Dic53—44no—
92C22⋊Dic53—42no—
93C4⋊Dic53—42no—
94#11719919—44no—
95#11719917—24no—
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×C53#12411yescharacteristic
109C23×C53C4⋊Q811yescharacteristic
110D4×C53—42no—
111D4×C53—42no—
112C2×C4×C53—21no—
113C2×C4×C53—21no—
114C2×C4×C53—22no—
115C2×C4×C53—21no—
116C2×C4×C53—21no—
117C2×C4×C53—21no—
118C2×C4×C53—42no—
119C2×C4×C53—42no—
120C2×C4×C53—42no—
121C2×C4×C53—41no—
122C2×C4×C53—41no—
123C2×C4×C53—41no—
124C2×C4×C53—81no—
125C2×C4×C53—82no—
126C2×C4×C53—82no—
127C2×C4×C53—82no—
128C2×C4×C53#9911yescharacteristic
129C2×C4×C53#9911yescharacteristic
130C2×C4×C53C22⋊Q812yes—
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—44no—
143#13293—531nomaximal, Sylow
144C22×C53—21no—
145C22×C53—21no—
146C22×C53—21no—
147C22×C53—22no—
148C22×C53—21no—
149C22×C53—42no—
150C22×C53—41no—
151C22×C53—41no—
152C22×C53—42no—
153C22×C53—41no—
154C22×C53—41no—
155C22×C53—41no—
156C22×C53—41no—
157C22×C53—81no—
158C22×C53#39111yescharacteristic
159C22×C53C2×#9911yescharacteristic
160C22×C53#41011yescharacteristic
161C4×C53—21no—
162C4×C53—22no—
163C4×C53—41no—
164C4×C53—41no—
165C4×C53—41no—
166C4×C53—162no—
167C4×C53AΣL(1,𝔽16)12yes—
168Dic53—44no—
169Dic53—82no—
170#2409—531no—
171#1799D5311yescharacteristic
172C4×#100—531no—
173#1390—534no—
174C2×C53—21no—
175C2×C53—21no—
176C2×C53—41no—
177C2×C53—42no—
178C2×C53—41no—
179C2×C53#199611yescharacteristic
180C2×C53C2×AΣL(1,𝔽16)11yescharacteristic
181C2×C53#212311yescharacteristic
182C23×Q8—531no—
183C2×C22⋊Q8—1061no—
184C2×C4×D4C2×D5311yescharacteristic
185C2×#100—532no—
186C2×#99C2×D5312yes—
187C4×M16—1061no—
188#395—1061no—
189#393—1061no—
190#391—1061no—
191#386—532no—
192#385—532no—
193#380—21no—
194#380—531no—
195#343—1061no—
196#342—21no—
197C53#1329311yescharacteristic, complemented, Sylow
198C22×Q8—1061no—
199C22×Q8—1061no—
200C22×Q8—2121no—
201C22×Q8—2121no—
202C22×Q8—2124no—
203C22×D4C22×D5311yescharacteristic
204C23×C4—532no—
205C23×C4C4×D5311yescharacteristic
206C23×C4C22⋊D5311yescharacteristic
207C2×M16—1062no—
208C22⋊Q8—2124no—
209C4×D4—41no—
210C4×D4—41no—
211C2×C4⋊C4—41no—
212C2×C4⋊C4—1062no—
213C2×C4⋊C4—1061no—
214C2×C4⋊C4D21211yescharacteristic
215C2×C22⋊C4—22no—
216C2×C22⋊C4—532no—
217C2×C22⋊C4—1062no—
218C2×C22⋊C4C4×D5311yescharacteristic
219C2×C22⋊C4C22⋊D5311yescharacteristic
220C2×C42D21211yescharacteristic
221#100—534no—
222#100—1062no—
223#99—24no—
224#97—2121no—
225C4×C8—2121no—
226#95—41no—
227#95—1062no—
228#95—1061no—
229#95—1062no—
230C24C2×C4×D5311yescharacteristic
231C24C2×C22⋊D5311yescharacteristic
232C2×Q8—1064no—
233C2×Q8—2122no—
234C2×Q8—2122no—
235C2×Q8—2124no—
236C2×Q8—2124no—
237C2×Q8—2128no—
238C2×Q8—2124no—
239C2×Q8—2121no—
240C2×Q8—2121no—
241C2×D4—41no—
242C2×D4—41no—
243C2×D4#1171994712yes—
244C2×D4Q8×D5312yes—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—42no—
250C22×C4—42no—
251C22×C4—42no—
252C22×C4—1062no—
253C22×C4—1062no—
254C22×C4—2124no—
255C22×C4—2122no—
256C22×C4#1171992111yescharacteristic
257C22×C4#1171992111yescharacteristic
258C22×C4#1171992111yescharacteristic
259C22×C4#1171992111yescharacteristic
260C22×C4C2×D21211yescharacteristic
261M16—1064no—
262M16—2122no—
263C2×C8—2122no—
264C4⋊C4—41no—
265C4⋊C4—82no—
266C4⋊C4—2122no—
267C4⋊C4—2124no—
268C22⋊C4—41no—
269C22⋊C4—41no—
270C22⋊C4—42no—
271C22⋊C4—42no—
272C22⋊C4—2124no—
273C22⋊C4—2122no—
274C42—21no—
275C42—21no—
276C23—21no—
277C23—21no—
278C23—21no—
279C23—21no—
280C23—41no—
281C23—41no—
282C23—41no—
283C23—41no—
284C23C2×#1171992111yescharacteristic
285C23#1313318212yes—
286C23#1313320411yescharacteristic
287C23#1313321511yescharacteristic
288Q8—2124no—
289Q8—2128no—
290Q8—2124no—
291D4—42no—
292D4—42no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—22no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—42no—
300C2×C4—42no—
301C2×C4—42no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—81no—
306C2×C4—82no—
307C2×C4—82no—
308C2×C4—82no—
309C2×C4—1064no—
310C2×C4—2122no—
311C2×C4—2122no—
312C2×C4—2122no—
313C2×C4—2122no—
314C2×C4—2124no—
315C2×C4—2122no—
316C2×C4—2124no—
317C2×C4#1313313111yescharacteristic
318C2×C4#1313313111yescharacteristic
319C2×C4#1313319712yes—
320C8—2124no—
321C22—21no—
322C22—21no—
323C22—21no—
324C22—22no—
325C22—21no—
326C22—42no—
327C22—41no—
328C22—41no—
329C22—42no—
330C22—41no—
331C22—41no—
332C22—41no—
333C22—41no—
334C22—81no—
335C22#1681052811yescharacteristic
336C22C2×#1313313111yescharacteristic
337C22#1681059611yescharacteristic
338C4—21no—
339C4—22no—
340C4—41no—
341C4—41no—
342C4—41no—
343C4—162no—
344C4—4242no—
345C4—2124no—
346C4#1681040912yes—
347C2—21no—
348C2—21no—
349C2—41no—
350C2—42no—
351C2—41no—
352C2#2027569911yescharacteristic
353C2C2×#1681040911yescharacteristic
354C2#2027636811yescharacteristic
subgroupG/H
centerC22C2×#13133131
commutatorC24×C53C22×C4
FrattiniC22×D4C22×D53
FittingC53×#1799C2
socleC22×C53C2×#99
prime radical, cosocleC22×D4×C53C23
hypercenter#1799D53
hypocenterC53#13293
LCSG ▷ C24×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #1799

Complements

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

NH ≅ G/Ntype
5. C23×Dic106C4semidirect
13. #16810495C4semidirect
14. #16810483C4semidirect
33. #13133134Q8semidirect
197. C53#13293semidirect

Maximal subgroups

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

subgroupindex
#202780802
C4×#131331342
#202728322
C53×#17992
#1329353