Finite Group Databasebeta

← #27284146

Subgroups — #27284146

Order 13568.

556 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20274695C211yesmaximal, characteristic
2#20274477C211yesmaximal, characteristic
3#20272558C212yesmaximal
4#20272558C212yesmaximal
5C2×C53×#387C211yesmaximal, characteristic
6C2×C4⋊#88322—21no—
7C2×C22⋊#88322C2212yes—
8C2×C22⋊#88322C2212yes—
9C2×#13133128—21no—
10#16810378—21no—
11#16810378—21no—
12C22×C4⋊C4×C53C411yescharacteristic
13C22×C4⋊C4×C53C411yescharacteristic
14C22×C42×C53C2211yescharacteristic
15C53×#387—21no—
16C53×#387—21no—
17C53×#387—21no—
18C53×#387—21no—
19C2×#95×C53C411yescharacteristic
20C2×#95×C53C411yescharacteristic
21C2×#95×C53C2211yescharacteristic
22C2×#95×C53C2211yescharacteristic
23C22×#88322—22no—
24C22×#88322—22no—
25C4⋊#88322—42no—
26C22⋊#88322—22no—
27C22⋊#88322—24no—
28C22⋊#88322Q814yes—
29#13133128—42no—
30C23×C4×C53C2×C411yescharacteristic
31C23×C4×C53C2×C411yescharacteristic
32C23×C4×C53C2×C411yescharacteristic
33C23×C4×C53C2×C411yescharacteristic
34C23×C4×C53C2×C411yescharacteristic
35C23×C4×C53C2×C411yescharacteristic
36C23×C4×C53C2311yescharacteristic
37C2×C4⋊C4×C53—21no—
38C2×C4⋊C4×C53—21no—
39C2×C4⋊C4×C53—21no—
40C2×C4⋊C4×C53—21no—
41C2×C4⋊C4×C53—22no—
42C2×C4⋊C4×C53—22no—
43C2×C4⋊C4×C53—42no—
44C2×C42×C53—42no—
45C2×C42×C53D411yescharacteristic
46C2×C42×C53D411yescharacteristic
47C2×C42×C53D411yescharacteristic
48C2×C42×C53D411yescharacteristic
49#95×C53—21no—
50#95×C53—21no—
51#95×C53—21no—
52#95×C53—21no—
53#95×C53—21no—
54#95×C53—21no—
55#95×C53—21no—
56#95×C53—21no—
57C24×C53C22×C411yescharacteristic
58C22×C4×C53—21no—
59C22×C4×C53—21no—
60C22×C4×C53—21no—
61C22×C4×C53—21no—
62C22×C4×C53—21no—
63C22×C4×C53—21no—
64C22×C4×C53—22no—
65C22×C4×C53—22no—
66C22×C4×C53—21no—
67C22×C4×C53—21no—
68C22×C4×C53—22no—
69C22×C4×C53—22no—
70C22×C4×C53—41no—
71C22×C4×C53—41no—
72C22×C4×C53—41no—
73C22×C4×C53—41no—
74C22×C4×C53—41no—
75C22×C4×C53—41no—
76C22×C4×C53—41no—
77C22×C4×C53—41no—
78C22×C4×C53—41no—
79C22×C4×C53—41no—
80C22×C4×C53—41no—
81C22×C4×C53—41no—
82C22×C4×C53—41no—
83C22×C4×C53C22⋊C411yescharacteristic
84C22×C4×C53C22⋊C411yescharacteristic
85C22×C4×C53C22⋊C411yescharacteristic
86C22×C4×C53C22⋊C411yescharacteristic
87C22×C4×C53M1612yes—
88C22×C4×C53M1612yes—
89C22×C4×C53C2×D411yescharacteristic
90C22×C4×C53C2×D411yescharacteristic
91C22×C4×C53C2×Q812yes—
92C22×C4×C53#4112yes—
93C4⋊C4×C53—24no—
94C4⋊C4×C53—24no—
95C4⋊C4×C53—42no—
96C4⋊C4×C53—42no—
97C42×C53—22no—
98C42×C53—22no—
99C42×C53—41no—
100C42×C53—41no—
101C2×#88322—24no—
102C2×#88322—42no—
103C2×#88322—42no—
104C2×#88322—42no—
105C2×#88322—42no—
106C2×#88322—42no—
107C23×C53—21no—
108C23×C53—21no—
109C23×C53—21no—
110C23×C53—21no—
111C23×C53C2×C22⋊C411yescharacteristic
112C23×C53C4×Q812yes—
113C23×C53C2×M1611yescharacteristic
114C23×C53C2×M1611yescharacteristic
115C23×C53#13112yes—
116C2×C4×C53—21no—
117C2×C4×C53—21no—
118C2×C4×C53—21no—
119C2×C4×C53—21no—
120C2×C4×C53—21no—
121C2×C4×C53—21no—
122C2×C4×C53—21no—
123C2×C4×C53—21no—
124C2×C4×C53—22no—
125C2×C4×C53—22no—
126C2×C4×C53—21no—
127C2×C4×C53—21no—
128C2×C4×C53—21no—
129C2×C4×C53—21no—
130C2×C4×C53—21no—
131C2×C4×C53—21no—
132C2×C4×C53—21no—
133C2×C4×C53—42no—
134C2×C4×C53—42no—
135C2×C4×C53—44no—
136C2×C4×C53—42no—
137C2×C4×C53—42no—
138C2×C4×C53—42no—
139C2×C4×C53—42no—
140C2×C4×C53—41no—
141C2×C4×C53—41no—
142C2×C4×C53—41no—
143C2×C4×C53—41no—
144C2×C4×C53—42no—
145C2×C4×C53—42no—
146C2×C4×C53—41no—
147C2×C4×C53—41no—
148C2×C4×C53—41no—
149C2×C4×C53—41no—
150C2×C4×C53—41no—
151C2×C4×C53—41no—
152C2×C4×C53—82no—
153C2×C4×C53—82no—
154C2×C4×C53#9912yes—
155C2×C4×C53C22⋊Q812yes—
156C2×C4×C53#12411yescharacteristic
157C2×C4×C53C4⋊Q811yescharacteristic
158#88322—44no—
159#88322—82no—
160#9881—531nomaximal, Sylow
161C22×C53—21no—
162C22×C53—21no—
163C22×C53—21no—
164C22×C53—21no—
165C22×C53—22no—
166C22×C53—22no—
167C22×C53—22no—
168C22×C53—22no—
169C22×C53#39111yescharacteristic
170C22×C53#40711yescharacteristic
171C22×C53#40811yescharacteristic
172C22×C53C2×#9911yescharacteristic
173C22×C53#41011yescharacteristic
174C22×C53#41311yescharacteristic
175C22×C53#41311yescharacteristic
176C22×C53#44612yes—
177C22×C53#44612yes—
178C4×C53—22no—
179C4×C53—22no—
180C4×C53—21no—
181C4×C53—21no—
182C4×C53—42no—
183C4×C53—41no—
184C4×C53—41no—
185C4×C53—42no—
186C4×C53—41no—
187C4×C53—81no—
188C4×C53—81no—
189C2×#387D5311yescharacteristic
190#1797—531no—
191#1749—531no—
192#1280—532no—
193#1280—532no—
194C2×C53—22no—
195C2×C53—22no—
196C2×C53#146411yescharacteristic
197C2×C53#146511yescharacteristic
198C2×C53#146811yescharacteristic
199C2×C53#147311yescharacteristic
200C2×C53#198511yescharacteristic
201C2×C53#199611yescharacteristic
202C2×C53#199811yescharacteristic
203C22×C4⋊C4Dic5311yescharacteristic
204C22×C4⋊C4Dic5311yescharacteristic
205C22×C42C2×D5311yescharacteristic
206C2×C4⋊C8—1061no—
207C2×C22⋊C8—532no—
208C2×C22⋊C8—532no—
209C2×#97—1061no—
210#387—21no—
211#387—21no—
212#387—21no—
213#387—21no—
214C2×#95Dic5311yescharacteristic
215C2×#95Dic5311yescharacteristic
216C2×#95C2×D5311yescharacteristic
217C2×#95C2×D5311yescharacteristic
218#336—1061no—
219#336—1061no—
220C53#988111yescharacteristic, complemented, Sylow
221C23×C4C2×Dic5311yescharacteristic
222C23×C4C2×Dic5311yescharacteristic
223C23×C4C2×Dic5311yescharacteristic
224C23×C4C2×Dic5311yescharacteristic
225C23×C4C2×Dic5311yescharacteristic
226C23×C4C2×Dic5311yescharacteristic
227C23×C4C22×D5311yescharacteristic
228C22×C8—1062no—
229C22×C8—1062no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—42no—
237C2×C42—42no—
238C2×C42C22⋊D5311yescharacteristic
239C2×C42C22⋊D5311yescharacteristic
240C2×C42C22⋊D5311yescharacteristic
241C2×C42C22⋊D5311yescharacteristic
242C4⋊C8—2122no—
243C22⋊C8—534no—
244C22⋊C8—1062no—
245C22⋊C8—1064no—
246#97—2122no—
247#95—21no—
248#95—21no—
249#95—21no—
250#95—21no—
251#95—21no—
252#95—21no—
253#95—21no—
254#95—21no—
255C24C22×Dic5311yescharacteristic
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—41no—
269C22×C4—41no—
270C22×C4—41no—
271C22×C4—41no—
272C22×C4—41no—
273C22×C4—41no—
274C22×C4—41no—
275C22×C4—41no—
276C22×C4—41no—
277C22×C4—41no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4#1171991712yes—
282C22×C4#1171991712yes—
283C22×C4C22⋊Dic5311yescharacteristic
284C22×C4C22⋊Dic5311yescharacteristic
285C22×C4C22⋊Dic5311yescharacteristic
286C22×C4C22⋊Dic5311yescharacteristic
287C22×C4Q8×D5312yes—
288C22×C4#1171994912yes—
289C22×C4C2×C22⋊D5311yescharacteristic
290C22×C4C2×C22⋊D5311yescharacteristic
291C2×C8—1064no—
292C2×C8—2122no—
293C2×C8—2122no—
294C2×C8—2122no—
295C2×C8—2122no—
296C2×C8—2122no—
297C4⋊C4—24no—
298C4⋊C4—24no—
299C4⋊C4—42no—
300C4⋊C4—42no—
301C42—22no—
302C42—22no—
303C42—41no—
304C42—41no—
305C23—21no—
306C23—21no—
307C23—21no—
308C23—21no—
309C23C2×C22⋊Dic5311yescharacteristic
310C23Q8×Dic5312yes—
311C23C2×#1171991711yescharacteristic
312C23C2×#1171991711yescharacteristic
313C23#1313322412yes—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—22no—
321C2×C4—22no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—42no—
332C2×C4—42no—
333C2×C4—44no—
334C2×C4—42no—
335C2×C4—42no—
336C2×C4—42no—
337C2×C4—42no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—42no—
343C2×C4—42no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—82no—
351C2×C4—82no—
352C2×C4#1313313212yes—
353C2×C4#1313319312yes—
354C2×C4#1313320611yescharacteristic
355C2×C4#1313321511yescharacteristic
356C8—4242no—
357C8—2124no—
358C22—21no—
359C22—21no—
360C22—21no—
361C22—21no—
362C22—22no—
363C22—22no—
364C22—22no—
365C22—22no—
366C22#1681053011yescharacteristic
367C22#1681058711yescharacteristic
368C22#1681059011yescharacteristic
369C22C2×#1313313211yescharacteristic
370C22#1681059711yescharacteristic
371C22#1681060711yescharacteristic
372C22#1681060711yescharacteristic
373C22#1681075412yes—
374C22#1681075412yes—
375C4—22no—
376C4—21no—
377C4—22no—
378C4—21no—
379C4—42no—
380C4—41no—
381C4—41no—
382C4—42no—
383C4—41no—
384C4—81no—
385C4—81no—
386C2—22no—
387C2—22no—
388C2#2027304911yescharacteristic
389C2#2027305311yescharacteristic
390C2#2027306311yescharacteristic
391C2#2027308411yescharacteristic
392C2#2027565611yescharacteristic
393C2#2027570111yescharacteristic
394C2#2027570911yescharacteristic
subgroupG/H
centerC23C2×C22⋊Dic53
commutator, prime radicalC24×C53C22×C4
FrattiniC23×C4C22×D53
FittingC2×C53×#387C2
socleC23×C53C2×C22⋊C4
cosocleC23×C4×C53C23
hypercenterC2×#387D53
hypocenterC53#9881
LCSG ▷ C24×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×#387

Complements

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

NH ≅ G/Ntype
220. C53#9881semidirect

Maximal subgroups

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

subgroupindex
#202746952
#202744772
#202725582
C2×C53×#3872
#988153