Finite Group Databasebeta

← #27590816

Subgroups — #27590816

Order 13568.

1304 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20282990C211yesmaximal, characteristic
2#20282957C214yesmaximal
3#20282955C214yesmaximal
4#20282953C214yesmaximal
5#20282783C211yesmaximal, characteristic
6#20282782C212yesmaximal
7#20278023C214yesmaximal
8#20278022C214yesmaximal
9#20278022C214yesmaximal
10C4×#13133209C212yesmaximal
11C4×#125×C53C211yesmaximal, characteristic
12C2×#13133209C2212yes—
13#16810565C2214yes—
14#16810565C2214yes—
15#16810564C2214yes—
16#16810564C2214yes—
17#16810564C2214yes—
18#16810564C2214yes—
19#16810563C2214yes—
20#16810563C2214yes—
21#16810562C2214yes—
22#16810562C2214yes—
23#16810561C2214yes—
24#16810561C2214yes—
25#16810557C2214yes—
26#16810557C2214yes—
27#16810556C2214yes—
28#16810556C2214yes—
29#16810556C2214yes—
30#16810556C2214yes—
31#16810555C2214yes—
32#16810555C2214yes—
33#16810554C2214yes—
34#16810554C2214yes—
35#16810553C2214yes—
36#16810553C2214yes—
37#16810521C2212yes—
38#16810521C2212yes—
39#16810507C2214yes—
40#16810507C2214yes—
41#16810489C2212yes—
42C4×#11719919C2214yes—
43C4×#11719919C2214yes—
44C4×#11719919C2214yes—
45C4×#11719919C2214yes—
46C4×C4⋊Dic53C2212yes—
47C4×C4⋊Dic53C2214yes—
48C4×C4⋊Dic53C2212yes—
49#16810469C2214yes—
50C2×#125×C53C2211yescharacteristic
51C53×#389C2212yes—
52C53×#387C2214yes—
53C53×#382C2211yescharacteristic
54C4×C4⋊C4×C53C2212yes—
55C4×C4⋊C4×C53C2214yes—
56C43×C53C2211yescharacteristic
57#13133209—28no—
58C2×C4⋊Dic53C2312yes—
59C2×C4⋊Dic53C2312yes—
60C2×C4⋊Dic53C2312yes—
61C2×C4⋊Dic53C2312yes—
62C2×C4⋊Dic53C2314yes—
63C2×C4⋊Dic53C2314yes—
64C2×#11719919C2314yes—
65C2×#11719919C2314yes—
66C2×#11719919C2314yes—
67C2×#11719919C2314yes—
68C2×#11719919C2314yes—
69C2×#11719919C2314yes—
70C2×#11719919C2314yes—
71C2×#11719919C2314yes—
72C2×C4×Dic53C2314yes—
73C2×C4×Dic53C2314yes—
74#13133124C2314yes—
75#13133124C2314yes—
76#13133124C2314yes—
77#13133124C2314yes—
78#13133124C2314yes—
79#13133124C2314yes—
80#13133124C2314yes—
81#13133124C2314yes—
82#13133124C2314yes—
83#13133124C2314yes—
84#13133124C2314yes—
85#13133124C2314yes—
86#13133124C2314yes—
87#13133124C2314yes—
88#13133124C2314yes—
89#13133124C2314yes—
90#125×C53—24no—
91C2×C4⋊C4×C53C2314yes—
92C2×C4⋊C4×C53C2312yes—
93C2×C4⋊C4×C53C2312yes—
94C2×C4⋊C4×C53C2314yes—
95C2×C42×C53C2312yes—
96C2×C42×C53C2311yescharacteristic
97C2×C42×C53C2314yes—
98C2×C42×C53C2311yescharacteristic
99C2×C42×C53C2312yes—
100C2×C42×C53C2311yescharacteristic
101#95×C53C2314yes—
102#95×C53C2314yes—
103#95×C53C2314yes—
104C22×C4×C53C2411yescharacteristic
105C22×C4×C53C2414yes—
106C22×C4×C53C2412yes—
107C22×C4×C53C2414yes—
108C22×C4×C53C2412yes—
109C22×C4×C53C2411yescharacteristic
110C22×C4×C53C2411yescharacteristic
111C22×Dic53C2414yes—
112C22×Dic53C2414yes—
113C22×Dic53C2414yes—
114C22×Dic53C2414yes—
115C4⋊C4×C53—24no—
116C4⋊C4×C53—28no—
117C4⋊C4×C53—42no—
118C4⋊C4×C53—44no—
119C42×C53—24no—
120C42×C53—22no—
121C42×C53—28no—
122C42×C53—42no—
123C42×C53—41no—
124C42×C53#4114yes—
125C4⋊Dic53—24no—
126C4⋊Dic53—24no—
127C4⋊Dic53—42no—
128C4⋊Dic53—44no—
129C4⋊Dic53—42no—
130C4⋊Dic53—44no—
131#11719919—28no—
132#11719919—28no—
133#11719919—44no—
134#11719919—44no—
135#11719919—44no—
136#11719919—44no—
137#11719919—44no—
138#11719919—44no—
139C4×Dic53—28no—
140C4×Dic53—28no—
141C23×C53C2511yescharacteristic
142C2×C4×C53—22no—
143C2×C4×C53—21no—
144C2×C4×C53—22no—
145C2×C4×C53—21no—
146C2×C4×C53—24no—
147C2×C4×C53—24no—
148C2×C4×C53—24no—
149C2×C4×C53—22no—
150C2×C4×C53—21no—
151C2×C4×C53—22no—
152C2×C4×C53—22no—
153C2×C4×C53—24no—
154C2×C4×C53—24no—
155C2×C4×C53—24no—
156C2×C4×C53—22no—
157C2×C4×C53C2×#4112yes—
158C2×C4×C53C2×#4114yes—
159C2×C4×C532−1+414yes—
160C2×C4×C532−1+412yes—
161C2×Dic53—24no—
162C2×Dic53—24no—
163C2×Dic53—24no—
164C2×Dic53—24no—
165C2×Dic53—24no—
166C2×Dic53—24no—
167C2×Dic53—24no—
168C2×Dic53—24no—
169C2×Dic53—24no—
170C2×Dic53—24no—
171C2×Dic53—24no—
172C2×Dic53—24no—
173#36701—531nomaximal, Sylow
174C22×C53C22×#4111yescharacteristic
175C22×C53C2×2−1+412yes—
176C22×C53C2×2−1+411yescharacteristic
177C22×C53#58511yescharacteristic
178C22×C53#58512yes—
179C4×C53—24no—
180C4×C53—22no—
181C4×C53—44no—
182C4×C53—42no—
183C4×C53—44no—
184C4×C53—41no—
185C4×C53#56414yes—
186Dic53—44no—
187Dic53—44no—
188Dic53—44no—
189Dic53—44no—
190#2845—531no—
191#2839—534no—
192#2839—534no—
193#2839—534no—
194#2822—531no—
195#2822—532no—
196#2402—534no—
197#2402—534no—
198#2402—534no—
199C4×#125—532no—
200C4×#125D5311yescharacteristic
201C2×C53#346112yes—
202C2×C53#346111yescharacteristic
203C2×C53C2×#56411yescharacteristic
204C2×C53#353112yes—
205C2×C53#353111yescharacteristic
206C2×#125—532no—
207C2×#125C2×D5311yescharacteristic
208#400—534no—
209#400—534no—
210#400—534no—
211#400—534no—
212#400—534no—
213#400—534no—
214#400—534no—
215#400—534no—
216#400—534no—
217#400—534no—
218#400—534no—
219#400—534no—
220#398—534no—
221#398—534no—
222#398—534no—
223#398—534no—
224#398—534no—
225#398—534no—
226#398—534no—
227#398—534no—
228#398—534no—
229#398—534no—
230#398—534no—
231#398—534no—
232#389—532no—
233#389—532no—
234#389C2×D5312yes—
235#387—534no—
236#387—534no—
237#387C2×D5314yes—
238#382—532no—
239#382C2×D5311yescharacteristic
240C4×C4⋊C4—532no—
241C4×C4⋊C4—532no—
242C4×C4⋊C4—534no—
243C4×C4⋊C4—534no—
244C4×C4⋊C4—534no—
245C4×C4⋊C4—534no—
246C4×C4⋊C4—534no—
247C4×C4⋊C4C2×D5314yes—
248C4×C4⋊C4C2×D5312yes—
249#376—534no—
250C43C2×D5311yescharacteristic
251C53#3670111yescharacteristic, complemented, Sylow
252#125—24no—
253#125—1068no—
254C2×C4⋊C4—534no—
255C2×C4⋊C4—534no—
256C2×C4⋊C4—532no—
257C2×C4⋊C4—532no—
258C2×C4⋊C4—534no—
259C2×C4⋊C4—534no—
260C2×C4⋊C4—534no—
261C2×C4⋊C4—534no—
262C2×C4⋊C4—534no—
263C2×C4⋊C4—534no—
264C2×C4⋊C4—532no—
265C2×C4⋊C4—532no—
266C2×C4⋊C4—534no—
267C2×C4⋊C4—534no—
268C2×C4⋊C4C22×D5314yes—
269C2×C4⋊C4C22×D5312yes—
270C2×C4⋊C4C22×D5312yes—
271C2×C4⋊C4C22×D5314yes—
272C2×C42—534no—
273C2×C42—534no—
274C2×C42C22×D5312yes—
275C2×C42C22×D5311yescharacteristic
276C2×C42C22×D5314yes—
277C2×C42C22×D5311yescharacteristic
278C2×C42C22×D5312yes—
279C2×C42C22×D5311yescharacteristic
280#95—534no—
281#95—534no—
282#95—534no—
283#95—534no—
284#95—534no—
285#95—534no—
286#95—534no—
287#95—534no—
288#95—534no—
289#95—534no—
290#95—534no—
291#95—534no—
292#95—534no—
293#95—534no—
294#95—534no—
295#95—534no—
296#95C22×D5314yes—
297#95C22×D5314yes—
298#95C22×D5314yes—
299C22×C4—534no—
300C22×C4—534no—
301C22×C4—534no—
302C22×C4—534no—
303C22×C4C23×D5312yes—
304C22×C4C23×D5314yes—
305C22×C4C23×D5311yescharacteristic
306C22×C4C23×D5314yes—
307C22×C4C23×D5311yescharacteristic
308C22×C4C23×D5312yes—
309C22×C4C23×D5311yescharacteristic
310C4⋊C4—28no—
311C4⋊C4—24no—
312C4⋊C4—42no—
313C4⋊C4—44no—
314C4⋊C4—1068no—
315C4⋊C4—1068no—
316C4⋊C4—1064no—
317C4⋊C4—1064no—
318C4⋊C4—2124no—
319C4⋊C4—2124no—
320C4⋊C4—2124no—
321C4⋊C4—2124no—
322C4⋊C4—2124no—
323C4⋊C4—2124no—
324C4⋊C4—2122no—
325C4⋊C4—2122no—
326C4⋊C4—2124no—
327C4⋊C4—2124no—
328C42—24no—
329C42—28no—
330C42—22no—
331C42—41no—
332C42—42no—
333C42—1068no—
334C42—1068no—
335C42#1171994514yes—
336C23C24×D5311yescharacteristic
337C2×C4—22no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—22no—
341C2×C4—24no—
342C2×C4—22no—
343C2×C4—24no—
344C2×C4—24no—
345C2×C4—21no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—22no—
349C2×C4—24no—
350C2×C4—24no—
351C2×C4—24no—
352C2×C4—1064no—
353C2×C4—1064no—
354C2×C4—1064no—
355C2×C4—1064no—
356C2×C4—1064no—
357C2×C4—1064no—
358C2×C4—1064no—
359C2×C4—1064no—
360C2×C4—1064no—
361C2×C4—1064no—
362C2×C4—1064no—
363C2×C4—1064no—
364C2×C4C2×#1171994514yes—
365C2×C4C2×#1171994512yes—
366C2×C4#1313326414yes—
367C2×C4#1313326512yes—
368C22C22×#1171994511yescharacteristic
369C22C2×#1313326412yes—
370C22C2×#1313326511yescharacteristic
371C22#1681145911yescharacteristic
372C22#1681145912yes—
373C4—24no—
374C4—22no—
375C4—42no—
376C4—44no—
377C4—41no—
378C4—44no—
379C4—2124no—
380C4—2124no—
381C4—2124no—
382C4—2124no—
383C4C53⋊#56414yes—
384C2#2028814212yes—
385C2#2028814811yescharacteristic
386C2C2×C53⋊#56411yescharacteristic
387C2#2028886012yes—
388C2#2028886211yescharacteristic
subgroupG/H
centerC22×C4C23×D53
commutator, socle, prime radical, cosocleC23×C53C25
FrattiniC23C24×D53
FittingC4×#125×C53C2
hypercenterC4×#125D53
hypocenterC53#36701
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C22×C4 ◁ C4×#125

Complements

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

NH ≅ G/Ntype
251. C53#36701semidirect

Maximal subgroups

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

subgroupindex
#202829902
#202829572
#202829552
#202829532
#202827832
#202827822
#202780232
#202780222
C4×#131332092
C4×#125×C532
#3670153