Finite Group Databasebeta

← #27384602

Subgroups — #27384602

Order 13568.

854 conjugacy classes of subgroups.

The subgroup lattice has 402 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810685C211yesmaximal, characteristic
2C2×#16810682C211yesmaximal, characteristic
3#20277054C211yesmaximal, characteristic
4#20274868C212yesmaximal, complemented
5#20274830C212yesmaximal, complemented
6#20274698C212yesmaximal
7#20274692C212yesmaximal
8#20274572C211yesmaximal, characteristic, complemented
9#20274569C211yesmaximal, characteristic, complemented
10C4×#13133146C211yesmaximal, characteristic, complemented
11C53×#1768C211yesmaximal, characteristic, complemented
12C2×#13133174C2211yescharacteristic
13C2×#13133174C2211yescharacteristic
14C2×C42×D53C2211yescharacteristic
15#16810685—24no—
16#16810682—24no—
17C2×C4⋊#88322C2212yes—
18C2×#13133146C2212yes—
19C2×#13133146C2212yes—
20C2×#13133146C2212yes—
21C2×#13133146C2212yes—
22C2×#13133129C2212yes—
23C2×#13133129C2212yes—
24C2×#13133128C2211yescharacteristic
25C2×C4×#88322C2211yescharacteristic
26#16810474C412yes—
27#16810474C412yes—
28#16810470C412yes—
29#16810470C412yes—
30#16810469C2211yescharacteristic
31C42×Dic53C2211yescharacteristic
32C42×Dic53C2211yescharacteristic
33#16810378C2212yes—
34#16810377C2212yes—
35#16810377C2212yes—
36C2×C4⋊C8×C53C2212yes—
37C2×#97×C53C2211yescharacteristic
38C2×C4×C8×C53C2211yescharacteristic
39C53×#376C2211yescharacteristic
40C53×#336C2212yes—
41C22×C4×D53C2×C412yes—
42C22×C4×D53C2311yescharacteristic
43C22×#88322C2312yes—
44C22×#88322C2312yes—
45#13133174—24no—
46#13133174D414yes—
47#13133174Q814yes—
48C2×#11719919C2×C411yescharacteristic
49C2×#11719919C2×C411yescharacteristic
50C2×#11719919C2×C412yes—
51C2×#11719921C2×C411yescharacteristic
52C2×#11719921C2×C411yescharacteristic
53C2×#11719921C2×C412yes—
54C2×C4×Dic53C2×C412yes—
55C2×C4×Dic53C2×C412yes—
56C2×C4×Dic53C2×C412yes—
57C2×C4×Dic53C2×C412yes—
58C2×C4×Dic53C2×C412yes—
59C2×C4×Dic53C2311yescharacteristic
60C2×C4×Dic53C2311yescharacteristic
61C2×C4×Dic53C2311yescharacteristic
62C42×D53—24no—
63C4⋊#88322—24no—
64#13133146—24no—
65#13133146—24no—
66#13133146—24no—
67#13133146—24no—
68#13133129—24no—
69#13133129—24no—
70#13133128—22no—
71C4×#88322—22no—
72#13133124C2×C412yes—
73#13133124C2×C412yes—
74C22×C8×C53C2312yes—
75C22×C8×C53C2312yes—
76C2×C42×C53C2311yescharacteristic
77C2×C42×C53C2311yescharacteristic
78C2×C42×C53C2311yescharacteristic
79C4⋊C8×C53—24no—
80#97×C53—22no—
81C4×C8×C53—22no—
82#95×C53C2×C412yes—
83#95×C53C2×C412yes—
84C23×D53C22×C411yescharacteristic
85C22×C4×C53C22×C411yescharacteristic
86C22×C4×C53C22×C411yescharacteristic
87C22×C4×C53C22×C412yes—
88C22×C4×C53C22×C412yes—
89C22×C4×C53C2411yescharacteristic
90C22×Dic53C22×C412yes—
91C22×Dic53C22×C411yescharacteristic
92C22×Dic53C22×C411yescharacteristic
93C22×Dic53C22×C411yescharacteristic
94C22×Dic53C22×C412yes—
95C2×C4×D53—22no—
96C2×C4×D53—22no—
97C2×C4×D53—24no—
98C2×C4×D53—24no—
99C2×C4×D53—42no—
100C2×C4×D53C2×D412yes—
101C2×C4×D53C2×Q812yes—
102C2×C8×C53—22no—
103C2×C8×C53—22no—
104C2×C8×C53—22no—
105C2×C8×C53—22no—
106C2×C8×C53—22no—
107C2×C8×C53—22no—
108C42×C53—22no—
109C42×C53C2×D412yes—
110C42×C53C2×Q812yes—
111C42×C53#4114yes—
112#11719921—22no—
113#11719921—22no—
114#11719921C4⋊C418yes—
115#11719919—22no—
116#11719919—22no—
117#11719919C4⋊C418yes—
118C4×Dic53—24no—
119C4×Dic53—22no—
120C4×Dic53—24no—
121C4×Dic53—24no—
122C4×Dic53—42no—
123C4×Dic53—42no—
124C4×Dic53C2×D412yes—
125C4×Dic53C2×Q812yes—
126C4×Dic53#4114yes—
127C2×#88322—22no—
128C2×#88322—22no—
129C2×#88322—22no—
130C2×#88322—22no—
131C2×#88322—22no—
132C2×#88322—22no—
133C23×C53C23×C411yescharacteristic
134C22×D53—21no—
135C22×D53—21no—
136C22×D53—22no—
137C22×D53—22no—
138C22×D53C2×C4⋊C412yes—
139C2×C4×C53—22no—
140C2×C4×C53—22no—
141C2×C4×C53—22no—
142C2×C4×C53—22no—
143C2×C4×C53—21no—
144C2×C4×C53—21no—
145C2×C4×C53—22no—
146C2×C4×C53—22no—
147C2×C4×C53C2×C4⋊C414yes—
148C2×C4×C53#11714yes—
149C2×C4×C53C22×D411yescharacteristic
150C2×C4×C53C22×Q811yescharacteristic
151C2×C4×C53C2×#4112yes—
152C2×C4×C53C2×#4112yes—
153C2×Dic53—22no—
154C2×Dic53—21no—
155C2×Dic53—21no—
156C2×Dic53—22no—
157C2×Dic53—22no—
158C2×Dic53—22no—
159C2×Dic53—22no—
160C2×Dic53—22no—
161C2×Dic53C2×C4⋊C412yes—
162C2×Dic53C2×C4⋊C414yes—
163C2×Dic53#11712yes—
164C2×Dic53#11712yes—
165C2×Dic53#13114yes—
166C4×D53—24no—
167C4×D53—24no—
168C4×D53—48no—
169C8×C53—42no—
170C8×C53—42no—
171#88322—42no—
172#88322—42no—
173#20813—531nomaximal, Sylow
174C22×C53C22×C4⋊C411yescharacteristic
175C22×C53C2×#11711yescharacteristic
176C22×C53C2×#11711yescharacteristic
177C22×C53C2×#13112yes—
178C22×C53#56812yes—
179C2×D53—24no—
180C2×D53—22no—
181C2×D53—41no—
182C2×D53—41no—
183C2×D53—42no—
184C4×C53—24no—
185C4×C53—24no—
186C4×C53—41no—
187C4×C53—41no—
188C4×C53#53212yes—
189C4×C53#53312yes—
190Dic53—24no—
191Dic53—22no—
192Dic53—22no—
193Dic53—24no—
194Dic53—22no—
195C2×#433—531no—
196C2×#433—531no—
197C4×#117—531no—
198#1843—532no—
199#1831—532no—
200#1798—532no—
201#1797—532no—
202#1768—531no—
203#1768D5311yescharacteristic, complemented
204#1767—531no—
205C4×C4⋊C8—531no—
206C2×C53#228511yescharacteristic
207C2×C53#290612yes—
208C2×C53#292211yescharacteristic
209C2×C53#292311yescharacteristic
210C2×C53#292412yes—
211D53—42no—
212C2×#117—531no—
213C2×#117—531no—
214C22×C42—531no—
215#433—1064no—
216#433—1064no—
217C2×C4⋊C8—532no—
218C2×C4⋊C8—532no—
219C2×C4⋊C8—532no—
220C2×C4⋊C8—532no—
221C2×C4⋊C8—532no—
222C2×C4⋊C8C2×D5312yes—
223C2×C22⋊C8—532no—
224C2×C22⋊C8—532no—
225C2×#97—531no—
226C2×#97C2×D5311yescharacteristic
227C2×C4×C8—531no—
228C2×C4×C8C2×D5311yescharacteristic
229C4×C4⋊C4—532no—
230C4×C4⋊C4—532no—
231C4×C22⋊C4—532no—
232C4×C22⋊C4—532no—
233#376—531no—
234#376C2×D5311yescharacteristic
235C43—531no—
236C43—531no—
237#336—532no—
238#336—532no—
239#336—532no—
240#336C2×D5312yes—
241C53#2081311yescharacteristic, complemented, Sylow
242C23×C4—531no—
243C23×C4—532no—
244C22×C8—532no—
245C22×C8—532no—
246C22×C8C22×D5312yes—
247C22×C8C22×D5312yes—
248#117—534no—
249#117—534no—
250#117—1064no—
251C2×C4⋊C4—532no—
252C2×C4⋊C4—531no—
253C2×C4⋊C4—531no—
254C2×C22⋊C4—532no—
255C2×C22⋊C4—531no—
256C2×C22⋊C4—531no—
257C2×C42—531no—
258C2×C42—531no—
259C2×C42—532no—
260C2×C42—532no—
261C2×C42—532no—
262C2×C42—531no—
263C2×C42—532no—
264C2×C42—532no—
265C2×C42—1064no—
266C2×C42C22×D5311yescharacteristic
267C2×C42C22×D5311yescharacteristic
268C2×C42C22×D5311yescharacteristic
269C4⋊C8—24no—
270C4⋊C8—1064no—
271C4⋊C8—1064no—
272C4⋊C8—1064no—
273C4⋊C8—1064no—
274C4⋊C8—1064no—
275C22⋊C8—1064no—
276C22⋊C8—1064no—
277#97—22no—
278#97—1062no—
279C4×C8—22no—
280C4×C8—1062no—
281#95—532no—
282#95—532no—
283#95C4×D5312yes—
284#95C4×D5312yes—
285C24—531no—
286C22×C4—532no—
287C22×C4—531no—
288C22×C4—531no—
289C22×C4—532no—
290C22×C4—531no—
291C22×C4—532no—
292C22×C4—532no—
293C22×C4—1064no—
294C22×C4—1062no—
295C22×C4—1062no—
296C22×C4—1064no—
297C22×C4—2122no—
298C22×C4C2×C4×D5311yescharacteristic
299C22×C4C2×C4×D5311yescharacteristic
300C22×C4C2×C4×D5312yes—
301C22×C4C2×C4×D5312yes—
302C22×C4C23×D5311yescharacteristic
303C2×C8—22no—
304C2×C8—22no—
305C2×C8—22no—
306C2×C8—22no—
307C2×C8—22no—
308C2×C8—22no—
309C2×C8—1062no—
310C2×C8—1062no—
311C2×C8—1062no—
312C2×C8—1062no—
313C2×C8—1062no—
314C2×C8—1062no—
315C4⋊C4—538no—
316C4⋊C4—1062no—
317C4⋊C4—1062no—
318C22⋊C4—538no—
319C22⋊C4—1062no—
320C22⋊C4—1062no—
321C42—22no—
322C42—532no—
323C42—532no—
324C42—534no—
325C42—1064no—
326C42—1064no—
327C42—1064no—
328C42—1062no—
329C42—2122no—
330C42—2122no—
331C42#1171994514yes—
332C42D4×D5312yes—
333C42Q8×D5312yes—
334C23—532no—
335C23—1061no—
336C23—1061no—
337C23—1062no—
338C23—1062no—
339C23C22×C4×D5311yescharacteristic
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—22no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—534no—
349C2×C4—532no—
350C2×C4—532no—
351C2×C4—532no—
352C2×C4—534no—
353C2×C4—1061no—
354C2×C4—1061no—
355C2×C4—1062no—
356C2×C4—1062no—
357C2×C4—1062no—
358C2×C4—1062no—
359C2×C4—1062no—
360C2×C4—1062no—
361C2×C4—1064no—
362C2×C4—1064no—
363C2×C4—2128no—
364C2×C4C4⋊C4×D5314yes—
365C2×C4#1313317414yes—
366C2×C4C2×D4×D5311yescharacteristic
367C2×C4C2×Q8×D5311yescharacteristic
368C2×C4#41×D5312yes—
369C2×C4C2×#1171994512yes—
370C8—42no—
371C8—42no—
372C8—2122no—
373C8—2122no—
374C22—1062no—
375C22—1064no—
376C22—2121no—
377C22—2121no—
378C22—2122no—
379C22C2×C4⋊C4×D5311yescharacteristic
380C22#117×D5311yescharacteristic
381C22C2×#1313317411yescharacteristic
382C22#131×D5312yes—
383C22#1681138812yes—
384C4—24no—
385C4—24no—
386C4—41no—
387C4—41no—
388C4—1062no—
389C4—1064no—
390C4—1062no—
391C4—1062no—
392C4—1064no—
393C4#1681117012yes—
394C4#1681117712yes—
395C2—2122no—
396C2#2027711911yescharacteristic
397C2#2028328212yes—
398C2#2028335611yescharacteristic
399C2#2028336311yescharacteristic
400C2#2028337012yes—
subgroupG/H
center, FrattiniC22×C4C23×D53
commutator, socleC23×C53C23×C4
FittingC53×#1768C2
prime radicalC23×D53C22×C4
cosocleC22×C4×C53C24
hypercenter#1768D53
hypocenterC53#20813
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C22×C4 ◁ #1768

Complements

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

NH ≅ G/Ntype
4. #20274868C2semidirect
5. #20274830C2semidirect
8. #20274572C2semidirect
9. #20274569C2semidirect
10. C4×#13133146C2semidirect
11. C53×#1768C2semidirect
203. #1768D53semidirect
241. C53#20813semidirect

Maximal subgroups

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

subgroupindex
C2×#168106852
C2×#168106822
#202770542
#202748682
#202748302
#202746982
#202746922
#202745722
#202745692
C4×#131331462
C53×#17682
#2081353