Finite Group Databasebeta

← #27279336

Subgroups — #27279336

Order 13568.

432 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20283664C211yesmaximal, characteristic
2#20275660C211yesmaximal, characteristic
3#20273214C211yesmaximal, characteristic
4#20273118C211yesmaximal, characteristic
5#20272555C211yesmaximal, characteristic
6#20272552C211yesmaximal, characteristic
7C53×#3170C211yesmaximal, characteristic
8#16810754—21no—
9#16810754—22no—
10Q8×#88322—21no—
11#16810676C2211yescharacteristic
12#16810650—21no—
13C2×C4⋊#88322C2211yescharacteristic
14#16810579—21no—
15C4×#11719917C2211yescharacteristic
16C4×C4⋊Dic53C2211yescharacteristic
17#16810382—21no—
18#16810382—21no—
19#16810360—22no—
20Q8⋊#88322—21no—
21Q8⋊#88322—21no—
22C53×#533C2211yescharacteristic
23C53×#533C2211yescharacteristic
24C53×#500C411yescharacteristic, complemented
25C53×#494C411yescharacteristic, complemented
26C53×#487C411yescharacteristic, complemented
27C53×#487C411yescharacteristic, complemented
28C53×#467—22no—
29C4×Q16×C53—22no—
30C2×C4×C8×C53C2211yescharacteristic
31C2×#11719917D411yescharacteristic
32C2×#11719917D411yescharacteristic
33C22×#88322—21no—
34C2×C4⋊Dic53—21no—
35C2×C4×Dic53—21no—
36C4⋊#88322—22no—
37C4⋊#88322—22no—
38C4⋊#88322—21no—
39C4⋊#88322—21no—
40C4⋊#88322D411yescharacteristic
41C4⋊#88322D411yescharacteristic
42C22⋊#88322—21no—
43#13133128—22no—
44#13133128—21no—
45C4×#88322—22no—
46C4×#88322—21no—
47C4×#88322—21no—
48#13133124—21no—
49C2×Q16×C53—22no—
50C22×C8×C53—21no—
51C4⋊Q8×C53C2×C411yescharacteristic
52C4⋊Q8×C53C2×C411yescharacteristic
53C4⋊Q8×C53C2×C411yescharacteristic
54C4⋊Q8×C53C2×C411yescharacteristic
55#125×C53—21no—
56#125×C53—21no—
57C22⋊Q8×C53—21no—
58C22⋊Q8×C53—21no—
59C22⋊Q8×C53—21no—
60C22⋊Q8×C53D412yes—
61C4×Q8×C53—21no—
62C4×Q8×C53—21no—
63C4×Q8×C53—21no—
64C4×Q8×C53D412yes—
65#117×C53—21no—
66#117×C53—21no—
67C2×C42×C53C2311yescharacteristic
68#107×C53—22no—
69#103×C53—22no—
70#103×C53—22no—
71C4×C8×C53—21no—
72C4×C8×C53C2×C411yescharacteristic
73C4×C8×C53C2×C411yescharacteristic
74C2×Q8×C53—21no—
75C2×Q8×C53—21no—
76C2×Q8×C53—21no—
77C2×Q8×C53C22⋊C412yes—
78C22×C4×C53C2×D411yescharacteristic
79C22×C4×C53C2×D411yescharacteristic
80C22×C4×C53#4111yescharacteristic
81C22×Dic53—41no—
82Q16×C53—44no—
83C2×C8×C53—21no—
84C2×C8×C53—21no—
85C2×C8×C53—42no—
86C4⋊C4×C53—21no—
87C4⋊C4×C53—21no—
88C4⋊C4×C53—21no—
89C4⋊C4×C53—22no—
90C4⋊C4×C53—22no—
91C4⋊C4×C53—21no—
92C4⋊C4×C53—21no—
93C4⋊C4×C53—21no—
94C4⋊C4×C53—21no—
95C4⋊C4×C53—41no—
96C4⋊C4×C53—41no—
97C4⋊C4×C53C22⋊C412yes—
98C22⋊C4×C53—22no—
99C22⋊C4×C53—41no—
100C42×C53—22no—
101C42×C53—41no—
102C42×C53C22×C411yescharacteristic
103C42×C53C2×D411yescharacteristic
104C42×C53C2×D411yescharacteristic
105C42×C53#4111yescharacteristic
106C4⋊Dic53—22no—
107C4⋊Dic53—41no—
108C4×Dic53—41no—
109C4×Dic53—41no—
110#11719917—21no—
111#11719917—21no—
112#11719917—21no—
113#11719917Q1612yescomplemented
114C2×#88322—21no—
115C2×#88322—21no—
116C2×#88322—21no—
117C2×#88322—21no—
118C2×#88322—41no—
119C2×#88322—41no—
120C2×#88322—41no—
121C23×C53#12111yescharacteristic
122Q8×C53—42no—
123Q8×C53—42no—
124Q8×C53—41no—
125Q8×C53—41no—
126Q8×C53—41no—
127Q8×C53—41no—
128C2×C4×C53—21no—
129C2×C4×C53—21no—
130C2×C4×C53—22no—
131C2×C4×C53—22no—
132C2×C4×C53—41no—
133C2×C4×C53—41no—
134C2×C4×C53—41no—
135C2×C4×C53C2×C22⋊C411yescharacteristic
136C2×C4×C53C4×D411yescharacteristic
137C2×C4×C53C4×D411yescharacteristic
138C2×C4×C53#12011yescharacteristic
139C2×C4×C53#12111yescharacteristic
140C2×C4×C53#12311yescharacteristic
141C2×C4×C53C2×Q1611yescharacteristic
142C2×C4×C53#13511yescharacteristic
143C2×C4×C53#13711yescharacteristic
144C2×C4×C53#13711yescharacteristic
145C2×Dic53—42no—
146C2×Dic53—81no—
147C2×Dic53—81no—
148C8×C53—22no—
149C8×C53—41no—
150#88322—22no—
151#88322—41no—
152#88322—41no—
153#88322—41no—
154#88322—81no—
155#9056—531nomaximal, Sylow
156C22×C53—21no—
157C22×C53—21no—
158C22×C53#38611yescharacteristic
159C22×C53#46211yescharacteristic
160C22×C53#46411yescharacteristic
161C4×C53—21no—
162C4×C53—21no—
163C4×C53—21no—
164C4×C53—42no—
165C4×C53—42no—
166C4×C53—81no—
167C4×C53—81no—
168C4×C53C4×Q1611yescharacteristic
169C4×C53#44111yescharacteristic
170C4×C53#44311yescharacteristic
171C4×C53#44411yescharacteristic
172C4×C53#45111yescharacteristic
173C4×C53#45211yescharacteristic
174Dic53—82no—
175#3170D5311yescharacteristic
176#2962—531no—
177#1986—531no—
178#1503—531no—
179#1481—531no—
180#1279—531no—
181#1278—531no—
182C2×C53—21no—
183C2×C53—21no—
184C2×C53#187611yescharacteristic
185C2×C53#187811yescharacteristic
186C2×C53#188411yescharacteristic
187#533C2×D5311yescharacteristic
188#533C2×D5311yescharacteristic
189#500Dic5311yescharacteristic, complemented
190#494Dic5311yescharacteristic, complemented
191#487Dic5311yescharacteristic, complemented
192#487Dic5311yescharacteristic, complemented
193#467—22no—
194#446—1061no—
195#446—1062no—
196C8×Q8—1061no—
197C4×Q16—22no—
198#431—531no—
199#424—1061no—
200C2×C4⋊C8—531no—
201#405—1061no—
202C4×M16—531no—
203C2×C4×C8C2×D5311yescharacteristic
204C4×C4⋊C4—531no—
205#339—1061no—
206#339—1061no—
207#328—1062no—
208Q8⋊C8—1061no—
209Q8⋊C8—1061no—
210C53#905611yescharacteristic, complemented, Sylow
211C2×Q16—22no—
212C2×M16—531no—
213C2×M16—531no—
214C22×C8—21no—
215C22×C8—1061no—
216C4⋊Q8C2×Dic5311yescharacteristic
217C4⋊Q8C2×Dic5311yescharacteristic
218C4⋊Q8C2×Dic5311yescharacteristic
219C4⋊Q8C2×Dic5311yescharacteristic
220#125—21no—
221#125—21no—
222C22⋊Q8—21no—
223C22⋊Q8—21no—
224C22⋊Q8—21no—
225C22⋊Q8C22⋊D5312yes—
226C4×Q8—21no—
227C4×Q8—21no—
228C4×Q8—21no—
229C4×Q8C22⋊D5312yes—
230#117—21no—
231#117—21no—
232C2×C4⋊C4—1061no—
233C2×C42—1061no—
234C2×C42C22×D5311yescharacteristic
235#107—22no—
236C4⋊C8—531no—
237C4⋊C8—531no—
238C4⋊C8—1062no—
239C4⋊C8—1062no—
240C4⋊C8—1061no—
241C4⋊C8—1061no—
242#103—22no—
243#103—22no—
244C22⋊C8—1061no—
245#97—1062no—
246#97—1061no—
247C4×C8—21no—
248C4×C8—1062no—
249C4×C8—1061no—
250C4×C8—1061no—
251C4×C8C2×Dic5311yescharacteristic
252C4×C8C2×Dic5311yescharacteristic
253#95—1061no—
254C2×Q8—21no—
255C2×Q8—21no—
256C2×Q8—21no—
257C2×Q8C22⋊Dic5312yes—
258C22×C4—2121no—
259C22×C4D4×D5311yescharacteristic
260C22×C4#1171994711yescharacteristic
261C22×C4C2×C22⋊D5311yescharacteristic
262Q16—44no—
263M16—532no—
264M16—1061no—
265M16—1061no—
266M16—1061no—
267C2×C8—21no—
268C2×C8—21no—
269C2×C8—42no—
270C2×C8—1061no—
271C2×C8—1061no—
272C2×C8—1061no—
273C2×C8—1061no—
274C2×C8—2121no—
275C2×C8—2121no—
276C2×C8—2121no—
277C4⋊C4—21no—
278C4⋊C4—21no—
279C4⋊C4—21no—
280C4⋊C4—22no—
281C4⋊C4—22no—
282C4⋊C4—21no—
283C4⋊C4—21no—
284C4⋊C4—21no—
285C4⋊C4—21no—
286C4⋊C4—41no—
287C4⋊C4—41no—
288C4⋊C4—1062no—
289C4⋊C4—2121no—
290C4⋊C4C22⋊Dic5312yes—
291C22⋊C4—22no—
292C22⋊C4—41no—
293C42—22no—
294C42—41no—
295C42—2121no—
296C42—2121no—
297C42D4×D5311yescharacteristic
298C42#1171994711yescharacteristic
299C42C22×Dic5311yescharacteristic
300C42C2×C22⋊D5311yescharacteristic
301C23#1313318911yescharacteristic
302Q8—42no—
303Q8—42no—
304Q8—41no—
305Q8—41no—
306Q8—41no—
307Q8—41no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—22no—
311C2×C4—22no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—4241no—
316C2×C4—4241no—
317C2×C4—2122no—
318C2×C4C2×C22⋊Dic5311yescharacteristic
319C2×C4D4×Dic5311yescharacteristic
320C2×C4D4×Dic5311yescharacteristic
321C2×C4#1313318511yescharacteristic
322C2×C4#1313318911yescharacteristic
323C2×C4#1313319811yescharacteristic
324C2×C4Q16×D5311yescharacteristic
325C2×C4#1313324011yescharacteristic
326C2×C4#1313324911yescharacteristic
327C2×C4#1313324911yescharacteristic
328C8—22no—
329C8—41no—
330C8—1062no—
331C8—4241no—
332C8—2121no—
333C8—2121no—
334C8—2121no—
335C22—21no—
336C22—21no—
337C22#1681050311yescharacteristic
338C22#1681083111yescharacteristic
339C22#1681084711yescharacteristic
340C4—21no—
341C4—21no—
342C4—21no—
343C4—42no—
344C4—42no—
345C4—81no—
346C4—81no—
347C4—4242no—
348C4Q16×Dic5311yescharacteristic
349C4#1681073011yescharacteristic
350C4#1681074011yescharacteristic
351C4#1681074511yescharacteristic
352C4#1681077711yescharacteristic
353C4#1681078411yescharacteristic
354C2—21no—
355C2—21no—
356C2#2027504411yescharacteristic
357C2#2027505811yescharacteristic
358C2#2027509911yescharacteristic
subgroupG/H
centerC2×C4#13133185
commutatorC42×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#3170C2
socleC22×C53#386
prime radicalC23×C53#121
cosocleC2×C42×C53C23
hypercenter#3170D53
hypocenterC53#9056
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #3170

Complements

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

NH ≅ G/Ntype
24. C53×#500C4semidirect
25. C53×#494C4semidirect
26. C53×#487C4semidirect
27. C53×#487C4semidirect
113. #11719917Q16semidirect
189. #500Dic53semidirect
190. #494Dic53semidirect
191. #487Dic53semidirect
192. #487Dic53semidirect
210. C53#9056semidirect

Maximal subgroups

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

subgroupindex
#202836642
#202756602
#202732142
#202731182
#202725552
#202725522
C53×#31702
#905653