Finite Group Databasebeta

← #27279551

Subgroups — #27279551

Order 13568.

488 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20283798C211yesmaximal, characteristic, complemented
2C2×#16810725C211yesmaximal, characteristic
3#20273111C211yesmaximal, characteristic, complemented
4#20273105C211yesmaximal, characteristic
5#20273035C211yesmaximal, characteristic
6#20272546C211yesmaximal, characteristic, complemented
7C53×#1281C211yesmaximal, characteristic, complemented
8C22×#11719913C411yescharacteristic
9C2×C4×Dic106C2211yescharacteristic
10C2×C4×D212C2211yescharacteristic
11#16810753—22no—
12#16810750—22no—
13#16810725—22no—
14#16810725—22no—
15#16810677—22no—
16#16810676C2211yescharacteristic
17#16810676C2211yescharacteristic
18C2×#13133151C411yescharacteristic
19C2×#13133142C411yescharacteristic
20C2×#13133139C411yescharacteristic
21#16810355—22no—
22#16810350—22no—
23#16810346—22no—
24C53×#431C2211yescharacteristic
25C53×#431C2211yescharacteristic
26C2×#97×C53C2211yescharacteristic
27C22×Dic106C2×C411yescharacteristic
28C22×D212C2×C411yescharacteristic
29C22×C4×D53—21no—
30C2×#11719913—22no—
31C2×#11719913—22no—
32C2×#11719913—22no—
33C4×Dic106—21no—
34C4×Dic106—21no—
35C4×Dic106D412yes—
36C4×Dic106D412yes—
37C4×D212—21no—
38C4×D212—21no—
39C4×D212—21no—
40C4×D212—21no—
41C2×C4⋊Dic53C2×C411yescharacteristic
42C2×C4⋊Dic53C2×C411yescharacteristic
43C2×#11719919—21no—
44C2×#11719921—21no—
45C2×C4×Dic53—21no—
46#13133151—22no—
47C4⋊#88322—21no—
48C4⋊#88322—21no—
49#13133146—24no—
50#13133142—22no—
51#13133139—22no—
52C22⋊#88322—21no—
53C22⋊#88322—21no—
54#13133127—24no—
55C4×#88322—21no—
56C4×#88322D412yes—
57C22×C8×C53C2×C411yescharacteristic
58C22×C8×C53C2×C411yescharacteristic
59C2×C42×C53C2311yescharacteristic
60C4⋊C8×C53—21no—
61C4⋊C8×C53—21no—
62C22⋊C8×C53—21no—
63C22⋊C8×C53D412yes—
64#97×C53—21no—
65#97×C53—21no—
66C4×C8×C53—21no—
67C4×C8×C53—21no—
68C23×D53—21no—
69C22×C4×C53C22×C411yescharacteristic
70C22×C4×C53C2×D411yescharacteristic
71C22×C4×C53#4111yescharacteristic
72C22×Dic53—21no—
73C22×Dic53—21no—
74C22×Dic53—21no—
75C2×D212—21no—
76C2×D212—21no—
77C2×D212—21no—
78C2×D212—21no—
79C2×D212—21no—
80C2×D212M1612yes—
81C2×C4×D53—41no—
82C2×C4×D53—41no—
83C2×C4×D53—82no—
84C2×Dic106—21no—
85C2×Dic106—21no—
86C2×Dic106—21no—
87C2×Dic106—21no—
88C2×Dic106—21no—
89C2×Dic106C22⋊C412yes—
90C2×C8×C53—21no—
91C2×C8×C53—21no—
92C2×C8×C53—21no—
93C2×C8×C53—22no—
94C2×C8×C53—21no—
95C2×C8×C53—21no—
96C2×C8×C53—41no—
97C2×C8×C53M1612yes—
98C42×C53C2×D411yescharacteristic
99C42×C53C2×D411yescharacteristic
100C42×C53C2×D411yescharacteristic
101C42×C53#4111yescharacteristic
102#11719921—41no—
103#11719921—41no—
104C4⋊Dic53—21no—
105C4⋊Dic53—21no—
106C4⋊Dic53—21no—
107C4⋊Dic53C22⋊C412yes—
108#11719919—22no—
109#11719919—22no—
110C4×Dic53—22no—
111C4×Dic53—22no—
112C2×#88322—22no—
113C2×#88322—41no—
114#11719913—28no—
115C23×C53C4×D411yescharacteristic
116C22×D53—22no—
117C22×D53—41no—
118C22×D53—41no—
119C22×D53—81no—
120C22×D53—81no—
121C2×C4×C53—41no—
122C2×C4×C53—41no—
123C2×C4×C53C2×C22⋊C411yescharacteristic
124C2×C4×C53C4×D411yescharacteristic
125C2×C4×C53#12011yescharacteristic
126C2×C4×C53#12111yescharacteristic
127C2×C4×C53#12111yescharacteristic
128C2×C4×C53#12311yescharacteristic
129C2×C4×C53C2×M1611yescharacteristic
130C2×C4×C53C2×M1611yescharacteristic
131C2×C4×C53#13111yescharacteristic
132C2×C4×C53#13111yescharacteristic
133C2×Dic53—22no—
134C2×Dic53—22no—
135C2×Dic53—41no—
136C2×Dic53—41no—
137C2×Dic53—41no—
138C2×Dic53—81no—
139C2×Dic53—81no—
140D212—22no—
141D212—22no—
142D212—41no—
143D212—41no—
144C4×D53—82no—
145C4×D53—82no—
146Dic106—22no—
147Dic106—22no—
148Dic106—41no—
149Dic106—41no—
150C8×C53—22no—
151C8×C53—42no—
152C8×C53—41no—
153C8×C53—81no—
154#88322—42no—
155#88322—81no—
156#9094—531nomaximal, Sylow
157C22×C53—21no—
158C22×C53—21no—
159C22×C53#38611yescharacteristic
160C22×C53#43511yescharacteristic
161C22×C53#43611yescharacteristic
162C2×D53—42no—
163C2×D53—82no—
164C2×D53—81no—
165C2×D53—81no—
166C2×D53—82no—
167C4×C53—21no—
168C4×C53—21no—
169C4×C53—21no—
170C4×C53—21no—
171C4×C53#40711yescharacteristic
172C4×C53#40811yescharacteristic
173C4×C53#43511yescharacteristic
174C4×C53#43611yescharacteristic
175Dic53—42no—
176Dic53—42no—
177Dic53—81no—
178#2983—531no—
179C2×#440—531no—
180#1479—531no—
181#1478—531no—
182#1460—531no—
183#1281D5311yescharacteristic, complemented
184#1275—531no—
185C2×C53—21no—
186C2×C53—21no—
187C2×C53#187011yescharacteristic
188C2×C53#190311yescharacteristic
189C2×C53#190511yescharacteristic
190D53—82no—
191C22×SD16—531no—
192C2×C4×Q8—531no—
193C2×C4×D4—531no—
194#446—1062no—
195#446—1062no—
196#440—1062no—
197#440—1062no—
198#432—1062no—
199#431—531no—
200#431—531no—
201#431C2×D5311yescharacteristic
202#431C2×D5311yescharacteristic
203C2×#107—531no—
204C2×#103—531no—
205C2×#102—531no—
206C2×#97C2×D5311yescharacteristic
207Q8⋊C8—1062no—
208D4⋊C8—1062no—
209#323—1062no—
210C53#909411yescharacteristic, complemented, Sylow
211C22×Q8—531no—
212C22×D4—531no—
213C23×C4—1061no—
214C2×SD16—1062no—
215C2×SD16—1062no—
216C2×SD16—1062no—
217C22×C8C4×D5311yescharacteristic
218C22×C8C4×D5311yescharacteristic
219C4×Q8—532no—
220C4×Q8—532no—
221C4×Q8—1061no—
222C4×Q8—1061no—
223C4×D4—1061no—
224C4×D4—1061no—
225C4×D4—1061no—
226C4×D4—1061no—
227C2×C4⋊C4—531no—
228C2×C4⋊C4—531no—
229C2×C4⋊C4—1061no—
230C2×C22⋊C4—1061no—
231C2×C42—1061no—
232C2×C42C22×D5311yescharacteristic
233#107—1062no—
234C4⋊C8—21no—
235C4⋊C8—21no—
236C4⋊C8—1064no—
237C4⋊C8—1061no—
238C4⋊C8—1061no—
239#103—1062no—
240#102—1062no—
241C22⋊C8—21no—
242C22⋊C8—1061no—
243C22⋊C8—1061no—
244C22⋊C8D21212yes—
245#97—21no—
246#97—21no—
247#97—1064no—
248C4×C8—21no—
249C4×C8—21no—
250C4×C8—532no—
251C4×C8—1061no—
252C24—1061no—
253C2×Q8—532no—
254C2×Q8—1061no—
255C2×Q8—1061no—
256C2×Q8—1061no—
257C2×Q8—1061no—
258C2×Q8—1061no—
259C2×D4—532no—
260C2×D4—1061no—
261C2×D4—1061no—
262C2×D4—1061no—
263C2×D4—1061no—
264C2×D4—1061no—
265C22×C4—1061no—
266C22×C4—1061no—
267C22×C4—1061no—
268C22×C4—4242no—
269C22×C4—2121no—
270C22×C4—2121no—
271C22×C4C2×C4×D5311yescharacteristic
272C22×C4C2×D21211yescharacteristic
273C22×C4#1171994511yescharacteristic
274SD16—1068no—
275C2×C8—21no—
276C2×C8—21no—
277C2×C8—21no—
278C2×C8—22no—
279C2×C8—21no—
280C2×C8—21no—
281C2×C8—41no—
282C2×C8—1062no—
283C2×C8—2121no—
284C2×C8#1171991212yes—
285C4⋊C4—532no—
286C4⋊C4—1061no—
287C4⋊C4—1062no—
288C4⋊C4—1062no—
289C4⋊C4—1061no—
290C4⋊C4—1061no—
291C22⋊C4—2121no—
292C22⋊C4—2121no—
293C42—1062no—
294C42—1062no—
295C42D4×D5311yescharacteristic
296C42D4×D5311yescharacteristic
297C42D4×D5311yescharacteristic
298C42#1171994911yescharacteristic
299C23—1062no—
300C23—4241no—
301C23—4241no—
302C23—2121no—
303C23—2121no—
304C23C4×D21211yescharacteristic
305Q8—1062no—
306Q8—1062no—
307Q8—2121no—
308Q8—2121no—
309D4—1062no—
310D4—1062no—
311D4—2121no—
312D4—2121no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—1062no—
316C2×C4—1062no—
317C2×C4—4242no—
318C2×C4—4242no—
319C2×C4—4241no—
320C2×C4—4241no—
321C2×C4—2121no—
322C2×C4—2121no—
323C2×C4—2121no—
324C2×C4C22⋊C4×D5311yescharacteristic
325C2×C4#1313317911yescharacteristic
326C2×C4#1313318711yescharacteristic
327C2×C4#1313319011yescharacteristic
328C2×C4#1313319211yescharacteristic
329C2×C4#1313320211yescharacteristic
330C2×C4M16×D5311yescharacteristic
331C2×C4C2×#1171991211yescharacteristic
332C2×C4#1313322311yescharacteristic
333C2×C4#1313322511yescharacteristic
334C8—22no—
335C8—42no—
336C8—41no—
337C8—81no—
338C8—4241no—
339C8—2122no—
340C22—21no—
341C22—21no—
342C22—4242no—
343C22—4242no—
344C22—4241no—
345C22—4241no—
346C22—2122no—
347C22#1681050111yescharacteristic
348C22#1681069611yescharacteristic
349C22#1681070211yescharacteristic
350C4—21no—
351C4—21no—
352C4—21no—
353C4—21no—
354C4—4241no—
355C4—2122no—
356C4—2122no—
357C4#1681058511yescharacteristic
358C4#1681058811yescharacteristic
359C4#1681069311yescharacteristic
360C4#1681070011yescharacteristic
361C2—21no—
362C2—21no—
363C2—4242no—
364C2#2027501111yescharacteristic
365C2#2027519411yescharacteristic
366C2#2027520811yescharacteristic
subgroupG/H
centerC22#16810501
commutatorC22×C4×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#1281C2
socleC22×C53#386
prime radicalC22×D212C2×C4
cosocleC2×C42×C53C23
hypercenter#1281D53
hypocenterC53#9094
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1281

Complements

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

NH ≅ G/Ntype
1. #20283798C2semidirect
3. #20273111C2semidirect
6. #20272546C2semidirect
7. C53×#1281C2semidirect
183. #1281D53semidirect
210. C53#9094semidirect

Maximal subgroups

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

subgroupindex
#202837982
C2×#168107252
#202731112
#202731052
#202730352
#202725462
C53×#12812
#909453