Finite Group Databasebeta

← #27276970

Subgroups — #27276970

Order 13568.

462 conjugacy classes of subgroups.

The subgroup lattice has 386 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#13133225C211yesmaximal, characteristic
2#20273461C211yesmaximal, characteristic, complemented
3C2×#16810375C211yesmaximal, characteristic, complemented
4#20273103C211yesmaximal, characteristic
5#20273100C211yesmaximal, characteristic
6#20272566C211yesmaximal, characteristic, complemented
7C53×#1283C211yesmaximal, characteristic, complemented
8C2×#13133225C411yescharacteristic
9C2×#13133225C411yescharacteristic
10C4×#11719945C2211yescharacteristic
11#16810676C2211yescharacteristic
12C2×C4⋊#88322C2211yescharacteristic
13#16810581C411yescharacteristic
14#16810581C411yescharacteristic
15#16810580—21no—
16#16810580—21no—
17C4×#11719912—21no—
18C2×C4×#88322C2211yescharacteristic
19C4×C8×D53—21no—
20#16810375—21no—
21#16810375—21no—
22#16810375—21no—
23#16810373—21no—
24#16810355—21no—
25#16810353—21no—
26#16810352—21no—
27#16810350—21no—
28C53×#431C2211yescharacteristic
29C2×C4⋊C8×C53C2211yescharacteristic
30C4×M16×C53C2211yescharacteristic
31C2×#11719945C2×C411yescharacteristic
32#13133225—22no—
33#13133225—22no—
34#13133225—22no—
35#13133225C814yescomplemented
36C2×#11719912—21no—
37C2×#11719912—21no—
38C22×#88322—21no—
39C22×#88322C2×C411yescharacteristic
40C22×#88322C2×C411yescharacteristic
41C2×C8×D53—21no—
42C2×C8×D53—21no—
43C4×Dic106D411yescharacteristic
44C4×Dic106Q811yescharacteristic
45C4×D212D411yescharacteristic
46C4×D212Q811yescharacteristic
47C4×C22⋊D53—21no—
48C4×C22⋊D53—21no—
49#13133176C2×C411yescharacteristic
50#13133174—21no—
51C42×D53—21no—
52C4⋊#88322—21no—
53C4⋊#88322—21no—
54C4⋊#88322—21no—
55C22⋊#88322—21no—
56#13133127—21no—
57#13133127—21no—
58C4×#88322—21no—
59C4×#88322D411yescharacteristic
60C4×#88322D411yescharacteristic
61C4×#88322D411yescharacteristic
62C4×#88322D411yescharacteristic
63C8×Dic53—21no—
64C8×Dic53—21no—
65C2×M16×C53C2×C411yescharacteristic
66C2×M16×C53C2×C411yescharacteristic
67C22×C8×C53—21no—
68C2×C42×C53C2311yescharacteristic
69C4⋊C8×C53—21no—
70C4⋊C8×C53—21no—
71C4⋊C8×C53—21no—
72C22⋊C8×C53—21no—
73#97×C53—21no—
74C4×C8×C53—21no—
75C4×C8×C53—21no—
76C22×C4×C53C22×C411yescharacteristic
77C22×C4×C53#4111yescharacteristic
78C22×C4×C53#4111yescharacteristic
79C2×C22⋊D53—21no—
80#11719945—21no—
81#11719945—21no—
82#11719945C2×C812yes—
83#11719945M1612yes—
84C2×D212C4⋊C411yescharacteristic
85C2×C4×D53—21no—
86C2×C4×D53—21no—
87C2×C4×D53—21no—
88C2×Dic106C4⋊C411yescharacteristic
89M16×C53—21no—
90M16×C53—21no—
91M16×C53C2×C812yes—
92M16×C53M1612yes—
93C2×C8×C53—21no—
94C2×C8×C53—21no—
95C2×C8×C53—41no—
96C2×C8×C53—41no—
97C2×C8×C53—81no—
98C42×C53C2×D411yescharacteristic
99C42×C53C2×D411yescharacteristic
100C42×C53C2×D411yescharacteristic
101C42×C53C2×Q811yescharacteristic
102C22⋊Dic53—21no—
103#11719921—21no—
104#11719921—21no—
105C4⋊Dic53C4⋊C411yescharacteristic
106C4⋊Dic53C4⋊C411yescharacteristic
107#11719919—21no—
108#11719919—21no—
109C4×Dic53—21no—
110C4×Dic53—21no—
111C4×Dic53—21no—
112C2×#88322—21no—
113C2×#88322—21no—
114C2×#88322—41no—
115C2×#88322—41no—
116C2×#88322—81no—
117C2×#88322C22⋊C411yescharacteristic
118C2×#88322C22⋊C411yescharacteristic
119C2×#88322C22⋊C411yescharacteristic
120C2×#88322C22⋊C411yescharacteristic
121C2×#88322C2×C812yes—
122C2×#88322M1612yes—
123#11719912—24no—
124#11719912—24no—
125C8×D53—24no—
126C8×D53—24no—
127C23×C53#11711yescharacteristic
128C22×D53—21no—
129C2×C4×C53—21no—
130C2×C4×C53—21no—
131C2×C4×C53—41no—
132C2×C4×C53C2×C22⋊C411yescharacteristic
133C2×C4×C53C2×C4⋊C411yescharacteristic
134C2×C4×C53#12111yescharacteristic
135C2×C4×C53#12111yescharacteristic
136C2×C4×C53C22⋊Q811yescharacteristic
137C2×C4×C53C22⋊Q811yescharacteristic
138C2×C4×C53C22×C811yescharacteristic
139C2×C4×C53C2×M1611yescharacteristic
140C2×C4×C53C2×M1611yescharacteristic
141C2×C4×C53C2×M1611yescharacteristic
142C22⋊D53—22no—
143C22⋊D53—22no—
144C2×Dic53—21no—
145C2×Dic53—21no—
146C2×Dic53—21no—
147D212—21no—
148D212C4⋊C812yes—
149C4×D53—22no—
150C4×D53—22no—
151C4×D53—42no—
152C4×D53—41no—
153C4×D53—41no—
154Dic106—21no—
155Dic106C4⋊C812yes—
156C8×C53—22no—
157C8×C53—22no—
158C8×C53—81no—
159C8×C53—81no—
160#88322—21no—
161#88322—21no—
162#88322—81no—
163#88322—81no—
164#88322C22⋊C812yes—
165#88322C22⋊C812yes—
166#8605—531nomaximal, Sylow
167C22×C53—21no—
168C22×C53—21no—
169C22×C53#38011yescharacteristic
170C22×C53#43111yescharacteristic
171C22×C53#43211yescharacteristic
172C2×D53—22no—
173C2×D53—41no—
174C2×D53—41no—
175C4×C53—21no—
176C4×C53—21no—
177C4×C53—21no—
178C4×C53C2×C22⋊C811yescharacteristic
179C4×C53#40711yescharacteristic
180C4×C53C2×C4⋊C811yescharacteristic
181C4×C53#42311yescharacteristic
182C4×C53#47211yescharacteristic
183C4×C53#47311yescharacteristic
184Dic53—22no—
185Dic53—41no—
186Dic53—41no—
187C4×#131—531no—
188#1564—531no—
189C2×#335—531no—
190#1477—531no—
191#1477—531no—
192#1283—531no—
193#1283D5311yescharacteristic, complemented
194C2×C53—21no—
195C2×C53—21no—
196C2×C53#179711yescharacteristic
197C2×C53#181411yescharacteristic
198C2×C53#181611yescharacteristic
199D53—42no—
200C2×#131—531no—
201C2×#131—531no—
202C4×#41—531no—
203#431—531no—
204#431C2×D5311yescharacteristic
205C2×C4⋊C8—531no—
206C2×C4⋊C8C2×D5311yescharacteristic
207#405—531no—
208#405—531no—
209#405—1061no—
210#405—1061no—
211C4×M16—1061no—
212C4×M16C2×D5311yescharacteristic
213C2×C4×C8—531no—
214C2×C4×C8—1061no—
215#335—1061no—
216#335—1061no—
217#335—1061no—
218#334—1061no—
219Q8⋊C8—1061no—
220Q8⋊C8—1061no—
221D4⋊C8—1061no—
222D4⋊C8—1061no—
223C53#860511yescharacteristic, complemented, Sylow
224C2×#41—531no—
225#131—534no—
226#131—1062no—
227#131—1062no—
228#131—1062no—
229C2×M16—1061no—
230C2×M16—1061no—
231C2×M16C4×D5311yescharacteristic
232C2×M16C4×D5311yescharacteristic
233C22×C8—21no—
234C22×C8—531no—
235C22×C8—531no—
236C22×C8—1061no—
237C22×C8—1061no—
238C22×C8—1061no—
239C4×Q8—531no—
240C4×Q8—531no—
241C4×D4—531no—
242C4×D4—531no—
243C4×D4—1061no—
244C4×D4—1061no—
245#117—531no—
246#117—1061no—
247C2×C42—1061no—
248C2×C42C22×D5311yescharacteristic
249C4⋊C8—21no—
250C4⋊C8—21no—
251C4⋊C8—21no—
252C4⋊C8—1061no—
253C4⋊C8—1061no—
254C4⋊C8—1061no—
255C22⋊C8—21no—
256C22⋊C8—1061no—
257#97—21no—
258#97—1061no—
259#97—1061no—
260C4×C8—21no—
261C4×C8—21no—
262C4×C8—531no—
263C4×C8—531no—
264C4×C8—531no—
265C4×C8—531no—
266C4×C8—1061no—
267C4×C8—1061no—
268C4×C8—1061no—
269#41—532no—
270#41—532no—
271#41—1061no—
272#41—1061no—
273C2×Q8—531no—
274C2×D4—531no—
275C2×D4—1061no—
276C22×C4—1061no—
277C22×C4—1061no—
278C22×C4—1061no—
279C22×C4C2×C4×D5311yescharacteristic
280C22×C4#1171994511yescharacteristic
281C22×C4#1171994511yescharacteristic
282M16—21no—
283M16—21no—
284M16—1064no—
285M16—1064no—
286M16C8×D5312yes—
287M16#1171991212yes—
288C2×C8—21no—
289C2×C8—21no—
290C2×C8—41no—
291C2×C8—41no—
292C2×C8—81no—
293C2×C8—531no—
294C2×C8—531no—
295C2×C8—532no—
296C2×C8—532no—
297C2×C8—531no—
298C2×C8—531no—
299C2×C8—1064no—
300C2×C8—1064no—
301C2×C8—1061no—
302C2×C8—1061no—
303C2×C8—4241no—
304C2×C8—2121no—
305C2×C8—2121no—
306C4⋊C4—531no—
307C4⋊C4—531no—
308C4⋊C4—1061no—
309C4⋊C4—1061no—
310C22⋊C4—1061no—
311C22⋊C4—1061no—
312C22⋊C4—1061no—
313C42—1061no—
314C42—1061no—
315C42—1061no—
316C42D4×D5311yescharacteristic
317C42D4×D5311yescharacteristic
318C42D4×D5311yescharacteristic
319C42Q8×D5311yescharacteristic
320C23—1061no—
321C23#1313317411yescharacteristic
322Q8—532no—
323Q8—1061no—
324D4—532no—
325D4—1062no—
326D4—1062no—
327D4—1061no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—41no—
331C2×C4—1061no—
332C2×C4—1062no—
333C2×C4—1062no—
334C2×C4—1061no—
335C2×C4—1061no—
336C2×C4—2122no—
337C2×C4—2121no—
338C2×C4—2121no—
339C2×C4C22⋊C4×D5311yescharacteristic
340C2×C4C4⋊C4×D5311yescharacteristic
341C2×C4#1313319011yescharacteristic
342C2×C4#1313319011yescharacteristic
343C2×C4#1313319511yescharacteristic
344C2×C4#1313319511yescharacteristic
345C2×C4C2×C8×D5311yescharacteristic
346C2×C4M16×D5311yescharacteristic
347C2×C4M16×D5311yescharacteristic
348C2×C4C2×#1171991211yescharacteristic
349C8—22no—
350C8—22no—
351C8—81no—
352C8—81no—
353C8—532no—
354C8—532no—
355C8—1061no—
356C8—1061no—
357C8—4241no—
358C8—4241no—
359C22—21no—
360C22—21no—
361C22—1062no—
362C22—2121no—
363C22—2121no—
364C22#1681048211yescharacteristic
365C22#1681067411yescharacteristic
366C22#1681067811yescharacteristic
367C4—21no—
368C4—21no—
369C4—21no—
370C4—1062no—
371C4—2121no—
372C4—2121no—
373C4C22⋊C8×D5311yescharacteristic
374C4#1681058511yescharacteristic
375C4C4⋊C8×D5311yescharacteristic
376C4#1681064511yescharacteristic
377C4#1681089211yescharacteristic
378C4#1681089711yescharacteristic
379C2—21no—
380C2—21no—
381C2—2122no—
382C2#2027469411yescharacteristic
383C2#2027476811yescharacteristic
384C2#2027477811yescharacteristic
subgroupG/H
centerC2×C4#13133190
commutatorC2×C4×C53C22×C8
FrattiniC2×C42C22×D53
FittingC53×#1283C2
socleC22×C53#380
prime radicalC2×#11719945C2×C4
cosocleC2×C42×C53C23
hypercenter#1283D53
hypocenterC53#8605
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1283

Complements

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

NH ≅ G/Ntype
2. #20273461C2semidirect
3. C2×#16810375C2semidirect
6. #20272566C2semidirect
7. C53×#1283C2semidirect
35. #13133225C8semidirect
193. #1283D53semidirect
223. C53#8605semidirect

Maximal subgroups

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

subgroupindex
C4×#131332252
#202734612
C2×#168103752
#202731032
#202731002
#202725662
C53×#12832
#860553