Finite Group Databasebeta

← #27277075

Subgroups — #27277075

Order 13568.

456 conjugacy classes of subgroups.

The subgroup lattice has 380 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20283409C211yesmaximal, characteristic
2#20274744C211yesmaximal, characteristic, complemented
3#20273259C212yesmaximal
4#20272622C212yesmaximal, complemented
5C53×#1810C211yesmaximal, characteristic, complemented
6C2×#13133223C411yescharacteristic, complemented
7#16811134C2211yescharacteristic
8#16810694—21no—
9#16810694—21no—
10C8×C22⋊D53—21no—
11C8×C22⋊D53—21no—
12C2×#13133151—21no—
13#16810649C411yescharacteristic, complemented
14#16810589C411yescharacteristic, complemented
15#16810587C2211yescharacteristic
16#16810578C411yescharacteristic, complemented
17#16810483C2212yes—
18#16810386—21no—
19#16810361—22no—
20#16810358—22no—
21C23×C8×C53C2211yescharacteristic
22C2×#108×C53—21no—
23C53×#407C2212yes—
24C53×#341—21no—
25C2×#11719945C2×C411yescharacteristic
26#13133223—42no—
27C2×#11719917C2×C411yescharacteristic
28C2×#11719917C2×C411yescharacteristic
29C2×#11719912—21no—
30C2×C8×D53—21no—
31#13133200—21no—
32#13133196D411yescharacteristic
33#13133196Q811yescharacteristic
34#13133191D411yescharacteristic
35#13133191Q811yescharacteristic
36#13133186—21no—
37C4×C22⋊D53—21no—
38C4×C22⋊D53—21no—
39#13133176C2×C411yescharacteristic
40C2×C4⋊Dic53—22no—
41C2×C22⋊Dic53—22no—
42#13133151—42no—
43#13133146—21no—
44#13133146—21no—
45C22⋊#88322—21no—
46C22⋊#88322—21no—
47#13133129—21no—
48#13133129—21no—
49#13133127—21no—
50C8×Dic53—21no—
51#13133124—22no—
52C23×C4×C53C2311yescharacteristic
53C2×M16×C53—22no—
54C22×C8×C53—42no—
55C22×C8×C53C2×C411yescharacteristic
56C22×C8×C53C2×C411yescharacteristic
57C22×C8×C53D411yescharacteristic
58C22×C8×C53D411yescharacteristic
59C22×C8×C53D411yescharacteristic
60C22×C8×C53D411yescharacteristic
61#108×C53—42no—
62C22⋊C8×C53—22no—
63C22⋊C8×C53—22no—
64C24×C53#4111yescharacteristic
65C22×C4×C53—41no—
66C22×C4×C53—41no—
67C22×C4×C53C22×C411yescharacteristic
68C22×C4×C53C2×D411yescharacteristic
69C22×C4×C53C2×D411yescharacteristic
70C22×C4×C53C2×D411yescharacteristic
71C22×C4×C53C2×Q811yescharacteristic
72C22×C4×C53#4111yescharacteristic
73C2×C22⋊D53—21no—
74C2×C22⋊D53—21no—
75C2×C22⋊D53—21no—
76C22×Dic53—42no—
77#11719945—41no—
78#11719945—41no—
79C2×D212C4⋊C411yescharacteristic
80C2×C4×D53—21no—
81C2×Dic106C4⋊C411yescharacteristic
82M16×C53—42no—
83M16×C53—42no—
84C2×C8×C53—22no—
85C2×C8×C53—22no—
86C2×C8×C53—22no—
87C2×C8×C53—41no—
88C2×C8×C53—41no—
89C2×C8×C53—41no—
90C2×C8×C53—42no—
91C2×C8×C53C22⋊C411yescharacteristic
92C2×C8×C53C22⋊C411yescharacteristic
93C2×C8×C53C22⋊C411yescharacteristic
94C2×C8×C53C22⋊C411yescharacteristic
95C22⋊Dic53—21no—
96C22⋊Dic53—21no—
97C22⋊Dic53—21no—
98C22⋊Dic53—82no—
99#11719921—21no—
100#11719921—21no—
101C4⋊Dic53—42no—
102C4⋊Dic53—42no—
103C4⋊Dic53C4⋊C411yescharacteristic
104C4⋊Dic53C4⋊C411yescharacteristic
105#11719919—21no—
106#11719919—21no—
107C4×Dic53—21no—
108#11719917—41no—
109#11719917—41no—
110C2×#88322—21no—
111C2×#88322—21no—
112#11719912—42no—
113C8×D53—42no—
114C23×C53—21no—
115C23×C53—21no—
116C23×C53—41no—
117C23×C53—41no—
118C23×C53#11711yescharacteristic
119C23×C53#12111yescharacteristic
120C23×C53C22⋊Q811yescharacteristic
121C22×D53—21no—
122C2×C4×C53—21no—
123C2×C4×C53—21no—
124C2×C4×C53—21no—
125C2×C4×C53—21no—
126C2×C4×C53—21no—
127C2×C4×C53—21no—
128C2×C4×C53—41no—
129C2×C4×C53—41no—
130C2×C4×C53—41no—
131C2×C4×C53C2×C22⋊C411yescharacteristic
132C2×C4×C53C2×C4⋊C411yescharacteristic
133C2×C4×C53#12111yescharacteristic
134C2×C4×C53C22⋊Q811yescharacteristic
135C22⋊D53—42no—
136C22⋊D53—42no—
137C22⋊D53—41no—
138C22⋊D53—41no—
139C2×Dic53—21no—
140C2×Dic53—21no—
141C2×Dic53—21no—
142C2×Dic53—82no—
143C2×Dic53—82no—
144C2×Dic53—82no—
145D212—41no—
146C4×D53—41no—
147C4×D53—41no—
148Dic106—41no—
149C8×C53—21no—
150C8×C53—21no—
151C8×C53—21no—
152C8×C53—82no—
153C8×C53#9912yes—
154#88322—41no—
155#88322—41no—
156#8628—531nomaximal, Sylow
157C22×C53—21no—
158C22×C53—21no—
159C22×C53—21no—
160C22×C53—21no—
161C22×C53—21no—
162C22×C53—21no—
163C22×C53—21no—
164C22×C53—21no—
165C22×C53—42no—
166C22×C53—42no—
167C22×C53#38011yescharacteristic
168C22×C53#47211yescharacteristic
169C22×C53#47311yescharacteristic
170C2×D53—41no—
171C2×D53—41no—
172C2×D53—41no—
173C4×C53—21no—
174C4×C53—21no—
175C4×C53—21no—
176C4×C53C2×#9911yescharacteristic
177C4×C53#41011yescharacteristic
178Dic53—41no—
179Dic53—41no—
180Dic53—41no—
181Dic53—162no—
182#2929—531no—
183#1810D5311yescharacteristic, complemented
184#1808—531no—
185#1513—532no—
186#1305—532no—
187C2×C53—21no—
188C2×C53—21no—
189C2×C53—41no—
190C2×C53—41no—
191C2×C53#179911yescharacteristic
192C2×C53#181411yescharacteristic
193C2×C53#181611yescharacteristic
194D53—81no—
195C2×#131—531no—
196C23×C8C2×D5311yescharacteristic
197#525—531no—
198#435—1061no—
199#435—1061no—
200C8×D4—1061no—
201C8×D4—1061no—
202C2×#108—21no—
203C2×#107—1061no—
204#424—531no—
205#408—531no—
206#407—531no—
207#407C2×D5312yes—
208#405—531no—
209#380—532no—
210#341—21no—
211#340—1061no—
212#328—1062no—
213#327—1062no—
214C53#862811yescharacteristic, complemented, Sylow
215C2×#41—531no—
216C23×C4C22×D5311yescharacteristic
217#131—2122no—
218C2×M16—22no—
219C2×M16—531no—
220C2×M16—531no—
221C2×M16—1061no—
222C22×C8—42no—
223C22×C8—1061no—
224C22×C8C4×D5311yescharacteristic
225C22×C8C4×D5311yescharacteristic
226C22×C8D21211yescharacteristic
227C22×C8D21211yescharacteristic
228C22×C8C22⋊D5311yescharacteristic
229C22×C8C22⋊D5311yescharacteristic
230#123—1061no—
231C22⋊Q8—531no—
232C22⋊Q8—531no—
233#121—531no—
234#121—531no—
235#120—1061no—
236C4×D4—1061no—
237C4×D4—1061no—
238#117—531no—
239C2×C4⋊C4—1062no—
240C2×C22⋊C4—1062no—
241#108—42no—
242#107—2122no—
243C4⋊C8—1061no—
244C4⋊C8—1061no—
245C22⋊C8—22no—
246C22⋊C8—22no—
247C22⋊C8—1061no—
248C22⋊C8—1061no—
249C22⋊C8—1061no—
250C22⋊C8—1061no—
251#97—1061no—
252C4×C8—1061no—
253#95—1062no—
254C24#1171994911yescharacteristic
255#41—2121no—
256#41—2121no—
257C2×Q8—531no—
258C2×D4—531no—
259C2×D4—1061no—
260C2×D4—1061no—
261C2×D4—1061no—
262C22×C4—41no—
263C22×C4—41no—
264C22×C4—1061no—
265C22×C4—2122no—
266C22×C4C2×C4×D5311yescharacteristic
267C22×C4C2×D21211yescharacteristic
268C22×C4D4×D5311yescharacteristic
269C22×C4#1171994711yescharacteristic
270C22×C4Q8×D5311yescharacteristic
271C22×C4C2×C22⋊D5311yescharacteristic
272M16—42no—
273M16—42no—
274M16—2122no—
275M16—2121no—
276M16—2121no—
277C2×C8—22no—
278C2×C8—22no—
279C2×C8—22no—
280C2×C8—41no—
281C2×C8—41no—
282C2×C8—41no—
283C2×C8—42no—
284C2×C8—1061no—
285C2×C8—1061no—
286C2×C8—2122no—
287C2×C8#1171992111yescharacteristic
288C2×C8#1171992111yescharacteristic
289C2×C8#1171992111yescharacteristic
290C2×C8#1171992111yescharacteristic
291C4⋊C4—531no—
292C4⋊C4—531no—
293C4⋊C4—1061no—
294C4⋊C4—1061no—
295C4⋊C4—2122no—
296C4⋊C4—2122no—
297C22⋊C4—1061no—
298C22⋊C4—1061no—
299C22⋊C4—1061no—
300C22⋊C4—1061no—
301C22⋊C4—1061no—
302C22⋊C4—4242no—
303C42—1061no—
304C23—21no—
305C23—21no—
306C23—41no—
307C23—41no—
308C23—1061no—
309C23#1313317511yescharacteristic
310C23#1313319211yescharacteristic
311C23#1313319311yescharacteristic
312Q8—2121no—
313D4—2121no—
314D4—2122no—
315D4—2122no—
316D4—2121no—
317D4—2121no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—1061no—
328C2×C4—1061no—
329C2×C4—1061no—
330C2×C4—4242no—
331C2×C4—4242no—
332C2×C4—4242no—
333C2×C4—2121no—
334C2×C4—2121no—
335C2×C4C2×#1171992111yescharacteristic
336C2×C4C4⋊C4×D5311yescharacteristic
337C2×C4#1313318811yescharacteristic
338C2×C4#1313319711yescharacteristic
339C8—21no—
340C8—21no—
341C8—21no—
342C8—82no—
343C8—2121no—
344C8—2121no—
345C8#1313313112yes—
346C22—21no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—21no—
353C22—21no—
354C22—42no—
355C22—42no—
356C22—2121no—
357C22—2121no—
358C22—2121no—
359C22#1681048011yescharacteristic
360C22#1681089411yescharacteristic
361C22#1681089911yescharacteristic
362C4—21no—
363C4—21no—
364C4—21no—
365C4—8482no—
366C4—2121no—
367C4—2121no—
368C4—2121no—
369C4C2×#1313313111yescharacteristic
370C4#1681059611yescharacteristic
371C2—21no—
372C2—21no—
373C2—41no—
374C2—41no—
375C2—4241no—
376C2#2027470011yescharacteristic
377C2#2027477011yescharacteristic
378C2#2027477611yescharacteristic
subgroupG/H
centerC22#16810480
commutatorC22×C4×C53C22×C4
FrattiniC23×C4C22×D53
FittingC53×#1810C2
socleC22×C53#380
prime radical#16811134C22
cosocleC23×C4×C53C23
hypercenter#1810D53
hypocenterC53#8628
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1810

Complements

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

NH ≅ G/Ntype
2. #20274744C2semidirect
4. #20272622C2semidirect
5. C53×#1810C2semidirect
6. C2×#13133223C4semidirect
13. #16810649C4semidirect
14. #16810589C4semidirect
16. #16810578C4semidirect
183. #1810D53semidirect
214. C53#8628semidirect

Maximal subgroups

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

subgroupindex
#202834092
#202747442
#202732592
#202726222
C53×#18102
#862853