Finite Group Databasebeta

← #27280620

Subgroups — #27280620

Order 13568.

496 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#20285836C211yesmaximal, characteristic
2#20283458C211yesmaximal, characteristic
3#20275560C211yesmaximal, characteristic, complemented
4#20273390C211yesmaximal, characteristic
5#20272612C211yesmaximal, characteristic, complemented
6#20272565C211yesmaximal, characteristic, complemented
7C2×C53×#332C211yesmaximal, characteristic, complemented
8C2×#13133223—21no—
9#16811173C2211yescharacteristic
10#16811026—21no—
11#16810971—21no—
12#16810936C414yes—
13#16810900—22no—
14#16810700—21no—
15#16810693—22no—
16#16810688—21no—
17#16810676C2211yescharacteristic
18#16810647C2211yescharacteristic
19C2×#13133136—21no—
20C2×#13133136—21no—
21#16810589—21no—
22C4×#11719917C2211yescharacteristic
23#16810368—21no—
24#16810366—21no—
25C2×C4⋊Q8×C53C2211yescharacteristic
26C2×C4⋊C8×C53C2211yescharacteristic
27C2×C4⋊C8×C53C2211yescharacteristic
28C53×#332—21no—
29C53×#332—21no—
30C2×#11719945—21no—
31#13133223—42no—
32#13133223—42no—
33C2×#11719917—21no—
34C2×#11719917D411yescharacteristic
35C2×#11719917D411yescharacteristic
36C2×#11719912—22no—
37C2×C8×D53—22no—
38#13133217D411yescharacteristic
39#13133214D411yescharacteristic
40#13133205—21no—
41#13133191—21no—
42#13133191C2×C412yes—
43C4×D212—21no—
44C4×D212D412yes—
45#13133150—24no—
46C4⋊#88322—21no—
47C4⋊#88322—21no—
48C4⋊#88322—21no—
49#13133137—24no—
50#13133136—42no—
51#13133136—42no—
52C22⋊#88322C2×C412yes—
53#13133129—22no—
54#13133129—22no—
55#13133128—21no—
56C4×#88322—21no—
57C4×#88322—21no—
58C22×Q8×C53—21no—
59C22×C8×C53—21no—
60C22×C8×C53—21no—
61C4⋊Q8×C53—21no—
62C4⋊Q8×C53—21no—
63C4⋊Q8×C53—21no—
64C4⋊Q8×C53—21no—
65C2×C4⋊C4×C53—21no—
66C2×C4⋊C4×C53C2×C412yes—
67C2×C42×C53C2311yescharacteristic
68C4⋊C8×C53—21no—
69C4⋊C8×C53—21no—
70C4⋊C8×C53—21no—
71C4⋊C8×C53D411yescharacteristic
72C4⋊C8×C53D411yescharacteristic
73C2×Q8×C53—41no—
74C2×Q8×C53—41no—
75C2×Q8×C53—41no—
76C2×Q8×C53—41no—
77C2×Q8×C53—41no—
78C2×Q8×C53—41no—
79C22×C4×C53—22no—
80C22×C4×C53C22×C411yescharacteristic
81C22×C4×C53C2×D411yescharacteristic
82C22×C4×C53#4111yescharacteristic
83C2×C22⋊D53—22no—
84#11719945—41no—
85#11719945—41no—
86#11719945—41no—
87#11719945—41no—
88C2×D212—21no—
89C2×D212—21no—
90C2×D212C22⋊C412yes—
91C2×C4×D53—22no—
92C2×Dic106—21no—
93C2×C8×C53—21no—
94C2×C8×C53—21no—
95C2×C8×C53—41no—
96C2×C8×C53—41no—
97C2×C8×C53—41no—
98C2×C8×C53—81no—
99C4⋊C4×C53—22no—
100C4⋊C4×C53—22no—
101C4⋊C4×C53—41no—
102C4⋊C4×C53—41no—
103C42×C53C2×D411yescharacteristic
104C42×C53C2×D411yescharacteristic
105C42×C53C2×D411yescharacteristic
106C42×C53#4111yescharacteristic
107#11719921—22no—
108#11719921—22no—
109C4⋊Dic53—21no—
110C4⋊Dic53C22⋊C412yes—
111#11719917—21no—
112#11719917—21no—
113#11719917—21no—
114#11719917—41no—
115#11719917—41no—
116#11719917D812yes—
117C2×#88322—21no—
118C2×#88322—22no—
119C2×#88322—21no—
120C2×#88322—41no—
121#11719912—44no—
122C8×D53—44no—
123C23×C53C4×D411yescharacteristic
124C22×D53—22no—
125Q8×C53—81no—
126Q8×C53—81no—
127Q8×C53—81no—
128Q8×C53—81no—
129C2×C4×C53—21no—
130C2×C4×C53—21no—
131C2×C4×C53—41no—
132C2×C4×C53—42no—
133C2×C4×C53—42no—
134C2×C4×C53—42no—
135C2×C4×C53C2×C22⋊C411yescharacteristic
136C2×C4×C53C4×D411yescharacteristic
137C2×C4×C53#12011yescharacteristic
138C2×C4×C53#12111yescharacteristic
139C2×C4×C53#12111yescharacteristic
140C2×C4×C53#12311yescharacteristic
141C2×C4×C53C2×D811yescharacteristic
142C2×C4×C53#13511yescharacteristic
143C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
144C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
145C22⋊D53—42no—
146C22⋊D53—42no—
147C2×Dic53—22no—
148D212—42no—
149D212—41no—
150D212—41no—
151D212—41no—
152D212—41no—
153C4×D53—42no—
154C4×D53—42no—
155Dic106—41no—
156Dic106—41no—
157C8×C53—41no—
158C8×C53—41no—
159C8×C53—81no—
160#88322—22no—
161#88322—42no—
162#88322—41no—
163#88322—81no—
164#9271—531nomaximal, Sylow
165C22×C53—21no—
166C22×C53—21no—
167C22×C53#38611yescharacteristic
168C22×C53C4×D811yescharacteristic
169C22×C53#44211yescharacteristic
170C2×D53—42no—
171C2×D53—42no—
172C2×D53—42no—
173C4×C53—21no—
174C4×C53—21no—
175C4×C53—21no—
176C4×C53—82no—
177C4×C53#44711yescharacteristic
178C4×C53#44911yescharacteristic
179C4×C53#45911yescharacteristic
180C4×C53#46311yescharacteristic
181C4×C53#47111yescharacteristic
182C4×C53#47311yescharacteristic
183Dic53—42no—
184#3226—531no—
185#2935—531no—
186#1966—531no—
187#1547—531no—
188C2×#332D5311yescharacteristic, complemented
189#1301—531no—
190#1283—531no—
191C2×C53—21no—
192C2×C53—21no—
193C2×C53#187911yescharacteristic
194C2×C53#190811yescharacteristic
195C2×C53#191511yescharacteristic
196D53—82no—
197C2×#131—1061no—
198#532—531no—
199C2×C4⋊Q8C2×D5311yescharacteristic
200#500—1061no—
201#486—1061no—
202#480—534no—
203#474—1062no—
204#436—1061no—
205#435—1062no—
206C8×D4—1061no—
207#431—531no—
208#423—531no—
209C2×C4⋊C8C2×D5311yescharacteristic
210C2×C4⋊C8C2×D5311yescharacteristic
211C2×#101—1061no—
212C2×#101—1061no—
213#408—1061no—
214C4×M16—531no—
215#332—21no—
216#332—21no—
217#332—1061no—
218#331—1061no—
219C53#927111yescharacteristic, complemented, Sylow
220C2×#41—1061no—
221C22×Q8—21no—
222#131—2122no—
223#131—2122no—
224C2×M16—531no—
225C2×M16—531no—
226C2×M16—1062no—
227C2×M16—1061no—
228C22×C8—21no—
229C22×C8—21no—
230C22×C8—1062no—
231C4⋊Q8—21no—
232C4⋊Q8—21no—
233C4⋊Q8—21no—
234C4⋊Q8—21no—
235C4⋊Q8—531no—
236#127—531no—
237#124—1061no—
238#121—532no—
239#121—1061no—
240C4×D4—532no—
241C4×D4—1061no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4C4×D5312yes—
244C2×C42C22×D5311yescharacteristic
245#107—1064no—
246C4⋊C8—21no—
247C4⋊C8—21no—
248C4⋊C8—21no—
249C4⋊C8—1061no—
250C4⋊C8—1061no—
251C4⋊C8—1061no—
252C4⋊C8C22⋊D5311yescharacteristic
253C4⋊C8C22⋊D5311yescharacteristic
254#102—1064no—
255#101—2122no—
256#101—2122no—
257C22⋊C8—532no—
258C22⋊C8—1062no—
259C22⋊C8—1062no—
260#97—1061no—
261C4×C8—1061no—
262C4×C8—1061no—
263#41—2121no—
264#41—2121no—
265#41—2121no—
266#41—2121no—
267C2×Q8—41no—
268C2×Q8—41no—
269C2×Q8—41no—
270C2×Q8—41no—
271C2×Q8—41no—
272C2×Q8—41no—
273C2×Q8—1061no—
274C2×D4—532no—
275C2×D4—1062no—
276C2×D4—1061no—
277C2×D4—1061no—
278C22×C4—22no—
279C22×C4—1062no—
280C22×C4C2×C4×D5311yescharacteristic
281C22×C4D4×D5311yescharacteristic
282C22×C4#1171994711yescharacteristic
283M16—532no—
284M16—1061no—
285M16—1061no—
286M16—1061no—
287M16—2124no—
288M16—2121no—
289M16—2121no—
290C2×C8—21no—
291C2×C8—21no—
292C2×C8—41no—
293C2×C8—41no—
294C2×C8—41no—
295C2×C8—81no—
296C2×C8—1061no—
297C2×C8—1062no—
298C2×C8—1061no—
299C2×C8—2124no—
300C2×C8—2121no—
301C4⋊C4—22no—
302C4⋊C4—22no—
303C4⋊C4—41no—
304C4⋊C4—41no—
305C4⋊C4—532no—
306C4⋊C4—1061no—
307C22⋊C4—1062no—
308C22⋊C4—1062no—
309C42#1171994511yescharacteristic
310C42D4×D5311yescharacteristic
311C42D4×D5311yescharacteristic
312C42C2×C22⋊D5311yescharacteristic
313C23—1062no—
314C23#1313317711yescharacteristic
315Q8—81no—
316Q8—81no—
317Q8—81no—
318Q8—81no—
319Q8—2121no—
320Q8—2121no—
321D4—2122no—
322D4—2122no—
323D4—2122no—
324D4—2121no—
325D4—2121no—
326D4—2121no—
327D4—2121no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—41no—
331C2×C4—42no—
332C2×C4—42no—
333C2×C4—42no—
334C2×C4—1062no—
335C2×C4—2122no—
336C2×C4—2122no—
337C2×C4C22⋊C4×D5311yescharacteristic
338C2×C4C4×C22⋊D5311yescharacteristic
339C2×C4#1313318511yescharacteristic
340C2×C4#1313318811yescharacteristic
341C2×C4#1313319011yescharacteristic
342C2×C4#1313319911yescharacteristic
343C2×C4D8×D5311yescharacteristic
344C2×C4#1313323811yescharacteristic
345C2×C4#1313324111yescharacteristic
346C2×C4#1313324411yescharacteristic
347C8—41no—
348C8—41no—
349C8—81no—
350C8—1062no—
351C8—4241no—
352C8—2122no—
353C8—2121no—
354C22—21no—
355C22—21no—
356C22—2122no—
357C22—2122no—
358C22—2122no—
359C22#1681049911yescharacteristic
360C22#1681070311yescharacteristic
361C22#1681073211yescharacteristic
362C4—21no—
363C4—21no—
364C4—21no—
365C4—82no—
366C4—2122no—
367C4#1681075511yescharacteristic
368C4#1681076711yescharacteristic
369C4#1681081711yescharacteristic
370C4#1681083711yescharacteristic
371C4#1681088311yescharacteristic
372C4#1681089511yescharacteristic
373C2—21no—
374C2—21no—
375C2—4242no—
376C2#2027506011yescharacteristic
377C2#2027523311yescharacteristic
378C2#2027526411yescharacteristic
subgroupG/H
centerC22#16810499
commutatorC22×C4×C53C22×C4
FrattiniC2×C42C22×D53
FittingC2×C53×#332C2
socleC22×C53#386
prime radical#16811173C22
cosocleC2×C42×C53C23
hypercenterC2×#332D53
hypocenterC53#9271
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#332

Complements

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

NH ≅ G/Ntype
3. #20275560C2semidirect
5. #20272612C2semidirect
6. #20272565C2semidirect
7. C2×C53×#332C2semidirect
188. C2×#332D53semidirect
219. C53#9271semidirect

Maximal subgroups

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

subgroupindex
#202858362
#202834582
#202755602
#202733902
#202726122
#202725652
C2×C53×#3322
#927153