Finite Group Databasebeta

← #27280191

Subgroups — #27280191

Order 13568.

456 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20283682C211yesmaximal, characteristic
2#20274735C211yesmaximal, characteristic
3#20273214C211yesmaximal, characteristic, complemented
4#20273170C211yesmaximal, characteristic, complemented
5#20272612C211yesmaximal, characteristic, complemented
6#20272554C211yesmaximal, characteristic, complemented
7C53×#3076C211yesmaximal, characteristic
8#16810754—21no—
9#16810701—21no—
10#16810695—22no—
11#16810676—21no—
12#16810676C2211yescharacteristic
13#16810647C2211yescharacteristic
14C2×#13133144—21no—
15C4×#11719917—21no—
16C2×C4×#88322C2211yescharacteristic
17C4×C4⋊Dic53C2211yescharacteristic
18#16810382—21no—
19#16810360—22no—
20Q8⋊#88322—21no—
21Q8⋊#88322—21no—
22C53×#533C2211yescharacteristic
23C4×#41×C53C2211yescharacteristic
24C53×#479—21no—
25C53×#478—21no—
26C53×#477C411yescharacteristic, complemented
27C53×#476C411yescharacteristic, complemented
28C53×#475C411yescharacteristic, complemented
29C53×#474C411yescharacteristic, complemented
30C53×#427—21no—
31C2×C4⋊C8×C53C2211yescharacteristic
32C53×#416—21no—
33C2×#11719917—22no—
34C2×#11719917—21no—
35C22×#88322—21no—
36C2×C4⋊Dic53—21no—
37C2×C4×Dic53—21no—
38C4⋊#88322—22no—
39C4⋊#88322—21no—
40C4⋊#88322—21no—
41C4⋊#88322—21no—
42#13133144—42no—
43C22⋊#88322—22no—
44C22⋊#88322D412yes—
45#13133128—22no—
46C4×#88322—21no—
47C4×#88322—21no—
48C4×#88322D411yescharacteristic
49C4×#88322D411yescharacteristic
50#13133124—21no—
51C2×#41×C53D411yescharacteristic
52C22×C8×C53—21no—
53C4⋊Q8×C53C2×C411yescharacteristic
54C4⋊Q8×C53C2×C411yescharacteristic
55#125×C53—21no—
56C22⋊Q8×C53—21no—
57C22⋊Q8×C53—21no—
58C4×Q8×C53—21no—
59C4×Q8×C53—21no—
60C4×Q8×C53C2×C411yescharacteristic
61C4×Q8×C53D411yescharacteristic
62C4×D4×C53—21no—
63C4×D4×C53—21no—
64C4×D4×C53C2×C411yescharacteristic
65C4×D4×C53D411yescharacteristic
66#117×C53—21no—
67#117×C53—21no—
68#117×C53D411yescharacteristic
69C2×C42×C53—21no—
70C2×C42×C53C2311yescharacteristic
71#107×C53—21no—
72#107×C53—21no—
73#106×C53—21no—
74#106×C53—21no—
75C4⋊C8×C53—21no—
76C4⋊C8×C53C2×C411yescharacteristic
77C4⋊C8×C53C2×C411yescharacteristic
78#103×C53—21no—
79#103×C53—21no—
80#102×C53—21no—
81#102×C53—21no—
82#41×C53—21no—
83#41×C53—21no—
84#41×C53—21no—
85#41×C53—21no—
86C2×Q8×C53—21no—
87C2×Q8×C53—21no—
88C2×Q8×C53C22⋊C411yescharacteristic
89C2×D4×C53—21no—
90C2×D4×C53C22⋊C411yescharacteristic
91C22×C4×C53—21no—
92C22×C4×C53—21no—
93C22×C4×C53—21no—
94C22×C4×C53C2×D411yescharacteristic
95C22×C4×C53C2×D411yescharacteristic
96C22×C4×C53#4111yescharacteristic
97C22×Dic53—41no—
98C2×C8×C53—21no—
99C2×C8×C53—21no—
100C2×C8×C53—41no—
101C2×C8×C53—41no—
102C4⋊C4×C53—21no—
103C4⋊C4×C53—21no—
104C4⋊C4×C53—21no—
105C4⋊C4×C53—21no—
106C4⋊C4×C53—21no—
107C4⋊C4×C53—21no—
108C4⋊C4×C53—41no—
109C4⋊C4×C53—41no—
110C4⋊C4×C53C22⋊C411yescharacteristic
111C4⋊C4×C53C22⋊C411yescharacteristic
112C22⋊C4×C53—21no—
113C22⋊C4×C53—21no—
114C22⋊C4×C53—21no—
115C22⋊C4×C53—41no—
116C42×C53—21no—
117C42×C53—21no—
118C42×C53—21no—
119C42×C53—41no—
120C42×C53C22×C411yescharacteristic
121C42×C53C2×D411yescharacteristic
122C42×C53C2×D411yescharacteristic
123C42×C53#4111yescharacteristic
124C4⋊Dic53—22no—
125C4⋊Dic53—41no—
126C4×Dic53—41no—
127C4×Dic53—41no—
128#11719917—44no—
129#11719917—41no—
130#11719917—41no—
131C2×#88322—21no—
132C2×#88322—21no—
133C2×#88322—22no—
134C2×#88322—42no—
135C2×#88322—41no—
136C23×C53—21no—
137C23×C53#12111yescharacteristic
138Q8×C53—21no—
139Q8×C53—21no—
140Q8×C53—41no—
141Q8×C53—41no—
142Q8×C53—81no—
143D4×C53—21no—
144D4×C53—21no—
145D4×C53—41no—
146D4×C53—41no—
147C2×C4×C53—21no—
148C2×C4×C53—21no—
149C2×C4×C53—21no—
150C2×C4×C53—21no—
151C2×C4×C53—21no—
152C2×C4×C53—41no—
153C2×C4×C53—42no—
154C2×C4×C53—41no—
155C2×C4×C53—41no—
156C2×C4×C53—41no—
157C2×C4×C53—41no—
158C2×C4×C53—41no—
159C2×C4×C53—41no—
160C2×C4×C53C2×C22⋊C411yescharacteristic
161C2×C4×C53C4×D411yescharacteristic
162C2×C4×C53C4×D411yescharacteristic
163C2×C4×C53#12011yescharacteristic
164C2×C4×C53#12111yescharacteristic
165C2×C4×C53#12311yescharacteristic
166C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
167C2×C4×C53#13711yescharacteristic
168C2×C4×C53#13711yescharacteristic
169C2×C4×C53#13711yescharacteristic
170C2×Dic53—42no—
171C2×Dic53—81no—
172C2×Dic53—81no—
173C8×C53—41no—
174C8×C53—41no—
175#88322—22no—
176#88322—41no—
177#88322—42no—
178#88322—81no—
179#9209—531nomaximal, Sylow
180C22×C53—21no—
181C22×C53—21no—
182C22×C53—41no—
183C22×C53—41no—
184C22×C53—41no—
185C22×C53#38611yescharacteristic
186C22×C53#46811yescharacteristic
187C22×C53#47011yescharacteristic
188C4×C53—21no—
189C4×C53—21no—
190C4×C53—21no—
191C4×C53—41no—
192C4×C53—41no—
193C4×C53—41no—
194C4×C53—81no—
195C4×C53—81no—
196C4×C53#41711yescharacteristic
197C4×C53#41911yescharacteristic
198C4×C53#44311yescharacteristic
199C4×C53#44411yescharacteristic
200C4×C53#46811yescharacteristic
201C4×C53#47011yescharacteristic
202Dic53—82no—
203#3076D5311yescharacteristic
204#2965—531no—
205#1806—531no—
206#1503—531no—
207#1493—531no—
208#1301—531no—
209#1279—531no—
210C2×C53—21no—
211C2×C53—21no—
212C2×C53—81no—
213C2×C53#187611yescharacteristic
214C2×C53#189411yescharacteristic
215C2×C53#190611yescharacteristic
216#533C2×D5311yescharacteristic
217C4×#41C2×D5311yescharacteristic
218#479—21no—
219#478—21no—
220#477Dic5311yescharacteristic, complemented
221#476Dic5311yescharacteristic, complemented
222#475Dic5311yescharacteristic, complemented
223#474Dic5311yescharacteristic, complemented
224#446—1061no—
225#436—1061no—
226#435—1062no—
227#431—531no—
228#431—1061no—
229#427—21no—
230#423—531no—
231C2×C4⋊C8C2×D5311yescharacteristic
232C2×#104—1061no—
233#416—21no—
234C4×M16—1061no—
235C2×C4×C8—531no—
236C4×C4⋊C4—531no—
237#339—1061no—
238#328—1062no—
239Q8⋊C8—1061no—
240Q8⋊C8—1061no—
241C53#920911yescharacteristic, complemented, Sylow
242C2×#41C22⋊D5311yescharacteristic
243C2×M16—1062no—
244C2×M16—1061no—
245C22×C8—21no—
246C22×C8—1061no—
247C4⋊Q8C2×Dic5311yescharacteristic
248C4⋊Q8C2×Dic5311yescharacteristic
249#125—21no—
250C22⋊Q8—21no—
251C22⋊Q8—21no—
252C4×Q8—21no—
253C4×Q8—21no—
254C4×Q8C2×Dic5311yescharacteristic
255C4×Q8C22⋊D5311yescharacteristic
256C4×D4—21no—
257C4×D4—21no—
258C4×D4C2×Dic5311yescharacteristic
259C4×D4C22⋊D5311yescharacteristic
260#117—21no—
261#117—21no—
262#117C22⋊D5311yescharacteristic
263C2×C4⋊C4—1061no—
264C2×C42—21no—
265C2×C42—1061no—
266C2×C42C22×D5311yescharacteristic
267#107—21no—
268#107—21no—
269#106—21no—
270#106—21no—
271C4⋊C8—21no—
272C4⋊C8—1062no—
273C4⋊C8—1061no—
274C4⋊C8—1061no—
275C4⋊C8—1061no—
276C4⋊C8C2×Dic5311yescharacteristic
277C4⋊C8C2×Dic5311yescharacteristic
278#104—2122no—
279#103—21no—
280#103—21no—
281#102—21no—
282#102—21no—
283C22⋊C8—532no—
284C22⋊C8—1062no—
285#97—1062no—
286C4×C8—531no—
287C4×C8—531no—
288C4×C8—1061no—
289C4×C8—1061no—
290#95—1061no—
291#41—21no—
292#41—21no—
293#41—21no—
294#41—21no—
295C2×Q8—21no—
296C2×Q8—21no—
297C2×Q8C22⋊Dic5311yescharacteristic
298C2×D4—21no—
299C2×D4C22⋊Dic5311yescharacteristic
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—2121no—
304C22×C4D4×D5311yescharacteristic
305C22×C4#1171994711yescharacteristic
306C22×C4C2×C22⋊D5311yescharacteristic
307M16—2124no—
308M16—2121no—
309M16—2121no—
310C2×C8—21no—
311C2×C8—21no—
312C2×C8—41no—
313C2×C8—41no—
314C2×C8—1061no—
315C2×C8—1061no—
316C2×C8—1062no—
317C2×C8—2122no—
318C2×C8—2121no—
319C4⋊C4—21no—
320C4⋊C4—21no—
321C4⋊C4—21no—
322C4⋊C4—21no—
323C4⋊C4—21no—
324C4⋊C4—21no—
325C4⋊C4—41no—
326C4⋊C4—41no—
327C4⋊C4—1062no—
328C4⋊C4—2121no—
329C4⋊C4C22⋊Dic5311yescharacteristic
330C4⋊C4C22⋊Dic5311yescharacteristic
331C22⋊C4—21no—
332C22⋊C4—21no—
333C22⋊C4—21no—
334C22⋊C4—41no—
335C42—21no—
336C42—21no—
337C42—21no—
338C42—41no—
339C42—2121no—
340C42—2121no—
341C42D4×D5311yescharacteristic
342C42#1171994711yescharacteristic
343C42C22×Dic5311yescharacteristic
344C42C2×C22⋊D5311yescharacteristic
345C23—21no—
346C23#1313318911yescharacteristic
347Q8—21no—
348Q8—21no—
349Q8—41no—
350Q8—41no—
351Q8—81no—
352D4—21no—
353D4—21no—
354D4—41no—
355D4—41no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—41no—
362C2×C4—42no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—4241no—
370C2×C4—4241no—
371C2×C4—2122no—
372C2×C4C2×C22⋊Dic5311yescharacteristic
373C2×C4D4×Dic5311yescharacteristic
374C2×C4D4×Dic5311yescharacteristic
375C2×C4#1313318511yescharacteristic
376C2×C4#1313318911yescharacteristic
377C2×C4#1313319811yescharacteristic
378C2×C4#1313324211yescharacteristic
379C2×C4#1313324711yescharacteristic
380C2×C4#1313324811yescharacteristic
381C2×C4#1313324811yescharacteristic
382C8—41no—
383C8—41no—
384C8—1062no—
385C8—4241no—
386C8—2121no—
387C8—2122no—
388C22—21no—
389C22—21no—
390C22—41no—
391C22—41no—
392C22—41no—
393C22#1681050311yescharacteristic
394C22#1681087111yescharacteristic
395C22#1681087911yescharacteristic
396C4—21no—
397C4—21no—
398C4—21no—
399C4—41no—
400C4—41no—
401C4—41no—
402C4—81no—
403C4—81no—
404C4—4242no—
405C4#1681062411yescharacteristic
406C4#1681063211yescharacteristic
407C4#1681074011yescharacteristic
408C4#1681074511yescharacteristic
409C4#1681087111yescharacteristic
410C4#1681087911yescharacteristic
411C2—21no—
412C2—21no—
413C2—81no—
414C2#2027504411yescharacteristic
415C2#2027514911yescharacteristic
416C2#2027521811yescharacteristic
subgroupG/H
centerC2×C4#13133189
commutatorC42×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#3076C2
socleC22×C53#386
prime radicalC2×#41×C53D4
cosocleC2×C42×C53C23
hypercenter#3076D53
hypocenterC53#9209
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #3076

Complements

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

NH ≅ G/Ntype
3. #20273214C2semidirect
4. #20273170C2semidirect
5. #20272612C2semidirect
6. #20272554C2semidirect
26. C53×#477C4semidirect
27. C53×#476C4semidirect
28. C53×#475C4semidirect
29. C53×#474C4semidirect
220. #477Dic53semidirect
221. #476Dic53semidirect
222. #475Dic53semidirect
223. #474Dic53semidirect
241. C53#9209semidirect

Maximal subgroups

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

subgroupindex
#202836822
#202747352
#202732142
#202731702
#202726122
#202725542
C53×#30762
#920953