Finite Group Databasebeta

← #27276244

Subgroups — #27276244

Order 13568.

766 conjugacy classes of subgroups.

The subgroup lattice has 446 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20275514C211yesmaximal, characteristic, complemented
2#20273415C211yesmaximal, characteristic, complemented
3#20273410C211yesmaximal, characteristic, complemented
4#20273330C211yesmaximal, characteristic, complemented
5#20273326C211yesmaximal, characteristic, complemented
6#20273056C211yesmaximal, characteristic, complemented
7C53×#3506C211yesmaximal, characteristic
8#16810680C2211yescharacteristic
9#16810680C2211yescharacteristic
10C2×C4⋊#88322C2211yescharacteristic
11C2×C22⋊#88322—22no—
12C2×#13133128C2211yescharacteristic
13#16810372—21no—
14#16810369—21no—
15#16810369—21no—
16#16810367—21no—
17#16810365—22no—
18#16810363—22no—
19#16810347—22no—
20C53×#559—22no—
21C53×#556—21no—
22C53×#555—21no—
23C53×#553—22no—
24C53×#550—21no—
25C53×#547—21no—
26C53×#538—22no—
27C53×#535C412yes—
28C53×#533C2211yescharacteristic
29C53×#532C2211yescharacteristic
30C53×#525—22no—
31C53×#520C412yes—
32C2×C4×D4×C53C2211yescharacteristic
33C22×#88322—21no—
34C22×#88322—21no—
35C4⋊#88322—21no—
36C4⋊#88322—21no—
37C4⋊#88322—21no—
38C4⋊#88322—21no—
39C22⋊#88322—21no—
40C22⋊#88322—21no—
41C22⋊#88322—44no—
42#13133128—21no—
43#13133128—21no—
44#13133128—21no—
45#13133128—21no—
46C2×#41×C53—22no—
47C2×#41×C53C2×C412yes—
48C22×D4×C53C2×C411yescharacteristic
49C23×C4×C53D412yes—
50C4⋊Q8×C53D411yescharacteristic
51C4⋊Q8×C53D411yescharacteristic
52C4⋊Q8×C53D411yescharacteristic
53#127×C53—22no—
54#127×C53D411yescharacteristic
55#126×C53—24no—
56#125×C53D412yes—
57#124×C53—24no—
58#124×C53—22no—
59#124×C53D412yes—
60#123×C53—24no—
61#123×C53—24no—
62C22⋊Q8×C53—22no—
63C22⋊Q8×C53—21no—
64C22⋊Q8×C53—21no—
65#121×C53—22no—
66#121×C53—24no—
67#121×C53—22no—
68#121×C53—22no—
69#121×C53—21no—
70#121×C53—21no—
71#120×C53—24no—
72C4×Q8×C53—21no—
73C4×Q8×C53—21no—
74C4×Q8×C53—22no—
75C4×D4×C53—24no—
76C4×D4×C53—21no—
77C4×D4×C53—21no—
78C4×D4×C53—21no—
79C4×D4×C53—21no—
80C4×D4×C53—21no—
81C4×D4×C53—21no—
82C4×D4×C53—22no—
83C4×D4×C53—24no—
84#117×C53—22no—
85#117×C53—22no—
86#117×C53C2×C412yes—
87C2×C4⋊C4×C53—22no—
88C2×C4⋊C4×C53C2×C411yescharacteristic
89C2×C22⋊C4×C53D412yes—
90C2×C42×C53C2311yescharacteristic
91C24×C53C22⋊C412yes—
92#41×C53—44no—
93#41×C53—82no—
94#41×C53—82no—
95C2×Q8×C53—21no—
96C2×Q8×C53—21no—
97C2×Q8×C53C22⋊C412yes—
98C2×D4×C53—22no—
99C2×D4×C53—24no—
100C2×D4×C53—24no—
101C2×D4×C53—21no—
102C2×D4×C53—21no—
103C2×D4×C53—21no—
104C2×D4×C53—21no—
105C2×D4×C53—42no—
106C2×D4×C53C22⋊C412yes—
107C22×C4×C53—24no—
108C22×C4×C53—22no—
109C22×C4×C53—22no—
110C22×C4×C53—22no—
111C22×C4×C53—42no—
112C22×C4×C53—42no—
113C22×C4×C53C22⋊C412yes—
114C22×C4×C53C22×C411yescharacteristic
115C22×C4×C53C2×D411yescharacteristic
116C22×C4×C53C2×D411yescharacteristic
117C4⋊C4×C53—24no—
118C4⋊C4×C53—21no—
119C4⋊C4×C53—21no—
120C4⋊C4×C53—21no—
121C4⋊C4×C53—21no—
122C4⋊C4×C53—22no—
123C4⋊C4×C53—21no—
124C4⋊C4×C53—22no—
125C4⋊C4×C53—21no—
126C4⋊C4×C53C22⋊C412yes—
127C4⋊C4×C53C22⋊C412yes—
128C22⋊C4×C53—24no—
129C22⋊C4×C53—24no—
130C22⋊C4×C53—24no—
131C22⋊C4×C53—22no—
132C22⋊C4×C53—22no—
133C22⋊C4×C53—22no—
134C22⋊C4×C53—22no—
135C22⋊C4×C53—22no—
136C42×C53—24no—
137C42×C53—22no—
138C42×C53C2×D411yescharacteristic
139C42×C53C2×D411yescharacteristic
140C42×C53C2×D411yescharacteristic
141C42×C53C2×D411yescharacteristic
142C2×#88322—22no—
143C2×#88322—41no—
144C2×#88322—41no—
145C2×#88322—42no—
146C2×#88322—41no—
147C2×#88322—41no—
148C2×#88322—41no—
149C23×C53—22no—
150C23×C53—22no—
151C23×C53—22no—
152C23×C53—22no—
153C23×C53—42no—
154C23×C53—42no—
155C23×C53C2×C22⋊C411yescharacteristic
156Q8×C53—42no—
157Q8×C53—41no—
158Q8×C53—41no—
159Q8×C53—81no—
160D4×C53—42no—
161D4×C53—42no—
162D4×C53—42no—
163D4×C53—44no—
164D4×C53—41no—
165D4×C53—41no—
166D4×C53—81no—
167D4×C53—84no—
168D4×C53—84no—
169C2×C4×C53—22no—
170C2×C4×C53—21no—
171C2×C4×C53—21no—
172C2×C4×C53—22no—
173C2×C4×C53—22no—
174C2×C4×C53—22no—
175C2×C4×C53—22no—
176C2×C4×C53—41no—
177C2×C4×C53—44no—
178C2×C4×C53—42no—
179C2×C4×C53—42no—
180C2×C4×C53—42no—
181C2×C4×C53—82no—
182C2×C4×C53—84no—
183C2×C4×C53C2×C22⋊C411yescharacteristic
184C2×C4×C53C2×C22⋊C411yescharacteristic
185C2×C4×C53#12011yescharacteristic
186C2×C4×C53#12011yescharacteristic
187C2×C4×C53#12011yescharacteristic
188C2×C4×C53#12011yescharacteristic
189#88322—42no—
190#88322—81no—
191#88322—81no—
192#88322—81no—
193#8458—531nomaximal, Sylow
194C22×C53—21no—
195C22×C53—21no—
196C22×C53—44no—
197C22×C53—42no—
198C22×C53—42no—
199C22×C53—42no—
200C22×C53—42no—
201C22×C53—42no—
202C22×C53—42no—
203C22×C53—82no—
204C22×C53#37911yescharacteristic
205C22×C53#41312yes—
206C4×C53—21no—
207C4×C53—21no—
208C4×C53—21no—
209C4×C53—21no—
210C4×C53—21no—
211C4×C53—42no—
212C4×C53—42no—
213C4×C53—42no—
214C4×C53—42no—
215C4×C53#45311yescharacteristic
216C4×C53#45511yescharacteristic
217#3506D5311yescharacteristic
218#1955—531no—
219#1552—531no—
220#1551—531no—
221#1531—531no—
222#1530—531no—
223#1466—531no—
224C2×C53—22no—
225C2×C53—82no—
226C2×C53—82no—
227C2×C53#178511yescharacteristic
228C2×C53#179211yescharacteristic
229C2×C53#179411yescharacteristic
230#559—22no—
231#556—21no—
232#555—21no—
233#553—22no—
234#550—21no—
235#547—21no—
236#538—22no—
237#535Dic5312yes—
238#533C2×D5311yescharacteristic
239#532C2×D5311yescharacteristic
240#525—22no—
241#520Dic5312yes—
242C2×C4×D4C2×D5311yescharacteristic
243#432—531no—
244#432—531no—
245C2×C4⋊C8—531no—
246C2×C22⋊C8—1062no—
247C2×#97—531no—
248#333—1061no—
249#332—1061no—
250#332—1061no—
251#331—1061no—
252#330—1062no—
253#329—1062no—
254#323—1062no—
255C53#845811yescharacteristic, complemented, Sylow
256C2×#41—22no—
257C2×#41C2×Dic5312yes—
258C22×D4C2×Dic5311yescharacteristic
259C23×C4C22⋊D5312yes—
260C22×C8—1061no—
261C22×C8—1061no—
262C4⋊Q8C22⋊D5311yescharacteristic
263C4⋊Q8C22⋊D5311yescharacteristic
264C4⋊Q8C22⋊D5311yescharacteristic
265#127—22no—
266#127C22⋊D5311yescharacteristic
267#126—24no—
268#125C22⋊D5312yes—
269#124—24no—
270#124—22no—
271#124C22⋊D5312yes—
272#123—24no—
273#123—24no—
274C22⋊Q8—22no—
275C22⋊Q8—21no—
276C22⋊Q8—21no—
277#121—22no—
278#121—24no—
279#121—21no—
280#121—21no—
281#121—22no—
282#121—22no—
283#120—24no—
284C4×Q8—22no—
285C4×Q8—21no—
286C4×Q8—21no—
287C4×D4—24no—
288C4×D4—22no—
289C4×D4—24no—
290C4×D4—21no—
291C4×D4—21no—
292C4×D4—21no—
293C4×D4—21no—
294C4×D4—21no—
295C4×D4—21no—
296#117—22no—
297#117—22no—
298#117C2×Dic5312yes—
299C2×C4⋊C4—22no—
300C2×C4⋊C4C2×Dic5311yescharacteristic
301C2×C22⋊C4C22⋊D5312yes—
302C2×C42C22×D5311yescharacteristic
303C4⋊C8—1061no—
304C4⋊C8—1061no—
305C4⋊C8—1061no—
306C4⋊C8—1061no—
307C22⋊C8—1061no—
308C22⋊C8—1061no—
309C22⋊C8—2124no—
310#97—1061no—
311#97—1061no—
312#97—1061no—
313#97—1061no—
314C24C22⋊Dic5312yes—
315#41—44no—
316#41—82no—
317#41—82no—
318C2×Q8—21no—
319C2×Q8—21no—
320C2×Q8C22⋊Dic5312yes—
321C2×D4—22no—
322C2×D4—24no—
323C2×D4—24no—
324C2×D4—21no—
325C2×D4—21no—
326C2×D4—21no—
327C2×D4—21no—
328C2×D4—42no—
329C2×D4C22⋊Dic5312yes—
330C22×C4—24no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4C22⋊Dic5312yes—
337C22×C4C22×Dic5311yescharacteristic
338C22×C4C2×C22⋊D5311yescharacteristic
339C22×C4C2×C22⋊D5311yescharacteristic
340C2×C8—1062no—
341C2×C8—2122no—
342C2×C8—2121no—
343C2×C8—2121no—
344C2×C8—2121no—
345C2×C8—2121no—
346C2×C8—2121no—
347C4⋊C4—24no—
348C4⋊C4—21no—
349C4⋊C4—21no—
350C4⋊C4—21no—
351C4⋊C4—21no—
352C4⋊C4—22no—
353C4⋊C4—21no—
354C4⋊C4—22no—
355C4⋊C4—21no—
356C4⋊C4C22⋊Dic5312yes—
357C4⋊C4C22⋊Dic5312yes—
358C22⋊C4—24no—
359C22⋊C4—24no—
360C22⋊C4—24no—
361C22⋊C4—22no—
362C22⋊C4—22no—
363C22⋊C4—22no—
364C22⋊C4—22no—
365C22⋊C4—22no—
366C42—24no—
367C42—22no—
368C42C2×C22⋊D5311yescharacteristic
369C42C2×C22⋊D5311yescharacteristic
370C42C2×C22⋊D5311yescharacteristic
371C42C2×C22⋊D5311yescharacteristic
372C23—22no—
373C23—22no—
374C23—22no—
375C23—22no—
376C23—42no—
377C23—42no—
378C23C2×C22⋊Dic5311yescharacteristic
379Q8—42no—
380Q8—41no—
381Q8—41no—
382Q8—81no—
383D4—42no—
384D4—42no—
385D4—42no—
386D4—44no—
387D4—41no—
388D4—41no—
389D4—81no—
390D4—84no—
391D4—84no—
392C2×C4—22no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—22no—
396C2×C4—22no—
397C2×C4—22no—
398C2×C4—22no—
399C2×C4—41no—
400C2×C4—44no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—82no—
405C2×C4—84no—
406C2×C4C2×C22⋊Dic5311yescharacteristic
407C2×C4C2×C22⋊Dic5311yescharacteristic
408C2×C4#1313318611yescharacteristic
409C2×C4#1313318611yescharacteristic
410C2×C4#1313318611yescharacteristic
411C2×C4#1313318611yescharacteristic
412C8—4241no—
413C8—4241no—
414C8—4241no—
415C8—2122no—
416C22—21no—
417C22—21no—
418C22—44no—
419C22—42no—
420C22—42no—
421C22—42no—
422C22—42no—
423C22—42no—
424C22—42no—
425C22—82no—
426C22#1681047911yescharacteristic
427C22#1681060712yes—
428C4—21no—
429C4—21no—
430C4—21no—
431C4—21no—
432C4—21no—
433C4—42no—
434C4—42no—
435C4—42no—
436C4—42no—
437C4#1681079111yescharacteristic
438C4#1681080111yescharacteristic
439C2—22no—
440C2—82no—
441C2—82no—
442C2#2027464311yescharacteristic
443C2#2027466911yescharacteristic
444C2#2027467911yescharacteristic
subgroupG/H
centerC22#16810479
commutatorC22×C4×C53C22×C4
FrattiniC2×C42C22×D53
Fitting, prime radicalC53×#3506C2
socleC22×C53#379
cosocleC2×C42×C53C23
hypercenter#3506D53
hypocenterC53#8458
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3506

Complements

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

NH ≅ G/Ntype
1. #20275514C2semidirect
2. #20273415C2semidirect
3. #20273410C2semidirect
4. #20273330C2semidirect
5. #20273326C2semidirect
6. #20273056C2semidirect
255. C53#8458semidirect

Maximal subgroups

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

subgroupindex
#202755142
#202734152
#202734102
#202733302
#202733262
#202730562
C53×#35062
#845853