Finite Group Databasebeta

← #27285176

Subgroups — #27285176

Order 13568.

702 conjugacy classes of subgroups.

The subgroup lattice has 452 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810971C211yesmaximal, characteristic, complemented
2C2×#16810818C211yesmaximal, characteristic
3#20275642C211yesmaximal, characteristic, complemented
4C2×#16810369C211yesmaximal, characteristic, complemented
5C2×D4⋊#88322C211yesmaximal, characteristic, complemented
6#20272556C211yesmaximal, characteristic, complemented
7C53×#2547C211yesmaximal, characteristic, complemented
8C22×#11719922—21no—
9C2×#13133214C2211yescharacteristic
10#16810971—22no—
11#16810971—22no—
12#16810818—22no—
13#16810818—22no—
14C2×C4⋊#88322C2211yescharacteristic
15C2×C4⋊#88322C2211yescharacteristic
16C2×#13133137—22no—
17C2×#13133137—21no—
18C2×#13133137—22no—
19C2×C4×#88322C2211yescharacteristic
20#16810369—22no—
21D4⋊#88322—22no—
22C2×C4⋊Q8×C53C2211yescharacteristic
23C2×#124×C53—21no—
24C2×C4×D4×C53C2211yescharacteristic
25C53×#397—22no—
26C53×#391—21no—
27C53×#380—22no—
28C4×C4⋊C4×C53C2211yescharacteristic
29C22×D212—21no—
30C22×D212—21no—
31C22×D212—21no—
32C2×#11719922—24no—
33C2×#11719922—42no—
34C2×#11719922—42no—
35C22×#88322—21no—
36C22×#88322—21no—
37C22×#88322—21no—
38#13133214—21no—
39#13133214—21no—
40#13133214—21no—
41#13133214D412yes—
42C4⋊#88322—21no—
43C4⋊#88322—21no—
44C4⋊#88322—21no—
45C4⋊#88322D412yes—
46#13133137—24no—
47#13133137—28no—
48#13133137—44no—
49C4×#88322—21no—
50C4×#88322D412yes—
51C22×Q8×C53—21no—
52C22×D4×C53D411yescharacteristic
53C23×C4×C53—21no—
54C4⋊Q8×C53—21no—
55C4⋊Q8×C53—21no—
56C4⋊Q8×C53—21no—
57C4⋊Q8×C53—21no—
58#124×C53—42no—
59#124×C53—42no—
60C4×D4×C53—21no—
61C4×D4×C53—21no—
62C4×D4×C53—21no—
63C4×D4×C53—21no—
64C2×C4⋊C4×C53—21no—
65C2×C4⋊C4×C53D411yescharacteristic
66C2×C4⋊C4×C53D412yes—
67C2×C4⋊C4×C53D412yes—
68C2×C22⋊C4×C53—24no—
69C2×C22⋊C4×C53—21no—
70C2×C42×C53—21no—
71C2×C42×C53C2311yescharacteristic
72#95×C53—24no—
73#95×C53—21no—
74C24×C53—21no—
75C23×D53—41no—
76C2×Q8×C53—41no—
77C2×Q8×C53—41no—
78C2×Q8×C53—41no—
79C2×Q8×C53—41no—
80C2×Q8×C53—41no—
81C2×Q8×C53—41no—
82C2×D4×C53—21no—
83C2×D4×C53—21no—
84C2×D4×C53—21no—
85C2×D4×C53—21no—
86C2×D4×C53—21no—
87C2×D4×C53D812yes—
88C22×C4×C53—21no—
89C22×C4×C53—22no—
90C22×C4×C53—22no—
91C22×C4×C53—41no—
92C22×C4×C53—41no—
93C22×C4×C53—42no—
94C22×C4×C53—42no—
95C22×C4×C53C2×D411yescharacteristic
96C22×C4×C53C2×D411yescharacteristic
97C22×C4×C53C2×D411yescharacteristic
98C2×D212—22no—
99C2×D212—22no—
100C2×D212—22no—
101C2×D212—44no—
102C2×D212—41no—
103C2×D212—41no—
104C2×D212—42no—
105C2×D212—42no—
106C2×D212—41no—
107C2×D212—41no—
108C2×D212—41no—
109C2×D212—41no—
110C2×D212—41no—
111C4⋊C4×C53—21no—
112C4⋊C4×C53—22no—
113C4⋊C4×C53—21no—
114C4⋊C4×C53—22no—
115C4⋊C4×C53—22no—
116C4⋊C4×C53—22no—
117C4⋊C4×C53—41no—
118C4⋊C4×C53—41no—
119C22⋊C4×C53—48no—
120C22⋊C4×C53—41no—
121C22⋊C4×C53—41no—
122C42×C53—41no—
123C42×C53—41no—
124C42×C53C2×D411yescharacteristic
125C42×C53C2×D411yescharacteristic
126C42×C53C2×D411yescharacteristic
127C42×C53#4111yescharacteristic
128#11719922—48no—
129C2×#88322—22no—
130C2×#88322—22no—
131C2×#88322—42no—
132C2×#88322—41no—
133C2×#88322—41no—
134C2×#88322—41no—
135C2×#88322—42no—
136C23×C53—22no—
137C23×C53—41no—
138C23×C53—42no—
139C23×C53—41no—
140C23×C53—42no—
141C23×C53#12011yescharacteristic
142C22×D53—42no—
143C22×D53—81no—
144C22×D53—81no—
145C22×D53—82no—
146C22×D53—82no—
147Q8×C53—81no—
148Q8×C53—81no—
149Q8×C53—81no—
150Q8×C53—81no—
151D4×C53—22no—
152D4×C53—22no—
153D4×C53—41no—
154D4×C53—41no—
155C2×C4×C53—22no—
156C2×C4×C53—41no—
157C2×C4×C53—42no—
158C2×C4×C53—42no—
159C2×C4×C53—42no—
160C2×C4×C53—42no—
161C2×C4×C53—42no—
162C2×C4×C53—42no—
163C2×C4×C53—42no—
164C2×C4×C53—82no—
165C2×C4×C53—81no—
166C2×C4×C53—81no—
167C2×C4×C53#12011yescharacteristic
168C2×C4×C53#12011yescharacteristic
169C2×C4×C53#12111yescharacteristic
170C2×C4×C53#12111yescharacteristic
171C2×C4×C53#12111yescharacteristic
172C2×C4×C53#12711yescharacteristic
173C2×C4×C53C2×D811yescharacteristic
174C2×C4×C53#13511yescharacteristic
175C2×C4×C53#13512yes—
176C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
177C2×C4×C53AGL(1,ℤ/8)12yes—
178C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
179D212—44no—
180D212—42no—
181D212—42no—
182D212—44no—
183D212—82no—
184D212—81no—
185D212—81no—
186D212—81no—
187D212—81no—
188#88322—42no—
189#88322—42no—
190#88322—81no—
191#10050—531nomaximal, Sylow
192C22×C53—44no—
193C22×C53—42no—
194C22×C53—82no—
195C22×C53—81no—
196C22×C53—81no—
197C22×C53#39211yescharacteristic
198C22×C53#44711yescharacteristic
199C22×C53#44912yes—
200C22×C53#44911yescharacteristic
201C22×C53#45412yes—
202C2×D53—82no—
203C2×D53—82no—
204C2×D53—82no—
205C2×D53—162no—
206C2×D53—162no—
207C4×C53—21no—
208C4×C53—21no—
209C4×C53—82no—
210C4×C53—82no—
211C4×C53—81no—
212C4×C53#44711yescharacteristic
213C4×C53#44911yescharacteristic
214C4×C53#46312yes—
215C4×C53#46611yescharacteristic
216C4×C53#46911yescharacteristic
217C4×C53#49511yescharacteristic
218C4×C53#49611yescharacteristic
219C2×#486—531no—
220C2×#459—531no—
221#2547D5311yescharacteristic, complemented
222#1982—531no—
223C2×#332—531no—
224C2×D4⋊C8—531no—
225#1279—531no—
226C2×C53—82no—
227C2×C53#162512yes—
228C2×C53#162812yes—
229C2×C53#199911yescharacteristic
230C2×C53#200311yescharacteristic
231C2×C53#201211yescharacteristic
232D53—162no—
233C22×D8—1061no—
234C2×C4⋊Q8C2×D5311yescharacteristic
235C2×#127—531no—
236C2×#124—21no—
237C2×C4×D4C2×D5311yescharacteristic
238#486—1062no—
239#486—1062no—
240#459—1062no—
241#459—1062no—
242C2×C4⋊C8—531no—
243C2×C4⋊C8—531no—
244C2×#102—1062no—
245C2×#102—1061no—
246C2×#102—1062no—
247C2×C4×C8—531no—
248#397—22no—
249#391—21no—
250#380—22no—
251C4×C4⋊C4C2×D5311yescharacteristic
252#332—1062no—
253D4⋊C8—1062no—
254C53#1005011yescharacteristic, complemented, Sylow
255C22×Q8—21no—
256C22×D4—1061no—
257C22×D4—1061no—
258C22×D4—1061no—
259C22×D4C22⋊D5311yescharacteristic
260C23×C4—21no—
261C2×D8—1064no—
262C2×D8—2122no—
263C2×D8—2122no—
264C22×C8—1061no—
265C22×C8—1061no—
266C22×C8—1061no—
267C4⋊Q8—21no—
268C4⋊Q8—21no—
269C4⋊Q8—21no—
270C4⋊Q8—21no—
271#127—532no—
272#127—1061no—
273#127—1061no—
274#127—1061no—
275#124—42no—
276#124—42no—
277C4×D4—21no—
278C4×D4—21no—
279C4×D4—21no—
280C4×D4—21no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4C22⋊D5311yescharacteristic
283C2×C4⋊C4C22⋊D5312yes—
284C2×C4⋊C4C22⋊D5312yes—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—21no—
287C2×C42—21no—
288C2×C42C22×D5311yescharacteristic
289C4⋊C8—532no—
290C4⋊C8—1061no—
291C4⋊C8—1061no—
292C4⋊C8—1061no—
293#102—1064no—
294#102—1068no—
295#102—2124no—
296C4×C8—532no—
297C4×C8—1061no—
298#95—24no—
299#95—21no—
300C24—21no—
301C24—2121no—
302C2×Q8—41no—
303C2×Q8—41no—
304C2×Q8—41no—
305C2×Q8—41no—
306C2×Q8—41no—
307C2×Q8—41no—
308C2×D4—21no—
309C2×D4—21no—
310C2×D4—21no—
311C2×D4—21no—
312C2×D4—21no—
313C2×D4—1062no—
314C2×D4—1062no—
315C2×D4—1062no—
316C2×D4—2124no—
317C2×D4—2121no—
318C2×D4—2121no—
319C2×D4—2121no—
320C2×D4—2121no—
321C2×D4—2121no—
322C2×D4—2122no—
323C2×D4—2122no—
324C2×D4—2121no—
325C2×D4—2121no—
326C2×D4#1171992212yes—
327C22×C4—21no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—41no—
331C22×C4—41no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4C2×C22⋊D5311yescharacteristic
335C22×C4C2×C22⋊D5311yescharacteristic
336C22×C4C2×C22⋊D5311yescharacteristic
337D8—2128no—
338C2×C8—1062no—
339C2×C8—1062no—
340C2×C8—2121no—
341C2×C8—2122no—
342C2×C8—2122no—
343C2×C8—2121no—
344C2×C8—2121no—
345C4⋊C4—21no—
346C4⋊C4—22no—
347C4⋊C4—21no—
348C4⋊C4—22no—
349C4⋊C4—22no—
350C4⋊C4—22no—
351C4⋊C4—41no—
352C4⋊C4—41no—
353C22⋊C4—48no—
354C22⋊C4—41no—
355C22⋊C4—41no—
356C42—41no—
357C42—41no—
358C42D4×D5311yescharacteristic
359C42D4×D5311yescharacteristic
360C42D4×D5311yescharacteristic
361C42#1171994711yescharacteristic
362C23—22no—
363C23—42no—
364C23—41no—
365C23—41no—
366C23—42no—
367C23—4241no—
368C23—4242no—
369C23—4241no—
370C23—4242no—
371C23—2122no—
372C23#1313318611yescharacteristic
373Q8—81no—
374Q8—81no—
375Q8—81no—
376Q8—81no—
377D4—22no—
378D4—22no—
379D4—41no—
380D4—41no—
381D4—4242no—
382D4—4241no—
383D4—4241no—
384D4—4241no—
385D4—4241no—
386D4—2124no—
387D4—2122no—
388D4—2122no—
389D4—2124no—
390C2×C4—22no—
391C2×C4—42no—
392C2×C4—42no—
393C2×C4—42no—
394C2×C4—42no—
395C2×C4—42no—
396C2×C4—41no—
397C2×C4—42no—
398C2×C4—42no—
399C2×C4—81no—
400C2×C4—82no—
401C2×C4—81no—
402C2×C4#1313318511yescharacteristic
403C2×C4#1313318511yescharacteristic
404C2×C4#1313318811yescharacteristic
405C2×C4#1313318911yescharacteristic
406C2×C4#1313318911yescharacteristic
407C2×C4#1313321311yescharacteristic
408C2×C4C2×#1171992211yescharacteristic
409C2×C4#1313323711yescharacteristic
410C2×C4#1313323712yes—
411C2×C4#1313324211yescharacteristic
412C2×C4#1313324212yes—
413C2×C4#1313324311yescharacteristic
414C8—4241no—
415C8—2122no—
416C8—2122no—
417C22—44no—
418C22—42no—
419C22—81no—
420C22—81no—
421C22—82no—
422C22—8482no—
423C22—8482no—
424C22—4242no—
425C22—4242no—
426C22—4242no—
427C22#1681053111yescharacteristic
428C22#1681075711yescharacteristic
429C22#1681076812yes—
430C22#1681076811yescharacteristic
431C22#1681079612yes—
432C4—21no—
433C4—21no—
434C4—81no—
435C4—82no—
436C4—82no—
437C4#1681075611yescharacteristic
438C4#1681076611yescharacteristic
439C4#1681083512yes—
440C4#1681085711yescharacteristic
441C4#1681087411yescharacteristic
442C4#1681100511yescharacteristic
443C4#1681101311yescharacteristic
444C2—82no—
445C2—8482no—
446C2#2027380412yes—
447C2#2027382512yes—
448C2#2027571111yescharacteristic
449C2#2027573511yescharacteristic
450C2#2027579211yescharacteristic
subgroupG/H
centerC23#13133186
commutator, cosocleC2×C42×C53C23
FrattiniC2×C42C22×D53
FittingC53×#2547C2
socleC23×C53#120
prime radicalC2×#16810818C2
hypercenter#2547D53
hypocenterC53#10050
LCSG ▷ C2×C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2547

Complements

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

NH ≅ G/Ntype
1. C2×#16810971C2semidirect
3. #20275642C2semidirect
4. C2×#16810369C2semidirect
5. C2×D4⋊#88322C2semidirect
6. #20272556C2semidirect
7. C53×#2547C2semidirect
221. #2547D53semidirect
254. C53#10050semidirect

Maximal subgroups

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

subgroupindex
C2×#168109712
C2×#168108182
#202756422
C2×#168103692
C2×D4⋊#883222
#202725562
C53×#25472
#1005053