Finite Group Databasebeta

← #27285174

Subgroups — #27285174

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×#16810969C211yesmaximal, characteristic, complemented
2#20279707C211yesmaximal, characteristic, complemented
3#20275643C211yesmaximal, characteristic, complemented
4C2×#16810370C211yesmaximal, characteristic, complemented
5C2×D4⋊#88322C211yesmaximal, characteristic, complemented
6#20272554C211yesmaximal, characteristic, complemented
7C2×C53×#459C211yesmaximal, characteristic
8C2×#13133217C2211yescharacteristic
9C2×#13133204—21no—
10#16810969—22no—
11#16810969—22no—
12C2×C4⋊#88322C2211yescharacteristic
13C2×#13133138—22no—
14C2×#13133138—22no—
15C2×C4×#88322C2211yescharacteristic
16#16810548—22no—
17#16810528—21no—
18#16810483—22no—
19C4×C4⋊Dic53C2211yescharacteristic
20#16810370—22no—
21D4⋊#88322—22no—
22C22×D8×C53—21no—
23C2×#127×C53C2211yescharacteristic
24C2×C4×D4×C53C2211yescharacteristic
25C53×#459—22no—
26C53×#459—22no—
27C2×C4⋊C8×C53C2211yescharacteristic
28C2×#102×C53—21no—
29C22×Dic106—21no—
30C22×#88322—21no—
31C22×#88322—21no—
32#13133217—21no—
33#13133217—21no—
34#13133217—21no—
35#13133217—21no—
36#13133204—42no—
37#13133204—42no—
38C2×C4⋊Dic53—21no—
39C2×C4⋊Dic53D412yes—
40C2×C4⋊Dic53D412yes—
41C2×C22⋊Dic53—24no—
42C2×C4×Dic53—21no—
43C4⋊#88322—21no—
44C4⋊#88322—21no—
45#13133138—28no—
46#13133138—44no—
47C4×#88322—21no—
48C4×#88322D412yes—
49#13133124—24no—
50#13133124—21no—
51C22×D4×C53—21no—
52C22×D4×C53—21no—
53C22×D4×C53—21no—
54C22×D4×C53D411yescharacteristic
55C23×C4×C53—21no—
56C2×D8×C53—24no—
57C2×D8×C53—42no—
58C2×D8×C53—42no—
59C22×C8×C53—21no—
60#127×C53—21no—
61#127×C53—21no—
62#127×C53—21no—
63#127×C53D412yes—
64C4×D4×C53—21no—
65C4×D4×C53—21no—
66C4×D4×C53—21no—
67C4×D4×C53—21no—
68C2×C4⋊C4×C53D411yescharacteristic
69C2×C22⋊C4×C53—21no—
70C2×C42×C53C2311yescharacteristic
71C4⋊C8×C53—21no—
72C4⋊C8×C53D412yes—
73#102×C53—24no—
74C24×C53—21no—
75C24×C53—41no—
76C2×D4×C53—21no—
77C2×D4×C53—22no—
78C2×D4×C53—22no—
79C2×D4×C53—21no—
80C2×D4×C53—22no—
81C2×D4×C53—21no—
82C2×D4×C53—21no—
83C2×D4×C53—21no—
84C2×D4×C53—44no—
85C2×D4×C53—41no—
86C2×D4×C53—41no—
87C2×D4×C53—41no—
88C2×D4×C53—41no—
89C2×D4×C53—42no—
90C2×D4×C53—41no—
91C2×D4×C53—41no—
92C2×D4×C53—41no—
93C2×D4×C53—42no—
94C2×D4×C53D812yes—
95C22×C4×C53—21no—
96C22×C4×C53—41no—
97C22×C4×C53—41no—
98C22×C4×C53—42no—
99C22×C4×C53—42no—
100C22×C4×C53C2×D411yescharacteristic
101C22×C4×C53C2×D411yescharacteristic
102C22×C4×C53C2×D411yescharacteristic
103C22×Dic53—22no—
104C22×Dic53—22no—
105C2×Dic106—41no—
106C2×Dic106—41no—
107C2×Dic106—41no—
108C2×Dic106—41no—
109C2×Dic106—41no—
110C2×Dic106—41no—
111D8×C53—48no—
112C2×C8×C53—22no—
113C2×C8×C53—41no—
114C2×C8×C53—41no—
115C4⋊C4×C53—21no—
116C4⋊C4×C53—21no—
117C22⋊C4×C53—41no—
118C22⋊C4×C53—41no—
119C42×C53C2×D411yescharacteristic
120C42×C53C2×D411yescharacteristic
121C42×C53C2×D411yescharacteristic
122C42×C53#4111yescharacteristic
123C22⋊Dic53—48no—
124C4⋊Dic53—22no—
125C4⋊Dic53—22no—
126C4⋊Dic53—22no—
127C4⋊Dic53—22no—
128C4⋊Dic53—41no—
129C4⋊Dic53—41no—
130C4×Dic53—41no—
131C4×Dic53—41no—
132C2×#88322—22no—
133C2×#88322—42no—
134C2×#88322—41no—
135C2×#88322—42no—
136C23×C53—22no—
137C23×C53—41no—
138C23×C53—42no—
139C23×C53—41no—
140C23×C53—42no—
141C23×C53—42no—
142C23×C53—81no—
143C23×C53—82no—
144C23×C53—81no—
145C23×C53—82no—
146C23×C53#12011yescharacteristic
147D4×C53—22no—
148D4×C53—22no—
149D4×C53—44no—
150D4×C53—42no—
151D4×C53—41no—
152D4×C53—42no—
153D4×C53—44no—
154D4×C53—41no—
155D4×C53—82no—
156D4×C53—81no—
157D4×C53—81no—
158D4×C53—81no—
159D4×C53—81no—
160C2×C4×C53—22no—
161C2×C4×C53—41no—
162C2×C4×C53—42no—
163C2×C4×C53—82no—
164C2×C4×C53—81no—
165C2×C4×C53—81no—
166C2×C4×C53#12011yescharacteristic
167C2×C4×C53#12011yescharacteristic
168C2×C4×C53#12111yescharacteristic
169C2×C4×C53#12111yescharacteristic
170C2×C4×C53#12111yescharacteristic
171C2×C4×C53#12711yescharacteristic
172C2×C4×C53C2×D811yescharacteristic
173C2×C4×C53#13512yes—
174C2×C4×C53#13511yescharacteristic
175C2×C4×C53AGL(1,ℤ/8)12yes—
176C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
177C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
178C2×Dic53—42no—
179C2×Dic53—42no—
180C2×Dic53—42no—
181C2×Dic53—42no—
182C2×Dic53—42no—
183C2×Dic53—42no—
184Dic106—81no—
185Dic106—81no—
186Dic106—81no—
187Dic106—81no—
188C8×C53—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—82no—
198C22×C53—82no—
199C22×C53—82no—
200C22×C53—162no—
201C22×C53—162no—
202C22×C53#39211yescharacteristic
203C22×C53#44711yescharacteristic
204C22×C53#44912yes—
205C22×C53#44911yescharacteristic
206C22×C53#45412yes—
207C4×C53—21no—
208C4×C53—21no—
209C4×C53—81no—
210C4×C53#44711yescharacteristic
211C4×C53#44911yescharacteristic
212C4×C53#46312yes—
213C4×C53#46611yescharacteristic
214C4×C53#46911yescharacteristic
215C4×C53#49511yescharacteristic
216C4×C53#49611yescharacteristic
217Dic53—82no—
218Dic53—82no—
219C2×#486—531no—
220C2×#459D5311yescharacteristic
221#2547—531no—
222#1982—531no—
223C2×#332—531no—
224C2×D4⋊C8—531no—
225#1279—531no—
226C2×C53—82no—
227C2×C53—162no—
228C2×C53#162512yes—
229C2×C53#162812yes—
230C2×C53#199911yescharacteristic
231C2×C53#200311yescharacteristic
232C2×C53#201211yescharacteristic
233C22×D8—21no—
234C2×C4⋊Q8—531no—
235C2×#127C2×D5311yescharacteristic
236C2×#124—1061no—
237C2×C4×D4C2×D5311yescharacteristic
238#486—1062no—
239#486—1062no—
240#459—22no—
241#459—22no—
242C2×C4⋊C8—531no—
243C2×C4⋊C8C2×D5311yescharacteristic
244C2×#102—21no—
245C2×#102—1062no—
246C2×#102—1062no—
247C2×C4×C8—531no—
248#397—1062no—
249#391—1061no—
250#380—1062no—
251C4×C4⋊C4—531no—
252#332—1062no—
253D4⋊C8—1062no—
254C53#1005011yescharacteristic, complemented, Sylow
255C22×Q8—1061no—
256C22×D4—21no—
257C22×D4—21no—
258C22×D4—21no—
259C22×D4C22⋊D5311yescharacteristic
260C23×C4—21no—
261C2×D8—24no—
262C2×D8—42no—
263C2×D8—42no—
264C22×C8—21no—
265C22×C8—1061no—
266C22×C8—1061no—
267C4⋊Q8—1061no—
268C4⋊Q8—1061no—
269C4⋊Q8—1061no—
270C4⋊Q8—1061no—
271#127—21no—
272#127—21no—
273#127—21no—
274#127C22⋊D5312yes—
275#124—2122no—
276#124—2122no—
277C4×D4—21no—
278C4×D4—21no—
279C4×D4—21no—
280C4×D4—21no—
281C2×C4⋊C4—532no—
282C2×C4⋊C4—532no—
283C2×C4⋊C4—1061no—
284C2×C4⋊C4C22⋊D5311yescharacteristic
285C2×C22⋊C4—21no—
286C2×C22⋊C4—1064no—
287C2×C42—1061no—
288C2×C42C22×D5311yescharacteristic
289C4⋊C8—21no—
290C4⋊C8—1061no—
291C4⋊C8—1061no—
292C4⋊C8C22⋊D5312yes—
293#102—24no—
294#102—1068no—
295#102—2124no—
296C4×C8—532no—
297C4×C8—1061no—
298#95—1064no—
299#95—1061no—
300C24—21no—
301C24—41no—
302C2×Q8—2121no—
303C2×Q8—2121no—
304C2×Q8—2121no—
305C2×Q8—2121no—
306C2×Q8—2121no—
307C2×Q8—2121no—
308C2×D4—22no—
309C2×D4—21no—
310C2×D4—22no—
311C2×D4—21no—
312C2×D4—22no—
313C2×D4—21no—
314C2×D4—21no—
315C2×D4—21no—
316C2×D4—44no—
317C2×D4—41no—
318C2×D4—41no—
319C2×D4—41no—
320C2×D4—41no—
321C2×D4—42no—
322C2×D4—41no—
323C2×D4—42no—
324C2×D4—41no—
325C2×D4—41no—
326C2×D4#1171992212yes—
327C22×C4—21no—
328C22×C4—41no—
329C22×C4—41no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—1062no—
333C22×C4—1062no—
334C22×C4D4×D5311yescharacteristic
335C22×C4D4×D5311yescharacteristic
336C22×C4C2×C22⋊D5311yescharacteristic
337D8—48no—
338C2×C8—22no—
339C2×C8—41no—
340C2×C8—41no—
341C2×C8—1062no—
342C2×C8—2121no—
343C2×C8—2122no—
344C2×C8—2122no—
345C4⋊C4—21no—
346C4⋊C4—21no—
347C4⋊C4—1062no—
348C4⋊C4—1062no—
349C4⋊C4—1062no—
350C4⋊C4—1062no—
351C4⋊C4—2121no—
352C4⋊C4—2121no—
353C22⋊C4—41no—
354C22⋊C4—41no—
355C22⋊C4—2128no—
356C42—2121no—
357C42—2121no—
358C42D4×D5311yescharacteristic
359C42#1171994711yescharacteristic
360C42C2×C22⋊D5311yescharacteristic
361C42C2×C22⋊D5311yescharacteristic
362C23—22no—
363C23—42no—
364C23—41no—
365C23—41no—
366C23—42no—
367C23—42no—
368C23—81no—
369C23—82no—
370C23—81no—
371C23—82no—
372C23#1313318511yescharacteristic
373Q8—4241no—
374Q8—4241no—
375Q8—4241no—
376Q8—4241no—
377D4—22no—
378D4—22no—
379D4—44no—
380D4—42no—
381D4—41no—
382D4—42no—
383D4—44no—
384D4—41no—
385D4—82no—
386D4—81no—
387D4—81no—
388D4—81no—
389D4—81no—
390C2×C4—22no—
391C2×C4—42no—
392C2×C4—41no—
393C2×C4—81no—
394C2×C4—82no—
395C2×C4—81no—
396C2×C4—2122no—
397C2×C4—2122no—
398C2×C4—2122no—
399C2×C4—2122no—
400C2×C4—2122no—
401C2×C4—2122no—
402C2×C4#1313318511yescharacteristic
403C2×C4#1313318611yescharacteristic
404C2×C4#1313318811yescharacteristic
405C2×C4#1313318911yescharacteristic
406C2×C4#1313318911yescharacteristic
407C2×C4#1313321311yescharacteristic
408C2×C4C2×#1171992211yescharacteristic
409C2×C4#1313323612yes—
410C2×C4#1313323711yescharacteristic
411C2×C4#1313324112yes—
412C2×C4#1313324211yescharacteristic
413C2×C4#1313324311yescharacteristic
414C8—42no—
415C8—4241no—
416C8—2122no—
417C22—44no—
418C22—42no—
419C22—81no—
420C22—81no—
421C22—82no—
422C22—82no—
423C22—82no—
424C22—82no—
425C22—162no—
426C22—162no—
427C22#1681053111yescharacteristic
428C22#1681075611yescharacteristic
429C22#1681076512yes—
430C22#1681076611yescharacteristic
431C22#1681079412yes—
432C4—21no—
433C4—21no—
434C4—81no—
435C4—4242no—
436C4—4242no—
437C4#1681075711yescharacteristic
438C4#1681076811yescharacteristic
439C4#1681083612yes—
440C4#1681085711yescharacteristic
441C4#1681087411yescharacteristic
442C4#1681100611yescharacteristic
443C4#1681101111yescharacteristic
444C2—82no—
445C2—162no—
446C2#2027380212yes—
447C2#2027382312yes—
448C2#2027571111yescharacteristic
449C2#2027573411yescharacteristic
450C2#2027579011yescharacteristic
subgroupG/H
centerC23#13133185
commutator, cosocleC2×C42×C53C23
FrattiniC2×C42C22×D53
Fitting, prime radicalC2×C53×#459C2
socleC23×C53#120
hypercenterC2×#459D53
hypocenterC53#10050
LCSG ▷ C2×C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#459

Complements

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

NH ≅ G/Ntype
1. C2×#16810969C2semidirect
2. #20279707C2semidirect
3. #20275643C2semidirect
4. C2×#16810370C2semidirect
5. C2×D4⋊#88322C2semidirect
6. #20272554C2semidirect
254. C53#10050semidirect

Maximal subgroups

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

subgroupindex
C2×#168109692
#202797072
#202756432
C2×#168103702
C2×D4⋊#883222
#202725542
C2×C53×#4592
#1005053