Finite Group Databasebeta

← #27301148

Subgroups — #27301148

Order 13568.

590 conjugacy classes of subgroups.

The subgroup lattice has 404 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×D848C211yesmaximal, characteristic
2C2×C4×D424C211yesmaximal, characteristic, complemented
3C2×#16810425C211yesmaximal, characteristic, complemented
4C2×#16810425C211yesmaximal, characteristic, complemented
5#20275581C211yesmaximal, characteristic, complemented
6#20272807C211yesmaximal, characteristic, complemented
7C53×#1374C211yesmaximal, characteristic, complemented
8C22×D424C2211yescharacteristic
9C22×D424C2211yescharacteristic
10C2×C4×D212—21no—
11C2×D848—22no—
12C2×D848—22no—
13C2×D848C414yescomplemented
14C4×D424—22no—
15C4×D424—22no—
16C2×#13133151C2211yescharacteristic
17C2×#13133151C2211yescharacteristic
18C2×#13133139—21no—
19C2×#13133139—21no—
20#16810525—21no—
21#16810425—22no—
22#16810425—22no—
23C22×C16×C53C2211yescharacteristic
24C22×C16×C53C2211yescharacteristic
25C2×C4×C8×C53C2211yescharacteristic
26C22×D212—21no—
27C22×D212—21no—
28C22×C4×D53—41no—
29C2×D424—21no—
30C2×D424—21no—
31C2×D424—21no—
32C2×D424—21no—
33C2×D424—21no—
34C2×D424—21no—
35C2×D424—21no—
36C2×D424—21no—
37C2×D424C2×C412yes—
38C2×D424C2×C412yescomplemented
39C2×D424D412yes—
40C2×D424D412yes—
41C4×D212—24no—
42C4×D212—42no—
43C2×C4⋊Dic53—21no—
44C2×C4⋊Dic53—21no—
45C2×#11719921—41no—
46C2×#11719921—41no—
47C2×#11719921—41no—
48D848—28no—
49#13133151—21no—
50#13133151—21no—
51#13133151—21no—
52#13133151D412yes—
53#13133139—24no—
54#13133139—42no—
55C22×C8×C53D411yescharacteristic
56C22×C8×C53C2311yescharacteristic
57C2×C42×C53D411yescharacteristic
58C2×C16×C53—21no—
59C2×C16×C53—21no—
60C2×C16×C53—21no—
61C2×C16×C53—21no—
62C2×C16×C53—21no—
63C2×C16×C53C2×C412yescomplemented
64C4×C8×C53—21no—
65C4×C8×C53—21no—
66C23×D53—41no—
67C23×D53—41no—
68C22×C4×C53D811yescharacteristic
69C22×C4×C53D811yescharacteristic
70C22×C4×C53C2×D411yescharacteristic
71C22×Dic53—41no—
72C22×Dic53—41no—
73C2×D212—22no—
74C2×D212—22no—
75C2×D212—22no—
76C2×D212—22no—
77C2×D212—41no—
78C2×D212—41no—
79C2×D212—41no—
80C2×D212—41no—
81C2×D212—41no—
82C2×D212—41no—
83C2×D212—41no—
84C2×D212—41no—
85C2×C4×D53—44no—
86C2×C4×D53—82no—
87C2×C4×D53—82no—
88C2×C8×C53—21no—
89C2×C8×C53—21no—
90C2×C8×C53—21no—
91C2×C8×C53C22×C411yescharacteristic
92C2×C8×C53C2×D411yescharacteristic
93C2×C8×C53C2×D411yescharacteristic
94C2×C8×C53C2×D411yescharacteristic
95C2×C8×C53#4111yescharacteristic
96C2×C8×C53#4111yescharacteristic
97C42×C53—21no—
98C42×C53—21no—
99#11719921—44no—
100#11719921—82no—
101#11719921—82no—
102C4⋊Dic53—22no—
103C4⋊Dic53—41no—
104C4⋊Dic53—41no—
105C4⋊Dic53—41no—
106D424—22no—
107D424—22no—
108D424—22no—
109D424—22no—
110D424—22no—
111D424—41no—
112D424—41no—
113D424C22⋊C414yescomplemented
114C16×C53—22no—
115C16×C53—41no—
116C23×C53C2×D811yescharacteristic
117C22×D53—42no—
118C22×D53—42no—
119C22×D53—42no—
120C22×D53—42no—
121C22×D53—81no—
122C22×D53—81no—
123C22×D53—81no—
124C22×D53—81no—
125C22×D53—81no—
126C22×D53—81no—
127C22×D53—81no—
128C22×D53—81no—
129C22×D53—81no—
130C22×D53—81no—
131C2×C4×C53—21no—
132C2×C4×C53—21no—
133C2×C4×C53—21no—
134C2×C4×C53—21no—
135C2×C4×C53—21no—
136C2×C4×C53D1612yescomplemented
137C2×C4×C53C4×D411yescharacteristic
138C2×C4×C53#12011yescharacteristic
139C2×C4×C53#12111yescharacteristic
140C2×C4×C53#13511yescharacteristic
141C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
142C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
143C2×Dic53—42no—
144C2×Dic53—81no—
145C2×Dic53—81no—
146C2×Dic53—81no—
147C2×Dic53—81no—
148C2×Dic53—81no—
149D212—24no—
150D212—42no—
151D212—42no—
152D212—42no—
153D212—42no—
154D212—42no—
155D212—81no—
156D212—81no—
157C4×D53—88no—
158C8×C53—41no—
159C8×C53C2×C22⋊C411yescharacteristic
160C8×C53C4×D411yescharacteristic
161C8×C53#12111yescharacteristic
162C8×C53#12311yescharacteristic
163#13012—531nomaximal, Sylow
164C22×C53C4×D811yescharacteristic
165C22×C53#44711yescharacteristic
166C22×C53#45911yescharacteristic
167C22×C53C2×D1611yescharacteristic
168C22×C53#50811yescharacteristic
169C22×C53#50911yescharacteristic
170C22×C53#50911yescharacteristic
171C2×D53—44no—
172C2×D53—82no—
173C2×D53—82no—
174C2×D53—82no—
175C2×D53—82no—
176C2×D53—82no—
177C2×D53—82no—
178C2×D53—82no—
179C2×D53—82no—
180C2×D53—82no—
181C2×D53—161no—
182C2×D53—161no—
183C2×D53—161no—
184C2×D53—161no—
185C4×C53—22no—
186C4×C53—41no—
187C4×C53#38611yescharacteristic
188C4×C53#44211yescharacteristic
189C4×C53#44911yescharacteristic
190C4×C53#46311yescharacteristic
191Dic53—82no—
192Dic53—161no—
193C22×D16—531no—
194C2×C4×D8—531no—
195C2×#357—531no—
196C2×#357—531no—
197#1970—531no—
198#1380—531no—
199#1374D5311yescharacteristic, complemented
200C2×C53#187911yescharacteristic
201C2×C53C4×D1611yescharacteristic
202C2×C53#217711yescharacteristic
203C2×C53#218411yescharacteristic
204C2×C53#218611yescharacteristic
205C2×C53#220611yescharacteristic
206C2×C53#221011yescharacteristic
207D53—84no—
208D53—162no—
209C22×D8—531no—
210C22×D8—531no—
211C2×C4×D4—1061no—
212C2×D16—534no—
213C2×D16—1062no—
214C2×D16—1062no—
215C22×C16C2×D5311yescharacteristic
216C22×C16C2×D5311yescharacteristic
217C4×D8—1062no—
218C4×D8—1062no—
219C2×#107—531no—
220C2×#107—531no—
221C2×#102—1061no—
222C2×#102—1061no—
223C2×C4×C8C2×D5311yescharacteristic
224#390—1061no—
225#357—1062no—
226#357—1062no—
227C53#1301211yescharacteristic, complemented, Sylow
228C22×D4—1061no—
229C22×D4—1061no—
230C23×C4—2121no—
231C2×D8—532no—
232C2×D8—532no—
233C2×D8—532no—
234C2×D8—532no—
235C2×D8—1061no—
236C2×D8—1061no—
237C2×D8—1061no—
238C2×D8—1061no—
239C2×D8—1061no—
240C2×D8—1061no—
241C2×D8—1061no—
242C2×D8—1061no—
243C22×C8D21211yescharacteristic
244C22×C8C22×D5311yescharacteristic
245C4×D4—1064no—
246C4×D4—2122no—
247C2×C4⋊C4—1061no—
248C2×C4⋊C4—1061no—
249C2×C22⋊C4—2121no—
250C2×C22⋊C4—2121no—
251C2×C22⋊C4—2121no—
252C2×C42D21211yescharacteristic
253D16—1068no—
254C2×C16—21no—
255C2×C16—21no—
256C2×C16—21no—
257C2×C16—21no—
258C2×C16—21no—
259C2×C16C4×D5312yescomplemented
260#107—532no—
261#107—1061no—
262#107—1061no—
263#107—1061no—
264#102—1064no—
265#102—2122no—
266C4×C8—21no—
267C4×C8—21no—
268C24—2121no—
269C24—2121no—
270C2×D4—1062no—
271C2×D4—1062no—
272C2×D4—1062no—
273C2×D4—1062no—
274C2×D4—2121no—
275C2×D4—2121no—
276C2×D4—2121no—
277C2×D4—2121no—
278C2×D4—2121no—
279C2×D4—2121no—
280C2×D4—2121no—
281C2×D4—2121no—
282C22×C4—4242no—
283C22×C4—4242no—
284C22×C4—2124no—
285C22×C4—2121no—
286C22×C4—2121no—
287C22×C4D42411yescharacteristic
288C22×C4D42411yescharacteristic
289C22×C4C2×D21211yescharacteristic
290D8—534no—
291D8—1062no—
292D8—1062no—
293D8—1062no—
294D8—1062no—
295D8—1062no—
296D8—2121no—
297D8—2121no—
298C2×C8—21no—
299C2×C8—21no—
300C2×C8—21no—
301C2×C8C2×C4×D5311yescharacteristic
302C2×C8#1171994511yescharacteristic
303C2×C8D4×D5311yescharacteristic
304C2×C8D4×D5311yescharacteristic
305C2×C8D4×D5311yescharacteristic
306C2×C8#1171994911yescharacteristic
307C4⋊C4—1062no—
308C4⋊C4—2121no—
309C4⋊C4—2121no—
310C4⋊C4—2121no—
311C22⋊C4—4242no—
312C22⋊C4—4242no—
313C22⋊C4—2124no—
314C42—21no—
315C42—21no—
316C16—22no—
317C16—41no—
318C23—4241no—
319C23—4241no—
320C23—4241no—
321C23—4241no—
322C23—4241no—
323C23—4241no—
324C23—4241no—
325C23—4241no—
326C23—4241no—
327C23—4241no—
328C23—2122no—
329C23—2122no—
330C23—2122no—
331C23—2122no—
332C23C2×D42411yescharacteristic
333D4—1064no—
334D4—4241no—
335D4—4241no—
336D4—2122no—
337D4—2122no—
338D4—2122no—
339D4—2122no—
340D4—2122no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—4248no—
347C2×C4—4241no—
348C2×C4—4241no—
349C2×C4—4241no—
350C2×C4—4241no—
351C2×C4—4241no—
352C2×C4—2122no—
353C2×C4D84812yescomplemented
354C2×C4C4×D21211yescharacteristic
355C2×C4#1313318711yescharacteristic
356C2×C4#1313319211yescharacteristic
357C2×C4#1313323911yescharacteristic
358C2×C4#1313324511yescharacteristic
359C2×C4#1313324511yescharacteristic
360C8—41no—
361C8C22⋊C4×D5311yescharacteristic
362C8#1313317911yescharacteristic
363C8#1313319011yescharacteristic
364C8#1313320211yescharacteristic
365C22—8481no—
366C22—8481no—
367C22—8481no—
368C22—8481no—
369C22—4242no—
370C22—4242no—
371C22—4242no—
372C22—4242no—
373C22—4242no—
374C22—4242no—
375C22—4242no—
376C22—4242no—
377C22—4242no—
378C22—2124no—
379C22C4×D42411yescharacteristic
380C22#1681075911yescharacteristic
381C22#1681081911yescharacteristic
382C22C2×D84811yescharacteristic
383C22#1681105511yescharacteristic
384C22#1681106111yescharacteristic
385C22#1681106111yescharacteristic
386C4—22no—
387C4—41no—
388C4—8481no—
389C4—4242no—
390C4#1681050111yescharacteristic
391C4#1681073611yescharacteristic
392C4#1681077011yescharacteristic
393C4#1681083911yescharacteristic
394C2—8482no—
395C2—4244no—
396C2#2027506211yescharacteristic
397C2C4×D84811yescharacteristic
398C2#2027659211yescharacteristic
399C2#2027662911yescharacteristic
400C2#2027664011yescharacteristic
401C2#2027675111yescharacteristic
402C2#2027677111yescharacteristic
subgroupG/H
centerC23C2×D424
commutatorC2×C8×C53C22×C4
FrattiniC22×C8C22×D53
FittingC53×#1374C2
socleC23×C53C2×D8
prime radicalC22×D848C2
cosocleC22×C8×C53C23
hypercenter#1374D53
hypocenterC53#13012
LCSG ▷ C2×C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×C4×C8 ◁ #1374

Complements

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

NH ≅ G/Ntype
2. C2×C4×D424C2semidirect
3. C2×#16810425C2semidirect
4. C2×#16810425C2semidirect
5. #20275581C2semidirect
6. #20272807C2semidirect
7. C53×#1374C2semidirect
13. C2×D848C4semidirect
38. C2×D424C2×C4semidirect
63. C2×C16×C53C2×C4semidirect
113. D424C22⋊C4semidirect
136. C2×C4×C53D16semidirect
199. #1374D53semidirect
227. C53#13012semidirect
259. C2×C16C4×D53semidirect
353. C2×C4D848semidirect

Maximal subgroups

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

subgroupindex
C22×D8482
C2×C4×D4242
C2×#168104252
#202755812
#202728072
C53×#13742
#1301253