Finite Group Databasebeta

← #27287296

Subgroups — #27287296

Order 13568.

446 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 446 conjugacy classes of subgroups — too many to draw. Only its 75 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20280888C211yesmaximal, characteristic, complemented
2#20273244C211yesmaximal, characteristic
3#20273231C211yesmaximal, characteristic
4#20272621C211yesmaximal, characteristic, complemented
5#20272601C211yesmaximal, characteristic, complemented
6#20272568C211yesmaximal, characteristic, complemented
7C53×#1964C211yesmaximal, characteristic
8#16810587C2211yescharacteristic
9#16810587C2211yescharacteristic
10#16810562—21no—
11#16810546—21no—
12#16810543—21no—
13#16810535—21no—
14#16810527—21no—
15#16810510—21no—
16#16810486C2211yescharacteristic
17C4×C22⋊Dic53C2211yescharacteristic
18#16810357—21no—
19#16810357—21no—
20#16810357—21no—
21#16810357—21no—
22C2×#121×C53C2211yescharacteristic
23C2×#99×C53—21no—
24C2×#99×C53—21no—
25C53×#380C2211yescharacteristic
26C4×C22⋊C4×C53C2211yescharacteristic
27C22×Dic106—21no—
28C2×#11719917—21no—
29C2×#11719917—21no—
30C2×C4⋊Dic53—21no—
31C2×#11719919—21no—
32C2×#11719919—21no—
33C2×C22⋊Dic53—21no—
34C2×C22⋊Dic53—21no—
35C2×C4×Dic53—21no—
36C22⋊#88322—21no—
37C22⋊#88322—21no—
38C22⋊#88322—21no—
39C22⋊#88322—21no—
40#13133124—21no—
41#13133124—21no—
42#13133124—21no—
43#13133124—21no—
44#13133124—21no—
45#13133124D411yescharacteristic
46#13133124D411yescharacteristic
47#13133124D411yescharacteristic
48#13133124D411yescharacteristic
49C22×D4×C53—21no—
50C22×D4×C53Q811yescharacteristic, complemented
51C23×C4×C53C2311yescharacteristic
52#121×C53—21no—
53#121×C53—21no—
54#121×C53—21no—
55#121×C53—21no—
56C2×C4⋊C4×C53—21no—
57C2×C4⋊C4×C53Q811yescharacteristic, complemented
58C2×C22⋊C4×C53—21no—
59C2×C22⋊C4×C53—21no—
60C2×C22⋊C4×C53D411yescharacteristic
61C2×C22⋊C4×C53D411yescharacteristic
62C2×C42×C53—21no—
63#99×C53—42no—
64#99×C53—42no—
65#95×C53—21no—
66#95×C53—21no—
67C24×C53—21no—
68C24×C53C2×D411yescharacteristic
69C2×D4×C53—21no—
70C2×D4×C53—21no—
71C2×D4×C53—41no—
72C2×D4×C53—41no—
73C2×D4×C53—41no—
74C2×D4×C53—41no—
75C2×D4×C53—41no—
76C2×D4×C53—41no—
77C2×D4×C53—81no—
78C22×C4×C53—21no—
79C22×C4×C53—21no—
80C22×C4×C53—21no—
81C22×C4×C53—41no—
82C22×C4×C53—81no—
83C22×C4×C53C2×D411yescharacteristic
84C22×C4×C53C2×D411yescharacteristic
85C22×C4×C53C2×Q811yescharacteristic
86C22×C4×C53#4111yescharacteristic
87C22×C4×C53#4111yescharacteristic
88C22×C4×C53#4111yescharacteristic
89C22×Dic53—21no—
90C22×Dic53—21no—
91C22×Dic53—21no—
92C22×Dic53—21no—
93C2×Dic106—42no—
94C2×Dic106—81no—
95C2×Dic106—81no—
96C4⋊C4×C53—21no—
97C4⋊C4×C53—21no—
98C4⋊C4×C53—81no—
99C22⋊C4×C53—22no—
100C22⋊C4×C53—41no—
101C22⋊C4×C53—41no—
102C22⋊C4×C53—41no—
103C22⋊C4×C53—41no—
104C22⋊C4×C53—41no—
105C42×C53—81no—
106C22⋊Dic53—42no—
107C22⋊Dic53—81no—
108C4⋊Dic53—81no—
109#11719919—42no—
110#11719919—81no—
111C4×Dic53—81no—
112#11719917—81no—
113#11719917—81no—
114C2×#88322—41no—
115C2×#88322—41no—
116C23×C53—21no—
117C23×C53—21no—
118C23×C53—41no—
119C23×C53—41no—
120C23×C53—41no—
121C23×C53—41no—
122C23×C53—41no—
123C23×C53—81no—
124C23×C53—81no—
125C23×C53#12011yescharacteristic
126C23×C53C22⋊Q811yescharacteristic
127C23×C53#12411yescharacteristic
128D4×C53—81no—
129D4×C53—81no—
130D4×C53—81no—
131D4×C53—81no—
132D4×C53—81no—
133D4×C53—81no—
134C2×C4×C53—41no—
135C2×C4×C53—41no—
136C2×C4×C53—41no—
137C2×C4×C53—41no—
138C2×C4×C53—41no—
139C2×C4×C53—41no—
140C2×C4×C53—41no—
141C2×C4×C53—41no—
142C2×C4×C53—41no—
143C2×C4×C53—41no—
144C2×C4×C53—41no—
145C2×C4×C53—41no—
146C2×C4×C53—42no—
147C2×C4×C53—41no—
148C2×C4×C53—41no—
149C2×C4×C53—41no—
150C2×C4×C53—81no—
151C2×C4×C53—81no—
152C2×C4×C53C22⋊Q811yescharacteristic
153C2×C4×C53C22⋊Q811yescharacteristic
154C2×C4×C53#12411yescharacteristic
155C2×C4×C53#12411yescharacteristic
156C2×Dic53—22no—
157C2×Dic53—41no—
158C2×Dic53—41no—
159C2×Dic53—41no—
160C2×Dic53—41no—
161C2×Dic53—41no—
162C2×Dic53—41no—
163C2×Dic53—41no—
164C2×Dic53—41no—
165C2×Dic53—41no—
166C2×Dic53—41no—
167C2×Dic53—41no—
168Dic106—84no—
169#88322—81no—
170#88322—81no—
171#10382—531nomaximal, Sylow
172C22×C53—21no—
173C22×C53—21no—
174C22×C53—21no—
175C22×C53—21no—
176C22×C53—21no—
177C22×C53—21no—
178C22×C53—21no—
179C22×C53—21no—
180C22×C53—81no—
181C22×C53—81no—
182C22×C53—81no—
183C22×C53—81no—
184C22×C53—81no—
185C22×C53—161no—
186C22×C53—161no—
187C22×C53#39311yescharacteristic
188C22×C53#45311yescharacteristic
189C22×C53#45411yescharacteristic
190C4×C53—21no—
191C4×C53—21no—
192C4×C53—21no—
193C4×C53—21no—
194C4×C53—81no—
195C4×C53—81no—
196C4×C53—82no—
197C4×C53—81no—
198Dic53—42no—
199Dic53—81no—
200Dic53—81no—
201Dic53—81no—
202#2665—531no—
203#1964D5311yescharacteristic
204#1510—531no—
205#1507—531no—
206#1305—531no—
207#1297—531no—
208#1284—531no—
209C2×C53—21no—
210C2×C53—21no—
211C2×C53—41no—
212C2×C53—41no—
213C2×C53—161no—
214C2×C53#202711yescharacteristic
215C2×C53#203111yescharacteristic
216C2×C53#203211yescharacteristic
217C2×#121C2×D5311yescharacteristic
218C2×#99—21no—
219C2×#99—21no—
220#407—531no—
221#407—531no—
222#400—1061no—
223#396—1061no—
224#395—1061no—
225#393—1061no—
226#391—1061no—
227#387—1061no—
228#381—531no—
229#380C2×D5311yescharacteristic
230C4×C22⋊C4—531no—
231C4×C22⋊C4C2×D5311yescharacteristic
232#327—1061no—
233#327—1061no—
234#327—1061no—
235#327—1061no—
236C53#1038211yescharacteristic, complemented, Sylow
237C22×Q8—1061no—
238C22×D4—21no—
239C22×D4Dic10611yescharacteristic, complemented
240C23×C4C22×D5311yescharacteristic
241C2×M16—1061no—
242C2×M16—1061no—
243#121—21no—
244#121—21no—
245#121—21no—
246#121—21no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—1061no—
249C2×C4⋊C4—1061no—
250C2×C4⋊C4—1061no—
251C2×C4⋊C4Dic10611yescharacteristic, complemented
252C2×C22⋊C4—21no—
253C2×C22⋊C4—21no—
254C2×C22⋊C4—1061no—
255C2×C22⋊C4—1061no—
256C2×C22⋊C4C22⋊D5311yescharacteristic
257C2×C22⋊C4C22⋊D5311yescharacteristic
258C2×C42—21no—
259C2×C42—1061no—
260#99—42no—
261#99—42no—
262C22⋊C8—1061no—
263C22⋊C8—1061no—
264C22⋊C8—1061no—
265C22⋊C8—1061no—
266#95—21no—
267#95—21no—
268#95—531no—
269#95—531no—
270#95—531no—
271#95—531no—
272#95—1061no—
273#95—1061no—
274#95—1061no—
275#95—1061no—
276#95—1061no—
277C24—21no—
278C24C2×C22⋊D5311yescharacteristic
279C2×Q8—4241no—
280C2×Q8—4241no—
281C2×Q8—2122no—
282C2×D4—21no—
283C2×D4—21no—
284C2×D4—41no—
285C2×D4—41no—
286C2×D4—41no—
287C2×D4—41no—
288C2×D4—41no—
289C2×D4—41no—
290C2×D4—81no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—41no—
295C22×C4—81no—
296C22×C4—1061no—
297C22×C4—1061no—
298C22×C4—1061no—
299C22×C4—1061no—
300C22×C4C2×Dic10611yescharacteristic
301C22×C4#1171994511yescharacteristic
302C22×C4D4×D5311yescharacteristic
303C22×C4D4×D5311yescharacteristic
304C22×C4#1171994711yescharacteristic
305C22×C4#1171994711yescharacteristic
306M16—4241no—
307M16—4241no—
308C2×C8—2121no—
309C2×C8—2121no—
310C4⋊C4—21no—
311C4⋊C4—21no—
312C4⋊C4—81no—
313C4⋊C4—4241no—
314C4⋊C4—4241no—
315C4⋊C4—2122no—
316C22⋊C4—22no—
317C22⋊C4—41no—
318C22⋊C4—41no—
319C22⋊C4—41no—
320C22⋊C4—41no—
321C22⋊C4—41no—
322C22⋊C4—4241no—
323C22⋊C4—2122no—
324C42—81no—
325C42—4241no—
326C23—21no—
327C23—21no—
328C23—41no—
329C23—41no—
330C23—41no—
331C23—41no—
332C23—41no—
333C23—81no—
334C23—81no—
335C23#1313318511yescharacteristic
336C23#1313319611yescharacteristic
337C23#1313320411yescharacteristic
338Q8—4244no—
339D4—81no—
340D4—81no—
341D4—81no—
342D4—81no—
343D4—81no—
344D4—81no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—42no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—81no—
362C2×C4—81no—
363C2×C4—1062no—
364C2×C4—2121no—
365C2×C4—2121no—
366C2×C4—2121no—
367C2×C4—2121no—
368C2×C4—2121no—
369C2×C4—2121no—
370C2×C4—2121no—
371C2×C4—2121no—
372C2×C4—2121no—
373C2×C4—2121no—
374C2×C4—2121no—
375C2×C4#1313319411yescharacteristic
376C2×C4#1313319411yescharacteristic
377C2×C4#1313320311yescharacteristic
378C2×C4#1313320311yescharacteristic
379C8—4241no—
380C8—4241no—
381C22—21no—
382C22—21no—
383C22—21no—
384C22—21no—
385C22—21no—
386C22—21no—
387C22—21no—
388C22—21no—
389C22—81no—
390C22—81no—
391C22—81no—
392C22—81no—
393C22—81no—
394C22—161no—
395C22—161no—
396C22#1681053511yescharacteristic
397C22#1681079011yescharacteristic
398C22#1681079511yescharacteristic
399C4—21no—
400C4—21no—
401C4—21no—
402C4—21no—
403C4—81no—
404C4—81no—
405C4—82no—
406C4—81no—
407C4—4241no—
408C4—4241no—
409C4—4241no—
410C4—2122no—
411C2—21no—
412C2—21no—
413C2—41no—
414C2—41no—
415C2—161no—
416C2#2027587111yescharacteristic
417C2#2027588911yescharacteristic
418C2#2027589211yescharacteristic
subgroupG/H
centerC22#16810535
commutator, cosocleC23×C4×C53C23
FrattiniC23×C4C22×D53
FittingC53×#1964C2
socleC22×C53#393
prime radicalC2×#121×C53C22
hypercenter#1964D53
hypocenterC53#10382
LCSG ▷ C23×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1964

Complements

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

NH ≅ G/Ntype
1. #20280888C2semidirect
4. #20272621C2semidirect
5. #20272601C2semidirect
6. #20272568C2semidirect
50. C22×D4×C53Q8semidirect
57. C2×C4⋊C4×C53Q8semidirect
236. C53#10382semidirect
239. C22×D4Dic106semidirect
251. C2×C4⋊C4Dic106semidirect

Maximal subgroups

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

subgroupindex
#202808882
#202732442
#202732312
#202726212
#202726012
#202725682
C53×#19642
#1038253