Finite Group Databasebeta

← #27280409

Subgroups — #27280409

Order 13568.

452 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 452 conjugacy classes of subgroups — too many to draw. Only its 93 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
1C2×#16810961C211yesmaximal, characteristic
2#20283369C211yesmaximal, characteristic
3#20274812C211yesmaximal, characteristic
4#20273235C211yesmaximal, characteristic
5#20272591C211yesmaximal, characteristic
6#20272561C211yesmaximal, characteristic
7C53×#1472C211yesmaximal, characteristic
8C2×#13133176C2211yescharacteristic
9#16810961—21no—
10#16810961—21no—
11#16810961—21no—
12#16810961—21no—
13#16810681—22no—
14#16810677—21no—
15#16810673—21no—
16C2×#13133150C411yescharacteristic, complemented
17C2×#13133148C411yescharacteristic, complemented
18#16810649—21no—
19C2×#13133141C411yescharacteristic, complemented
20C2×#13133141C411yescharacteristic, complemented
21#16810587C2211yescharacteristic
22C2×C22⋊#88322C2211yescharacteristic
23#16810578—21no—
24C4×C22⋊Dic53C2211yescharacteristic
25#16810384—21no—
26#16810384—21no—
27#16810360—22no—
28C2×C22⋊Q8×C53C2211yescharacteristic
29C53×#407C2211yescharacteristic
30C2×C22⋊C8×C53C2211yescharacteristic
31C53×#324—21no—
32C53×#324—21no—
33C2×#11719917—21no—
34C22×#88322C2×C411yescharacteristic
35C22×#88322C2×C411yescharacteristic
36#13133176—21no—
37#13133176—21no—
38#13133176D411yescharacteristic
39#13133176D411yescharacteristic
40#13133176D411yescharacteristic
41#13133176D411yescharacteristic
42C2×C4⋊Dic53C2×C411yescharacteristic
43C2×C4⋊Dic53C2×C411yescharacteristic
44C2×C22⋊Dic53—21no—
45C2×C22⋊Dic53—21no—
46C2×C4×Dic53—21no—
47C2×C4×Dic53—21no—
48#13133150—21no—
49#13133150—21no—
50#13133148—21no—
51#13133148—21no—
52#13133146—22no—
53#13133146—22no—
54#13133141—21no—
55#13133141—21no—
56#13133141—21no—
57#13133141—21no—
58C22⋊#88322—21no—
59C22⋊#88322—21no—
60C22⋊#88322—21no—
61C22⋊#88322D412yes—
62#13133127—22no—
63C8×Dic53—22no—
64#13133124—21no—
65C22×Q8×C53C2×C411yescharacteristic
66C23×C4×C53C2311yescharacteristic
67C2×M16×C53—21no—
68C22×C8×C53—21no—
69C22⋊Q8×C53—21no—
70C22⋊Q8×C53—21no—
71C22⋊Q8×C53—21no—
72C22⋊Q8×C53—21no—
73C2×C4⋊C4×C53—21no—
74C2×C4⋊C4×C53C2×C411yescharacteristic
75C2×C22⋊C4×C53—21no—
76C22⋊C8×C53—21no—
77C22⋊C8×C53—21no—
78C22⋊C8×C53—21no—
79C22⋊C8×C53D411yescharacteristic
80C22⋊C8×C53D411yescharacteristic
81C24×C53#4111yescharacteristic
82C2×Q8×C53—21no—
83C2×Q8×C53—21no—
84C2×Q8×C53—41no—
85C2×Q8×C53—41no—
86C22×C4×C53—21no—
87C22×C4×C53—21no—
88C22×C4×C53—41no—
89C22×C4×C53—41no—
90C22×C4×C53C22×C411yescharacteristic
91C22×C4×C53C2×D411yescharacteristic
92C22×C4×C53C2×D411yescharacteristic
93C22×C4×C53C2×D411yescharacteristic
94C22×C4×C53C2×D411yescharacteristic
95C22×C4×C53#4111yescharacteristic
96C22×Dic53—21no—
97C22×Dic53—21no—
98C22×Dic53—41no—
99M16×C53—41no—
100M16×C53—41no—
101C2×C8×C53—21no—
102C2×C8×C53—21no—
103C2×C8×C53—41no—
104C2×C8×C53—41no—
105C2×C8×C53—41no—
106C4⋊C4×C53—21no—
107C4⋊C4×C53—21no—
108C4⋊C4×C53—41no—
109C4⋊C4×C53—41no—
110C22⋊C4×C53—41no—
111C22⋊C4×C53—41no—
112C22⋊Dic53—21no—
113C22⋊Dic53—21no—
114C22⋊Dic53—21no—
115C22⋊Dic53—21no—
116C22⋊Dic53—22no—
117C22⋊Dic53—41no—
118C4⋊Dic53—21no—
119C4⋊Dic53—21no—
120C4⋊Dic53C22⋊C411yescharacteristic
121C4⋊Dic53C22⋊C411yescharacteristic
122C4⋊Dic53C22⋊C411yescharacteristic
123C4⋊Dic53C22⋊C411yescharacteristic
124C4×Dic53—21no—
125C4×Dic53—21no—
126C4×Dic53—21no—
127C4×Dic53—21no—
128C4×Dic53—41no—
129C4×Dic53—41no—
130#11719917—41no—
131#11719917—41no—
132C2×#88322—21no—
133C2×#88322—21no—
134C2×#88322—22no—
135C2×#88322—41no—
136C2×#88322—41no—
137C23×C53—21no—
138C23×C53—21no—
139C23×C53—41no—
140C23×C53—41no—
141C23×C53C4×D411yescharacteristic
142C23×C53#12111yescharacteristic
143C23×C53#12311yescharacteristic
144Q8×C53—41no—
145Q8×C53—41no—
146Q8×C53—81no—
147C2×C4×C53—21no—
148C2×C4×C53—21no—
149C2×C4×C53—21no—
150C2×C4×C53—21no—
151C2×C4×C53—21no—
152C2×C4×C53—21no—
153C2×C4×C53—41no—
154C2×C4×C53—41no—
155C2×C4×C53—41no—
156C2×C4×C53—41no—
157C2×C4×C53—41no—
158C2×C4×C53—41no—
159C2×C4×C53—41no—
160C2×C4×C53—81no—
161C2×C4×C53C2×C22⋊C411yescharacteristic
162C2×C4×C53C4×D411yescharacteristic
163C2×C4×C53#12011yescharacteristic
164C2×C4×C53#12111yescharacteristic
165C2×Dic53—21no—
166C2×Dic53—21no—
167C2×Dic53—21no—
168C2×Dic53—21no—
169C2×Dic53—41no—
170C2×Dic53—41no—
171C2×Dic53—42no—
172C2×Dic53—81no—
173C2×Dic53—81no—
174C2×Dic53—81no—
175C8×C53—41no—
176C8×C53—41no—
177C8×C53—81no—
178#88322—42no—
179#88322—41no—
180#88322—41no—
181#9240—531nomaximal, Sylow
182C22×C53—21no—
183C22×C53—21no—
184C22×C53—21no—
185C22×C53—21no—
186C22×C53—21no—
187C22×C53—21no—
188C22×C53—21no—
189C22×C53—21no—
190C22×C53—81no—
191C22×C53—81no—
192C22×C53#38611yescharacteristic
193C22×C53#44311yescharacteristic
194C22×C53#44411yescharacteristic
195C4×C53—21no—
196C4×C53—21no—
197C4×C53—21no—
198C4×C53—81no—
199C4×C53—81no—
200C4×C53#41011yescharacteristic
201C4×C53#41011yescharacteristic
202Dic53—41no—
203Dic53—41no—
204Dic53—41no—
205Dic53—41no—
206Dic53—82no—
207C2×#485—531no—
208#2924—531no—
209#1826—531no—
210#1508—531no—
211#1472D5311yescharacteristic
212#1293—531no—
213#1282—531no—
214C2×C53—21no—
215C2×C53—21no—
216C2×C53—41no—
217C2×C53—41no—
218C2×C53#187611yescharacteristic
219C2×C53#189911yescharacteristic
220C2×C53#190211yescharacteristic
221C2×C22⋊Q8C2×D5311yescharacteristic
222C2×#117—531no—
223#485—1061no—
224#485—1061no—
225#485—1061no—
226#485—1061no—
227#433—1062no—
228#432—1061no—
229#431—1061no—
230C2×#107—531no—
231C2×#106—531no—
232#424—1061no—
233C2×#103—531no—
234C2×#103—531no—
235#407—531no—
236#407C2×D5311yescharacteristic
237C2×C22⋊C8—531no—
238C2×C22⋊C8C2×D5311yescharacteristic
239#405—1061no—
240C4×C22⋊C4—531no—
241#339—1061no—
242#339—1061no—
243#328—1062no—
244#324—21no—
245#324—21no—
246C53#924011yescharacteristic, complemented, Sylow
247C22×Q8C4×D5311yescharacteristic
248C23×C4C22×D5311yescharacteristic
249C2×M16—21no—
250C2×M16—1061no—
251C22×C8—21no—
252C22×C8—531no—
253C22×C8—531no—
254C22⋊Q8—21no—
255C22⋊Q8—21no—
256C22⋊Q8—21no—
257C22⋊Q8—21no—
258#117—531no—
259#117—531no—
260#117—531no—
261#117—531no—
262#117—1061no—
263#117—1061no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—531no—
266C2×C4⋊C4—531no—
267C2×C4⋊C4C4×D5311yescharacteristic
268C2×C22⋊C4—21no—
269C2×C22⋊C4—1061no—
270C2×C22⋊C4—1061no—
271C2×C42—1061no—
272C2×C42—1061no—
273#107—1061no—
274#107—1061no—
275#106—1061no—
276#106—1061no—
277C4⋊C8—1062no—
278C4⋊C8—1062no—
279#103—1061no—
280#103—1061no—
281#103—1061no—
282#103—1061no—
283C22⋊C8—21no—
284C22⋊C8—21no—
285C22⋊C8—21no—
286C22⋊C8—532no—
287C22⋊C8—1061no—
288C22⋊C8—1061no—
289C22⋊C8—1061no—
290C22⋊C8C22⋊D5311yescharacteristic
291C22⋊C8C22⋊D5311yescharacteristic
292#97—1062no—
293C4×C8—1062no—
294#95—1061no—
295C24#1171994511yescharacteristic
296C2×Q8—21no—
297C2×Q8—21no—
298C2×Q8—41no—
299C2×Q8—41no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—41no—
303C22×C4—41no—
304C22×C4—1061no—
305C22×C4—1061no—
306C22×C4—2121no—
307C22×C4C2×C4×D5311yescharacteristic
308C22×C4D4×D5311yescharacteristic
309C22×C4D4×D5311yescharacteristic
310C22×C4D4×D5311yescharacteristic
311C22×C4#1171994711yescharacteristic
312C22×C4C2×C22⋊D5311yescharacteristic
313M16—41no—
314M16—41no—
315M16—2121no—
316M16—2121no—
317C2×C8—21no—
318C2×C8—21no—
319C2×C8—41no—
320C2×C8—41no—
321C2×C8—41no—
322C2×C8—1061no—
323C2×C8—1061no—
324C2×C8—1062no—
325C2×C8—2121no—
326C2×C8—2121no—
327C4⋊C4—21no—
328C4⋊C4—21no—
329C4⋊C4—41no—
330C4⋊C4—41no—
331C4⋊C4—531no—
332C4⋊C4—531no—
333C4⋊C4—531no—
334C4⋊C4—531no—
335C4⋊C4—1061no—
336C4⋊C4—1061no—
337C22⋊C4—41no—
338C22⋊C4—41no—
339C22⋊C4—1061no—
340C22⋊C4—1061no—
341C22⋊C4—1061no—
342C22⋊C4—1061no—
343C22⋊C4—1062no—
344C22⋊C4—2121no—
345C42—1061no—
346C42—1061no—
347C42—1061no—
348C42—1061no—
349C42—2121no—
350C42—2121no—
351C23—21no—
352C23—21no—
353C23—41no—
354C23—41no—
355C23C4×C22⋊D5311yescharacteristic
356C23#1313319011yescharacteristic
357C23#1313319911yescharacteristic
358Q8—41no—
359Q8—41no—
360Q8—81no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—81no—
375C2×C4—1061no—
376C2×C4—1061no—
377C2×C4—1061no—
378C2×C4—1061no—
379C2×C4—4241no—
380C2×C4—4241no—
381C2×C4—4241no—
382C2×C4—2122no—
383C2×C4—2121no—
384C2×C4—2121no—
385C2×C4C22⋊C4×D5311yescharacteristic
386C2×C4#1313317711yescharacteristic
387C2×C4#1313318511yescharacteristic
388C2×C4#1313318811yescharacteristic
389C8—41no—
390C8—41no—
391C8—81no—
392C8—2122no—
393C8—2121no—
394C8—2121no—
395C22—21no—
396C22—21no—
397C22—21no—
398C22—21no—
399C22—21no—
400C22—21no—
401C22—21no—
402C22—21no—
403C22—81no—
404C22—81no—
405C22#1681049911yescharacteristic
406C22#1681073811yescharacteristic
407C22#1681074311yescharacteristic
408C4—21no—
409C4—21no—
410C4—21no—
411C4—81no—
412C4—81no—
413C4—4242no—
414C4—2121no—
415C4—2121no—
416C4—2121no—
417C4—2121no—
418C4#1681059511yescharacteristic
419C4#1681059511yescharacteristic
420C2—21no—
421C2—21no—
422C2—41no—
423C2—41no—
424C2#2027504511yescharacteristic
425C2#2027517011yescharacteristic
426C2#2027519111yescharacteristic
subgroupG/H
centerC22#16810499
commutatorC22×C4×C53C22×C4
FrattiniC23×C4C22×D53
FittingC53×#1472C2
socleC22×C53#386
prime radicalC24×C53#41
cosocleC23×C4×C53C23
hypercenter#1472D53
hypocenterC53#9240
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1472

Complements

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

NH ≅ G/Ntype
16. C2×#13133150C4semidirect
17. C2×#13133148C4semidirect
19. C2×#13133141C4semidirect
20. C2×#13133141C4semidirect
246. C53#9240semidirect

Maximal subgroups

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

subgroupindex
C2×#168109612
#202833692
#202748122
#202732352
#202725912
#202725612
C53×#14722
#924053