Finite Group Databasebeta

← #27281120

Subgroups — #27281120

Order 13568.

478 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 478 conjugacy classes of subgroups — too many to draw. Only its 99 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#20285536C211yesmaximal, characteristic
2#20275699C211yesmaximal, characteristic
3#20273247C211yesmaximal, characteristic
4#20273235C211yesmaximal, characteristic
5#20272621C211yesmaximal, characteristic
6#20272567C211yesmaximal, characteristic
7C53×#2378C211yesmaximal, characteristic
8C2×#13133196C2211yescharacteristic
9#16810953—21no—
10#16810951C412yescomplemented
11#16810879—22no—
12#16810779—21no—
13#16810778C412yescomplemented
14#16810587C2211yescharacteristic
15#16810587C2211yescharacteristic
16C4×C22⋊Dic53C2211yescharacteristic
17#16810389—21no—
18#16810389—21no—
19#16810360—22no—
20#16810359—21no—
21#16810359—21no—
22C2×C22⋊Q8×C53C2211yescharacteristic
23C53×#391—21no—
24C53×#389—21no—
25C53×#388—21no—
26C53×#388—21no—
27C53×#387—21no—
28C53×#387—21no—
29C53×#381C2211yescharacteristic
30C53×#380C2211yescharacteristic
31C22×Dic106C2×C411yescharacteristic
32C2×#11719925—22no—
33C2×#11719917—21no—
34C2×#11719917—21no—
35#13133196—21no—
36#13133196—21no—
37#13133196—21no—
38#13133196—21no—
39C2×C4⋊Dic53C2×C411yescharacteristic
40C2×#11719919—21no—
41C2×C22⋊Dic53—21no—
42C2×C22⋊Dic53—21no—
43C2×C4×Dic53—21no—
44#13133148—22no—
45#13133141—22no—
46#13133140—22no—
47C22⋊#88322—21no—
48C22⋊#88322—21no—
49C22⋊#88322—21no—
50C22⋊#88322C2×C412yes—
51#13133124—21no—
52C22×Q8×C53D411yescharacteristic
53C23×C4×C53C2311yescharacteristic
54C22⋊Q8×C53—21no—
55C22⋊Q8×C53—21no—
56C22⋊Q8×C53C2×C412yes—
57C22⋊Q8×C53D412yes—
58C2×C4⋊C4×C53—21no—
59C2×C4⋊C4×C53—21no—
60C2×C4⋊C4×C53—21no—
61C2×C4⋊C4×C53—21no—
62C2×C4⋊C4×C53D411yescharacteristic
63C2×C22⋊C4×C53—21no—
64C2×C22⋊C4×C53—21no—
65C2×C22⋊C4×C53D411yescharacteristic
66C2×C22⋊C4×C53D411yescharacteristic
67C2×C42×C53—21no—
68C2×C42×C53—21no—
69#95×C53—21no—
70#95×C53—21no—
71#95×C53—21no—
72#95×C53—21no—
73#95×C53—21no—
74#95×C53—21no—
75#95×C53D411yescharacteristic
76#95×C53D411yescharacteristic
77C24×C53C2×D411yescharacteristic
78C2×Q8×C53—21no—
79C2×Q8×C53—81no—
80C2×Q8×C53C22⋊C412yes—
81C22×C4×C53—21no—
82C22×C4×C53—21no—
83C22×C4×C53—21no—
84C22×C4×C53—21no—
85C22×C4×C53—21no—
86C22×C4×C53—21no—
87C22×C4×C53—81no—
88C22×C4×C53C22×C411yescharacteristic
89C22×C4×C53C2×D411yescharacteristic
90C22×C4×C53C2×D411yescharacteristic
91C22×C4×C53C2×D411yescharacteristic
92C22×C4×C53#4111yescharacteristic
93C22×C4×C53#4111yescharacteristic
94C22×Dic53—21no—
95C22×Dic53—21no—
96C22×Dic53—41no—
97C2×Dic106—21no—
98C2×Dic106—21no—
99C2×Dic106—81no—
100C4⋊C4×C53—22no—
101C4⋊C4×C53—22no—
102C4⋊C4×C53—21no—
103C4⋊C4×C53—81no—
104C4⋊C4×C53—81no—
105C4⋊C4×C53—81no—
106C4⋊C4×C53C22⋊C412yes—
107C22⋊C4×C53—22no—
108C22⋊C4×C53—22no—
109C22⋊C4×C53—41no—
110C22⋊C4×C53—41no—
111C22⋊C4×C53—81no—
112C42×C53—81no—
113C42×C53—81no—
114C22⋊Dic53—22no—
115C22⋊Dic53—41no—
116C22⋊Dic53—41no—
117C22⋊Dic53—41no—
118#11719925—82no—
119C4⋊Dic53—21no—
120C4⋊Dic53—21no—
121#11719919—41no—
122#11719919—41no—
123C4×Dic53—81no—
124#11719917—81no—
125#11719917—81no—
126C2×#88322—22no—
127C2×#88322—41no—
128C23×C53—21no—
129C23×C53—21no—
130C23×C53—41no—
131C23×C53—41no—
132C23×C53C4×D411yescharacteristic
133C23×C53#12011yescharacteristic
134C23×C53#12111yescharacteristic
135Q8×C53—42no—
136Q8×C53—81no—
137C2×C4×C53—22no—
138C2×C4×C53—22no—
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—41no—
147C2×C4×C53—41no—
148C2×C4×C53—41no—
149C2×C4×C53—41no—
150C2×C4×C53—41no—
151C2×C4×C53—41no—
152C2×C4×C53—41no—
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—41no—
161C2×C4×C53C2×C22⋊C411yescharacteristic
162C2×C4×C53C4×D411yescharacteristic
163C2×C4×C53#12111yescharacteristic
164C2×C4×C53#12311yescharacteristic
165C2×Dic53—42no—
166C2×Dic53—41no—
167C2×Dic53—41no—
168C2×Dic53—81no—
169C2×Dic53—81no—
170C2×Dic53—81no—
171C2×Dic53—81no—
172Dic106—42no—
173Dic106—81no—
174#88322—42no—
175#88322—81no—
176#9342—531nomaximal, Sylow
177C22×C53—21no—
178C22×C53—21no—
179C22×C53—21no—
180C22×C53—21no—
181C22×C53—21no—
182C22×C53—21no—
183C22×C53—21no—
184C22×C53—21no—
185C22×C53—81no—
186C22×C53—81no—
187C22×C53#38611yescharacteristic
188C22×C53#45511yescharacteristic
189C22×C53#45611yescharacteristic
190C4×C53—21no—
191C4×C53—21no—
192C4×C53—21no—
193C4×C53—21no—
194C4×C53—42no—
195C4×C53—42no—
196C4×C53—81no—
197C4×C53—81no—
198C4×C53—81no—
199C4×C53—81no—
200Dic53—82no—
201Dic53—81no—
202Dic53—81no—
203#3195—531no—
204#2378D5311yescharacteristic
205#1996—531no—
206#1511—531no—
207#1508—531no—
208#1305—531no—
209#1284—531no—
210C2×C53—21no—
211C2×C53—21no—
212C2×C53—41no—
213C2×C53—41no—
214C2×C53#188211yescharacteristic
215C2×C53#190511yescharacteristic
216C2×C53#191311yescharacteristic
217C2×C22⋊Q8—531no—
218C2×C22⋊Q8C2×D5311yescharacteristic
219#483—532no—
220#483—1061no—
221#470—1062no—
222#451—532no—
223#451—1061no—
224#407—531no—
225#407—531no—
226#391—21no—
227#389—21no—
228#388—21no—
229#388—21no—
230#387—21no—
231#387—21no—
232#381C2×D5311yescharacteristic
233#380C2×D5311yescharacteristic
234C4×C22⋊C4—531no—
235#342—1061no—
236#342—1061no—
237#328—1062no—
238#328—1061no—
239#328—1061no—
240C53#934211yescharacteristic, complemented, Sylow
241C22×Q8—531no—
242C22×Q8C22⋊D5311yescharacteristic
243C23×C4C22×D5311yescharacteristic
244C2×Q16—1062no—
245C2×M16—1061no—
246C2×M16—1061no—
247C22⋊Q8—21no—
248C22⋊Q8—21no—
249C22⋊Q8—1061no—
250C22⋊Q8—1061no—
251C22⋊Q8—1061no—
252C22⋊Q8—1061no—
253C22⋊Q8C4×D5312yes—
254C22⋊Q8D21212yes—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—531no—
260C2×C4⋊C4—1061no—
261C2×C4⋊C4C22⋊D5311yescharacteristic
262C2×C22⋊C4—21no—
263C2×C22⋊C4—21no—
264C2×C22⋊C4—1061no—
265C2×C22⋊C4—1061no—
266C2×C22⋊C4C22⋊D5311yescharacteristic
267C2×C22⋊C4C22⋊D5311yescharacteristic
268C2×C42—21no—
269C2×C42—21no—
270C2×C42—1061no—
271#106—1062no—
272#103—1062no—
273#103—1062no—
274C22⋊C8—532no—
275C22⋊C8—1061no—
276C22⋊C8—1061no—
277C22⋊C8—1061no—
278#95—21no—
279#95—21no—
280#95—21no—
281#95—21no—
282#95—21no—
283#95—21no—
284#95—1061no—
285#95C22⋊D5311yescharacteristic
286#95C22⋊D5311yescharacteristic
287C24C2×C22⋊D5311yescharacteristic
288C2×Q8—21no—
289C2×Q8—81no—
290C2×Q8—1061no—
291C2×Q8—1061no—
292C2×Q8—4241no—
293C2×Q8#1171992112yes—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—81no—
301C22×C4—1061no—
302C22×C4—1061no—
303C22×C4—2121no—
304C22×C4C2×C4×D5311yescharacteristic
305C22×C4C2×D21211yescharacteristic
306C22×C4#1171994511yescharacteristic
307C22×C4#1171994511yescharacteristic
308C22×C4C2×C22⋊D5311yescharacteristic
309C22×C4C2×C22⋊D5311yescharacteristic
310Q16—4242no—
311M16—4241no—
312M16—4241no—
313C2×C8—1062no—
314C2×C8—2121no—
315C4⋊C4—22no—
316C4⋊C4—22no—
317C4⋊C4—21no—
318C4⋊C4—81no—
319C4⋊C4—81no—
320C4⋊C4—81no—
321C4⋊C4—1061no—
322C4⋊C4—1061no—
323C4⋊C4—2121no—
324C4⋊C4—2121no—
325C4⋊C4#1171992112yes—
326C22⋊C4—22no—
327C22⋊C4—22no—
328C22⋊C4—41no—
329C22⋊C4—41no—
330C22⋊C4—81no—
331C22⋊C4—1062no—
332C22⋊C4—2121no—
333C22⋊C4—2121no—
334C22⋊C4—2121no—
335C42—81no—
336C42—81no—
337C42—4241no—
338C23—21no—
339C23—21no—
340C23—41no—
341C23—41no—
342C23C4×C22⋊D5311yescharacteristic
343C23#1313318611yescharacteristic
344C23#1313319111yescharacteristic
345Q8—42no—
346Q8—81no—
347Q8—4241no—
348Q8—2122no—
349C2×C4—22no—
350C2×C4—22no—
351C2×C4—41no—
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—41no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—4241no—
374C2×C4—4241no—
375C2×C4—4241no—
376C2×C4—4241no—
377C2×C4—2122no—
378C2×C4—2121no—
379C2×C4—2121no—
380C2×C4C2×#1171992111yescharacteristic
381C2×C4C4×C22⋊D5311yescharacteristic
382C2×C4#1313319111yescharacteristic
383C2×C4#1313320011yescharacteristic
384C8—4241no—
385C8—2122no—
386C22—21no—
387C22—21no—
388C22—21no—
389C22—21no—
390C22—21no—
391C22—21no—
392C22—21no—
393C22—21no—
394C22—81no—
395C22—81no—
396C22#1681050211yescharacteristic
397C22#1681080111yescharacteristic
398C22#1681080611yescharacteristic
399C4—21no—
400C4—21no—
401C4—21no—
402C4—21no—
403C4—42no—
404C4—42no—
405C4—81no—
406C4—81no—
407C4—81no—
408C4—81no—
409C4—4242no—
410C4—4241no—
411C4—4241no—
412C2—21no—
413C2—21no—
414C2—41no—
415C2—41no—
416C2#2027508411yescharacteristic
417C2#2027520911yescharacteristic
418C2#2027525711yescharacteristic
subgroupG/H
centerC22#16810502
commutatorC22×C4×C53C22×C4
FrattiniC23×C4C22×D53
FittingC53×#2378C2
socleC22×C53#386
prime radicalC24×C53C2×D4
cosocleC23×C4×C53C23
hypercenter#2378D53
hypocenterC53#9342
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2378

Complements

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

NH ≅ G/Ntype
10. #16810951C4semidirect
13. #16810778C4semidirect
240. C53#9342semidirect

Maximal subgroups

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

subgroupindex
#202855362
#202756992
#202732472
#202732352
#202726212
#202725672
C53×#23782
#934253