Finite Group Databasebeta

← #27281260

Subgroups — #27281260

Order 13568.

500 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 103 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#20284663C211yesmaximal, characteristic
2#20275699C211yesmaximal, characteristic
3#20274814C211yesmaximal, characteristic
4#20273234C211yesmaximal, characteristic
5#20272621C211yesmaximal, characteristic
6#20272601C211yesmaximal, characteristic
7C53×#2347C211yesmaximal, characteristic
8C2×#13133196C2211yescharacteristic
9#16810964—22no—
10#16810958—21no—
11#16810953—21no—
12#16810666C412yescomplemented
13#16810631C412yescomplemented
14#16810587C2211yescharacteristic
15#16810587C2211yescharacteristic
16C4×C22⋊Dic53C2211yescharacteristic
17#16810389—22no—
18#16810382—22no—
19#16810359—21no—
20#16810359—21no—
21C2×#117×C53C2211yescharacteristic
22C53×#389—21no—
23C53×#388—22no—
24C53×#383—21no—
25C53×#380C2211yescharacteristic
26C4×C4⋊C4×C53—21no—
27C4×C4⋊C4×C53—21no—
28C4×C22⋊C4×C53C2211yescharacteristic
29C22×Dic106C2×C411yescharacteristic
30C2×#11719917—21no—
31C2×#11719917C2×C412yes—
32#13133196—21no—
33#13133196—21no—
34#13133196—21no—
35#13133196—21no—
36C2×C4⋊Dic53C2×C411yescharacteristic
37C2×#11719919—21no—
38C2×C22⋊Dic53—21no—
39C2×C22⋊Dic53—21no—
40C2×C4×Dic53—21no—
41#13133150—22no—
42#13133148—22no—
43#13133140—22no—
44#13133140—22no—
45C22⋊#88322—21no—
46C22⋊#88322—21no—
47C22⋊#88322—21no—
48C22⋊#88322—21no—
49#13133124—21no—
50C23×C4×C53C2311yescharacteristic
51#117×C53—21no—
52#117×C53—21no—
53#117×C53C2×C412yes—
54#117×C53D412yes—
55C2×C4⋊C4×C53—22no—
56C2×C4⋊C4×C53—21no—
57C2×C4⋊C4×C53D411yescharacteristic
58C2×C4⋊C4×C53D411yescharacteristic
59C2×C22⋊C4×C53—21no—
60C2×C22⋊C4×C53—21no—
61C2×C22⋊C4×C53D412yes—
62C2×C42×C53—22no—
63C2×C42×C53—21no—
64C2×C42×C53—21no—
65#95×C53—22no—
66#95×C53—22no—
67#95×C53—21no—
68#95×C53D411yescharacteristic
69#95×C53D411yescharacteristic
70C24×C53C2×D411yescharacteristic
71C22×C4×C53—21no—
72C22×C4×C53—21no—
73C22×C4×C53—21no—
74C22×C4×C53—21no—
75C22×C4×C53—22no—
76C22×C4×C53—42no—
77C22×C4×C53C22×C411yescharacteristic
78C22×C4×C53C2×D411yescharacteristic
79C22×C4×C53C2×D411yescharacteristic
80C22×C4×C53C2×D411yescharacteristic
81C22×C4×C53#4111yescharacteristic
82C22×C4×C53#4111yescharacteristic
83C22×Dic53—21no—
84C22×Dic53—21no—
85C22×Dic53—41no—
86C2×Dic106—21no—
87C2×Dic106—21no—
88C2×Dic106—42no—
89C4⋊C4×C53—21no—
90C4⋊C4×C53—21no—
91C4⋊C4×C53—42no—
92C4⋊C4×C53—44no—
93C4⋊C4×C53C22⋊C412yes—
94C4⋊C4×C53C22⋊C412yes—
95C22⋊C4×C53—22no—
96C22⋊C4×C53—22no—
97C22⋊C4×C53—42no—
98C22⋊C4×C53—42no—
99C42×C53—22no—
100C42×C53—22no—
101C42×C53—42no—
102C42×C53—82no—
103C22⋊Dic53—22no—
104C22⋊Dic53—41no—
105C22⋊Dic53—41no—
106C22⋊Dic53—41no—
107C4⋊Dic53—21no—
108C4⋊Dic53—21no—
109#11719919—41no—
110#11719919—41no—
111C4×Dic53—42no—
112#11719917—42no—
113#11719917—42no—
114C2×#88322—22no—
115C2×#88322—41no—
116C23×C53—21no—
117C23×C53—21no—
118C23×C53—41no—
119C23×C53—41no—
120C23×C53C4×D411yescharacteristic
121C23×C53#12011yescharacteristic
122C23×C53#12111yescharacteristic
123C2×C4×C53—22no—
124C2×C4×C53—22no—
125C2×C4×C53—22no—
126C2×C4×C53—22no—
127C2×C4×C53—22no—
128C2×C4×C53—22no—
129C2×C4×C53—41no—
130C2×C4×C53—41no—
131C2×C4×C53—41no—
132C2×C4×C53—41no—
133C2×C4×C53—42no—
134C2×C4×C53—41no—
135C2×C4×C53—41no—
136C2×C4×C53—41no—
137C2×C4×C53—41no—
138C2×C4×C53—41no—
139C2×C4×C53—42no—
140C2×C4×C53—42no—
141C2×C4×C53—42no—
142C2×C4×C53—41no—
143C2×C4×C53C2×C22⋊C411yescharacteristic
144C2×C4×C53C4×D411yescharacteristic
145C2×C4×C53#12111yescharacteristic
146C2×C4×C53#12311yescharacteristic
147C2×Dic53—42no—
148C2×Dic53—41no—
149C2×Dic53—41no—
150C2×Dic53—81no—
151C2×Dic53—81no—
152C2×Dic53—81no—
153C2×Dic53—81no—
154Dic106—42no—
155Dic106—81no—
156#88322—42no—
157#88322—81no—
158#9362—531nomaximal, Sylow
159C22×C53—21no—
160C22×C53—21no—
161C22×C53—21no—
162C22×C53—21no—
163C22×C53—21no—
164C22×C53—21no—
165C22×C53—21no—
166C22×C53—21no—
167C22×C53—81no—
168C22×C53—81no—
169C22×C53#38611yescharacteristic
170C22×C53#45511yescharacteristic
171C22×C53#45611yescharacteristic
172C4×C53—21no—
173C4×C53—21no—
174C4×C53—21no—
175C4×C53—42no—
176C4×C53—42no—
177C4×C53—42no—
178C4×C53—82no—
179C4×C53—81no—
180C4×C53#41012yes—
181Dic53—82no—
182Dic53—81no—
183Dic53—81no—
184#3095—531no—
185#2347D5311yescharacteristic
186#1996—531no—
187#1826—531no—
188#1508—531no—
189#1305—531no—
190#1297—531no—
191C2×C53—21no—
192C2×C53—21no—
193C2×C53—41no—
194C2×C53—41no—
195C2×C53#188211yescharacteristic
196C2×C53#189911yescharacteristic
197C2×C53#190611yescharacteristic
198C2×C22⋊Q8—531no—
199C2×#117C2×D5311yescharacteristic
200#485—1062no—
201#484—1061no—
202#483—1061no—
203#428—532no—
204#419—532no—
205#407—531no—
206#407—531no—
207#389—21no—
208#388—22no—
209#383—21no—
210#380C2×D5311yescharacteristic
211C4×C4⋊C4—21no—
212C4×C4⋊C4—21no—
213C4×C22⋊C4—531no—
214C4×C22⋊C4C2×D5311yescharacteristic
215#342—1062no—
216#339—1062no—
217#328—1061no—
218#328—1061no—
219C53#936211yescharacteristic, complemented, Sylow
220C22×Q8—531no—
221C23×C4C22×D5311yescharacteristic
222C2×M16—532no—
223C2×M16—1061no—
224C22⋊Q8—1061no—
225C22⋊Q8—1061no—
226C22⋊Q8—1061no—
227C22⋊Q8—1061no—
228#117—21no—
229#117—21no—
230#117C4×D5312yes—
231#117D21212yes—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—531no—
235C2×C4⋊C4—1061no—
236C2×C4⋊C4C22⋊D5311yescharacteristic
237C2×C4⋊C4C22⋊D5311yescharacteristic
238C2×C22⋊C4—21no—
239C2×C22⋊C4—21no—
240C2×C22⋊C4—1061no—
241C2×C22⋊C4—1061no—
242C2×C22⋊C4C22⋊D5312yes—
243C2×C42—21no—
244C2×C42—21no—
245C2×C42—22no—
246C2×C42—1061no—
247#107—1062no—
248#106—1062no—
249#103—1062no—
250#103—1062no—
251C22⋊C8—1061no—
252C22⋊C8—1061no—
253C22⋊C8—1061no—
254C22⋊C8—1061no—
255#95—22no—
256#95—22no—
257#95—21no—
258#95—1061no—
259#95C22⋊D5311yescharacteristic
260#95C22⋊D5311yescharacteristic
261C24C2×C22⋊D5311yescharacteristic
262C2×Q8—1061no—
263C2×Q8—1061no—
264C2×Q8—2122no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—22no—
270C22×C4—42no—
271C22×C4—1061no—
272C22×C4—1061no—
273C22×C4—2121no—
274C22×C4C2×C4×D5311yescharacteristic
275C22×C4C2×D21211yescharacteristic
276C22×C4#1171994511yescharacteristic
277C22×C4#1171994511yescharacteristic
278C22×C4C2×C22⋊D5311yescharacteristic
279C22×C4C2×C22⋊D5311yescharacteristic
280M16—2122no—
281M16—2122no—
282C2×C8—1062no—
283C2×C8—2121no—
284C4⋊C4—21no—
285C4⋊C4—21no—
286C4⋊C4—42no—
287C4⋊C4—44no—
288C4⋊C4—1061no—
289C4⋊C4—1061no—
290C4⋊C4—2121no—
291C4⋊C4—2121no—
292C4⋊C4#1171992112yes—
293C4⋊C4#1171992112yes—
294C22⋊C4—22no—
295C22⋊C4—22no—
296C22⋊C4—42no—
297C22⋊C4—42no—
298C22⋊C4—1062no—
299C22⋊C4—2121no—
300C22⋊C4—2121no—
301C22⋊C4—2121no—
302C42—22no—
303C42—22no—
304C42—42no—
305C42—82no—
306C42—2122no—
307C23—21no—
308C23—21no—
309C23—41no—
310C23—41no—
311C23C4×C22⋊D5311yescharacteristic
312C23#1313318611yescharacteristic
313C23#1313319111yescharacteristic
314Q8—4241no—
315Q8—2122no—
316C2×C4—22no—
317C2×C4—22no—
318C2×C4—22no—
319C2×C4—22no—
320C2×C4—22no—
321C2×C4—22no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—42no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—42no—
333C2×C4—42no—
334C2×C4—42no—
335C2×C4—41no—
336C2×C4—4241no—
337C2×C4—4241no—
338C2×C4—4241no—
339C2×C4—4241no—
340C2×C4—2122no—
341C2×C4—2121no—
342C2×C4—2121no—
343C2×C4C2×#1171992111yescharacteristic
344C2×C4C4×C22⋊D5311yescharacteristic
345C2×C4#1313319111yescharacteristic
346C2×C4#1313320011yescharacteristic
347C8—4241no—
348C8—2122no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—21no—
353C22—21no—
354C22—21no—
355C22—21no—
356C22—21no—
357C22—81no—
358C22—81no—
359C22#1681050211yescharacteristic
360C22#1681080111yescharacteristic
361C22#1681080611yescharacteristic
362C4—21no—
363C4—21no—
364C4—21no—
365C4—42no—
366C4—42no—
367C4—42no—
368C4—82no—
369C4—81no—
370C4—4242no—
371C4—4241no—
372C4—4241no—
373C4#1681059612yes—
374C2—21no—
375C2—21no—
376C2—41no—
377C2—41no—
378C2#2027508411yescharacteristic
379C2#2027517311yescharacteristic
380C2#2027521611yescharacteristic
subgroupG/H
centerC22#16810502
commutatorC22×C4×C53C22×C4
FrattiniC23×C4C22×D53
FittingC53×#2347C2
socleC22×C53#386
prime radicalC24×C53C2×D4
cosocleC23×C4×C53C23
hypercenter#2347D53
hypocenterC53#9362
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2347

Complements

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

NH ≅ G/Ntype
12. #16810666C4semidirect
13. #16810631C4semidirect
219. C53#9362semidirect

Maximal subgroups

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

subgroupindex
#202846632
#202756992
#202748142
#202732342
#202726212
#202726012
C53×#23472
#936253