Finite Group Databasebeta

← #27281840

Subgroups — #27281840

Order 13568.

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

normal subgroups only

This group has 554 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
1#20279207C211yesmaximal, characteristic
2#20277316C211yesmaximal, characteristic, complemented
3#20274793C211yesmaximal, characteristic, complemented
4#20273245C211yesmaximal, characteristic
5#20273243C211yesmaximal, characteristic
6#20272568C211yesmaximal, characteristic, complemented
7C53×#1284C211yesmaximal, characteristic, complemented
8C2×#13133191C2211yescharacteristic
9C22×C4⋊Dic53C2211yescharacteristic
10#16810587C2211yescharacteristic
11#16810550—22no—
12#16810525C412yescomplemented
13#16810508—22no—
14#16810506—22no—
15#16810500—21no—
16#16810500—21no—
17#16810489C412yescomplemented
18#16810486C2211yescharacteristic
19#16810481—21no—
20C22⋊C4×Dic53—21no—
21#16810391—22no—
22#16810358—22no—
23#16810356—22no—
24C53×#407C2211yescharacteristic
25C53×#381C2211yescharacteristic
26C4×C22⋊C4×C53C2211yescharacteristic
27C22×D212C2×C411yescharacteristic
28C22×C22⋊D53—21no—
29C23×Dic53D411yescharacteristic
30C23×Dic53D411yescharacteristic
31C2×#11719917—21no—
32#13133191—21no—
33#13133191—21no—
34#13133191—21no—
35#13133191—21no—
36C2×C4⋊Dic53—21no—
37C2×C4⋊Dic53—21no—
38C2×C4⋊Dic53—21no—
39C2×C4⋊Dic53—21no—
40C2×C4⋊Dic53C2×C411yescharacteristic
41C2×C4⋊Dic53C2×C412yes—
42C2×C4⋊Dic53D412yes—
43C2×#11719919—22no—
44C2×C22⋊Dic53—21no—
45C2×#11719921—21no—
46C2×#11719921—22no—
47C2×#11719921—22no—
48C2×C4×Dic53—22no—
49C22⋊#88322—21no—
50C22⋊#88322—21no—
51#13133124—22no—
52#13133124—22no—
53#13133124—21no—
54#13133124—22no—
55#13133124—22no—
56#13133124D412yes—
57C23×C4×C53C2311yescharacteristic
58C2×M16×C53—21no—
59C2×C22⋊C4×C53—21no—
60C2×C22⋊C4×C53—21no—
61C2×C42×C53C2×C412yes—
62C22⋊C8×C53—21no—
63C22⋊C8×C53—21no—
64#95×C53—21no—
65#95×C53—21no—
66#95×C53D412yes—
67C24×C53C2×D411yescharacteristic
68C23×D53—21no—
69C22×C4×C53—22no—
70C22×C4×C53—22no—
71C22×C4×C53—42no—
72C22×C4×C53C22×C411yescharacteristic
73C22×C4×C53C2×D411yescharacteristic
74C22×C4×C53C2×D411yescharacteristic
75C22×C4×C53C2×D411yescharacteristic
76C22×C4×C53#4111yescharacteristic
77C22×C4×C53#4111yescharacteristic
78C2×C22⋊D53—44no—
79C2×C22⋊D53—41no—
80C2×C22⋊D53—41no—
81C22×Dic53—21no—
82C22×Dic53—21no—
83C22×Dic53—21no—
84C22×Dic53—21no—
85C22×Dic53—21no—
86C22×Dic53—22no—
87C22×Dic53—81no—
88C22×Dic53—81no—
89C22×Dic53C22⋊C412yes—
90C22×Dic53C22⋊C412yes—
91C2×D212—21no—
92C2×D212—21no—
93C2×D212—42no—
94M16×C53—42no—
95C2×C8×C53—41no—
96C22⋊C4×C53—42no—
97C22⋊C4×C53—42no—
98C42×C53—42no—
99C22⋊Dic53—42no—
100#11719921—41no—
101#11719921—41no—
102#11719921—82no—
103#11719921—82no—
104C4⋊Dic53—22no—
105C4⋊Dic53—22no—
106C4⋊Dic53—22no—
107C4⋊Dic53—22no—
108C4⋊Dic53—21no—
109C4⋊Dic53—21no—
110#11719919—82no—
111C4×Dic53—82no—
112#11719917—42no—
113C2×#88322—41no—
114C23×C53—22no—
115C23×C53—22no—
116C23×C53—41no—
117C23×C53C2×C22⋊C411yescharacteristic
118C23×C53#12011yescharacteristic
119C23×C53#12311yescharacteristic
120C22×D53—42no—
121C22×D53—42no—
122C22×D53—41no—
123C22×D53—42no—
124C2×C4×C53—22no—
125C2×C4×C53—22no—
126C2×C4×C53—41no—
127C2×C4×C53—41no—
128C2×C4×C53—41no—
129C2×C4×C53—41no—
130C2×C4×C53—42no—
131C2×C4×C53—42no—
132C2×C4×C53—42no—
133C2×C4×C53—42no—
134C2×C4×C53—42no—
135C2×C4×C53—42no—
136C2×C4×C53C4×D411yescharacteristic
137C2×C4×C53C4×D411yescharacteristic
138C2×C4×C53#12111yescharacteristic
139C2×C4×C53#12111yescharacteristic
140C22⋊D53—82no—
141C22⋊D53—82no—
142C2×Dic53—22no—
143C2×Dic53—22no—
144C2×Dic53—42no—
145C2×Dic53—42no—
146C2×Dic53—42no—
147C2×Dic53—42no—
148C2×Dic53—42no—
149C2×Dic53—42no—
150C2×Dic53—42no—
151C2×Dic53—41no—
152C2×Dic53—42no—
153C2×Dic53—81no—
154C2×Dic53—81no—
155D212—81no—
156D212—81no—
157C8×C53—81no—
158#88322—81no—
159#9465—531nomaximal, Sylow
160C22×C53—21no—
161C22×C53—21no—
162C22×C53—21no—
163C22×C53—21no—
164C22×C53—21no—
165C22×C53—21no—
166C22×C53—21no—
167C22×C53—44no—
168C22×C53#38611yescharacteristic
169C22×C53#41012yes—
170C22×C53#45412yes—
171C2×D53—84no—
172C2×D53—81no—
173C2×D53—81no—
174C2×D53—81no—
175C4×C53—21no—
176C4×C53—21no—
177C4×C53—21no—
178C4×C53—21no—
179C4×C53—82no—
180C4×C53—82no—
181Dic53—42no—
182Dic53—42no—
183Dic53—82no—
184Dic53—81no—
185#2504—531no—
186#2313—531no—
187#1821—531no—
188#1510—531no—
189#1510—531no—
190#1284—531no—
191#1284D5311yescharacteristic, complemented
192C2×C53—22no—
193C2×C53—22no—
194C2×C53—41no—
195C2×C53#188811yescharacteristic
196C2×C53#189911yescharacteristic
197C2×C53#191111yescharacteristic
198D53—161no—
199C2×#121—531no—
200C22×C4⋊C4—531no—
201#407—531no—
202#407C2×D5311yescharacteristic
203#397—1062no—
204#390—532no—
205#387—1062no—
206#387—1062no—
207#386—1061no—
208#386—1061no—
209#382—532no—
210#381—531no—
211#381C2×D5311yescharacteristic
212#380—1061no—
213C4×C22⋊C4—1061no—
214C4×C22⋊C4C2×D5311yescharacteristic
215#343—1062no—
216#327—1062no—
217#327—1062no—
218C53#946511yescharacteristic, complemented, Sylow
219C22×D4—531no—
220C22×D4—1061no—
221C23×C4—531no—
222C23×C4—531no—
223C23×C4C22×D5311yescharacteristic
224C2×M16—21no—
225C2×M16—1061no—
226#121—1061no—
227#121—1061no—
228#121—1061no—
229#121—1061no—
230C2×C4⋊C4—532no—
231C2×C4⋊C4—532no—
232C2×C4⋊C4—531no—
233C2×C4⋊C4—1061no—
234C2×C4⋊C4—1061no—
235C2×C4⋊C4—1061no—
236C2×C4⋊C4—1061no—
237C2×C4⋊C4—1062no—
238C2×C22⋊C4—21no—
239C2×C22⋊C4—21no—
240C2×C22⋊C4—1062no—
241C2×C22⋊C4—1062no—
242C2×C22⋊C4—1061no—
243C2×C22⋊C4—1061no—
244C2×C42—1062no—
245C2×C42C4×D5312yes—
246C22⋊C8—21no—
247C22⋊C8—21no—
248C22⋊C8—1061no—
249C22⋊C8—1061no—
250#95—21no—
251#95—21no—
252#95—532no—
253#95—1062no—
254#95—1062no—
255#95—1062no—
256#95—1062no—
257#95—1061no—
258#95D21212yes—
259C24—1061no—
260C24D4×D5311yescharacteristic
261C2×D4—1061no—
262C2×D4—1061no—
263C2×D4—2122no—
264C2×D4—2124no—
265C2×D4—2121no—
266C2×D4—2121no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—42no—
270C22×C4—532no—
271C22×C4—532no—
272C22×C4—1061no—
273C22×C4—1061no—
274C22×C4—1061no—
275C22×C4—1061no—
276C22×C4—1061no—
277C22×C4—1062no—
278C22×C4—4241no—
279C22×C4—4241no—
280C22×C4C2×C4×D5311yescharacteristic
281C22×C4C2×D21211yescharacteristic
282C22×C4#1171994511yescharacteristic
283C22×C4D4×D5311yescharacteristic
284C22×C4D4×D5311yescharacteristic
285C22×C4#1171994911yescharacteristic
286M16—42no—
287M16—2122no—
288C2×C8—41no—
289C2×C8—2121no—
290C4⋊C4—1062no—
291C4⋊C4—1062no—
292C4⋊C4—1062no—
293C4⋊C4—1062no—
294C4⋊C4—1061no—
295C4⋊C4—1061no—
296C4⋊C4—4242no—
297C22⋊C4—42no—
298C22⋊C4—42no—
299C22⋊C4—4242no—
300C22⋊C4—4242no—
301C22⋊C4—2122no—
302C22⋊C4—2121no—
303C22⋊C4—2121no—
304C42—42no—
305C42—4242no—
306C23—22no—
307C23—22no—
308C23—41no—
309C23—2122no—
310C23—2122no—
311C23—2121no—
312C23—2122no—
313C23C22⋊C4×D5311yescharacteristic
314C23#1313318711yescharacteristic
315C23#1313320211yescharacteristic
316D4—4241no—
317D4—4241no—
318D4—4242no—
319D4—4242no—
320C2×C4—22no—
321C2×C4—22no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—42no—
327C2×C4—42no—
328C2×C4—42no—
329C2×C4—42no—
330C2×C4—42no—
331C2×C4—42no—
332C2×C4—1062no—
333C2×C4—1062no—
334C2×C4—4241no—
335C2×C4—4241no—
336C2×C4—2122no—
337C2×C4—2122no—
338C2×C4—2122no—
339C2×C4—2121no—
340C2×C4—2122no—
341C2×C4—2122no—
342C2×C4—2122no—
343C2×C4—2122no—
344C2×C4—2122no—
345C2×C4#1313317911yescharacteristic
346C2×C4C4×D21211yescharacteristic
347C2×C4#1313319011yescharacteristic
348C2×C4#1313319211yescharacteristic
349C8—81no—
350C8—4241no—
351C22—21no—
352C22—21no—
353C22—21no—
354C22—21no—
355C22—21no—
356C22—21no—
357C22—21no—
358C22—44no—
359C22—4244no—
360C22—4241no—
361C22—4241no—
362C22—4241no—
363C22#1681050111yescharacteristic
364C22#1681059512yes—
365C22#1681079712yes—
366C4—21no—
367C4—21no—
368C4—21no—
369C4—21no—
370C4—82no—
371C4—82no—
372C4—4242no—
373C4—4241no—
374C4—2122no—
375C4—2122no—
376C2—22no—
377C2—22no—
378C2—41no—
379C2—8481no—
380C2#2027511711yescharacteristic
381C2#2027517211yescharacteristic
382C2#2027524611yescharacteristic
subgroupG/H
centerC22#16810501
commutatorC22×C4×C53C22×C4
FrattiniC23×C4C22×D53
FittingC53×#1284C2
socleC22×C53#386
prime radicalC2×#13133191C22
cosocleC23×C4×C53C23
hypercenter#1284D53
hypocenterC53#9465
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1284

Complements

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

NH ≅ G/Ntype
2. #20277316C2semidirect
3. #20274793C2semidirect
6. #20272568C2semidirect
7. C53×#1284C2semidirect
12. #16810525C4semidirect
17. #16810489C4semidirect
191. #1284D53semidirect
218. C53#9465semidirect

Maximal subgroups

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

subgroupindex
#202792072
#202773162
#202747932
#202732452
#202732432
#202725682
C53×#12842
#946553