Finite Group Databasebeta

← #28299491

Subgroups — #28299491

Order 13568.

606 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 606 conjugacy classes of subgroups — too many to draw. Only its 112 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#20290605C211yesmaximal, characteristic, complemented
2C4×C22⋊#88324C211yesmaximal, characteristic, complemented
3#20289534C212yesmaximal, complemented
4#20289532C212yesmaximal, complemented
5#20277221C211yesmaximal, characteristic
6C2×C22⋊#88324—21no—
7C2×C22⋊#88324—21no—
8C2×C4×#88324—21no—
9#16811489—21no—
10#16811489C2212yes—
11#16811489C2212yes—
12C2×#13133191C411yescharacteristic
13C2×#13133179C411yescharacteristic
14C2×#13133177—21no—
15C22×C4×Dic53C2211yescharacteristic
16#16810524C2211yescharacteristic
17#16810519C2211yescharacteristic
18#16810512—21no—
19#16810500—21no—
20C4⋊C4×Dic53C411yescharacteristic
21#16810470—21no—
22C53×#380C411yescharacteristic
23C22×#88324—22no—
24C22×#88324—22no—
25C22⋊#88324—24no—
26C22⋊#88324—42no—
27C4×#88324—42no—
28C22×D212C2×C411yescharacteristic
29C22×C22⋊D53—21no—
30C23×Dic53D411yescharacteristic
31C23×Dic53D411yescharacteristic
32C22×C4×D53—21no—
33#13133191—22no—
34#13133191C814yescomplemented
35#13133179—42no—
36#13133177—44no—
37C2×C4⋊Dic53C2×C411yescharacteristic
38C2×#11719919—21no—
39C2×#11719921—21no—
40C2×#11719921—21no—
41C2×#11719921—21no—
42C2×#11719921—21no—
43C2×C4×Dic53—21no—
44C2×C4×Dic53—42no—
45C2×C4×Dic53C2×C411yescharacteristic
46C2×C4×Dic53D411yescharacteristic
47C2×C4×Dic53D411yescharacteristic
48C2×C4×Dic53C2311yescharacteristic
49#13133124—21no—
50#13133124—21no—
51C23×C4×C53C2×C411yescharacteristic
52C2×C4⋊C4×C53C2×C411yescharacteristic
53C2×C4⋊C4×C53C2×C411yescharacteristic
54C2×C22⋊C4×C53—21no—
55#95×C53—21no—
56C24×C53C22⋊C411yescharacteristic
57C23×D53—21no—
58C22×C4×C53—21no—
59C22×C4×C53—21no—
60C22×C4×C53—41no—
61C22×C4×C53—41no—
62C22×C4×C53C22⋊C411yescharacteristic
63C22×C4×C53C2×C812yes—
64C22×C4×C53M1612yes—
65C22×C4×C53C22×C411yescharacteristic
66C2×C22⋊D53—24no—
67C2×C22⋊D53—42no—
68C2×C22⋊D53—42no—
69C22×Dic53—21no—
70C22×Dic53—22no—
71C22×Dic53—22no—
72C22×Dic53—22no—
73C22×Dic53—22no—
74C22×Dic53—41no—
75C22×Dic53—41no—
76C22×Dic53C22⋊C411yescharacteristic
77C22×Dic53C22⋊C411yescharacteristic
78C22×Dic53C2×D411yescharacteristic
79C22×Dic53C2×D411yescharacteristic
80C2×D212—21no—
81C2×D212—41no—
82C2×D212—41no—
83C2×D212C2×C812yes—
84C2×C4×D53—42no—
85C2×C4×D53—42no—
86C2×C4×D53—42no—
87C2×#88324—24no—
88C2×#88324—42no—
89C2×#88324—42no—
90C2×#88324—42no—
91C2×#88324—42no—
92C2×#88324—42no—
93C4⋊C4×C53—41no—
94C4⋊C4×C53—41no—
95C22⋊C4×C53—42no—
96#11719921—24no—
97#11719921—42no—
98#11719921—42no—
99#11719921—42no—
100C4⋊Dic53—21no—
101C4⋊Dic53C2×C812yes—
102#11719919—42no—
103C4×Dic53—22no—
104C4×Dic53—22no—
105C4×Dic53—41no—
106C4×Dic53—41no—
107C4×Dic53—42no—
108C23×C53—21no—
109C23×C53—21no—
110C23×C53—21no—
111C23×C53—21no—
112C23×C53—21no—
113C23×C53—21no—
114C23×C53C22⋊C812yes—
115C23×C53C2×C22⋊C411yescharacteristic
116C22×D53—22no—
117C22×D53—41no—
118C22×D53—41no—
119C22×D53—41no—
120C22×D53—41no—
121C22×D53—41no—
122C22×D53—41no—
123C2×C4×C53—21no—
124C2×C4×C53—21no—
125C2×C4×C53—21no—
126C2×C4×C53—21no—
127C2×C4×C53—42no—
128C2×C4×C53—42no—
129C2×C4×C53—41no—
130C2×C4×C53—41no—
131C2×C4×C53—41no—
132C2×C4×C53—41no—
133C2×C4×C53—41no—
134C2×C4×C53—41no—
135C2×C4×C53C22⋊C812yes—
136C2×C4×C53C22×C811yescharacteristic
137C2×C4×C53C2×M1611yescharacteristic
138C22⋊D53—48no—
139C2×Dic53—21no—
140C2×Dic53—21no—
141C2×Dic53—21no—
142C2×Dic53—21no—
143C2×Dic53—21no—
144C2×Dic53—21no—
145C2×Dic53—21no—
146C2×Dic53—21no—
147C2×Dic53—22no—
148C2×Dic53—44no—
149C2×Dic53—44no—
150C2×Dic53—41no—
151C2×Dic53—41no—
152C2×Dic53#9912yes—
153C2×Dic53#10412yes—
154C2×Dic53#13512yes—
155C2×Dic53AGL(1,ℤ/8)12yes—
156D212—42no—
157D212—42no—
158C4×D53—84no—
159#88324—44no—
160#88324—82no—
161#7582—531nomaximal, Sylow
162C22×C53—22no—
163C22×C53—22no—
164C22×C53—22no—
165C22×C53—41no—
166C22×C53—41no—
167C22×C53—41no—
168C22×C53—41no—
169C22×C53C2×C22⋊C811yescharacteristic
170C22×C53C2×#9911yescharacteristic
171C22×C53#41311yescharacteristic
172C22×C53#41611yescharacteristic
173C22×C53#41811yescharacteristic
174C22×C53C2×#10411yescharacteristic
175C22×C53#42111yescharacteristic
176C2×D53—42no—
177C2×D53—42no—
178C2×D53—42no—
179C2×D53—81no—
180C2×D53—81no—
181C2×D53—81no—
182C2×D53—81no—
183C4×C53—21no—
184C4×C53—21no—
185C4×C53—22no—
186C4×C53—81no—
187C4×C53—81no—
188Dic53—22no—
189Dic53—22no—
190Dic53—22no—
191Dic53—22no—
192Dic53—42no—
193C4×#121—531no—
194#1817—531no—
195C4×C22⋊C8—531no—
196#1271—532no—
197#1270—532no—
198C2×C53—42no—
199C2×C53#145911yescharacteristic
200C2×C53#147611yescharacteristic
201C2×C53#147811yescharacteristic
202C2×C53#150711yescharacteristic
203C2×C53#151611yescharacteristic
204C2×C53#152911yescharacteristic
205C2×C53#155311yescharacteristic
206D53—82no—
207C2×#121—531no—
208C2×C4×D4—531no—
209C2×C4×D4—1061no—
210C22×C42—531no—
211C2×C22⋊C8—1061no—
212C2×C22⋊C8—1061no—
213C2×C4×C8—1061no—
214#390—531no—
215#389—531no—
216#388—1061no—
217#386—1061no—
218#380#615711yescharacteristic
219C4×C4⋊C4—531no—
220C4×C22⋊C4—1061no—
221#336—532no—
222#336—532no—
223#336—1061no—
224C53#758211yescharacteristic, complemented, Sylow
225C22×D4—531no—
226C22×D4—1061no—
227C23×C4—531no—
228C23×C4—531no—
229C23×C4—1061no—
230C23×C4C2×#615711yescharacteristic
231C22×C8—1062no—
232C22×C8—1062no—
233#121—534no—
234#121—1062no—
235C4×D4—2122no—
236C4×D4—2124no—
237C2×C4⋊C4—531no—
238C2×C4⋊C4—1061no—
239C2×C4⋊C4C2×#615711yescharacteristic
240C2×C4⋊C4C2×#615711yescharacteristic
241C2×C22⋊C4—21no—
242C2×C22⋊C4—1061no—
243C2×C22⋊C4—1061no—
244C2×C22⋊C4—1061no—
245C2×C22⋊C4—1061no—
246C2×C42—531no—
247C2×C42—531no—
248C2×C42—531no—
249C2×C42—531no—
250C2×C42—1061no—
251C2×C42—2122no—
252C22⋊C8—1064no—
253C22⋊C8—2122no—
254C4×C8—2122no—
255#95—21no—
256#95—1061no—
257#95—1061no—
258C24—1061no—
259C24C22⋊#615711yescharacteristic
260C2×D4—532no—
261C2×D4—1064no—
262C2×D4—1061no—
263C2×D4—2121no—
264C2×D4—2121no—
265C2×D4—2122no—
266C2×D4—2122no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—41no—
270C22×C4—41no—
271C22×C4—531no—
272C22×C4—531no—
273C22×C4—531no—
274C22×C4—531no—
275C22×C4—1062no—
276C22×C4—1062no—
277C22×C4—1062no—
278C22×C4—1062no—
279C22×C4—1061no—
280C22×C4—2122no—
281C22×C4—2122no—
282C22×C4—2122no—
283C22×C4—2121no—
284C22×C4—2121no—
285C22×C4D53⋊C812yes—
286C22×C4#1171993612yes—
287C22×C4C22⋊#615711yescharacteristic
288C22×C4C22×#615711yescharacteristic
289C2×C8—1064no—
290C2×C8—2122no—
291C2×C8—2122no—
292C2×C8—2122no—
293C2×C8—2122no—
294C2×C8—2122no—
295C4⋊C4—41no—
296C4⋊C4—41no—
297C4⋊C4—532no—
298C4⋊C4—1061no—
299C4⋊C4—2122no—
300C22⋊C4—42no—
301C22⋊C4—1064no—
302C22⋊C4—2122no—
303C22⋊C4—2122no—
304C22⋊C4—2122no—
305C42—1062no—
306C42—1062no—
307C42—2121no—
308C42—2121no—
309C42—2122no—
310C23—21no—
311C23—21no—
312C23—21no—
313C23—21no—
314C23—21no—
315C23—21no—
316C23—1062no—
317C23—2121no—
318C23—2121no—
319C23—2121no—
320C23—2121no—
321C23—2121no—
322C23—2121no—
323C23#1313327412yes—
324C23C2×C22⋊#615711yescharacteristic
325D4—2122no—
326D4—2122no—
327D4—2128no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—42no—
333C2×C4—42no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—532no—
341C2×C4—532no—
342C2×C4—532no—
343C2×C4—532no—
344C2×C4—1061no—
345C2×C4—1061no—
346C2×C4—1061no—
347C2×C4—1061no—
348C2×C4—1061no—
349C2×C4—1061no—
350C2×C4—1061no—
351C2×C4—1061no—
352C2×C4—1062no—
353C2×C4—4244no—
354C2×C4—2124no—
355C2×C4—2124no—
356C2×C4—2121no—
357C2×C4—2121no—
358C2×C4#1313327412yes—
359C2×C4C2×D53⋊C811yescharacteristic
360C2×C4C2×#1171993611yescharacteristic
361C8—4242no—
362C8—2124no—
363C22—22no—
364C22—22no—
365C22—22no—
366C22—41no—
367C22—41no—
368C22—41no—
369C22—41no—
370C22—4241no—
371C22—4241no—
372C22—4241no—
373C22—4241no—
374C22—2122no—
375C22—2122no—
376C22—2122no—
377C22C2×#1313327411yescharacteristic
378C22D53⋊#9911yescharacteristic
379C22#1681161011yescharacteristic
380C22#1681161311yescharacteristic
381C22#1681161511yescharacteristic
382C22D53⋊#10411yescharacteristic
383C22#1681162211yescharacteristic
384C4—22no—
385C4—21no—
386C4—21no—
387C4—81no—
388C4—81no—
389C4—1062no—
390C4—1062no—
391C4—1062no—
392C4—1062no—
393C4—2122no—
394C2—42no—
395C2—4242no—
396C2#2028995311yescharacteristic
397C2#2029000811yescharacteristic
398C2#2029001611yescharacteristic
399C2#2029011011yescharacteristic
400C2#2029014211yescharacteristic
401C2#2029018311yescharacteristic
402C2#2029023911yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6157
commutatorC2×C4×C53C22×C8
FrattiniC22×C4C22×#6157
FittingC53×#380C4
socleC23×C53C2×C22⋊C4
prime radicalC2×#13133191C4
cosocleC2×C4×Dic53C23
hypercenter#380#6157
hypocenterC53#7582
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ #380

Complements

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

NH ≅ G/Ntype
1. #20290605C2semidirect
2. C4×C22⋊#88324C2semidirect
3. #20289534C2semidirect
4. #20289532C2semidirect
34. #13133191C8semidirect
224. C53#7582semidirect

Maximal subgroups

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

subgroupindex
#202906052
C4×C22⋊#883242
#202895342
#202895322
#202772212
#758253