Finite Group Databasebeta

← #28299555

Subgroups — #28299555

Order 13568.

504 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 504 conjugacy classes of subgroups — too many to draw. Only its 110 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
2#20290569C211yesmaximal, characteristic
3#20289536C212yesmaximal, complemented
4#20289534C212yesmaximal, complemented
5#20277243C211yesmaximal, characteristic
6C2×C4⋊#88324—21no—
7C2×C22⋊#88324—21no—
8C2×C22⋊#88324—21no—
9#16811489—21no—
10#16811489C2212yes—
11#16811489C2212yes—
12C2×#13133196C411yescharacteristic
13C2×#13133182C411yescharacteristic
14C22×C4×Dic53C2211yescharacteristic
15#16810528C2211yescharacteristic
16#16810519C2211yescharacteristic
17#16810505—21no—
18#16810495—21no—
19C4⋊C4×Dic53C411yescharacteristic
20#16810474—21no—
21C22⋊C4×Dic53—21no—
22C53×#380C411yescharacteristic
23C22×#88324—22no—
24C22×#88324—22no—
25C4⋊#88324—42no—
26C22⋊#88324—42no—
27C22⋊#88324—42no—
28C22×Dic106C2×C411yescharacteristic
29C23×Dic53D411yescharacteristic
30C23×Dic53D411yescharacteristic
31#13133196—22no—
32#13133196C814yescomplemented
33#13133182—42no—
34C2×C4⋊Dic53C2×C411yescharacteristic
35C2×#11719919—21no—
36C2×#11719919—21no—
37C2×#11719919—21no—
38C2×C22⋊Dic53—21no—
39C2×C4×Dic53—21no—
40C2×C4×Dic53—21no—
41C2×C4×Dic53—42no—
42C2×C4×Dic53C2×C411yescharacteristic
43C2×C4×Dic53D411yescharacteristic
44C2×C4×Dic53D411yescharacteristic
45C2×C4×Dic53C2311yescharacteristic
46#13133124—21no—
47#13133124—21no—
48#13133124—21no—
49#13133124—21no—
50C23×C4×C53C2×C411yescharacteristic
51C2×C4⋊C4×C53C2×C411yescharacteristic
52C2×C4⋊C4×C53C2×C411yescharacteristic
53C2×C22⋊C4×C53—21no—
54#95×C53—21no—
55C24×C53C22⋊C411yescharacteristic
56C22×C4×C53—21no—
57C22×C4×C53—21no—
58C22×C4×C53—41no—
59C22×C4×C53—41no—
60C22×C4×C53C22⋊C411yescharacteristic
61C22×C4×C53C2×C812yes—
62C22×C4×C53M1612yes—
63C22×C4×C53C22×C411yescharacteristic
64C22×Dic53—21no—
65C22×Dic53—21no—
66C22×Dic53—22no—
67C22×Dic53—21no—
68C22×Dic53—22no—
69C22×Dic53—21no—
70C22×Dic53—41no—
71C22×Dic53—41no—
72C22×Dic53—41no—
73C22×Dic53C22⋊C411yescharacteristic
74C22×Dic53C22⋊C411yescharacteristic
75C22×Dic53C2×D411yescharacteristic
76C22×Dic53C2×D411yescharacteristic
77C2×Dic106—21no—
78C2×Dic106—41no—
79C2×Dic106—41no—
80C2×Dic106C2×C812yes—
81C2×#88324—42no—
82C2×#88324—42no—
83C2×#88324—42no—
84C2×#88324—42no—
85C2×#88324—42no—
86C2×#88324—42no—
87C4⋊C4×C53—41no—
88C4⋊C4×C53—41no—
89C22⋊C4×C53—42no—
90C22⋊Dic53—24no—
91C4⋊Dic53—21no—
92C4⋊Dic53C2×C812yes—
93#11719919—24no—
94#11719919—42no—
95#11719919—41no—
96#11719919—41no—
97C4×Dic53—22no—
98C4×Dic53—22no—
99C4×Dic53—41no—
100C4×Dic53—41no—
101C4×Dic53—42no—
102C4×Dic53—42no—
103C23×C53—21no—
104C23×C53—21no—
105C23×C53—21no—
106C23×C53—21no—
107C23×C53—21no—
108C23×C53—21no—
109C23×C53C22⋊C812yes—
110C23×C53C2×C22⋊C411yescharacteristic
111C2×C4×C53—21no—
112C2×C4×C53—21no—
113C2×C4×C53—21no—
114C2×C4×C53—21no—
115C2×C4×C53—42no—
116C2×C4×C53—42no—
117C2×C4×C53—41no—
118C2×C4×C53—41no—
119C2×C4×C53—41no—
120C2×C4×C53—41no—
121C2×C4×C53—41no—
122C2×C4×C53—41no—
123C2×C4×C53C22⋊C812yes—
124C2×C4×C53C22×C811yescharacteristic
125C2×C4×C53C2×M1611yescharacteristic
126C2×Dic53—21no—
127C2×Dic53—21no—
128C2×Dic53—21no—
129C2×Dic53—21no—
130C2×Dic53—21no—
131C2×Dic53—21no—
132C2×Dic53—22no—
133C2×Dic53—22no—
134C2×Dic53—21no—
135C2×Dic53—21no—
136C2×Dic53—21no—
137C2×Dic53—44no—
138C2×Dic53—42no—
139C2×Dic53—42no—
140C2×Dic53—41no—
141C2×Dic53—41no—
142C2×Dic53—41no—
143C2×Dic53—41no—
144C2×Dic53#9912yes—
145C2×Dic53AGL(1,ℤ/8)11yescharacteristic
146C2×Dic53#13711yescharacteristic
147C2×Dic53#13712yes—
148Dic106—42no—
149Dic106—42no—
150#88324—82no—
151#88324—82no—
152#7600—531nomaximal, Sylow
153C22×C53—22no—
154C22×C53—22no—
155C22×C53—22no—
156C22×C53—41no—
157C22×C53—41no—
158C22×C53—41no—
159C22×C53—41no—
160C22×C53C2×C22⋊C811yescharacteristic
161C22×C53C2×#9911yescharacteristic
162C22×C53#41311yescharacteristic
163C22×C53#41711yescharacteristic
164C22×C53#41911yescharacteristic
165C22×C53#42111yescharacteristic
166C22×C53#42111yescharacteristic
167C4×C53—21no—
168C4×C53—21no—
169C4×C53—22no—
170C4×C53—81no—
171C4×C53—81no—
172Dic53—22no—
173Dic53—22no—
174Dic53—21no—
175Dic53—21no—
176Dic53—42no—
177Dic53—42no—
178Dic53—41no—
179C4×C22⋊Q8—531no—
180#1817—531no—
181#1797—531no—
182#1272—532no—
183#1271—532no—
184C2×C53—42no—
185C2×C53#145911yescharacteristic
186C2×C53#147711yescharacteristic
187C2×C53#147911yescharacteristic
188C2×C53#151111yescharacteristic
189C2×C53#151311yescharacteristic
190C2×C53#153511yescharacteristic
191C2×C53#155911yescharacteristic
192C2×C22⋊Q8—531no—
193C2×C4×Q8—531no—
194C22×C42—531no—
195C2×C4⋊C8—1061no—
196C2×C22⋊C8—1061no—
197C2×C22⋊C8—1061no—
198#391—531no—
199#389—531no—
200#387—1061no—
201#385—1061no—
202#380#615711yescharacteristic
203C4×C4⋊C4—531no—
204C4×C4⋊C4—1061no—
205C4×C22⋊C4—1061no—
206#336—532no—
207#336—532no—
208#336—1061no—
209C53#760011yescharacteristic, complemented, Sylow
210C22×Q8—531no—
211C23×C4—531no—
212C23×C4—531no—
213C23×C4C2×#615711yescharacteristic
214C22×C8—1062no—
215C22×C8—1062no—
216C22⋊Q8—534no—
217C22⋊Q8—1062no—
218C4×Q8—2122no—
219C2×C4⋊C4—531no—
220C2×C4⋊C4—1061no—
221C2×C4⋊C4—1061no—
222C2×C4⋊C4—1061no—
223C2×C4⋊C4C2×#615711yescharacteristic
224C2×C4⋊C4C2×#615711yescharacteristic
225C2×C22⋊C4—21no—
226C2×C22⋊C4—1061no—
227C2×C42—531no—
228C2×C42—531no—
229C2×C42—531no—
230C2×C42—531no—
231C2×C42—1061no—
232C2×C42—1061no—
233C2×C42—2122no—
234C4⋊C8—2122no—
235C22⋊C8—2122no—
236C22⋊C8—2122no—
237#95—21no—
238#95—1061no—
239#95—1061no—
240#95—1061no—
241#95—1061no—
242C24C22⋊#615711yescharacteristic
243C2×Q8—532no—
244C2×Q8—1061no—
245C2×Q8—2121no—
246C2×Q8—2121no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—41no—
250C22×C4—41no—
251C22×C4—531no—
252C22×C4—531no—
253C22×C4—531no—
254C22×C4—531no—
255C22×C4—1061no—
256C22×C4—1062no—
257C22×C4—1061no—
258C22×C4—1062no—
259C22×C4—1061no—
260C22×C4—1061no—
261C22×C4—2121no—
262C22×C4—2121no—
263C22×C4—2121no—
264C22×C4D53⋊C812yes—
265C22×C4#1171993612yes—
266C22×C4C22⋊#615711yescharacteristic
267C22×C4C22×#615711yescharacteristic
268C2×C8—2122no—
269C2×C8—2122no—
270C2×C8—2122no—
271C2×C8—2122no—
272C2×C8—2122no—
273C2×C8—2122no—
274C4⋊C4—41no—
275C4⋊C4—41no—
276C4⋊C4—532no—
277C4⋊C4—1064no—
278C4⋊C4—1061no—
279C4⋊C4—2121no—
280C4⋊C4—2121no—
281C4⋊C4—2122no—
282C22⋊C4—42no—
283C22⋊C4—1064no—
284C42—1062no—
285C42—1062no—
286C42—2121no—
287C42—2121no—
288C42—2122no—
289C42—2122no—
290C23—21no—
291C23—21no—
292C23—21no—
293C23—21no—
294C23—21no—
295C23—21no—
296C23#1313327412yes—
297C23C2×C22⋊#615711yescharacteristic
298Q8—2122no—
299Q8—2122no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—42no—
305C2×C4—42no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—532no—
313C2×C4—531no—
314C2×C4—531no—
315C2×C4—532no—
316C2×C4—1061no—
317C2×C4—1061no—
318C2×C4—1061no—
319C2×C4—1061no—
320C2×C4—1061no—
321C2×C4—1061no—
322C2×C4—1062no—
323C2×C4—1062no—
324C2×C4—1061no—
325C2×C4—1061no—
326C2×C4—1061no—
327C2×C4—2124no—
328C2×C4—2122no—
329C2×C4—2122no—
330C2×C4—2121no—
331C2×C4—2121no—
332C2×C4—2121no—
333C2×C4—2121no—
334C2×C4#1313327412yes—
335C2×C4C2×D53⋊C811yescharacteristic
336C2×C4C2×#1171993611yescharacteristic
337C8—4242no—
338C8—4242no—
339C22—22no—
340C22—22no—
341C22—22no—
342C22—41no—
343C22—41no—
344C22—41no—
345C22—41no—
346C22C2×#1313327411yescharacteristic
347C22D53⋊#9911yescharacteristic
348C22#1681161011yescharacteristic
349C22#1681161411yescharacteristic
350C22#1681161611yescharacteristic
351C22#1681162011yescharacteristic
352C22#1681162111yescharacteristic
353C4—22no—
354C4—21no—
355C4—21no—
356C4—81no—
357C4—81no—
358C4—1062no—
359C4—1062no—
360C4—1061no—
361C4—1061no—
362C4—2121no—
363C4—2122no—
364C4—2122no—
365C2—42no—
366C2#2028995311yescharacteristic
367C2#2029001211yescharacteristic
368C2#2029001911yescharacteristic
369C2#2029012411yescharacteristic
370C2#2029013111yescharacteristic
371C2#2029020011yescharacteristic
372C2#2029025111yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6157
commutatorC2×C4×C53C22×C8
FrattiniC22×C4C22×#6157
FittingC53×#380C4
socleC23×C53C2×C22⋊C4
prime radicalC24×C53C22⋊C4
cosocleC2×C4×Dic53C23
hypercenter#380#6157
hypocenterC53#7600
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
3. #20289536C2semidirect
4. #20289534C2semidirect
32. #13133196C8semidirect
209. C53#7600semidirect

Maximal subgroups

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

subgroupindex
#202906052
#202905692
#202895362
#202895342
#202772432
#760053