Finite Group Databasebeta

← #28299910

Subgroups — #28299910

Order 13568.

470 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 470 conjugacy classes of subgroups — too many to draw. Only its 92 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#20290569C211yesmaximal, characteristic
2#20290441C211yesmaximal, characteristic
3#20289535C212yesmaximal, complemented
4#20289534C212yesmaximal, complemented
5#20283049C211yesmaximal, characteristic
6C2×C4⋊#88324—21no—
7C2×C22⋊#88324—21no—
8C2×C22⋊#88324—21no—
9C2×#13133272—21no—
10#16811489C2212yes—
11#16811489C2212yes—
12C22×C4×Dic53C2211yescharacteristic
13#16810530C411yescharacteristic
14#16810530C2211yescharacteristic
15#16810519C411yescharacteristic
16#16810519C2211yescharacteristic
17#16810504—21no—
18#16810504—21no—
19#16810495—21no—
20#16810484—21no—
21#16810483C411yescharacteristic
22C2×C22⋊Q8×C53C411yescharacteristic
23C22×#88324—22no—
24C22×#88324—22no—
25C4⋊#88324—42no—
26C22⋊#88324—24no—
27C22⋊#88324—42no—
28#13133272—42no—
29C23×Dic53D411yescharacteristic
30C23×Dic53D411yescharacteristic
31C2×C4⋊Dic53C2×C411yescharacteristic
32C2×C4⋊Dic53C2×C411yescharacteristic
33C2×#11719919—21no—
34C2×#11719919—21no—
35C2×C22⋊Dic53—21no—
36C2×C4×Dic53—42no—
37C2×C4×Dic53C2×C411yescharacteristic
38C2×C4×Dic53D411yescharacteristic
39C2×C4×Dic53D411yescharacteristic
40C2×C4×Dic53C2311yescharacteristic
41#13133124—21no—
42#13133124—21no—
43#13133124—21no—
44#13133124—21no—
45#13133124—21no—
46#13133124—21no—
47#13133124—21no—
48C22×Q8×C53C2×C411yescharacteristic
49C23×C4×C53C2×C411yescharacteristic
50C22⋊Q8×C53—42no—
51C2×C4⋊C4×C53—21no—
52C2×C4⋊C4×C53C2×C411yescharacteristic
53C2×C22⋊C4×C53—21no—
54C24×C53C22⋊C411yescharacteristic
55C2×Q8×C53—22no—
56C2×Q8×C53—41no—
57C2×Q8×C53—41no—
58C22×C4×C53—22no—
59C22×C4×C53—22no—
60C22×C4×C53—21no—
61C22×C4×C53—21no—
62C22×C4×C53—41no—
63C22×C4×C53C22⋊C411yescharacteristic
64C22×C4×C53C22×C411yescharacteristic
65C22×Dic53—22no—
66C22×Dic53—21no—
67C22×Dic53—21no—
68C22×Dic53—22no—
69C22×Dic53—21no—
70C22×Dic53—21no—
71C22×Dic53—41no—
72C22×Dic53—41no—
73C22×Dic53—41no—
74C22×Dic53C22⋊C411yescharacteristic
75C22×Dic53C22⋊C411yescharacteristic
76C22×Dic53C2×D411yescharacteristic
77C22×Dic53C2×D411yescharacteristic
78C2×#88324—24no—
79C2×#88324—42no—
80C2×#88324—42no—
81C2×#88324—42no—
82C2×#88324—42no—
83C2×#88324—42no—
84C4⋊C4×C53—41no—
85C4⋊C4×C53—42no—
86C22⋊C4×C53—42no—
87C22⋊Dic53—42no—
88C4⋊Dic53—22no—
89C4⋊Dic53—41no—
90#11719919—42no—
91#11719919—41no—
92#11719919—41no—
93C4×Dic53—22no—
94C4×Dic53—22no—
95C4×Dic53—41no—
96C4×Dic53—41no—
97C23×C53—21no—
98C23×C53—21no—
99C23×C53—21no—
100C23×C53—21no—
101C23×C53—21no—
102C23×C53—21no—
103C23×C53#10412yes—
104C23×C53C2×C22⋊C411yescharacteristic
105Q8×C53—82no—
106C2×C4×C53—21no—
107C2×C4×C53—21no—
108C2×C4×C53—21no—
109C2×C4×C53—21no—
110C2×C4×C53—44no—
111C2×C4×C53—41no—
112C2×C4×C53—41no—
113C2×C4×C53—41no—
114C2×C4×C53—41no—
115C2×C4×C53—41no—
116C2×C4×C53—41no—
117C2×C4×C53#9912yes—
118C2×C4×C53#13112yes—
119C2×Dic53—21no—
120C2×Dic53—21no—
121C2×Dic53—21no—
122C2×Dic53—21no—
123C2×Dic53—21no—
124C2×Dic53—21no—
125C2×Dic53—21no—
126C2×Dic53—21no—
127C2×Dic53—21no—
128C2×Dic53—21no—
129C2×Dic53—44no—
130C2×Dic53—42no—
131C2×Dic53—42no—
132C2×Dic53—41no—
133C2×Dic53—41no—
134C2×Dic53—41no—
135C2×Dic53—41no—
136C2×Dic53—41no—
137C2×Dic53—41no—
138C2×Dic53AGL(1,ℤ/8)11yescharacteristic
139C2×Dic53#13711yescharacteristic
140C2×Dic53#13712yes—
141#88324—44no—
142#88324—82no—
143#7701—531nomaximal, Sylow
144C22×C53—22no—
145C22×C53—22no—
146C22×C53—22no—
147C22×C53—41no—
148C22×C53—41no—
149C22×C53—41no—
150C22×C53—41no—
151C22×C53#40811yescharacteristic
152C22×C53C2×#9911yescharacteristic
153C22×C53#41311yescharacteristic
154C22×C53#41711yescharacteristic
155C22×C53#41911yescharacteristic
156C22×C53C2×#10411yescharacteristic
157C22×C53#42111yescharacteristic
158C4×C53—22no—
159C4×C53—22no—
160C4×C53—81no—
161C4×C53—81no—
162Dic53—22no—
163Dic53—21no—
164Dic53—21no—
165Dic53—41no—
166Dic53—41no—
167Dic53—81no—
168Dic53—81no—
169#2855—531no—
170#1797—531no—
171#1749—531no—
172#1272—532no—
173#1271—532no—
174C2×C53—42no—
175C2×C53#146811yescharacteristic
176C2×C53#148311yescharacteristic
177C2×C53#149711yescharacteristic
178C2×C53#150811yescharacteristic
179C2×C53#151311yescharacteristic
180C2×C53#153211yescharacteristic
181C2×C53#155911yescharacteristic
182C2×C22⋊Q8#615711yescharacteristic
183C22×C42—531no—
184C2×C4⋊C8—1061no—
185C2×C22⋊C8—1061no—
186C2×C22⋊C8—1061no—
187C2×#97—1061no—
188#391—531no—
189#391—531no—
190#389—531no—
191#389—531no—
192#387—1061no—
193#387—1061no—
194#385—1061no—
195#381—1061no—
196#380—531no—
197#336—532no—
198#336—532no—
199C53#770111yescharacteristic, complemented, Sylow
200C22×Q8C2×#615711yescharacteristic
201C23×C4—531no—
202C23×C4—531no—
203C23×C4C2×#615711yescharacteristic
204C22×C8—1062no—
205C22×C8—1062no—
206C22⋊Q8—42no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—531no—
209C2×C4⋊C4—531no—
210C2×C4⋊C4—1061no—
211C2×C4⋊C4—1061no—
212C2×C4⋊C4C2×#615711yescharacteristic
213C2×C22⋊C4—21no—
214C2×C22⋊C4—1061no—
215C2×C42—531no—
216C2×C42—531no—
217C2×C42—531no—
218C2×C42—531no—
219C2×C42—2122no—
220C4⋊C8—2122no—
221C22⋊C8—1064no—
222C22⋊C8—2122no—
223#97—2122no—
224#95—1061no—
225#95—1061no—
226#95—1061no—
227#95—1061no—
228#95—1061no—
229#95—1061no—
230#95—1061no—
231C24C22⋊#615711yescharacteristic
232C2×Q8—22no—
233C2×Q8—41no—
234C2×Q8—41no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—41no—
240C22×C4—531no—
241C22×C4—531no—
242C22×C4—531no—
243C22×C4—531no—
244C22×C4—1061no—
245C22×C4—1061no—
246C22×C4—1062no—
247C22×C4—1062no—
248C22×C4—1061no—
249C22×C4—1061no—
250C22×C4—2121no—
251C22×C4—2121no—
252C22×C4—2121no—
253C22×C4C22⋊#615711yescharacteristic
254C22×C4C22×#615711yescharacteristic
255C2×C8—1064no—
256C2×C8—2122no—
257C2×C8—2122no—
258C2×C8—2122no—
259C2×C8—2122no—
260C2×C8—2122no—
261C4⋊C4—41no—
262C4⋊C4—42no—
263C4⋊C4—1062no—
264C4⋊C4—2121no—
265C4⋊C4—2121no—
266C4⋊C4—2121no—
267C4⋊C4—2122no—
268C22⋊C4—42no—
269C22⋊C4—2122no—
270C42—1062no—
271C42—1062no—
272C42—2121no—
273C42—2121no—
274C23—21no—
275C23—21no—
276C23—21no—
277C23—21no—
278C23—21no—
279C23—21no—
280C23#1313328212yes—
281C23C2×C22⋊#615711yescharacteristic
282Q8—82no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—44no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—531no—
295C2×C4—531no—
296C2×C4—532no—
297C2×C4—1061no—
298C2×C4—1061no—
299C2×C4—1061no—
300C2×C4—1061no—
301C2×C4—1061no—
302C2×C4—1061no—
303C2×C4—1061no—
304C2×C4—1061no—
305C2×C4—1061no—
306C2×C4—1061no—
307C2×C4—2124no—
308C2×C4—2122no—
309C2×C4—2122no—
310C2×C4—2121no—
311C2×C4—2121no—
312C2×C4—2121no—
313C2×C4—2121no—
314C2×C4—2121no—
315C2×C4—2121no—
316C2×C4#1313327512yes—
317C2×C4#1313330412yes—
318C8—4242no—
319C8—2124no—
320C22—22no—
321C22—22no—
322C22—22no—
323C22—41no—
324C22—41no—
325C22—41no—
326C22—41no—
327C22#1681159611yescharacteristic
328C22C2×#1313327511yescharacteristic
329C22#1681161011yescharacteristic
330C22#1681161411yescharacteristic
331C22#1681161611yescharacteristic
332C22C2×#1313328211yescharacteristic
333C22#1681162011yescharacteristic
334C4—22no—
335C4—22no—
336C4—81no—
337C4—81no—
338C4—1062no—
339C4—1061no—
340C4—1061no—
341C4—4241no—
342C4—4241no—
343C4—2121no—
344C4—2121no—
345C2—42no—
346C2#2028998011yescharacteristic
347C2#2029003211yescharacteristic
348C2#2029007711yescharacteristic
349C2#2029011511yescharacteristic
350C2#2029013011yescharacteristic
351C2#2029019311yescharacteristic
352C2#2029025111yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6157
commutatorC22×C4×C53C22×C4
FrattiniC22×C4C22×#6157
FittingC2×C22⋊Q8×C53C4
socleC23×C53C2×C22⋊C4
prime radicalC24×C53C22⋊C4
cosocleC2×C4×Dic53C23
hypercenterC2×C22⋊Q8#6157
hypocenterC53#7701
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
3. #20289535C2semidirect
4. #20289534C2semidirect
199. C53#7701semidirect

Maximal subgroups

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

subgroupindex
#202905692
#202904412
#202895352
#202895342
#202830492
#770153