Finite Group Databasebeta

← #28300003

Subgroups — #28300003

Order 13568.

494 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 494 conjugacy classes of subgroups — too many to draw. Only its 100 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#20290606C211yesmaximal, characteristic
2#20290569C211yesmaximal, characteristic
3#20289536C212yesmaximal, complemented
4#20289535C212yesmaximal, complemented
5#20283063C211yesmaximal, characteristic
6C2×C4⋊#88324—21no—
7C2×C22⋊#88324—21no—
8C2×C22⋊#88324—21no—
9#16811489—21no—
10#16811489C2212yes—
11#16811489C2212yes—
12C2×#13133215C411yescharacteristic
13C2×#13133196C411yescharacteristic
14C22×C4×Dic53C2211yescharacteristic
15#16810530C2211yescharacteristic
16#16810528C2211yescharacteristic
17#16810518—21no—
18#16810518—21no—
19#16810492C411yescharacteristic
20#16810481—21no—
21#16810481—21no—
22C2×C22⋊Q8×C53C411yescharacteristic
23C22×#88324—22no—
24C22×#88324—22no—
25C4⋊#88324—42no—
26C22⋊#88324—42no—
27C22⋊#88324—42no—
28C22×Dic106C2×C411yescharacteristic
29C23×Dic53D411yescharacteristic
30C23×Dic53D411yescharacteristic
31#13133215—42no—
32#13133196—42no—
33C2×C4⋊Dic53C2×C411yescharacteristic
34C2×#11719919—21no—
35C2×#11719919—21no—
36C2×#11719919—21no—
37C2×#11719919—21no—
38C2×#11719919—21no—
39C2×C22⋊Dic53—21no—
40C2×C4×Dic53—42no—
41C2×C4×Dic53C2×C411yescharacteristic
42C2×C4×Dic53D411yescharacteristic
43C2×C4×Dic53D411yescharacteristic
44C2×C4×Dic53C2311yescharacteristic
45#13133124—21no—
46#13133124—21no—
47#13133124—21no—
48#13133124—21no—
49C22×Q8×C53C2×C411yescharacteristic
50C23×C4×C53C2×C411yescharacteristic
51C22⋊Q8×C53—42no—
52C2×C4⋊C4×C53—21no—
53C2×C4⋊C4×C53C2×C411yescharacteristic
54C2×C22⋊C4×C53—21no—
55C24×C53C22⋊C411yescharacteristic
56C2×Q8×C53—22no—
57C2×Q8×C53—41no—
58C2×Q8×C53—41no—
59C22×C4×C53—22no—
60C22×C4×C53—22no—
61C22×C4×C53—21no—
62C22×C4×C53—21no—
63C22×C4×C53—41no—
64C22×C4×C53C22⋊C411yescharacteristic
65C22×C4×C53C22×C411yescharacteristic
66C22×Dic53—22no—
67C22×Dic53—22no—
68C22×Dic53—21no—
69C22×Dic53—21no—
70C22×Dic53—41no—
71C22×Dic53—41no—
72C22×Dic53—41no—
73C22×Dic53C22⋊C411yescharacteristic
74C22×Dic53C22⋊C411yescharacteristic
75C22×Dic53SD1612yes—
76C22×Dic53Q1612yes—
77C22×Dic53C2×D411yescharacteristic
78C22×Dic53C2×D411yescharacteristic
79C2×Dic106—22no—
80C2×Dic106—41no—
81C2×Dic106—41no—
82C2×#88324—42no—
83C2×#88324—42no—
84C2×#88324—42no—
85C2×#88324—42no—
86C2×#88324—42no—
87C2×#88324—42no—
88C4⋊C4×C53—41no—
89C4⋊C4×C53—42no—
90C22⋊C4×C53—42no—
91C22⋊Dic53—42no—
92C4⋊Dic53—41no—
93#11719919—42no—
94#11719919—42no—
95#11719919—42no—
96#11719919—42no—
97#11719919—42no—
98C4×Dic53—22no—
99C4×Dic53—22no—
100C4×Dic53—41no—
101C4×Dic53—41no—
102C23×C53—21no—
103C23×C53—21no—
104C23×C53—21no—
105C23×C53—21no—
106C23×C53—21no—
107C23×C53—21no—
108C23×C53#10312yes—
109C23×C53C2×C22⋊C411yescharacteristic
110Q8×C53—82no—
111C2×C4×C53—21no—
112C2×C4×C53—21no—
113C2×C4×C53—21no—
114C2×C4×C53—21no—
115C2×C4×C53—44no—
116C2×C4×C53—41no—
117C2×C4×C53—41no—
118C2×C4×C53—41no—
119C2×C4×C53—41no—
120C2×C4×C53—41no—
121C2×C4×C53—41no—
122C2×C4×C53#10012yes—
123C2×C4×C53#13112yes—
124C2×Dic53—21no—
125C2×Dic53—21no—
126C2×Dic53—21no—
127C2×Dic53—21no—
128C2×Dic53—21no—
129C2×Dic53—21no—
130C2×Dic53—21no—
131C2×Dic53—21no—
132C2×Dic53—42no—
133C2×Dic53—42no—
134C2×Dic53—44no—
135C2×Dic53—41no—
136C2×Dic53—41no—
137C2×Dic53—41no—
138C2×Dic53—41no—
139C2×Dic53—41no—
140C2×Dic53—41no—
141C2×Dic53#9912yes—
142C2×Dic53#10312yes—
143C2×Dic53C2×SD1611yescharacteristic
144C2×Dic53C2×Q1611yescharacteristic
145C2×Dic53#13712yes—
146Dic106—82no—
147#88324—82no—
148#88324—82no—
149#7727—531nomaximal, Sylow
150C22×C53—22no—
151C22×C53—22no—
152C22×C53—22no—
153C22×C53—41no—
154C22×C53—41no—
155C22×C53—41no—
156C22×C53—41no—
157C22×C53#40811yescharacteristic
158C22×C53C2×#9911yescharacteristic
159C22×C53C2×#10011yescharacteristic
160C22×C53C2×#10311yescharacteristic
161C22×C53#41911yescharacteristic
162C22×C53#42111yescharacteristic
163C22×C53#42111yescharacteristic
164C4×C53—22no—
165C4×C53—22no—
166C4×C53—81no—
167C4×C53—81no—
168Dic53—22no—
169Dic53—22no—
170Dic53—21no—
171Dic53—22no—
172Dic53—21no—
173Dic53—81no—
174Dic53—81no—
175#2857—531no—
176#1817—531no—
177#1797—531no—
178#1272—532no—
179#1272—532no—
180C2×C53—42no—
181C2×C53#147111yescharacteristic
182C2×C53#148111yescharacteristic
183C2×C53#149711yescharacteristic
184C2×C53#151111yescharacteristic
185C2×C53#152011yescharacteristic
186C2×C53#153611yescharacteristic
187C2×C53#154811yescharacteristic
188C2×C4⋊Q8—531no—
189C2×C22⋊Q8—531no—
190C2×C22⋊Q8#615711yescharacteristic
191C22×C42—531no—
192C2×C4⋊C8—1061no—
193C2×C22⋊C8—1061no—
194C2×C22⋊C8—1061no—
195#391—531no—
196#391—531no—
197#389—1061no—
198#389—1061no—
199#384—531no—
200#380—1061no—
201#380—1061no—
202#336—532no—
203#336—532no—
204#336—1061no—
205C53#772711yescharacteristic, complemented, Sylow
206C22×Q8—531no—
207C22×Q8C2×#615711yescharacteristic
208C23×C4—531no—
209C23×C4—531no—
210C23×C4C2×#615711yescharacteristic
211C22×C8—1062no—
212C22×C8—1062no—
213C4⋊Q8—2122no—
214C22⋊Q8—42no—
215C22⋊Q8—2122no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—531no—
218C2×C4⋊C4—1061no—
219C2×C4⋊C4—1061no—
220C2×C4⋊C4—1061no—
221C2×C4⋊C4—1061no—
222C2×C4⋊C4—1061no—
223C2×C4⋊C4C2×#615711yescharacteristic
224C2×C22⋊C4—21no—
225C2×C22⋊C4—1061no—
226C2×C42—531no—
227C2×C42—531no—
228C2×C42—531no—
229C2×C42—531no—
230C2×C42—2122no—
231C4⋊C8—2122no—
232C22⋊C8—2122no—
233C22⋊C8—2122no—
234#95—1061no—
235#95—1061no—
236#95—1061no—
237#95—1061no—
238C24C22⋊#615711yescharacteristic
239C2×Q8—22no—
240C2×Q8—41no—
241C2×Q8—41no—
242C2×Q8—1062no—
243C2×Q8—2121no—
244C2×Q8—2121no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—41no—
250C22×C4—531no—
251C22×C4—531no—
252C22×C4—531no—
253C22×C4—531no—
254C22×C4—532no—
255C22×C4—532no—
256C22×C4—1062no—
257C22×C4—1062no—
258C22×C4—1061no—
259C22×C4—1061no—
260C22×C4—2121no—
261C22×C4—2121no—
262C22×C4—2121no—
263C22×C4C22⋊#615711yescharacteristic
264C22×C4C22×#615711yescharacteristic
265C2×C8—2122no—
266C2×C8—2122no—
267C2×C8—2122no—
268C2×C8—2122no—
269C2×C8—2122no—
270C2×C8—2122no—
271C4⋊C4—41no—
272C4⋊C4—42no—
273C4⋊C4—2121no—
274C4⋊C4—2122no—
275C4⋊C4—2122no—
276C4⋊C4—2122no—
277C4⋊C4—2122no—
278C4⋊C4—2122no—
279C22⋊C4—42no—
280C22⋊C4—2122no—
281C42—1062no—
282C42—1062no—
283C42—2121no—
284C42—2121no—
285C23—21no—
286C23—21no—
287C23—21no—
288C23—21no—
289C23—21no—
290C23—21no—
291C23#1313328012yes—
292C23C2×C22⋊#615711yescharacteristic
293Q8—82no—
294Q8—4242no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—44no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—532no—
307C2×C4—532no—
308C2×C4—531no—
309C2×C4—531no—
310C2×C4—532no—
311C2×C4—1061no—
312C2×C4—1061no—
313C2×C4—1061no—
314C2×C4—1061no—
315C2×C4—1061no—
316C2×C4—1061no—
317C2×C4—1061no—
318C2×C4—1061no—
319C2×C4—2122no—
320C2×C4—2122no—
321C2×C4—2124no—
322C2×C4—2121no—
323C2×C4—2121no—
324C2×C4—2121no—
325C2×C4—2121no—
326C2×C4—2121no—
327C2×C4—2121no—
328C2×C4#1313327712yes—
329C2×C4#1313330412yes—
330C8—4242no—
331C8—4242no—
332C22—22no—
333C22—22no—
334C22—22no—
335C22—41no—
336C22—41no—
337C22—41no—
338C22—41no—
339C22#1681159611yescharacteristic
340C22D53⋊#9911yescharacteristic
341C22C2×#1313327711yescharacteristic
342C22C2×#1313328011yescharacteristic
343C22#1681161611yescharacteristic
344C22#1681162011yescharacteristic
345C22#1681162011yescharacteristic
346C4—22no—
347C4—22no—
348C4—81no—
349C4—81no—
350C4—1062no—
351C4—1062no—
352C4—1061no—
353C4—1062no—
354C4—1061no—
355C4—4241no—
356C4—4241no—
357C2—42no—
358C2#2028999111yescharacteristic
359C2#2029002311yescharacteristic
360C2#2029007711yescharacteristic
361C2#2029012411yescharacteristic
362C2#2029015311yescharacteristic
363C2#2029020311yescharacteristic
364C2#2029022811yescharacteristic
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#7727
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. #20289536C2semidirect
4. #20289535C2semidirect
205. C53#7727semidirect

Maximal subgroups

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

subgroupindex
#202906062
#202905692
#202895362
#202895352
#202830632
#772753