Finite Group Databasebeta

← #27293676

Subgroups — #27293676

Order 13568.

414 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 414 conjugacy classes of subgroups — too many to draw. Only its 83 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#20279189C211yesmaximal, characteristic
2#20273045C211yesmaximal, characteristic
3#20272660C212yesmaximal, complemented
4#20272648C212yesmaximal, complemented
5C2×C53×#324C211yesmaximal, characteristic
6C2×#13133196C411yescharacteristic
7#16810587—21no—
8#16810554—21no—
9#16810554—21no—
10#16810528C411yescharacteristic
11#16810528C2211yescharacteristic
12#16810496—21no—
13#16810496—21no—
14#16810492C411yescharacteristic
15#16810492C2211yescharacteristic
16C4×C22⋊Dic53C411yescharacteristic
17#16810349—21no—
18#16810349C2212yes—
19C22×C4⋊C4×C53C2211yescharacteristic
20C2×C22⋊C8×C53—21no—
21C53×#324—21no—
22C53×#324C2212yes—
23C22×Dic106C2×C411yescharacteristic
24C2×#11719917—41no—
25#13133196—41no—
26#13133196—41no—
27C2×C4⋊Dic53C2×C411yescharacteristic
28C2×#11719919—21no—
29C2×#11719919—21no—
30C2×#11719919—21no—
31C2×#11719919—21no—
32C2×#11719919—21no—
33C2×C22⋊Dic53—21no—
34C2×C22⋊Dic53—21no—
35C2×C4×Dic53C2×C411yescharacteristic
36C2×C4×Dic53C2×C411yescharacteristic
37C22⋊#88322—22no—
38C22⋊#88322—41no—
39#13133124—21no—
40#13133124—21no—
41#13133124—21no—
42#13133124—21no—
43#13133124—21no—
44C23×C4×C53C2×C411yescharacteristic
45C23×C4×C53D411yescharacteristic
46C23×C4×C53D411yescharacteristic
47C22×C8×C53—41no—
48C2×C4⋊C4×C53—41no—
49C2×C4⋊C4×C53—41no—
50C2×C4⋊C4×C53C2×C411yescharacteristic
51C2×C4⋊C4×C53D411yescharacteristic
52C2×C4⋊C4×C53D411yescharacteristic
53C2×C4⋊C4×C53C2311yescharacteristic
54C22⋊C8×C53—22no—
55C22⋊C8×C53—41no—
56C24×C53C22⋊C411yescharacteristic
57C22×C4×C53—21no—
58C22×C4×C53—21no—
59C22×C4×C53—21no—
60C22×C4×C53—21no—
61C22×C4×C53—21no—
62C22×C4×C53—21no—
63C22×C4×C53—22no—
64C22×C4×C53—41no—
65C22×C4×C53—81no—
66C22×C4×C53—81no—
67C22×C4×C53C22⋊C411yescharacteristic
68C22×C4×C53C22⋊C411yescharacteristic
69C22×C4×C53C22⋊C411yescharacteristic
70C22×C4×C53C22×C411yescharacteristic
71C22×C4×C53C2×D411yescharacteristic
72C22×C4×C53C2×D411yescharacteristic
73C22×Dic53—21no—
74C22×Dic53—21no—
75C22×Dic53—21no—
76C22×Dic53—21no—
77C2×Dic106—22no—
78C2×Dic106—41no—
79C2×Dic106—41no—
80C2×C8×C53—42no—
81C2×C8×C53—82no—
82C4⋊C4×C53—41no—
83C4⋊C4×C53—41no—
84C4⋊C4×C53—41no—
85C4⋊C4×C53—41no—
86C22⋊Dic53—24no—
87C22⋊Dic53—41no—
88C22⋊Dic53—41no—
89C4⋊Dic53—41no—
90#11719919—42no—
91#11719919—81no—
92#11719919—81no—
93#11719919—81no—
94#11719919—81no—
95C4×Dic53—22no—
96C4×Dic53—41no—
97#11719917—82no—
98C2×#88322—42no—
99C23×C53—22no—
100C23×C53—21no—
101C23×C53—21no—
102C23×C53—41no—
103C23×C53#9911yescharacteristic
104C23×C53#9911yescharacteristic
105C23×C53C2×C22⋊C411yescharacteristic
106C2×C4×C53—21no—
107C2×C4×C53—21no—
108C2×C4×C53—21no—
109C2×C4×C53—21no—
110C2×C4×C53—21no—
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—42no—
118C2×C4×C53—42no—
119C2×C4×C53—41no—
120C2×C4×C53—41no—
121C2×C4×C53—41no—
122C2×C4×C53—41no—
123C2×C4×C53—81no—
124C2×C4×C53—81no—
125C2×C4×C53#13112yes—
126C2×Dic53—22no—
127C2×Dic53—22no—
128C2×Dic53—41no—
129C2×Dic53—41no—
130C2×Dic53—41no—
131C2×Dic53—41no—
132C2×Dic53—41no—
133C2×Dic53—41no—
134C2×Dic53—41no—
135C2×Dic53—41no—
136C2×Dic53—41no—
137C2×Dic53—41no—
138Dic106—82no—
139C8×C53—82no—
140#88322—82no—
141#11555—531nomaximal, Sylow
142C22×C53—21no—
143C22×C53—21no—
144C22×C53—21no—
145C22×C53—21no—
146C22×C53—21no—
147C22×C53—21no—
148C22×C53—21no—
149C22×C53—42no—
150C22×C53—42no—
151C22×C53#35612yes—
152C22×C53#40811yescharacteristic
153C22×C53C2×#9911yescharacteristic
154C22×C53#41311yescharacteristic
155C4×C53—22no—
156C4×C53—41no—
157C4×C53—41no—
158C4×C53—41no—
159C4×C53—41no—
160C4×C53—41no—
161Dic53—42no—
162Dic53—42no—
163Dic53—81no—
164Dic53—81no—
165#2502—531no—
166#1463—531no—
167C2×#324D5311yescharacteristic
168#1325—532no—
169#1319—532no—
170C2×C53—21no—
171C2×C53—21no—
172C2×C53—22no—
173C2×C53—41no—
174C2×C53#146811yescharacteristic
175C2×C53#212711yescharacteristic
176C2×C53C2×#35611yescharacteristic
177C2×C22⋊Q8—531no—
178C22×C4⋊C4C2×D5311yescharacteristic
179#407—1061no—
180C2×C22⋊C8—21no—
181#398—1061no—
182#398—1061no—
183#391—531no—
184#391—531no—
185#385—1061no—
186#385—1061no—
187#384—531no—
188#384—531no—
189C4×C22⋊C4—531no—
190#324—21no—
191#324—532no—
192#324—1061no—
193#324C2×D5312yes—
194C53#1155511yescharacteristic, complemented, Sylow
195C22×Q8—531no—
196C23×C4C4×D5311yescharacteristic
197C23×C4D21211yescharacteristic
198C23×C4C22⋊D5311yescharacteristic
199C2×M16—2121no—
200C22×C8—41no—
201C22⋊Q8—2121no—
202C22⋊Q8—2121no—
203C2×C4⋊C4—41no—
204C2×C4⋊C4—41no—
205C2×C4⋊C4—531no—
206C2×C4⋊C4—1061no—
207C2×C4⋊C4—1061no—
208C2×C4⋊C4—1061no—
209C2×C4⋊C4—1061no—
210C2×C4⋊C4—1061no—
211C2×C4⋊C4C4×D5311yescharacteristic
212C2×C4⋊C4D21211yescharacteristic
213C2×C4⋊C4C22⋊D5311yescharacteristic
214C2×C4⋊C4C22×D5311yescharacteristic
215C2×C22⋊C4—1061no—
216C2×C22⋊C4—1061no—
217C2×C42—531no—
218C2×C42—531no—
219C22⋊C8—22no—
220C22⋊C8—41no—
221C22⋊C8—1062no—
222C22⋊C8—2121no—
223#95—1061no—
224#95—1061no—
225#95—1061no—
226#95—1061no—
227#95—1061no—
228C24#1171992111yescharacteristic
229C2×Q8—1062no—
230C2×Q8—2121no—
231C2×Q8—2121no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—22no—
239C22×C4—41no—
240C22×C4—81no—
241C22×C4—81no—
242C22×C4—1061no—
243C22×C4—1061no—
244C22×C4—1061no—
245C22×C4—1061no—
246C22×C4#1171992111yescharacteristic
247C22×C4#1171992111yescharacteristic
248C22×C4#1171992111yescharacteristic
249C22×C4C2×C4×D5311yescharacteristic
250C22×C4C2×D21211yescharacteristic
251C22×C4C2×C22⋊D5311yescharacteristic
252M16—4242no—
253C2×C8—42no—
254C2×C8—82no—
255C2×C8—2122no—
256C4⋊C4—41no—
257C4⋊C4—41no—
258C4⋊C4—41no—
259C4⋊C4—41no—
260C4⋊C4—4241no—
261C4⋊C4—4241no—
262C4⋊C4—4241no—
263C4⋊C4—4241no—
264C4⋊C4—2121no—
265C4⋊C4—2122no—
266C22⋊C4—1064no—
267C22⋊C4—2121no—
268C22⋊C4—2121no—
269C42—1062no—
270C42—2121no—
271C23—21no—
272C23—21no—
273C23—22no—
274C23—41no—
275C23#1313313111yescharacteristic
276C23#1313313111yescharacteristic
277C23C2×#1171992111yescharacteristic
278Q8—4242no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—42no—
289C2×C4—42no—
290C2×C4—42no—
291C2×C4—42no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—81no—
297C2×C4—81no—
298C2×C4—1062no—
299C2×C4—1062no—
300C2×C4—2121no—
301C2×C4—2121no—
302C2×C4—2121no—
303C2×C4—2121no—
304C2×C4—2121no—
305C2×C4—2121no—
306C2×C4—2121no—
307C2×C4—2121no—
308C2×C4—2121no—
309C2×C4—2121no—
310C2×C4#1313322312yes—
311C8—82no—
312C8—4242no—
313C22—21no—
314C22—21no—
315C22—21no—
316C22—21no—
317C22—21no—
318C22—21no—
319C22—21no—
320C22—42no—
321C22—42no—
322C22#1681042112yes—
323C22#1681058911yescharacteristic
324C22C2×#1313313111yescharacteristic
325C22#1681060511yescharacteristic
326C4—22no—
327C4—41no—
328C4—41no—
329C4—41no—
330C4—41no—
331C4—41no—
332C4—4241no—
333C4—4241no—
334C4—2122no—
335C4—2122no—
336C2—21no—
337C2—21no—
338C2—22no—
339C2—41no—
340C2#2027306111yescharacteristic
341C2#2027638211yescharacteristic
342C2C2×#1681042111yescharacteristic
subgroupG/H
centerC22C2×#13133131
commutatorC22×C4×C53C22×C4
FrattiniC2×C4⋊C4C22×D53
FittingC2×C53×#324C2
socleC22×C53C2×#99
prime radicalC24×C53C22⋊C4
cosocleC2×C4⋊C4×C53C23
hypercenterC2×#324D53
hypocenterC53#11555
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C24 ◁ C22×C4⋊C4 ◁ C2×#324

Complements

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

NH ≅ G/Ntype
3. #20272660C2semidirect
4. #20272648C2semidirect
194. C53#11555semidirect

Maximal subgroups

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

subgroupindex
#202791892
#202730452
#202726602
#202726482
C2×C53×#3242
#1155553