Finite Group Databasebeta

← #27295026

Subgroups — #27295026

Order 13568.

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

normal subgroups only

This group has 412 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#20280095C211yesmaximal, characteristic
2#20274795C211yesmaximal, characteristic
3#20272746C212yesmaximal, complemented
4#20272722C212yesmaximal, complemented
5C2×C53×#324C211yesmaximal, characteristic
6C2×#13133196C411yescharacteristic
7#16810587—21no—
8#16810553C411yescharacteristic
9#16810553C2211yescharacteristic
10#16810543C411yescharacteristic
11#16810543C2211yescharacteristic
12#16810518—21no—
13#16810518—21no—
14#16810496—21no—
15#16810496—21no—
16C4×C22⋊Dic53C411yescharacteristic
17#16810392—21no—
18#16810392C2212yes—
19C22×C4⋊C4×C53C2211yescharacteristic
20C2×C22⋊C8×C53—21no—
21C53×#324—21no—
22C53×#324C2212yes—
23C22×Dic106C2×C411yescharacteristic
24C2×#11719917—22no—
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×Dic53—21no—
36C22⋊#88322—41no—
37C22⋊#88322—41no—
38#13133124—21no—
39#13133124—21no—
40#13133124—21no—
41#13133124—21no—
42#13133124C2×C411yescharacteristic
43#13133124C2×C411yescharacteristic
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—41no—
78C2×Dic106—81no—
79C2×C8×C53—42no—
80C2×C8×C53—82no—
81C4⋊C4×C53—41no—
82C4⋊C4×C53—41no—
83C4⋊C4×C53—41no—
84C4⋊C4×C53—41no—
85C22⋊Dic53—24no—
86C22⋊Dic53—41no—
87C22⋊Dic53—41no—
88C4⋊Dic53—41no—
89#11719919—42no—
90#11719919—42no—
91#11719919—42no—
92#11719919—81no—
93#11719919—81no—
94C4×Dic53—81no—
95#11719917—42no—
96#11719917—42no—
97C2×#88322—42no—
98C23×C53—22no—
99C23×C53—21no—
100C23×C53—21no—
101C23×C53—41no—
102C23×C53#9911yescharacteristic
103C23×C53#9911yescharacteristic
104C23×C53C2×C22⋊C411yescharacteristic
105C2×C4×C53—21no—
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—42no—
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—81no—
122C2×C4×C53—81no—
123C2×C4×C53—81no—
124C2×C4×C53#13512yes—
125C2×Dic53—22no—
126C2×Dic53—22no—
127C2×Dic53—41no—
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—
137Dic106—82no—
138C8×C53—82no—
139#88322—82no—
140#11854—531nomaximal, Sylow
141C22×C53—21no—
142C22×C53—21no—
143C22×C53—21no—
144C22×C53—21no—
145C22×C53—21no—
146C22×C53—21no—
147C22×C53—21no—
148C22×C53—42no—
149C22×C53—42no—
150C22×C53#35612yes—
151C22×C53C2×#9911yescharacteristic
152C22×C53#41611yescharacteristic
153C22×C53#42111yescharacteristic
154C4×C53—22no—
155C4×C53—41no—
156C4×C53—41no—
157C4×C53—41no—
158C4×C53—41no—
159C4×C53—41no—
160Dic53—42no—
161Dic53—42no—
162Dic53—81no—
163Dic53—81no—
164#2583—531no—
165#1821—531no—
166C2×#324D5311yescharacteristic
167#1354—532no—
168#1346—532no—
169C2×C53—21no—
170C2×C53—21no—
171C2×C53—22no—
172C2×C53—41no—
173C2×C53#151611yescharacteristic
174C2×C53#212111yescharacteristic
175C2×C53C2×#35611yescharacteristic
176C2×C22⋊Q8—531no—
177C22×C4⋊C4C2×D5311yescharacteristic
178#407—1061no—
179C2×C22⋊C8—21no—
180#398—531no—
181#398—531no—
182#395—531no—
183#395—531no—
184#389—1061no—
185#389—1061no—
186#385—1061no—
187#385—1061no—
188C4×C22⋊C4—531no—
189#343—532no—
190#343—1061no—
191#324—21no—
192#324C2×D5312yes—
193C53#1185411yescharacteristic, complemented, Sylow
194C22×Q8—531no—
195C23×C4C4×D5311yescharacteristic
196C23×C4D21211yescharacteristic
197C23×C4C22⋊D5311yescharacteristic
198C2×M16—1062no—
199C22×C8—41no—
200C22⋊Q8—2121no—
201C22⋊Q8—2121no—
202C2×C4⋊C4—41no—
203C2×C4⋊C4—41no—
204C2×C4⋊C4—531no—
205C2×C4⋊C4—1061no—
206C2×C4⋊C4—1061no—
207C2×C4⋊C4—1061no—
208C2×C4⋊C4—1061no—
209C2×C4⋊C4—1061no—
210C2×C4⋊C4C4×D5311yescharacteristic
211C2×C4⋊C4D21211yescharacteristic
212C2×C4⋊C4C22⋊D5311yescharacteristic
213C2×C4⋊C4C22×D5311yescharacteristic
214C2×C22⋊C4—1061no—
215C2×C22⋊C4—1061no—
216C2×C42—1061no—
217C22⋊C8—22no—
218C22⋊C8—41no—
219C22⋊C8—2121no—
220C22⋊C8—2121no—
221#95—531no—
222#95—531no—
223#95—1061no—
224#95—1061no—
225#95—1061no—
226#95—1061no—
227C24#1171992111yescharacteristic
228C2×Q8—4241no—
229C2×Q8—2121no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—22no—
237C22×C4—41no—
238C22×C4—81no—
239C22×C4—81no—
240C22×C4—1061no—
241C22×C4—1061no—
242C22×C4—1061no—
243C22×C4—1061no—
244C22×C4#1171992111yescharacteristic
245C22×C4#1171992111yescharacteristic
246C22×C4#1171992111yescharacteristic
247C22×C4C2×C4×D5311yescharacteristic
248C22×C4C2×D21211yescharacteristic
249C22×C4C2×C22⋊D5311yescharacteristic
250M16—2122no—
251M16—2122no—
252C2×C8—42no—
253C2×C8—82no—
254C2×C8—2122no—
255C4⋊C4—41no—
256C4⋊C4—41no—
257C4⋊C4—41no—
258C4⋊C4—41no—
259C4⋊C4—4241no—
260C4⋊C4—4241no—
261C4⋊C4—2121no—
262C4⋊C4—2122no—
263C4⋊C4—2122no—
264C4⋊C4—2122no—
265C22⋊C4—1064no—
266C22⋊C4—2121no—
267C22⋊C4—2121no—
268C42—4241no—
269C23—21no—
270C23—21no—
271C23—22no—
272C23—41no—
273C23#1313313111yescharacteristic
274C23#1313313111yescharacteristic
275C23C2×#1171992111yescharacteristic
276Q8—4242no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—81no—
294C2×C4—81no—
295C2×C4—81no—
296C2×C4—1062no—
297C2×C4—1062no—
298C2×C4—2121no—
299C2×C4—2121no—
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#1313323912yes—
309C8—82no—
310C8—4242no—
311C22—21no—
312C22—21no—
313C22—21no—
314C22—21no—
315C22—21no—
316C22—21no—
317C22—21no—
318C22—42no—
319C22—42no—
320C22#1681042112yes—
321C22C2×#1313313111yescharacteristic
322C22#1681062011yescharacteristic
323C22#1681063911yescharacteristic
324C4—22no—
325C4—41no—
326C4—41no—
327C4—41no—
328C4—41no—
329C4—41no—
330C4—4241no—
331C4—4241no—
332C4—2122no—
333C4—2122no—
334C2—21no—
335C2—21no—
336C2—22no—
337C2—41no—
338C2#2027327211yescharacteristic
339C2#2027635911yescharacteristic
340C2C2×#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#11854
LCSG ▷ C22×C4×C53 ▷ C23×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. #20272746C2semidirect
4. #20272722C2semidirect
193. C53#11854semidirect

Maximal subgroups

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

subgroupindex
#202800952
#202747952
#202727462
#202727222
C2×C53×#3242
#1185453