Finite Group Databasebeta

← #28303323

Subgroups — #28303323

Order 13568.

496 conjugacy classes of subgroups; only the 116 normal subgroups are shown.

normal subgroups only

This group has 496 conjugacy classes of subgroups — too many to draw. Only its 116 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
2C4×C22⋊#88324C211yesmaximal, characteristic
3#20289551C212yesmaximal, complemented
4#20289551C212yesmaximal, complemented
5#20277110C211yesmaximal, characteristic
6C2×C4⋊#88324C2212yes—
7C2×C22⋊#88324—21no—
8C2×C22⋊#88324—21no—
9C2×C4×#88324C2212yes—
10#16811489—21no—
11#16811489—21no—
12C22×C4×Dic53C2211yescharacteristic
13#16810490—21no—
14#16810488C411yescharacteristic
15#16810488C411yescharacteristic
16#16810484—21no—
17#16810483C411yescharacteristic
18C4⋊C4×Dic53C2211yescharacteristic
19C4⋊C4×Dic53C2211yescharacteristic
20C22⋊C4×Dic53—21no—
21#16810468—21no—
22C53×#380C411yescharacteristic
23C22×#88324—22no—
24C22×#88324—22no—
25C4⋊#88324—24no—
26C22⋊#88324—24no—
27C22⋊#88324—42no—
28C4×#88324—24no—
29C23×Dic53D411yescharacteristic
30C23×Dic53D411yescharacteristic
31C2×C4⋊Dic53C2×C411yescharacteristic
32C2×C4⋊Dic53C2×C411yescharacteristic
33C2×C22⋊Dic53—21no—
34C2×C4×Dic53—24no—
35C2×C4×Dic53—21no—
36C2×C4×Dic53—21no—
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—
48C23×C4×C53C2×C411yescharacteristic
49C2×C4⋊C4×C53C2×C411yescharacteristic
50C2×C4⋊C4×C53C2×C411yescharacteristic
51C2×C22⋊C4×C53—21no—
52#95×C53—21no—
53C24×C53C22⋊C411yescharacteristic
54C22×C4×C53—22no—
55C22×C4×C53—22no—
56C22×C4×C53—22no—
57C22×C4×C53—21no—
58C22×C4×C53—21no—
59C22×C4×C53C22⋊C411yescharacteristic
60C22×C4×C53C22×C411yescharacteristic
61C22×Dic53—22no—
62C22×Dic53—22no—
63C22×Dic53—22no—
64C22×Dic53—22no—
65C22×Dic53—21no—
66C22×Dic53—21no—
67C22×Dic53—21no—
68C22×Dic53—21no—
69C22×Dic53—41no—
70C22×Dic53C22⋊C411yescharacteristic
71C22×Dic53C22⋊C411yescharacteristic
72C22×Dic53C2×D411yescharacteristic
73C22×Dic53C2×D411yescharacteristic
74C2×#88324—24no—
75C2×#88324—42no—
76C2×#88324—42no—
77C2×#88324—42no—
78C2×#88324—42no—
79C2×#88324—42no—
80C4⋊C4×C53—22no—
81C4⋊C4×C53—22no—
82C22⋊C4×C53—42no—
83C22⋊Dic53—42no—
84C4⋊Dic53—22no—
85C4⋊Dic53—22no—
86C4×Dic53—22no—
87C4×Dic53—22no—
88C4×Dic53—22no—
89C4×Dic53—42no—
90C4×Dic53—41no—
91C4×Dic53—41no—
92C4×Dic53#4114yes—
93C23×C53—21no—
94C23×C53—21no—
95C23×C53—21no—
96C23×C53—21no—
97C23×C53—21no—
98C23×C53—21no—
99C23×C53#10412yes—
100C23×C53C2×C22⋊C411yescharacteristic
101C2×C4×C53—21no—
102C2×C4×C53—21no—
103C2×C4×C53—21no—
104C2×C4×C53—44no—
105C2×C4×C53—41no—
106C2×C4×C53—41no—
107C2×C4×C53—41no—
108C2×C4×C53—41no—
109C2×C4×C53—41no—
110C2×C4×C53—41no—
111C2×C4×C53#11712yes—
112C2×C4×C53#13112yes—
113C2×C4×C53#13112yes—
114C2×Dic53—21no—
115C2×Dic53—21no—
116C2×Dic53—21no—
117C2×Dic53—21no—
118C2×Dic53—21no—
119C2×Dic53—21no—
120C2×Dic53—21no—
121C2×Dic53—21no—
122C2×Dic53—44no—
123C2×Dic53—42no—
124C2×Dic53—42no—
125C2×Dic53—41no—
126C2×Dic53—41no—
127C2×Dic53—41no—
128C2×Dic53—41no—
129C2×Dic53—41no—
130C2×Dic53—41no—
131C2×Dic53#12312yes—
132C2×Dic53#12312yes—
133C2×Dic53#13512yes—
134C2×Dic53AGL(1,ℤ/8)11yescharacteristic
135C2×Dic53#13711yescharacteristic
136#88324—44no—
137#88324—82no—
138#8711—531nomaximal, Sylow
139C22×C53—22no—
140C22×C53—22no—
141C22×C53—22no—
142C22×C53—41no—
143C22×C53—41no—
144C22×C53—41no—
145C22×C53—41no—
146C22×C53#38111yescharacteristic
147C22×C53#40811yescharacteristic
148C22×C53#40811yescharacteristic
149C22×C53#41611yescharacteristic
150C22×C53#41711yescharacteristic
151C22×C53C2×#10411yescharacteristic
152C22×C53#42111yescharacteristic
153C4×C53—41no—
154C4×C53—81no—
155C4×C53—81no—
156C4×C53#43314yes—
157Dic53—22no—
158Dic53—41no—
159Dic53—41no—
160Dic53—81no—
161Dic53—81no—
162Dic53#48212yes—
163Dic53#48512yes—
164#2283—531no—
165#1797—531no—
166C4×C22⋊C8—531no—
167#1279—532no—
168#1279—532no—
169C2×C53—42no—
170C2×C53#148211yescharacteristic
171C2×C53#148311yescharacteristic
172C2×C53#149411yescharacteristic
173C2×C53#149511yescharacteristic
174C2×C53#181811yescharacteristic
175C2×C53#182511yescharacteristic
176C2×C53#182611yescharacteristic
177C22×C42—531no—
178C2×C4⋊C8—532no—
179C2×C22⋊C8—1061no—
180C2×C22⋊C8—1061no—
181C2×C4×C8—532no—
182#383—1061no—
183#382—531no—
184#382—531no—
185#381—1061no—
186#380—531no—
187#380#615711yescharacteristic
188C4×C4⋊C4—531no—
189C4×C4⋊C4—531no—
190C4×C22⋊C4—1061no—
191#376—1061no—
192#336—1061no—
193#336—1061no—
194C53#871111yescharacteristic, complemented, Sylow
195C23×C4—531no—
196C23×C4—531no—
197C23×C4C2×#615711yescharacteristic
198C22×C8—1062no—
199C22×C8—1062no—
200C2×C4⋊C4—531no—
201C2×C4⋊C4—531no—
202C2×C4⋊C4C2×#615711yescharacteristic
203C2×C4⋊C4C2×#615711yescharacteristic
204C2×C22⋊C4—21no—
205C2×C22⋊C4—1061no—
206C2×C42—531no—
207C2×C42—531no—
208C2×C42—531no—
209C2×C42—531no—
210C2×C42—1064no—
211C2×C42—1061no—
212C2×C42—1061no—
213C4⋊C8—1064no—
214C22⋊C8—1064no—
215C22⋊C8—2122no—
216C4×C8—1064no—
217#95—21no—
218#95—1061no—
219#95—1061no—
220#95—1061no—
221#95—1061no—
222#95—1061no—
223#95—1061no—
224#95—1061no—
225C24C22⋊#615711yescharacteristic
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—531no—
232C22×C4—531no—
233C22×C4—531no—
234C22×C4—531no—
235C22×C4—1061no—
236C22×C4—1062no—
237C22×C4—1062no—
238C22×C4—1062no—
239C22×C4—1061no—
240C22×C4—1062no—
241C22×C4—1061no—
242C22×C4—1061no—
243C22×C4—2121no—
244C22×C4C22⋊#615711yescharacteristic
245C22×C4C22×#615711yescharacteristic
246C2×C8—1064no—
247C2×C8—2122no—
248C2×C8—2122no—
249C2×C8—2122no—
250C2×C8—2122no—
251C2×C8—2122no—
252C4⋊C4—22no—
253C4⋊C4—22no—
254C4⋊C4—1062no—
255C4⋊C4—1062no—
256C22⋊C4—42no—
257C22⋊C4—2122no—
258C42—534no—
259C42—1062no—
260C42—1062no—
261C42—1062no—
262C42—2122no—
263C42—2121no—
264C42—2121no—
265C23—21no—
266C23—21no—
267C23—21no—
268C23—21no—
269C23—21no—
270C23—21no—
271C23#1313328112yes—
272C23C2×C22⋊#615711yescharacteristic
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—44no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—532no—
284C2×C4—532no—
285C2×C4—531no—
286C2×C4—531no—
287C2×C4—532no—
288C2×C4—1061no—
289C2×C4—1061no—
290C2×C4—1061no—
291C2×C4—1061no—
292C2×C4—1061no—
293C2×C4—1061no—
294C2×C4—1061no—
295C2×C4—1061no—
296C2×C4—2122no—
297C2×C4—2122no—
298C2×C4—2124no—
299C2×C4—2121no—
300C2×C4—2121no—
301C2×C4—2121no—
302C2×C4—2121no—
303C2×C4—2121no—
304C2×C4—2121no—
305C2×C4#1313329612yes—
306C2×C4#1313330412yes—
307C2×C4#1313330412yes—
308C8—4242no—
309C8—2124no—
310C22—22no—
311C22—22no—
312C22—22no—
313C22—41no—
314C22—41no—
315C22—41no—
316C22—41no—
317C22#1681156611yescharacteristic
318C22#1681159611yescharacteristic
319C22#1681159611yescharacteristic
320C22#1681161311yescharacteristic
321C22#1681161411yescharacteristic
322C22C2×#1313328111yescharacteristic
323C22#1681162211yescharacteristic
324C4—41no—
325C4—81no—
326C4—81no—
327C4—532no—
328C4—532no—
329C4—1062no—
330C4—4241no—
331C4—4241no—
332C4—2121no—
333C4—2121no—
334C4#1681164714yes—
335C2—42no—
336C2#2029002811yescharacteristic
337C2#2029003011yescharacteristic
338C2#2029006611yescharacteristic
339C2#2029007211yescharacteristic
340C2#2029060811yescharacteristic
341C2#2029062011yescharacteristic
342C2#2029062111yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6157
commutatorC22×C4×C53C22×C4
FrattiniC22×C4C22×#6157
FittingC53×#380C4
socleC23×C53C2×C22⋊C4
prime radicalC24×C53C22⋊C4
cosocleC2×C4×Dic53C23
hypercenter#380#6157
hypocenterC53#8711
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ #380

Complements

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

NH ≅ G/Ntype
3. #20289551C2semidirect
4. #20289551C2semidirect
194. C53#8711semidirect

Maximal subgroups

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

subgroupindex
#202905692
C4×C22⋊#883242
#202895512
#202771102
#871153