Finite Group Databasebeta

← #27294898

Subgroups — #27294898

Order 13568.

416 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 416 conjugacy classes of subgroups — too many to draw. Only its 91 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#20274711C211yesmaximal, characteristic, complemented
2#20273059C211yesmaximal, characteristic, complemented
3#20272742C212yesmaximal, complemented
4#20272718C212yesmaximal
5C53×#2398C211yesmaximal, characteristic
6#16810680—21no—
7#16810647—21no—
8C2×#13133136—21no—
9#16810392C2212yes—
10#16810349—21no—
11#16810349C2212yes—
12C2×C4×Q8×C53C2211yescharacteristic
13C2×C4×D4×C53C411yescharacteristic
14C53×#400C411yescharacteristic
15C53×#400C2211yescharacteristic
16C53×#397—21no—
17C53×#394C411yescharacteristic
18C53×#394C2211yescharacteristic
19C53×#393—21no—
20C4×C22⋊C4×C53—21no—
21C4×C22⋊C4×C53—21no—
22C53×#376C411yescharacteristic
23C2×#11719917—22no—
24C4⋊#88322—41no—
25C4⋊#88322—41no—
26C4⋊#88322—41no—
27#13133136—42no—
28C22⋊#88322—22no—
29#13133128—41no—
30C22×Q8×C53D411yescharacteristic
31C22×D4×C53C2×C411yescharacteristic
32C23×C4×C53—21no—
33C4×Q8×C53—41no—
34C4×Q8×C53—41no—
35C4×D4×C53—41no—
36C4×D4×C53—41no—
37C2×C4⋊C4×C53C2×C411yescharacteristic
38C2×C4⋊C4×C53C2×C411yescharacteristic
39C2×C4⋊C4×C53D411yescharacteristic
40C2×C4⋊C4×C53C2311yescharacteristic
41C2×C22⋊C4×C53—21no—
42C2×C22⋊C4×C53—21no—
43C2×C22⋊C4×C53—21no—
44C2×C22⋊C4×C53—21no—
45C2×C22⋊C4×C53—21no—
46C2×C42×C53—21no—
47C2×C42×C53C2×C411yescharacteristic
48C2×C42×C53D411yescharacteristic
49C2×C42×C53D411yescharacteristic
50#95×C53—21no—
51#95×C53—21no—
52#95×C53—21no—
53#95×C53—21no—
54#95×C53—21no—
55#95×C53C2×C411yescharacteristic
56#95×C53C2×C411yescharacteristic
57C24×C53—21no—
58C2×Q8×C53—21no—
59C2×Q8×C53—21no—
60C2×Q8×C53—41no—
61C2×Q8×C53—41no—
62C2×D4×C53—41no—
63C2×D4×C53—42no—
64C22×C4×C53—21no—
65C22×C4×C53—21no—
66C22×C4×C53—21no—
67C22×C4×C53—42no—
68C22×C4×C53—42no—
69C22×C4×C53—41no—
70C22×C4×C53—41no—
71C22×C4×C53C22⋊C411yescharacteristic
72C22×C4×C53C22⋊C411yescharacteristic
73C22×C4×C53C22⋊C411yescharacteristic
74C22×C4×C53C22⋊C411yescharacteristic
75C22×C4×C53C22×C411yescharacteristic
76C22×C4×C53C2×D411yescharacteristic
77C22×C4×C53C2×D411yescharacteristic
78C4⋊C4×C53—41no—
79C4⋊C4×C53—41no—
80C4⋊C4×C53—41no—
81C4⋊C4×C53—41no—
82C22⋊C4×C53—42no—
83C22⋊C4×C53—42no—
84C22⋊C4×C53—42no—
85C22⋊C4×C53—42no—
86C22⋊C4×C53—41no—
87C22⋊C4×C53—41no—
88C42×C53—21no—
89C42×C53—21no—
90C42×C53—21no—
91C42×C53—21no—
92C42×C53—21no—
93C42×C53—21no—
94C42×C53—42no—
95#11719917—42no—
96#11719917—42no—
97C2×#88322—42no—
98C2×#88322—42no—
99C23×C53—22no—
100C23×C53—41no—
101C23×C53—41no—
102C23×C53—41no—
103C23×C53—41no—
104C23×C53—41no—
105C23×C53—41no—
106C23×C53C2×C22⋊C411yescharacteristic
107Q8×C53—42no—
108Q8×C53—42no—
109D4×C53—42no—
110D4×C53—42no—
111C2×C4×C53—21no—
112C2×C4×C53—21no—
113C2×C4×C53—22no—
114C2×C4×C53—21no—
115C2×C4×C53—21no—
116C2×C4×C53—21no—
117C2×C4×C53—21no—
118C2×C4×C53—21no—
119C2×C4×C53—21no—
120C2×C4×C53—21no—
121C2×C4×C53—21no—
122C2×C4×C53—21no—
123C2×C4×C53—41no—
124C2×C4×C53—41no—
125C2×C4×C53—41no—
126C2×C4×C53—41no—
127C2×C4×C53—41no—
128C2×C4×C53—41no—
129C2×C4×C53—41no—
130C2×C4×C53—41no—
131C2×C4×C53—41no—
132C2×C4×C53—41no—
133C2×C4×C53—41no—
134C2×C4×C53—41no—
135C2×C4×C53—82no—
136C2×C4×C53—82no—
137C2×C4×C53#9911yescharacteristic
138C2×C4×C53#9911yescharacteristic
139C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
140C2×C4×C53#13711yescharacteristic
141#88322—82no—
142#88322—82no—
143#11829—531nomaximal, Sylow
144C22×C53—21no—
145C22×C53—21no—
146C22×C53—42no—
147C22×C53—42no—
148C22×C53—42no—
149C22×C53—81no—
150C22×C53—81no—
151C22×C53—81no—
152C22×C53—81no—
153C22×C53C2×#9911yescharacteristic
154C22×C53#41711yescharacteristic
155C22×C53#42111yescharacteristic
156C4×C53—21no—
157C4×C53—21no—
158C4×C53—21no—
159C4×C53—41no—
160C4×C53—41no—
161C4×C53—41no—
162C4×C53—41no—
163C4×C53—41no—
164C4×C53—81no—
165C4×C53—81no—
166C4×C53—81no—
167C4×C53#35512yes—
168#2398D5311yescharacteristic
169#1801—531no—
170#1467—531no—
171#1353—532no—
172#1345—532no—
173C2×C53—21no—
174C2×C53—21no—
175C2×C53—82no—
176C2×C53#151311yescharacteristic
177C2×C53#212011yescharacteristic
178C2×C53C2×#35511yescharacteristic
179C2×C4×Q8C2×D5311yescharacteristic
180C2×C4×D4Dic5311yescharacteristic
181#432—1061no—
182#423—1061no—
183C2×#101—1061no—
184#400Dic5311yescharacteristic
185#400C2×D5311yescharacteristic
186#397—21no—
187#394Dic5311yescharacteristic
188#394C2×D5311yescharacteristic
189#393—21no—
190C4×C22⋊C4—21no—
191C4×C22⋊C4—21no—
192#376Dic5311yescharacteristic
193#343—532no—
194#324—532no—
195#324—1061no—
196C53#1182911yescharacteristic, complemented, Sylow
197C22×Q8C22⋊D5311yescharacteristic
198C22×D4C2×Dic5311yescharacteristic
199C23×C4—21no—
200C2×M16—1062no—
201C4×Q8—41no—
202C4×Q8—41no—
203C4×D4—41no—
204C4×D4—41no—
205C2×C4⋊C4C2×Dic5311yescharacteristic
206C2×C4⋊C4C2×Dic5311yescharacteristic
207C2×C4⋊C4C22⋊D5311yescharacteristic
208C2×C4⋊C4C22×D5311yescharacteristic
209C2×C22⋊C4—21no—
210C2×C22⋊C4—21no—
211C2×C22⋊C4—21no—
212C2×C22⋊C4—21no—
213C2×C22⋊C4—21no—
214C2×C42—21no—
215C2×C42C2×Dic5311yescharacteristic
216C2×C42C22⋊D5311yescharacteristic
217C2×C42C22⋊D5311yescharacteristic
218C4⋊C8—2121no—
219C4⋊C8—2121no—
220C4⋊C8—2121no—
221#101—2122no—
222C22⋊C8—1062no—
223#97—2121no—
224#95—21no—
225#95—21no—
226#95—21no—
227#95—21no—
228#95—21no—
229#95C2×Dic5311yescharacteristic
230#95C2×Dic5311yescharacteristic
231C24—21no—
232C2×Q8—21no—
233C2×Q8—21no—
234C2×Q8—41no—
235C2×Q8—41no—
236C2×D4—41no—
237C2×D4—42no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—42no—
242C22×C4—42no—
243C22×C4—41no—
244C22×C4—41no—
245C22×C4C22⋊Dic5311yescharacteristic
246C22×C4C22⋊Dic5311yescharacteristic
247C22×C4C22⋊Dic5311yescharacteristic
248C22×C4C22⋊Dic5311yescharacteristic
249C22×C4C22×Dic5311yescharacteristic
250C22×C4C2×C22⋊D5311yescharacteristic
251C22×C4C2×C22⋊D5311yescharacteristic
252M16—2122no—
253M16—2122no—
254C2×C8—2122no—
255C2×C8—2122no—
256C4⋊C4—41no—
257C4⋊C4—41no—
258C4⋊C4—41no—
259C4⋊C4—41no—
260C22⋊C4—42no—
261C22⋊C4—42no—
262C22⋊C4—42no—
263C22⋊C4—42no—
264C22⋊C4—41no—
265C22⋊C4—41no—
266C42—21no—
267C42—21no—
268C42—21no—
269C42—21no—
270C42—21no—
271C42—21no—
272C42—42no—
273C23—22no—
274C23—41no—
275C23—41no—
276C23—41no—
277C23—41no—
278C23—41no—
279C23—41no—
280C23C2×C22⋊Dic5311yescharacteristic
281Q8—42no—
282Q8—42no—
283D4—42no—
284D4—42no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—22no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—82no—
310C2×C4—82no—
311C2×C4#1313313211yescharacteristic
312C2×C4#1313313211yescharacteristic
313C2×C4#1313324211yescharacteristic
314C2×C4#1313324711yescharacteristic
315C8—4242no—
316C8—4242no—
317C22—21no—
318C22—21no—
319C22—42no—
320C22—42no—
321C22—42no—
322C22—81no—
323C22—81no—
324C22—81no—
325C22—81no—
326C22C2×#1313313211yescharacteristic
327C22#1681062411yescharacteristic
328C22#1681064011yescharacteristic
329C4—21no—
330C4—21no—
331C4—21no—
332C4—41no—
333C4—41no—
334C4—41no—
335C4—41no—
336C4—41no—
337C4—81no—
338C4—81no—
339C4—81no—
340C4#1681042012yes—
341C2—21no—
342C2—21no—
343C2—82no—
344C2#2027325811yescharacteristic
345C2#2027635611yescharacteristic
346C2C2×#1681042011yescharacteristic
subgroupG/H
centerC22C2×#13133132
commutatorC22×C4×C53C22×C4
FrattiniC2×C4⋊C4C22×D53
FittingC53×#2398C2
socleC22×C53C2×#99
prime radicalC22×D4×C53C2×C4
cosocleC2×C4⋊C4×C53C23
hypercenter#2398D53
hypocenterC53#11829
LCSG ▷ C22×C4×C53 ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ #2398

Complements

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

NH ≅ G/Ntype
1. #20274711C2semidirect
2. #20273059C2semidirect
3. #20272742C2semidirect
196. C53#11829semidirect

Maximal subgroups

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

subgroupindex
#202747112
#202730592
#202727422
#202727182
C53×#23982
#1182953