Finite Group Databasebeta

← #27272434

Subgroups — #27272434

Order 13568.

432 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20274858C211yesmaximal, characteristic, complemented
2#20274461C211yesmaximal, characteristic, complemented
3#20272541C212yesmaximal, complemented
4#20272536C212yesmaximal
5C53×#2861C211yesmaximal, characteristic
6C2×C4×#88322—21no—
7#16810378—21no—
8#16810378—21no—
9#16810378—21no—
10#16810378C2212yes—
11#16810378C2212yes—
12C2×#125×C53C411yescharacteristic
13C2×#124×C53C411yescharacteristic
14C53×#391C411yescharacteristic
15C53×#391C2211yescharacteristic
16C53×#390C411yescharacteristic
17C53×#390C2211yescharacteristic
18C53×#388—21no—
19C53×#388—21no—
20C53×#388—21no—
21C53×#388—21no—
22C43×C53C2211yescharacteristic
23C22×#88322—22no—
24C22×#88322—22no—
25C4×#88322—24no—
26C22×Q8×C53C2×C411yescharacteristic
27C22×D4×C53C2×C411yescharacteristic
28#125×C53—42no—
29#124×C53—42no—
30C2×C4⋊C4×C53—21no—
31C2×C4⋊C4×C53—21no—
32C2×C4⋊C4×C53C2×C411yescharacteristic
33C2×C4⋊C4×C53C2×C411yescharacteristic
34C2×C22⋊C4×C53—21no—
35C2×C22⋊C4×C53—21no—
36C2×C22⋊C4×C53—21no—
37C2×C22⋊C4×C53—21no—
38C2×C22⋊C4×C53—21no—
39C2×C22⋊C4×C53—21no—
40C2×C42×C53C2×C411yescharacteristic
41C2×C42×C53C2×C411yescharacteristic
42C2×C42×C53D411yescharacteristic
43C2×C42×C53D411yescharacteristic
44C2×C42×C53D411yescharacteristic
45C2×C42×C53D411yescharacteristic
46C2×C42×C53C2311yescharacteristic
47#95×C53—21no—
48#95×C53—21no—
49#95×C53—21no—
50#95×C53—21no—
51C24×C53—21no—
52C2×Q8×C53—22no—
53C2×Q8×C53—41no—
54C2×Q8×C53—41no—
55C2×D4×C53—22no—
56C2×D4×C53—41no—
57C2×D4×C53—41no—
58C22×C4×C53—21no—
59C22×C4×C53—21no—
60C22×C4×C53—21no—
61C22×C4×C53C22⋊C411yescharacteristic
62C22×C4×C53C22⋊C411yescharacteristic
63C22×C4×C53C22⋊C411yescharacteristic
64C22×C4×C53C22⋊C411yescharacteristic
65C22×C4×C53C22×C411yescharacteristic
66C22×C4×C53C2×D411yescharacteristic
67C22×C4×C53C2×D411yescharacteristic
68C4⋊C4×C53—42no—
69C4⋊C4×C53—41no—
70C4⋊C4×C53—41no—
71C4⋊C4×C53—42no—
72C22⋊C4×C53—42no—
73C22⋊C4×C53—42no—
74C22⋊C4×C53—42no—
75C22⋊C4×C53—42no—
76C22⋊C4×C53—42no—
77C22⋊C4×C53—42no—
78C42×C53—22no—
79C42×C53—22no—
80C42×C53—22no—
81C42×C53—22no—
82C42×C53—22no—
83C42×C53—41no—
84C42×C53—41no—
85C2×#88322—24no—
86C2×#88322—42no—
87C2×#88322—42no—
88C2×#88322—42no—
89C2×#88322—42no—
90C2×#88322—42no—
91C23×C53—41no—
92C23×C53—41no—
93C23×C53—41no—
94C23×C53—41no—
95C23×C53—41no—
96C23×C53—41no—
97C23×C53—41no—
98C23×C53C2×C22⋊C411yescharacteristic
99Q8×C53—82no—
100D4×C53—82no—
101C2×C4×C53—21no—
102C2×C4×C53—21no—
103C2×C4×C53—21no—
104C2×C4×C53—21no—
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—21no—
115C2×C4×C53—21no—
116C2×C4×C53—21no—
117C2×C4×C53—41no—
118C2×C4×C53—41no—
119C2×C4×C53—41no—
120C2×C4×C53—41no—
121C2×C4×C53—41no—
122C2×C4×C53—41no—
123C2×C4×C53—41no—
124C2×C4×C53—41no—
125C2×C4×C53—41no—
126C2×C4×C53#10012yes—
127C2×C4×C53#10412yes—
128C2×C4×C53#13112yes—
129C2×C4×C53#13512yes—
130C2×C4×C53#13512yes—
131#88322—44no—
132#88322—82no—
133#7656—531nomaximal, Sylow
134C22×C53—81no—
135C22×C53—81no—
136C22×C53—81no—
137C22×C53—81no—
138C22×C53—81no—
139C22×C53—81no—
140C22×C53—81no—
141C22×C53#40811yescharacteristic
142C22×C53#41011yescharacteristic
143C22×C53C2×#10011yescharacteristic
144C22×C53#41611yescharacteristic
145C22×C53#41611yescharacteristic
146C22×C53C2×#10411yescharacteristic
147C22×C53#42111yescharacteristic
148C4×C53—22no—
149C4×C53—22no—
150C4×C53—22no—
151C4×C53—22no—
152C4×C53—22no—
153C4×C53—41no—
154C4×C53—41no—
155C4×C53—81no—
156C4×C53—81no—
157C4×C53—81no—
158#2861D5311yescharacteristic
159#1839—531no—
160#1744—531no—
161#1272—532no—
162#1270—532no—
163C2×C53—161no—
164C2×C53#146911yescharacteristic
165C2×C53#148211yescharacteristic
166C2×C53#149411yescharacteristic
167C2×C53#150011yescharacteristic
168C2×C53#152111yescharacteristic
169C2×C53#153011yescharacteristic
170C2×C53#154311yescharacteristic
171C2×#125Dic5311yescharacteristic
172C2×#124Dic5311yescharacteristic
173C2×C4×C8—1061no—
174#391Dic5311yescharacteristic
175#391C2×D5311yescharacteristic
176#390Dic5311yescharacteristic
177#390C2×D5311yescharacteristic
178#388—21no—
179#388—21no—
180#388—21no—
181#388—21no—
182C43C2×D5311yescharacteristic
183#336—532no—
184#336—532no—
185#336—1061no—
186#336—1061no—
187#336—1061no—
188C53#765611yescharacteristic, complemented, Sylow
189C22×Q8C2×Dic5311yescharacteristic
190C22×D4C2×Dic5311yescharacteristic
191C22×C8—1062no—
192C22×C8—1062no—
193#125—42no—
194#124—42no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4C2×Dic5311yescharacteristic
198C2×C4⋊C4C2×Dic5311yescharacteristic
199C2×C22⋊C4—21no—
200C2×C22⋊C4—21no—
201C2×C22⋊C4—21no—
202C2×C22⋊C4—21no—
203C2×C22⋊C4—21no—
204C2×C22⋊C4—21no—
205C2×C42C2×Dic5311yescharacteristic
206C2×C42C2×Dic5311yescharacteristic
207C2×C42C22⋊D5311yescharacteristic
208C2×C42C22⋊D5311yescharacteristic
209C2×C42C22⋊D5311yescharacteristic
210C2×C42C22⋊D5311yescharacteristic
211C2×C42C22×D5311yescharacteristic
212C4×C8—1064no—
213#95—21no—
214#95—21no—
215#95—21no—
216#95—21no—
217C24—21no—
218C2×Q8—22no—
219C2×Q8—41no—
220C2×Q8—41no—
221C2×D4—22no—
222C2×D4—41no—
223C2×D4—41no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4C22⋊Dic5311yescharacteristic
228C22×C4C22⋊Dic5311yescharacteristic
229C22×C4C22⋊Dic5311yescharacteristic
230C22×C4C22⋊Dic5311yescharacteristic
231C22×C4C22×Dic5311yescharacteristic
232C22×C4C2×C22⋊D5311yescharacteristic
233C22×C4C2×C22⋊D5311yescharacteristic
234C2×C8—1064no—
235C2×C8—2122no—
236C2×C8—2122no—
237C2×C8—2122no—
238C2×C8—2122no—
239C2×C8—2122no—
240C4⋊C4—41no—
241C4⋊C4—41no—
242C4⋊C4—42no—
243C4⋊C4—42no—
244C22⋊C4—42no—
245C22⋊C4—42no—
246C22⋊C4—42no—
247C22⋊C4—42no—
248C22⋊C4—42no—
249C22⋊C4—42no—
250C42—22no—
251C42—22no—
252C42—22no—
253C42—22no—
254C42—22no—
255C42—41no—
256C42—41no—
257C23—41no—
258C23—41no—
259C23—41no—
260C23—41no—
261C23—41no—
262C23—41no—
263C23—41no—
264C23C2×C22⋊Dic5311yescharacteristic
265Q8—82no—
266D4—82no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4#1313313412yes—
293C2×C4#1313314412yes—
294C2×C4#1313322412yes—
295C2×C4#1313323712yes—
296C2×C4#1313323712yes—
297C8—4242no—
298C8—2124no—
299C22—81no—
300C22—81no—
301C22—81no—
302C22—81no—
303C22—81no—
304C22—81no—
305C22—81no—
306C22#1681059011yescharacteristic
307C22#1681059711yescharacteristic
308C22C2×#1313313411yescharacteristic
309C22#1681061911yescharacteristic
310C22#1681061911yescharacteristic
311C22C2×#1313314411yescharacteristic
312C22#1681064011yescharacteristic
313C4—22no—
314C4—22no—
315C4—22no—
316C4—22no—
317C4—22no—
318C4—41no—
319C4—41no—
320C4—81no—
321C4—81no—
322C4—81no—
323C2—161no—
324C2#2027306711yescharacteristic
325C2#2027312311yescharacteristic
326C2#2027317511yescharacteristic
327C2#2027320111yescharacteristic
328C2#2027329211yescharacteristic
329C2#2027332611yescharacteristic
330C2#2027337711yescharacteristic
subgroupG/H
centerC23C2×C22⋊Dic53
commutatorC22×C4×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#2861C2
socleC23×C53C2×C22⋊C4
prime radicalC22×D4×C53C2×C4
cosocleC2×C42×C53C23
hypercenter#2861D53
hypocenterC53#7656
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C43 ◁ #2861

Complements

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

NH ≅ G/Ntype
1. #20274858C2semidirect
2. #20274461C2semidirect
3. #20272541C2semidirect
188. C53#7656semidirect

Maximal subgroups

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

subgroupindex
#202748582
#202744612
#202725412
#202725362
C53×#28612
#765653