Finite Group Databasebeta

← #27277723

Subgroups — #27277723

Order 13568.

480 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 480 conjugacy classes of subgroups — too many to draw. Only its 103 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
1C4×#13133132C211yesmaximal, characteristic
2#20273053C211yesmaximal, characteristic
3#20272601C212yesmaximal, complemented
4#20272567C212yesmaximal, complemented
5C53×#2353C211yesmaximal, characteristic
6C2×#13133132—21no—
7#16810587C2212yes—
8C4×C22⋊Dic53C2212yes—
9#16810390—21no—
10#16810349—21no—
11#16810347—21no—
12C2×C4×D4×C53C2211yescharacteristic
13C53×#388—21no—
14C53×#387—21no—
15C53×#386—21no—
16C53×#383C411yescharacteristic, complemented
17C53×#381C2211yescharacteristic
18C53×#380C411yescharacteristic, complemented
19C4×C22⋊C4×C53—21no—
20C4×C22⋊C4×C53C2211yescharacteristic
21C53×#376C411yescharacteristic, complemented
22C2×#95×C53C411yescharacteristic, complemented
23C2×#11719917—22no—
24C2×C22⋊Dic53—22no—
25C2×C4×Dic53—22no—
26#13133132—24no—
27C22⋊#88322—22no—
28C22⋊#88322—22no—
29#13133124—22no—
30C22×D4×C53D411yescharacteristic
31C23×C4×C53—21no—
32C23×C4×C53C2×C411yescharacteristic
33C23×C4×C53C2311yescharacteristic
34C4×D4×C53—22no—
35C4×D4×C53—22no—
36C2×C4⋊C4×C53—21no—
37C2×C4⋊C4×C53D411yescharacteristic
38C2×C22⋊C4×C53—21no—
39C2×C22⋊C4×C53—21no—
40C2×C22⋊C4×C53—21no—
41C2×C22⋊C4×C53D411yescharacteristic
42C2×C22⋊C4×C53D411yescharacteristic
43C2×C42×C53—21no—
44C2×C42×C53—21no—
45C2×C42×C53C2×C411yescharacteristic
46#95×C53—21no—
47#95×C53—21no—
48#95×C53—21no—
49#95×C53—21no—
50#95×C53—21no—
51#95×C53C2×C411yescharacteristic
52#95×C53C2×C411yescharacteristic
53#95×C53C2×C411yescharacteristic
54#95×C53C2×C411yescharacteristic
55C24×C53C22⋊C411yescharacteristic
56C24×C53C2×D411yescharacteristic
57C2×D4×C53—22no—
58C2×D4×C53—22no—
59C2×D4×C53—41no—
60C22×C4×C53—21no—
61C22×C4×C53—21no—
62C22×C4×C53—21no—
63C22×C4×C53—21no—
64C22×C4×C53—21no—
65C22×C4×C53—21no—
66C22×C4×C53—42no—
67C22×C4×C53—42no—
68C22×C4×C53—81no—
69C22×C4×C53—81no—
70C22×C4×C53—81no—
71C22×C4×C53C22⋊C411yescharacteristic
72C22×C4×C53C22⋊C411yescharacteristic
73C22×C4×C53C22⋊C411yescharacteristic
74C22×C4×C53C22×C411yescharacteristic
75C22×C4×C53C2×D411yescharacteristic
76C22×C4×C53#4112yes—
77C22×C4×C53#4112yes—
78C22×Dic53—42no—
79C4⋊C4×C53—22no—
80C4⋊C4×C53—81no—
81C22⋊C4×C53—22no—
82C22⋊C4×C53—22no—
83C22⋊C4×C53—42no—
84C22⋊C4×C53—42no—
85C22⋊C4×C53—81no—
86C42×C53—21no—
87C42×C53—21no—
88C42×C53—42no—
89C42×C53—81no—
90C22⋊Dic53—24no—
91C22⋊Dic53—42no—
92C4×Dic53—82no—
93#11719917—82no—
94C2×#88322—42no—
95C23×C53—21no—
96C23×C53—21no—
97C23×C53—21no—
98C23×C53—21no—
99C23×C53—41no—
100C23×C53—41no—
101C23×C53—41no—
102C23×C53—41no—
103C23×C53#10412yes—
104C23×C53C2×C22⋊C411yescharacteristic
105C23×C53#12312yes—
106D4×C53—44no—
107C2×C4×C53—22no—
108C2×C4×C53—21no—
109C2×C4×C53—21no—
110C2×C4×C53—21no—
111C2×C4×C53—21no—
112C2×C4×C53—21no—
113C2×C4×C53—41no—
114C2×C4×C53—41no—
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—41no—
122C2×C4×C53—41no—
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—81no—
131C2×C4×C53—81no—
132C2×C4×C53—84no—
133C2×C4×C53#11711yescharacteristic
134C2×C4×C53#11711yescharacteristic
135C2×C4×C53#12312yes—
136C2×Dic53—44no—
137C2×Dic53—82no—
138C2×Dic53—82no—
139#88322—82no—
140#8777—531nomaximal, Sylow
141C22×C53—21no—
142C22×C53—21no—
143C22×C53—22no—
144C22×C53—22no—
145C22×C53—22no—
146C22×C53—22no—
147C22×C53—42no—
148C22×C53—42no—
149C22×C53—41no—
150C22×C53—41no—
151C22×C53—81no—
152C22×C53—81no—
153C22×C53#38111yescharacteristic
154C22×C53C2×#10411yescharacteristic
155C22×C53#42111yescharacteristic
156C4×C53—22no—
157C4×C53—21no—
158C4×C53—21no—
159C4×C53—42no—
160C4×C53—41no—
161C4×C53—41no—
162C4×C53—41no—
163C4×C53—81no—
164C4×C53—81no—
165C4×C53—81no—
166Dic53—84no—
167#2353D5311yescharacteristic
168C4×#99—531no—
169#1465—531no—
170#1297—532no—
171#1284—532no—
172C2×C53—21no—
173C2×C53—21no—
174C2×C53—42no—
175C2×C53—42no—
176C2×C53#182611yescharacteristic
177C2×C53#182711yescharacteristic
178C2×C53#182811yescharacteristic
179C2×C4×D4C2×D5311yescharacteristic
180C2×#99—1061no—
181#407—532no—
182#388—21no—
183#387—21no—
184#386—21no—
185#383Dic5311yescharacteristic, complemented
186#381C2×D5311yescharacteristic
187#380Dic5311yescharacteristic, complemented
188C4×C22⋊C4—21no—
189C4×C22⋊C4—532no—
190C4×C22⋊C4C2×D5311yescharacteristic
191#376Dic5311yescharacteristic, complemented
192C2×#95Dic5311yescharacteristic, complemented
193#342—1061no—
194#324—1061no—
195#323—1061no—
196C53#877711yescharacteristic, complemented, Sylow
197C22×D4C22⋊D5311yescharacteristic
198C23×C4—21no—
199C23×C4C2×Dic5311yescharacteristic
200C23×C4C22×D5311yescharacteristic
201C2×M16—1062no—
202C4×D4—22no—
203C4×D4—22no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4C22⋊D5311yescharacteristic
206C2×C22⋊C4—21no—
207C2×C22⋊C4—21no—
208C2×C22⋊C4—21no—
209C2×C22⋊C4—1062no—
210C2×C22⋊C4C22⋊D5311yescharacteristic
211C2×C22⋊C4C22⋊D5311yescharacteristic
212C2×C42—21no—
213C2×C42—21no—
214C2×C42—1062no—
215C2×C42C2×Dic5311yescharacteristic
216#99—1064no—
217C22⋊C8—1062no—
218C22⋊C8—1062no—
219#95—21no—
220#95—21no—
221#95—21no—
222#95—21no—
223#95—21no—
224#95—1062no—
225#95C2×Dic5311yescharacteristic
226#95C2×Dic5311yescharacteristic
227#95C2×Dic5311yescharacteristic
228#95C2×Dic5311yescharacteristic
229C24C22⋊Dic5311yescharacteristic
230C24C2×C22⋊D5311yescharacteristic
231C2×D4—22no—
232C2×D4—22no—
233C2×D4—41no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—42no—
241C22×C4—42no—
242C22×C4—81no—
243C22×C4—81no—
244C22×C4—81no—
245C22×C4—2122no—
246C22×C4C22⋊Dic5311yescharacteristic
247C22×C4C22⋊Dic5311yescharacteristic
248C22×C4C22⋊Dic5311yescharacteristic
249C22×C4#1171994512yes—
250C22×C4#1171994512yes—
251C22×C4C22×Dic5311yescharacteristic
252C22×C4C2×C22⋊D5311yescharacteristic
253M16—4242no—
254C2×C8—2122no—
255C4⋊C4—22no—
256C4⋊C4—81no—
257C22⋊C4—22no—
258C22⋊C4—22no—
259C22⋊C4—42no—
260C22⋊C4—42no—
261C22⋊C4—81no—
262C22⋊C4—1064no—
263C22⋊C4—2122no—
264C42—21no—
265C42—21no—
266C42—42no—
267C42—81no—
268C42—4242no—
269C23—21no—
270C23—21no—
271C23—21no—
272C23—21no—
273C23—41no—
274C23—41no—
275C23—41no—
276C23—41no—
277C23#1313314412yes—
278C23C2×C22⋊Dic5311yescharacteristic
279C23#1313320012yes—
280D4—44no—
281C2×C4—22no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—41no—
288C2×C4—42no—
289C2×C4—42no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—81no—
305C2×C4—81no—
306C2×C4—84no—
307C2×C4—4242no—
308C2×C4—4242no—
309C2×C4—2124no—
310C2×C4#1313317611yescharacteristic
311C2×C4#1313317611yescharacteristic
312C2×C4#1313320012yes—
313C8—4242no—
314C22—21no—
315C22—21no—
316C22—22no—
317C22—22no—
318C22—22no—
319C22—22no—
320C22—42no—
321C22—42no—
322C22—41no—
323C22—41no—
324C22—81no—
325C22—81no—
326C22#1681048611yescharacteristic
327C22C2×#1313314411yescharacteristic
328C22#1681064011yescharacteristic
329C4—22no—
330C4—21no—
331C4—21no—
332C4—42no—
333C4—41no—
334C4—41no—
335C4—41no—
336C4—81no—
337C4—81no—
338C4—81no—
339C4—4244no—
340C2—21no—
341C2—21no—
342C2—42no—
343C2—42no—
344C2#2027481411yescharacteristic
345C2#2027481911yescharacteristic
346C2#2027482311yescharacteristic
subgroupG/H
centerC22#16810486
commutatorC22×C4×C53C22×C4
FrattiniC23×C4C22×D53
FittingC53×#2353C2
socleC22×C53#381
prime radicalC22×D4×C53D4
cosocleC23×C4×C53C23
hypercenter#2353D53
hypocenterC53#8777
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2353

Complements

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

NH ≅ G/Ntype
3. #20272601C2semidirect
4. #20272567C2semidirect
16. C53×#383C4semidirect
18. C53×#380C4semidirect
21. C53×#376C4semidirect
22. C2×#95×C53C4semidirect
185. #383Dic53semidirect
187. #380Dic53semidirect
191. #376Dic53semidirect
192. C2×#95Dic53semidirect
196. C53#8777semidirect

Maximal subgroups

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

subgroupindex
C4×#131331322
#202730532
#202726012
#202725672
C53×#23532
#877753