Finite Group Databasebeta

← #27295253

Subgroups — #27295253

Order 13568.

372 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 372 conjugacy classes of subgroups — too many to draw. Only its 79 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#20279926C211yesmaximal, characteristic
2#20274710C211yesmaximal, characteristic
3#20273037C211yesmaximal, characteristic
4#20272743C211yesmaximal, characteristic
5#20272742C211yesmaximal, characteristic
6#20272741C211yesmaximal, characteristic
7C53×#1353C211yesmaximal, characteristic
8C2×#13133216C411yescharacteristic
9C2×#13133182C2211yescharacteristic
10#16810673—21no—
11#16810646—21no—
12C2×#13133135—21no—
13#16810563C411yescharacteristic
14#16810562C2211yescharacteristic
15#16810561C411yescharacteristic
16#16810527—21no—
17#16810527—21no—
18#16810527—21no—
19#16810487—21no—
20#16810474C411yescharacteristic
21#16810392C2211yescharacteristic
22#16810349C2211yescharacteristic
23#16810348—21no—
24C53×#400C2211yescharacteristic
25C53×#343C2211yescharacteristic
26C53×#324C2211yescharacteristic
27C22×Dic106—21no—
28C22×Dic106D411yescharacteristic
29C2×#11719917—21no—
30#13133216—41no—
31#13133216—41no—
32#13133182—41no—
33#13133182—41no—
34C2×C4⋊Dic53C2×C411yescharacteristic
35C2×#11719919—21no—
36C2×#11719919—21no—
37C2×#11719919C2×C411yescharacteristic
38C2×#11719919D411yescharacteristic
39C2×C4×Dic53—21no—
40C2×C4×Dic53C2×C411yescharacteristic
41C2×C4×Dic53D411yescharacteristic
42C2×C4×Dic53D411yescharacteristic
43#13133146—41no—
44#13133146—41no—
45#13133146—41no—
46#13133135—42no—
47C22⋊#88322—21no—
48C8×Dic53—41no—
49#13133124—21no—
50#13133124—21no—
51#13133124—21no—
52#13133124—21no—
53#13133124—21no—
54#13133124—21no—
55#13133124C2×C411yescharacteristic
56C2×M16×C53—21no—
57C2×C4⋊C4×C53C2×C411yescharacteristic
58C2×C4⋊C4×C53C2311yescharacteristic
59C22⋊C8×C53—21no—
60#95×C53—21no—
61#95×C53—21no—
62#95×C53C2×C411yescharacteristic
63C22×C4×C53—21no—
64C22×C4×C53—21no—
65C22×C4×C53C22×C411yescharacteristic
66C22×C4×C53C2×D411yescharacteristic
67C22×C4×C53C2×D411yescharacteristic
68C22×Dic53—21no—
69C22×Dic53—21no—
70C22×Dic53C22⋊C411yescharacteristic
71C22×Dic53C22⋊C411yescharacteristic
72C22×Dic53C22⋊C411yescharacteristic
73C22×Dic53C22⋊C411yescharacteristic
74C2×Dic106—41no—
75C2×Dic106—41no—
76C2×Dic106—44no—
77C2×Dic106—41no—
78C2×Dic106—41no—
79C2×Dic106SD1612yes—
80C2×Dic106Q1612yes—
81M16×C53—41no—
82M16×C53—41no—
83C2×C8×C53—41no—
84C2×C8×C53—41no—
85C4⋊C4×C53—41no—
86C4⋊C4×C53—41no—
87C4⋊Dic53—41no—
88#11719919—41no—
89#11719919—41no—
90#11719919—42no—
91#11719919—42no—
92C4×Dic53—21no—
93C4×Dic53—21no—
94C4×Dic53—21no—
95C4×Dic53—21no—
96C4×Dic53—21no—
97C4×Dic53—21no—
98C4×Dic53—42no—
99#11719917—41no—
100#11719917—41no—
101C2×#88322—41no—
102C2×#88322—41no—
103C23×C53C2×C22⋊C411yescharacteristic
104C2×C4×C53—21no—
105C2×C4×C53—21no—
106C2×C4×C53—21no—
107C2×C4×C53—21no—
108C2×C4×C53—41no—
109C2×C4×C53—41no—
110C2×C4×C53—42no—
111C2×C4×C53—41no—
112C2×C4×C53—41no—
113C2×C4×C53—41no—
114C2×C4×C53—41no—
115C2×C4×C53C2×SD1611yescharacteristic
116C2×C4×C53C2×Q1611yescharacteristic
117C2×Dic53—21no—
118C2×Dic53—21no—
119C2×Dic53—21no—
120C2×Dic53—21no—
121C2×Dic53—22no—
122C2×Dic53—21no—
123C2×Dic53—41no—
124C2×Dic53—42no—
125C2×Dic53—41no—
126C2×Dic53—41no—
127C2×Dic53—41no—
128C2×Dic53—41no—
129C2×Dic53#9911yescharacteristic
130C2×Dic53#9911yescharacteristic
131C2×Dic53#10312yes—
132C2×Dic53#10312yes—
133Dic106—42no—
134Dic106—42no—
135Dic106—82no—
136Dic106—82no—
137C8×C53—81no—
138C8×C53—81no—
139#88322—81no—
140#88322—81no—
141#11907—531nomaximal, Sylow
142C22×C53—21no—
143C22×C53—21no—
144C22×C53C2×#9911yescharacteristic
145C22×C53C2×#10311yescharacteristic
146C22×C53#42111yescharacteristic
147C4×C53—21no—
148C4×C53—21no—
149C4×C53—41no—
150C4×C53—41no—
151C4×C53—81no—
152C4×C53—81no—
153Dic53—21no—
154Dic53—22no—
155Dic53—22no—
156Dic53—41no—
157Dic53—81no—
158Dic53—81no—
159Dic53#35512yes—
160#2568—531no—
161#1801—531no—
162#1461—531no—
163#1353—531no—
164#1353—531no—
165#1353—531no—
166#1353D5311yescharacteristic
167C2×C53—21no—
168C2×C53—21no—
169C2×C53#152011yescharacteristic
170C2×C53#212311yescharacteristic
171C2×C53C2×#35511yescharacteristic
172C2×C4⋊Q8—531no—
173C2×C4×Q8—531no—
174#431—1061no—
175#423—1061no—
176C2×#101—1061no—
177#400—531no—
178#400—531no—
179#400—531no—
180#400C2×D5311yescharacteristic
181#391—1061no—
182#391—1061no—
183#391—1061no—
184#382—1061no—
185C4×C4⋊C4—531no—
186#343—531no—
187#343C2×D5311yescharacteristic
188#324—531no—
189#324—1061no—
190#324C2×D5311yescharacteristic
191C53#1190711yescharacteristic, complemented, Sylow
192C22×Q8—531no—
193C22×Q8—1061no—
194C2×M16—21no—
195C2×M16—1061no—
196C4⋊Q8—2121no—
197C4⋊Q8—2121no—
198C4×Q8—2121no—
199C4×Q8—2121no—
200C2×C4⋊C4—531no—
201C2×C4⋊C4—531no—
202C2×C4⋊C4—531no—
203C2×C4⋊C4—1061no—
204C2×C4⋊C4—1061no—
205C2×C4⋊C4C4×D5311yescharacteristic
206C2×C4⋊C4C22×D5311yescharacteristic
207C2×C42—531no—
208C2×C42—531no—
209C2×C42—531no—
210C2×C42—1061no—
211C4⋊C8—2121no—
212C4⋊C8—2121no—
213C4⋊C8—2121no—
214#101—2122no—
215C22⋊C8—21no—
216C22⋊C8—1061no—
217C4×C8—2121no—
218#95—21no—
219#95—21no—
220#95—531no—
221#95—1061no—
222#95—1061no—
223#95—1061no—
224#95—1061no—
225#95—1061no—
226#95—1061no—
227#95C4×D5311yescharacteristic
228C2×Q8—532no—
229C2×Q8—532no—
230C2×Q8—2121no—
231C2×Q8—2121no—
232C2×Q8—2121no—
233C2×Q8—2121no—
234C2×Q8—2124no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—531no—
238C22×C4—531no—
239C22×C4—531no—
240C22×C4—531no—
241C22×C4—1061no—
242C22×C4—1061no—
243C22×C4C2×C4×D5311yescharacteristic
244C22×C4D4×D5311yescharacteristic
245C22×C4D4×D5311yescharacteristic
246M16—41no—
247M16—41no—
248M16—2121no—
249M16—2121no—
250C2×C8—41no—
251C2×C8—41no—
252C2×C8—2121no—
253C2×C8—2121no—
254C4⋊C4—41no—
255C4⋊C4—41no—
256C4⋊C4—2121no—
257C4⋊C4—2121no—
258C4⋊C4—2121no—
259C4⋊C4—2122no—
260C4⋊C4—2122no—
261C42—1061no—
262C42—1061no—
263C42—1061no—
264C42—1061no—
265C42—1061no—
266C42—1061no—
267C42—2122no—
268C23C22⋊C4×D5311yescharacteristic
269Q8—4242no—
270Q8—4242no—
271Q8—2122no—
272Q8—2122no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—42no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—531no—
285C2×C4—531no—
286C2×C4—532no—
287C2×C4—532no—
288C2×C4—1062no—
289C2×C4—1061no—
290C2×C4—1061no—
291C2×C4—1061no—
292C2×C4—1061no—
293C2×C4—1061no—
294C2×C4—2121no—
295C2×C4—2121no—
296C2×C4—2122no—
297C2×C4—2121no—
298C2×C4—2121no—
299C2×C4—2121no—
300C2×C4SD16×D5311yescharacteristic
301C2×C4Q16×D5311yescharacteristic
302C8—81no—
303C8—81no—
304C8—4241no—
305C8—4241no—
306C22—21no—
307C22—21no—
308C22#99×D5311yescharacteristic
309C22#103×D5311yescharacteristic
310C22#1681063811yescharacteristic
311C4—21no—
312C4—21no—
313C4—41no—
314C4—41no—
315C4—81no—
316C4—81no—
317C4—532no—
318C4—1062no—
319C4—1062no—
320C4—1061no—
321C4—4241no—
322C4—4241no—
323C4—2121no—
324C2—21no—
325C2—21no—
326C2#2027328511yescharacteristic
327C2#2027636711yescharacteristic
328C2#355×D5311yescharacteristic
subgroupG/H
centerC22#99×D53
commutatorC22×C4×C53C22×C4
FrattiniC2×C4⋊C4C22×D53
FittingC53×#1353C2
socleC22×C53C2×#99
prime radicalC23×C53C2×C22⋊C4
cosocleC2×C4⋊C4×C53C23
hypercenter#1353D53
hypocenterC53#11907
LCSG ▷ C22×C4×C53 ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #1353

Complements

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

NH ≅ G/Ntype
191. C53#11907semidirect

Maximal subgroups

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

subgroupindex
#202799262
#202747102
#202730372
#202727432
#202727422
#202727412
C53×#13532
#1190753