Finite Group Databasebeta

← #27295257

Subgroups — #27295257

Order 13568.

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

normal subgroups only

This group has 388 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#20279919C211yesmaximal, characteristic
2#20274710C211yesmaximal, characteristic
3#20273037C211yesmaximal, characteristic
4#20272746C211yesmaximal, characteristic
5#20272745C211yesmaximal, characteristic
6#20272744C211yesmaximal, characteristic
7C53×#1354C211yesmaximal, characteristic
8C2×#13133215C411yescharacteristic
9C2×#13133182C2211yescharacteristic
10#16810673—21no—
11#16810646—21no—
12C2×#13133135—21no—
13#16810544C411yescharacteristic
14#16810543C2211yescharacteristic
15#16810542C411yescharacteristic
16#16810527—21no—
17#16810520—21no—
18#16810518—21no—
19#16810492—21no—
20#16810474C411yescharacteristic
21#16810392C2211yescharacteristic
22#16810349C2211yescharacteristic
23#16810348—21no—
24C53×#395C2211yescharacteristic
25C53×#343C2211yescharacteristic
26C53×#324C2211yescharacteristic
27C22×Dic106C2×C411yescharacteristic
28C22×Dic106D411yescharacteristic
29C2×#11719917—21no—
30#13133215—41no—
31#13133215—41no—
32#13133182—41no—
33#13133182—41no—
34C2×C4⋊Dic53—21no—
35C2×#11719919—21no—
36C2×#11719919—21no—
37C2×#11719919—21no—
38C2×#11719919—21no—
39C2×#11719919—21no—
40C2×#11719919C2×C411yescharacteristic
41C2×#11719919D411yescharacteristic
42C2×C4×Dic53—21no—
43C2×C4×Dic53C2×C411yescharacteristic
44C2×C4×Dic53D411yescharacteristic
45C2×C4×Dic53D411yescharacteristic
46#13133146—41no—
47#13133146—41no—
48#13133146—41no—
49#13133135—42no—
50C22⋊#88322—21no—
51C8×Dic53—41no—
52#13133124—21no—
53#13133124—21no—
54#13133124—21no—
55#13133124C2×C411yescharacteristic
56C22×Q8×C53C2×C411yescharacteristic
57C2×M16×C53—21no—
58C2×C4⋊C4×C53—21no—
59C2×C4⋊C4×C53C2311yescharacteristic
60C22⋊C8×C53—21no—
61#95×C53—21no—
62#95×C53C2×C411yescharacteristic
63C2×Q8×C53—41no—
64C2×Q8×C53—81no—
65C22×C4×C53—21no—
66C22×C4×C53—21no—
67C22×C4×C53C22×C411yescharacteristic
68C22×C4×C53C2×D411yescharacteristic
69C22×C4×C53C2×D411yescharacteristic
70C22×Dic53—21no—
71C22×Dic53—21no—
72C22×Dic53C22⋊C411yescharacteristic
73C22×Dic53C22⋊C411yescharacteristic
74C22×Dic53C22⋊C411yescharacteristic
75C22×Dic53C22⋊C411yescharacteristic
76C2×Dic106—41no—
77C2×Dic106—41no—
78C2×Dic106—41no—
79C2×Dic106—81no—
80C2×Dic106SD1612yes—
81C2×Dic106Q1612yes—
82M16×C53—41no—
83M16×C53—41no—
84C2×C8×C53—41no—
85C2×C8×C53—41no—
86C4⋊C4×C53—41no—
87C4⋊C4×C53—81no—
88C4⋊Dic53—81no—
89#11719919—24no—
90#11719919—41no—
91#11719919—41no—
92#11719919—42no—
93#11719919—41no—
94#11719919—41no—
95#11719919—81no—
96#11719919—81no—
97C4×Dic53—21no—
98C4×Dic53—21no—
99C4×Dic53—21no—
100C4×Dic53—21no—
101C4×Dic53—21no—
102C4×Dic53—21no—
103C4×Dic53—42no—
104#11719917—41no—
105#11719917—41no—
106C2×#88322—41no—
107C2×#88322—41no—
108C23×C53C2×C22⋊C411yescharacteristic
109Q8×C53—82no—
110C2×C4×C53—21no—
111C2×C4×C53—21no—
112C2×C4×C53—22no—
113C2×C4×C53—21no—
114C2×C4×C53—21no—
115C2×C4×C53—41no—
116C2×C4×C53—41no—
117C2×C4×C53—41no—
118C2×C4×C53—41no—
119C2×C4×C53—41no—
120C2×C4×C53—41no—
121C2×C4×C53—41no—
122C2×C4×C53C2×SD1611yescharacteristic
123C2×C4×C53C2×Q1611yescharacteristic
124C2×Dic53—21no—
125C2×Dic53—21no—
126C2×Dic53—21no—
127C2×Dic53—21no—
128C2×Dic53—22no—
129C2×Dic53—22no—
130C2×Dic53—21no—
131C2×Dic53—41no—
132C2×Dic53—41no—
133C2×Dic53—41no—
134C2×Dic53—41no—
135C2×Dic53—41no—
136C2×Dic53—41no—
137C2×Dic53#9911yescharacteristic
138C2×Dic53#9911yescharacteristic
139C2×Dic53#10312yes—
140C2×Dic53#10312yes—
141Dic106—42no—
142Dic106—42no—
143Dic106—82no—
144C8×C53—81no—
145C8×C53—81no—
146#88322—81no—
147#88322—81no—
148#11908—531nomaximal, Sylow
149C22×C53—21no—
150C22×C53—21no—
151C22×C53C2×#9911yescharacteristic
152C22×C53C2×#10311yescharacteristic
153C22×C53#42111yescharacteristic
154C4×C53—21no—
155C4×C53—21no—
156C4×C53—42no—
157C4×C53—41no—
158C4×C53—41no—
159C4×C53—81no—
160Dic53—21no—
161Dic53—22no—
162Dic53—22no—
163Dic53—42no—
164Dic53—41no—
165Dic53—81no—
166Dic53#35612yes—
167#2567—531no—
168#1801—531no—
169#1461—531no—
170#1354—531no—
171#1354—531no—
172#1354—531no—
173#1354D5311yescharacteristic
174C2×C53—21no—
175C2×C53—21no—
176C2×C53#152011yescharacteristic
177C2×C53#212211yescharacteristic
178C2×C53C2×#35611yescharacteristic
179C2×C4⋊Q8—531no—
180C2×C4×Q8—531no—
181#431—1061no—
182#423—1061no—
183C2×#101—1061no—
184#395—531no—
185#395—531no—
186#395—531no—
187#395C2×D5311yescharacteristic
188#391—1061no—
189#389—1061no—
190#389—1061no—
191#384—1061no—
192C4×C4⋊C4—531no—
193#343—531no—
194#343C2×D5311yescharacteristic
195#324—531no—
196#324—1061no—
197#324C2×D5311yescharacteristic
198C53#1190811yescharacteristic, complemented, Sylow
199C22×Q8—531no—
200C22×Q8—531no—
201C22×Q8C4×D5311yescharacteristic
202C2×M16—21no—
203C2×M16—1061no—
204C4⋊Q8—2121no—
205C4⋊Q8—2121no—
206C4×Q8—2121no—
207C4×Q8—2121no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—531no—
210C2×C4⋊C4—531no—
211C2×C4⋊C4—1061no—
212C2×C4⋊C4—1061no—
213C2×C4⋊C4—1061no—
214C2×C4⋊C4—1061no—
215C2×C4⋊C4—1061no—
216C2×C4⋊C4—1061no—
217C2×C4⋊C4C22×D5311yescharacteristic
218C2×C42—531no—
219C2×C42—531no—
220C2×C42—531no—
221C2×C42—1061no—
222C4⋊C8—2121no—
223C4⋊C8—2121no—
224C4⋊C8—2121no—
225#101—2122no—
226C22⋊C8—21no—
227C22⋊C8—1061no—
228C4×C8—2121no—
229#95—21no—
230#95—531no—
231#95—1061no—
232#95—1061no—
233#95—1061no—
234#95C4×D5311yescharacteristic
235C2×Q8—41no—
236C2×Q8—81no—
237C2×Q8—532no—
238C2×Q8—532no—
239C2×Q8—4241no—
240C2×Q8—2121no—
241C2×Q8—2121no—
242C2×Q8—2121no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—531no—
246C22×C4—531no—
247C22×C4—531no—
248C22×C4—531no—
249C22×C4—1061no—
250C22×C4—1061no—
251C22×C4C2×C4×D5311yescharacteristic
252C22×C4D4×D5311yescharacteristic
253C22×C4D4×D5311yescharacteristic
254M16—41no—
255M16—41no—
256M16—2121no—
257M16—2121no—
258C2×C8—41no—
259C2×C8—41no—
260C2×C8—2121no—
261C2×C8—2121no—
262C4⋊C4—41no—
263C4⋊C4—81no—
264C4⋊C4—1064no—
265C4⋊C4—4241no—
266C4⋊C4—4241no—
267C4⋊C4—4241no—
268C4⋊C4—2121no—
269C4⋊C4—2121no—
270C4⋊C4—2121no—
271C4⋊C4—2121no—
272C4⋊C4—2122no—
273C42—1061no—
274C42—1061no—
275C42—1061no—
276C42—1061no—
277C42—1061no—
278C42—1061no—
279C42—2122no—
280C23C22⋊C4×D5311yescharacteristic
281Q8—82no—
282Q8—4242no—
283Q8—2122no—
284Q8—2122no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—22no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—531no—
298C2×C4—531no—
299C2×C4—532no—
300C2×C4—532no—
301C2×C4—1062no—
302C2×C4—1061no—
303C2×C4—1061no—
304C2×C4—1061no—
305C2×C4—1061no—
306C2×C4—1062no—
307C2×C4—1061no—
308C2×C4—2121no—
309C2×C4—2121no—
310C2×C4—2121no—
311C2×C4—2121no—
312C2×C4—2121no—
313C2×C4—2121no—
314C2×C4SD16×D5311yescharacteristic
315C2×C4Q16×D5311yescharacteristic
316C8—81no—
317C8—81no—
318C8—4241no—
319C8—4241no—
320C22—21no—
321C22—21no—
322C22#99×D5311yescharacteristic
323C22#103×D5311yescharacteristic
324C22#1681063811yescharacteristic
325C4—21no—
326C4—21no—
327C4—42no—
328C4—41no—
329C4—41no—
330C4—81no—
331C4—532no—
332C4—1062no—
333C4—1062no—
334C4—1061no—
335C4—4241no—
336C4—2122no—
337C4—2121no—
338C2—21no—
339C2—21no—
340C2#2027328511yescharacteristic
341C2#2027636311yescharacteristic
342C2#356×D5311yescharacteristic
subgroupG/H
centerC22#99×D53
commutatorC22×C4×C53C22×C4
FrattiniC2×C4⋊C4C22×D53
FittingC53×#1354C2
socleC22×C53C2×#99
prime radicalC23×C53C2×C22⋊C4
cosocleC2×C4⋊C4×C53C23
hypercenter#1354D53
hypocenterC53#11908
LCSG ▷ C22×C4×C53 ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #1354

Complements

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

NH ≅ G/Ntype
198. C53#11908semidirect

Maximal subgroups

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

subgroupindex
#202799192
#202747102
#202730372
#202727462
#202727452
#202727442
C53×#13542
#1190853