Finite Group Databasebeta

← #27294816

Subgroups — #27294816

Order 13568.

382 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 382 conjugacy classes of subgroups — too many to draw. Only its 69 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#20280535C211yesmaximal, characteristic
2#20274791C211yesmaximal, characteristic
3#20273046C211yesmaximal, characteristic
4#20272737C211yesmaximal, characteristic
5#20272736C211yesmaximal, characteristic
6#20272735C211yesmaximal, characteristic
7C53×#1351C211yesmaximal, characteristic
8C2×#13133215C411yescharacteristic, complemented
9C2×#13133173C2211yescharacteristic
10#16810677—21no—
11#16810564—21no—
12#16810559—21no—
13#16810544C411yescharacteristic, complemented
14#16810542C411yescharacteristic, complemented
15#16810535C2211yescharacteristic
16#16810520—21no—
17#16810487C411yescharacteristic, complemented
18#16810484—21no—
19#16810481—21no—
20#16810390C2211yescharacteristic
21#16810389—21no—
22#16810348—21no—
23#16810347C2211yescharacteristic
24C53×#393C2211yescharacteristic
25C53×#342C2211yescharacteristic
26C53×#323C2211yescharacteristic
27C22×Dic106C2×C411yescharacteristic
28C23×Dic53D411yescharacteristic
29#13133215—41no—
30#13133215—41no—
31#13133173—41no—
32#13133173—41no—
33C2×C4⋊Dic53—21no—
34C2×#11719919—21no—
35C2×#11719919—21no—
36C2×#11719919—21no—
37C2×#11719919—21no—
38C2×#11719919C2×C411yescharacteristic
39C2×#11719919D411yescharacteristic
40C2×C22⋊Dic53—21no—
41C2×C22⋊Dic53—21no—
42C2×C22⋊Dic53D411yescharacteristic
43C2×C4×Dic53C2×C411yescharacteristic
44C2×C4×Dic53D411yescharacteristic
45#13133146—41no—
46C22⋊#88322—21no—
47#13133127—41no—
48#13133124—21no—
49#13133124—21no—
50#13133124—21no—
51#13133124—21no—
52#13133124—41no—
53#13133124—41no—
54#13133124C2×C411yescharacteristic
55C22×Q8×C53C2×C411yescharacteristic
56C2×C22⋊C4×C53—21no—
57C2×C22⋊C4×C53—21no—
58C2×C22⋊C4×C53C2311yescharacteristic
59C22⋊C8×C53—21no—
60#95×C53—21no—
61#95×C53C2×C411yescharacteristic
62C24×C53C2×D411yescharacteristic
63C2×Q8×C53—41no—
64C2×Q8×C53—81no—
65C22×C4×C53—21no—
66C22×C4×C53—21no—
67C22×C4×C53—41no—
68C22×C4×C53C22×C411yescharacteristic
69C22×C4×C53C2×D411yescharacteristic
70C22×Dic53—21no—
71C22×Dic53—21no—
72C22×Dic53—21no—
73C22×Dic53—21no—
74C22×Dic53—21no—
75C22×Dic53—21no—
76C22×Dic53—41no—
77C22×Dic53—41no—
78C22×Dic53C22⋊C411yescharacteristic
79C22×Dic53C22⋊C411yescharacteristic
80C22×Dic53C22⋊C411yescharacteristic
81C22×Dic53C22⋊C411yescharacteristic
82C2×Dic106—41no—
83C2×Dic106—81no—
84C2×C8×C53—41no—
85C22⋊C4×C53—41no—
86C22⋊C4×C53—41no—
87C22⋊C4×C53—41no—
88C22⋊C4×C53—81no—
89C22⋊Dic53—41no—
90C22⋊Dic53—41no—
91C22⋊Dic53—41no—
92C22⋊Dic53—81no—
93C4⋊Dic53—81no—
94#11719919—41no—
95#11719919—41no—
96#11719919—42no—
97#11719919—42no—
98#11719919—41no—
99#11719919—41no—
100#11719919—81no—
101C4×Dic53—21no—
102C4×Dic53—21no—
103C4×Dic53—21no—
104C4×Dic53—21no—
105C2×#88322—41no—
106C23×C53—21no—
107C23×C53—21no—
108C23×C53—41no—
109C23×C53—41no—
110C23×C53C2×C22⋊C411yescharacteristic
111C23×C53#13711yescharacteristic
112C23×C53#13711yescharacteristic
113Q8×C53—82no—
114C2×C4×C53—21no—
115C2×C4×C53—21no—
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×C53—41no—
123C2×C4×C53—41no—
124C2×C4×C53—81no—
125C2×C4×C53—81no—
126C2×C4×C53—81no—
127C2×Dic53—21no—
128C2×Dic53—21no—
129C2×Dic53—21no—
130C2×Dic53—21no—
131C2×Dic53—21no—
132C2×Dic53—21no—
133C2×Dic53—21no—
134C2×Dic53—41no—
135C2×Dic53—41no—
136C2×Dic53—41no—
137C2×Dic53—41no—
138C2×Dic53—41no—
139C2×Dic53—41no—
140C2×Dic53—41no—
141C2×Dic53—41no—
142C2×Dic53—41no—
143C2×Dic53—41no—
144C2×Dic53—41no—
145C2×Dic53—41no—
146C2×Dic53—81no—
147C2×Dic53—81no—
148C2×Dic53—81no—
149C2×Dic53#9911yescharacteristic
150C2×Dic53#9911yescharacteristic
151Dic106—82no—
152C8×C53—81no—
153#88322—81no—
154#11812—531nomaximal, Sylow
155C22×C53—21no—
156C22×C53—21no—
157C22×C53—21no—
158C22×C53—21no—
159C22×C53—41no—
160C22×C53—41no—
161C22×C53—41no—
162C22×C53—41no—
163C22×C53—81no—
164C22×C53C2×#9911yescharacteristic
165C22×C53#41911yescharacteristic
166C22×C53#42111yescharacteristic
167C4×C53—41no—
168C4×C53—41no—
169C4×C53—81no—
170C4×C53—81no—
171C4×C53—161no—
172Dic53—21no—
173Dic53—21no—
174Dic53—41no—
175Dic53—41no—
176Dic53—41no—
177Dic53—81no—
178Dic53—81no—
179Dic53—161no—
180#2628—531no—
181#1820—531no—
182#1464—531no—
183#1351—531no—
184#1351—531no—
185#1351—531no—
186#1351D5311yescharacteristic
187C2×C53—21no—
188C2×C53—21no—
189C2×C53—41no—
190C2×C53—41no—
191C2×C53#151111yescharacteristic
192C2×C53#212211yescharacteristic
193C2×C53#212911yescharacteristic
194C2×C4⋊Q8—531no—
195C2×#117—531no—
196#432—1061no—
197#400—1061no—
198#399—1061no—
199#395—531no—
200#395—531no—
201#393—531no—
202#393C2×D5311yescharacteristic
203#389—1061no—
204#382—531no—
205#381—1061no—
206#380—1061no—
207#342—531no—
208#342—1061no—
209#342C2×D5311yescharacteristic
210#324—1061no—
211#323—531no—
212#323C2×D5311yescharacteristic
213C53#1181211yescharacteristic, complemented, Sylow
214C22×Q8—531no—
215C22×Q8C4×D5311yescharacteristic
216C23×C4—531no—
217C4⋊Q8—2121no—
218C4⋊Q8—2121no—
219#117—2121no—
220#117—2121no—
221C2×C4⋊C4—531no—
222C2×C4⋊C4—531no—
223C2×C4⋊C4—1061no—
224C2×C4⋊C4—1061no—
225C2×C4⋊C4—1061no—
226C2×C4⋊C4—1061no—
227C2×C4⋊C4—1061no—
228C2×C22⋊C4—21no—
229C2×C22⋊C4—21no—
230C2×C22⋊C4—531no—
231C2×C22⋊C4—1061no—
232C2×C22⋊C4—1061no—
233C2×C22⋊C4C22×D5311yescharacteristic
234C2×C42—531no—
235C2×C42—531no—
236C4⋊C8—2121no—
237C22⋊C8—21no—
238C22⋊C8—1061no—
239#97—2121no—
240#95—21no—
241#95—531no—
242#95—1061no—
243#95—1061no—
244#95—1061no—
245#95—1061no—
246#95—2121no—
247#95—2121no—
248#95C4×D5311yescharacteristic
249C24D4×D5311yescharacteristic
250C2×Q8—41no—
251C2×Q8—81no—
252C2×Q8—4241no—
253C2×Q8—2121no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—41no—
257C22×C4—531no—
258C22×C4—531no—
259C22×C4—531no—
260C22×C4—531no—
261C22×C4—1061no—
262C22×C4—1061no—
263C22×C4—1061no—
264C22×C4—1061no—
265C22×C4—1061no—
266C22×C4—1061no—
267C22×C4—2121no—
268C22×C4—2121no—
269C22×C4C2×C4×D5311yescharacteristic
270C22×C4D4×D5311yescharacteristic
271C2×C8—41no—
272C2×C8—2121no—
273C4⋊C4—4241no—
274C4⋊C4—4241no—
275C4⋊C4—2121no—
276C4⋊C4—2121no—
277C4⋊C4—2121no—
278C4⋊C4—2121no—
279C4⋊C4—2122no—
280C4⋊C4—2122no—
281C22⋊C4—41no—
282C22⋊C4—41no—
283C22⋊C4—41no—
284C22⋊C4—81no—
285C22⋊C4—4241no—
286C22⋊C4—2121no—
287C22⋊C4—2121no—
288C22⋊C4—2121no—
289C42—1061no—
290C42—1061no—
291C42—1061no—
292C42—1061no—
293C23—21no—
294C23—21no—
295C23—41no—
296C23—41no—
297C23C22⋊C4×D5311yescharacteristic
298C23#1313324611yescharacteristic
299C23#1313324911yescharacteristic
300Q8—82no—
301Q8—4242no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—81no—
313C2×C4—81no—
314C2×C4—81no—
315C2×C4—531no—
316C2×C4—531no—
317C2×C4—1061no—
318C2×C4—1061no—
319C2×C4—1061no—
320C2×C4—1061no—
321C2×C4—1061no—
322C2×C4—1061no—
323C2×C4—1061no—
324C2×C4—4241no—
325C2×C4—4241no—
326C2×C4—4241no—
327C2×C4—2121no—
328C2×C4—2121no—
329C2×C4—2121no—
330C2×C4—2121no—
331C2×C4—2121no—
332C2×C4—2121no—
333C2×C4—2121no—
334C2×C4—2121no—
335C2×C4—2121no—
336C2×C4—2121no—
337C2×C4—2121no—
338C2×C4—2121no—
339C8—81no—
340C8—4241no—
341C22—21no—
342C22—21no—
343C22—21no—
344C22—21no—
345C22—41no—
346C22—41no—
347C22—41no—
348C22—41no—
349C22—81no—
350C22#99×D5311yescharacteristic
351C22#1681063011yescharacteristic
352C22#1681063811yescharacteristic
353C4—41no—
354C4—41no—
355C4—81no—
356C4—81no—
357C4—161no—
358C4—8481no—
359C4—1061no—
360C4—1061no—
361C4—4241no—
362C4—4241no—
363C4—2121no—
364C4—2121no—
365C4—2121no—
366C2—21no—
367C2—21no—
368C2—41no—
369C2—41no—
370C2#2027324611yescharacteristic
371C2#2027636311yescharacteristic
372C2#2027638811yescharacteristic
subgroupG/H
centerC22#99×D53
commutatorC22×C4×C53C22×C4
FrattiniC2×C22⋊C4C22×D53
FittingC53×#1351C2
socleC22×C53C2×#99
prime radicalC24×C53C2×D4
cosocleC2×C22⋊C4×C53C23
hypercenter#1351D53
hypocenterC53#11812
LCSG ▷ C22×C4×C53 ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23 ◁ C2×C22⋊C4 ◁ #1351

Complements

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

NH ≅ G/Ntype
8. C2×#13133215C4semidirect
13. #16810544C4semidirect
14. #16810542C4semidirect
17. #16810487C4semidirect
213. C53#11812semidirect

Maximal subgroups

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

subgroupindex
#202805352
#202747912
#202730462
#202727372
#202727362
#202727352
C53×#13512
#1181253