Finite Group Databasebeta

← #27285407

Subgroups — #27285407

Order 13568.

484 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 484 conjugacy classes of subgroups — too many to draw. Only its 97 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
1C2×#16811029C211yesmaximal, characteristic
2#20282782C211yesmaximal, characteristic
3#20275689C211yesmaximal, characteristic
4C2×#16810365C211yesmaximal, characteristic
5C2×#16810364C211yesmaximal, characteristic
6#20272597C211yesmaximal, characteristic
7C2×C53×#491C211yesmaximal, characteristic
8C2×#13133209C2211yescharacteristic
9#16811029—22no—
10#16811029—22no—
11C2×#13133150—21no—
12C2×#13133148—21no—
13C2×#13133128C2211yescharacteristic
14C2×#13133128C2211yescharacteristic
15#16810562—21no—
16#16810561—21no—
17#16810556—22no—
18#16810554—21no—
19#16810553—21no—
20#16810489C2211yescharacteristic
21#16810385—22no—
22#16810365—22no—
23#16810364—22no—
24C2×C4⋊Q8×C53C2211yescharacteristic
25C2×#125×C53C2211yescharacteristic
26C53×#491—22no—
27C53×#491—22no—
28C2×#103×C53—21no—
29C2×#103×C53—21no—
30C2×#97×C53C2211yescharacteristic
31C22×#88322—21no—
32C22×#88322—21no—
33#13133209—21no—
34#13133209—21no—
35#13133209—21no—
36#13133209—21no—
37C2×C4⋊Dic53D411yescharacteristic
38C2×C4⋊Dic53D411yescharacteristic
39C2×C4⋊Dic53D412yes—
40C2×#11719919—22no—
41C2×#11719919—21no—
42C2×#11719919—21no—
43#13133150—24no—
44#13133148—24no—
45#13133128—21no—
46#13133128—21no—
47#13133128—21no—
48#13133128D412yes—
49#13133124—22no—
50#13133124—22no—
51#13133124—22no—
52#13133124—21no—
53#13133124—21no—
54C22×Q8×C53—21no—
55C22×C8×C53—21no—
56C4⋊Q8×C53—21no—
57C4⋊Q8×C53—21no—
58C4⋊Q8×C53—21no—
59C4⋊Q8×C53D412yes—
60#125×C53—21no—
61#125×C53—21no—
62#125×C53—21no—
63#125×C53—21no—
64C2×C4⋊C4×C53—21no—
65C2×C4⋊C4×C53—21no—
66C2×C4⋊C4×C53—21no—
67C2×C4⋊C4×C53—21no—
68C2×C4⋊C4×C53D411yescharacteristic
69C2×C4⋊C4×C53D411yescharacteristic
70C2×C42×C53C2311yescharacteristic
71#103×C53—24no—
72#103×C53—24no—
73#97×C53—21no—
74#97×C53D412yes—
75C2×Q8×C53—22no—
76C2×Q8×C53—44no—
77C2×Q8×C53—41no—
78C22×C4×C53—21no—
79C22×C4×C53—21no—
80C22×C4×C53—41no—
81C22×C4×C53C2×D411yescharacteristic
82C22×C4×C53C2×D411yescharacteristic
83C22×C4×C53C2×D411yescharacteristic
84C22×Dic53—22no—
85C22×Dic53—21no—
86C22×Dic53—21no—
87C2×C8×C53—22no—
88C2×C8×C53—42no—
89C4⋊C4×C53—21no—
90C4⋊C4×C53—21no—
91C4⋊C4×C53—22no—
92C4⋊C4×C53—22no—
93C4⋊C4×C53—21no—
94C4⋊C4×C53—21no—
95C4⋊C4×C53—41no—
96C4⋊C4×C53—41no—
97C4⋊C4×C53—41no—
98C4⋊C4×C53—41no—
99C4⋊C4×C53—41no—
100C4⋊C4×C53—41no—
101C42×C53C2×D411yescharacteristic
102C42×C53C2×D411yescharacteristic
103C42×C53C2×D411yescharacteristic
104C42×C53#4111yescharacteristic
105C4⋊Dic53—21no—
106C4⋊Dic53—21no—
107C4⋊Dic53—21no—
108C4⋊Dic53—21no—
109C4⋊Dic53—42no—
110#11719919—41no—
111#11719919—41no—
112#11719919—41no—
113#11719919—41no—
114#11719919—82no—
115C2×#88322—24no—
116C2×#88322—22no—
117C2×#88322—22no—
118C2×#88322—41no—
119C2×#88322—41no—
120C23×C53#12011yescharacteristic
121Q8×C53—44no—
122Q8×C53—82no—
123C2×C4×C53—22no—
124C2×C4×C53—21no—
125C2×C4×C53—22no—
126C2×C4×C53—21no—
127C2×C4×C53—41no—
128C2×C4×C53—41no—
129C2×C4×C53—42no—
130C2×C4×C53—41no—
131C2×C4×C53—42no—
132C2×C4×C53—41no—
133C2×C4×C53—82no—
134C2×C4×C53#12011yescharacteristic
135C2×C4×C53#12011yescharacteristic
136C2×C4×C53#12111yescharacteristic
137C2×C4×C53#12111yescharacteristic
138C2×C4×C53#12111yescharacteristic
139C2×C4×C53#12711yescharacteristic
140C2×Dic53—41no—
141C2×Dic53—41no—
142C2×Dic53—42no—
143C2×Dic53—42no—
144C2×Dic53—41no—
145C2×Dic53—42no—
146C2×Dic53—42no—
147C2×Dic53—41no—
148C8×C53—42no—
149#88322—24no—
150#88322—42no—
151#10084—531nomaximal, Sylow
152C22×C53#39211yescharacteristic
153C22×C53#45412yes—
154C22×C53#45511yescharacteristic
155C22×C53#45611yescharacteristic
156C22×C53#45612yes—
157C4×C53—21no—
158C4×C53—21no—
159C4×C53—21no—
160C4×C53—21no—
161C4×C53—21no—
162C4×C53—21no—
163C4×C53—82no—
164C4×C53—81no—
165C4×C53—81no—
166Dic53—82no—
167Dic53—81no—
168Dic53—81no—
169C2×#501—531no—
170C2×#491D5311yescharacteristic
171#2822—531no—
172#1993—531no—
173C2×#330—531no—
174C2×#330—531no—
175#1295—531no—
176C2×C53#166312yes—
177C2×C53#166412yes—
178C2×C53#200311yescharacteristic
179C2×C53#200511yescharacteristic
180C2×C53#200711yescharacteristic
181C2×C4⋊Q8C2×D5311yescharacteristic
182C2×#125—531no—
183C2×#125C2×D5311yescharacteristic
184#501—1062no—
185#501—1062no—
186#491—22no—
187#491—22no—
188C2×#107—1061no—
189C2×#106—1061no—
190C2×#103—21no—
191C2×#103—21no—
192C2×#97—531no—
193C2×#97—531no—
194C2×#97C2×D5311yescharacteristic
195#400—1061no—
196#400—1061no—
197#398—1061no—
198#398—1062no—
199#398—1061no—
200#382—531no—
201#340—1062no—
202#330—1062no—
203#330—1062no—
204C53#1008411yescharacteristic, complemented, Sylow
205C22×Q8—21no—
206C22×C8—21no—
207C22×C8—1061no—
208C22×C8—1061no—
209C4⋊Q8—21no—
210C4⋊Q8—21no—
211C4⋊Q8—21no—
212C4⋊Q8C22⋊D5312yes—
213#125—21no—
214#125—21no—
215#125—21no—
216#125—21no—
217#125—1061no—
218#125—1061no—
219#125—1061no—
220#125—1061no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—531no—
226C2×C4⋊C4—531no—
227C2×C4⋊C4—532no—
228C2×C4⋊C4—1062no—
229C2×C4⋊C4—1061no—
230C2×C4⋊C4—1061no—
231C2×C4⋊C4C22⋊D5311yescharacteristic
232C2×C4⋊C4C22⋊D5311yescharacteristic
233C2×C42C22×D5311yescharacteristic
234#107—1064no—
235#106—1064no—
236#103—24no—
237#103—24no—
238#97—21no—
239#97—532no—
240#97—1061no—
241#97—1061no—
242#97—1061no—
243#97C22⋊D5312yes—
244#95—1062no—
245#95—1062no—
246#95—1062no—
247#95—1061no—
248#95—1061no—
249C2×Q8—22no—
250C2×Q8—44no—
251C2×Q8—41no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—41no—
255C22×C4—1062no—
256C22×C4—1061no—
257C22×C4—1061no—
258C22×C4D4×D5311yescharacteristic
259C22×C4D4×D5311yescharacteristic
260C22×C4C2×C22⋊D5311yescharacteristic
261C2×C8—22no—
262C2×C8—42no—
263C2×C8—1064no—
264C2×C8—1062no—
265C2×C8—1062no—
266C2×C8—2121no—
267C2×C8—2121no—
268C4⋊C4—22no—
269C4⋊C4—21no—
270C4⋊C4—21no—
271C4⋊C4—22no—
272C4⋊C4—21no—
273C4⋊C4—21no—
274C4⋊C4—41no—
275C4⋊C4—41no—
276C4⋊C4—41no—
277C4⋊C4—41no—
278C4⋊C4—41no—
279C4⋊C4—41no—
280C4⋊C4—1061no—
281C4⋊C4—1061no—
282C4⋊C4—1061no—
283C4⋊C4—1061no—
284C4⋊C4—4242no—
285C4⋊C4—2122no—
286C4⋊C4—2121no—
287C4⋊C4—2121no—
288C4⋊C4—2121no—
289C4⋊C4—2121no—
290C42D4×D5311yescharacteristic
291C42#1171994711yescharacteristic
292C42C2×C22⋊D5311yescharacteristic
293C42C2×C22⋊D5311yescharacteristic
294C23#1313318511yescharacteristic
295Q8—44no—
296Q8—82no—
297C2×C4—22no—
298C2×C4—21no—
299C2×C4—22no—
300C2×C4—21no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—42no—
304C2×C4—41no—
305C2×C4—42no—
306C2×C4—41no—
307C2×C4—82no—
308C2×C4—2121no—
309C2×C4—2121no—
310C2×C4—2122no—
311C2×C4—2122no—
312C2×C4—2121no—
313C2×C4—2122no—
314C2×C4—2121no—
315C2×C4—2122no—
316C2×C4#1313318511yescharacteristic
317C2×C4#1313318611yescharacteristic
318C2×C4#1313318811yescharacteristic
319C2×C4#1313318911yescharacteristic
320C2×C4#1313318911yescharacteristic
321C2×C4#1313321311yescharacteristic
322C8—42no—
323C8—1064no—
324C8—2122no—
325C22#1681053111yescharacteristic
326C22#1681079512yes—
327C22#1681079911yescharacteristic
328C22#1681080412yes—
329C22#1681080411yescharacteristic
330C4—21no—
331C4—21no—
332C4—21no—
333C4—21no—
334C4—21no—
335C4—21no—
336C4—82no—
337C4—81no—
338C4—81no—
339C4—4241no—
340C4—4241no—
341C4—4242no—
342C2#2027404112yes—
343C2#2027404612yes—
344C2#2027573711yescharacteristic
345C2#2027574811yescharacteristic
346C2#2027576011yescharacteristic
subgroupG/H
centerC23#13133185
commutator, cosocleC2×C42×C53C23
FrattiniC2×C42C22×D53
FittingC2×C53×#491C2
socle, prime radicalC23×C53#120
hypercenterC2×#491D53
hypocenterC53#10084
LCSG ▷ C2×C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#491

Complements

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

NH ≅ G/Ntype
204. C53#10084semidirect

Maximal subgroups

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

subgroupindex
C2×#168110292
#202827822
#202756892
C2×#168103652
C2×#168103642
#202725972
C2×C53×#4912
#1008453