Finite Group Databasebeta

← #27437715

Subgroups — #27437715

Order 13568.

1052 conjugacy classes of subgroups; only the 191 normal subgroups are shown.

normal subgroups only

This group has 1052 conjugacy classes of subgroups — too many to draw. Only its 191 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×#16811236C211yesmaximal, characteristic
2C2×#16811002C211yesmaximal, characteristic, complemented
3#20280482C211yesmaximal, characteristic
4#20275724C214yesmaximal
5#20275681C211yesmaximal, characteristic, complemented
6#20275535C211yesmaximal, characteristic, complemented
7#20275053C214yesmaximal, complemented
8#20274484C211yesmaximal, characteristic
9C2×C53×#491C211yesmaximal, characteristic, complemented
10C22×#11719925C2214yes—
11#16811236—24no—
12#16811236—24no—
13C2×#13133217C2211yescharacteristic
14C2×#13133212—22no—
15C2×#13133208—22no—
16C2×#13133197C2212yes—
17C2×#13133193C2212yes—
18C2×#13133182C2212yes—
19C2×#13133174C2211yescharacteristic
20#16811002—24no—
21C2×#13133142C2214yes—
22C2×#13133129C2212yes—
23C2×C4×#88322C2211yescharacteristic
24#16810554—22no—
25#16810549—22no—
26#16810530C2212yes—
27#16810489C2211yescharacteristic
28#16810385C2214yes—
29#16810377C2212yes—
30C2×C4⋊Q8×C53C2211yescharacteristic
31C2×#125×C53C2211yescharacteristic
32C53×#491—24no—
33C2×#103×C53C2214yes—
34C2×#97×C53C2211yescharacteristic
35C22×Dic106C2312yes—
36C22×C4×D53C2311yescharacteristic
37C2×#11719925—28no—
38C2×#11719925—28no—
39C2×#11719925—44no—
40C22×#88322C2312yes—
41#13133217—22no—
42#13133217D414yes—
43#13133212—216no—
44#13133208—216no—
45#13133197—24no—
46#13133197—24no—
47#13133193—44no—
48#13133182—24no—
49#13133182—24no—
50#13133174—22no—
51#13133174—22no—
52C2×C4⋊Dic53—22no—
53C2×C4⋊Dic53—22no—
54C2×C4⋊Dic53C2312yes—
55C2×#11719919—22no—
56C2×#11719919—22no—
57C2×#11719919D412yes—
58C2×#11719921—22no—
59C2×#11719921—22no—
60C2×#11719921D412yes—
61C2×C4×Dic53—22no—
62C2×C4×Dic53C2311yescharacteristic
63#13133142—44no—
64#13133129—24no—
65C4×#88322—22no—
66#13133124—22no—
67#13133124—24no—
68C22×Q8×C53C2312yes—
69C22×C8×C53C2312yes—
70C4⋊Q8×C53—22no—
71C4⋊Q8×C53—22no—
72#125×C53—22no—
73#125×C53D414yes—
74C2×C4⋊C4×C53—22no—
75C2×C4⋊C4×C53—22no—
76C2×C4⋊C4×C53C2312yes—
77C2×C42×C53C2311yescharacteristic
78#103×C53—28no—
79#97×C53—22no—
80C23×D53C2×D411yescharacteristic
81C2×Q8×C53—22no—
82C2×Q8×C53—22no—
83C2×Q8×C53—24no—
84C2×Q8×C53—24no—
85C22×C4×C53—22no—
86C22×C4×C53—22no—
87C22×C4×C53C2×D412yes—
88C22×C4×C53C2411yescharacteristic
89C22×Dic53—22no—
90C22×Dic53—22no—
91C22×Dic53C2×D412yes—
92C22×Dic53C2×D411yescharacteristic
93C2×C4×D53—21no—
94C2×C4×D53—21no—
95C2×C4×D53—22no—
96C2×C4×D53—22no—
97C2×Dic106—24no—
98C2×Dic106—24no—
99C2×Dic106—22no—
100C2×Dic106C2×D414yes—
101C2×C8×C53—22no—
102C2×C8×C53—22no—
103C2×C8×C53—22no—
104C4⋊C4×C53—28no—
105C4⋊C4×C53—22no—
106C4⋊C4×C53—44no—
107C4⋊C4×C53C2×D414yes—
108C42×C53C2×D412yes—
109C42×C53C2×D412yes—
110#11719925—416no—
111#11719921—28no—
112#11719921—24no—
113#11719921—44no—
114C4⋊Dic53—28no—
115C4⋊Dic53—28no—
116C4⋊Dic53—24no—
117#11719919—28no—
118#11719919—24no—
119#11719919—44no—
120C4×Dic53—28no—
121C4×Dic53—21no—
122C4×Dic53—21no—
123C2×#88322—22no—
124C2×#88322—22no—
125C2×#88322#4114yes—
126C23×C53C22×D411yescharacteristic
127C22×D53—21no—
128C22×D53—21no—
129C22×D53—21no—
130C22×D53—24no—
131Q8×C53—44no—
132Q8×C53—44no—
133C2×C4×C53—24no—
134C2×C4×C53—24no—
135C2×C4×C53—44no—
136C2×C4×C53—42no—
137C2×C4×C53—42no—
138C2×C4×C53—42no—
139C2×C4×C53#12014yes—
140C2×C4×C53C22×D411yescharacteristic
141C2×C4×C53C22×D411yescharacteristic
142C2×C4×C53C2×#4112yes—
143C2×C4×C53H(5,𝔽2)12yes—
144C2×Dic53—24no—
145C2×Dic53—21no—
146C2×Dic53—22no—
147C2×Dic53—21no—
148C2×Dic53—21no—
149C2×Dic53—24no—
150C2×Dic53—42no—
151C2×Dic53—42no—
152C2×Dic53—42no—
153C2×Dic53—44no—
154C4×D53—42no—
155C4×D53—42no—
156Dic106—28no—
157Dic106—44no—
158C8×C53—42no—
159#88322—24no—
160#25609—531nomaximal, Sylow
161C22×C53C2×#12011yescharacteristic
162C22×C53#53211yescharacteristic
163C22×C53#53511yescharacteristic
164C22×C53#57814yes—
165C2×D53—42no—
166C2×D53—41no—
167C2×D53—42no—
168C2×D53—42no—
169C4×C53—24no—
170C4×C53—44no—
171C4×C53—82no—
172C4×C53#54612yes—
173C4×C53#55012yes—
174Dic53—44no—
175Dic53—41no—
176Dic53—42no—
177Dic53—82no—
178C2×#541—531no—
179C2×#494—531no—
180C2×#491D5311yescharacteristic, complemented
181#2621—531no—
182#2001—534no—
183#1991—531no—
184#1960—531no—
185#1878—534no—
186#1750—531no—
187C2×C53#243111yescharacteristic
188C2×C53#301112yes—
189C2×C53#317712yes—
190C2×C53#322312yes—
191D53—81no—
192C22×Q16—534no—
193#541—1064no—
194#541—1064no—
195C2×C4⋊Q8—531no—
196C2×C4⋊Q8C2×D5311yescharacteristic
197C2×#126—1062no—
198C2×#125—1062no—
199C2×#125C2×D5311yescharacteristic
200C2×C22⋊Q8—532no—
201C2×C22⋊Q8—532no—
202C2×C4×Q8—532no—
203C2×#117—531no—
204#494—1064no—
205#491—24no—
206C2×#103—534no—
207C2×#103C2×D5314yes—
208C2×C22⋊C8—532no—
209C2×#97C2×D5311yescharacteristic
210C2×C4×C8—531no—
211#398—1062no—
212#397—1062no—
213#391—532no—
214#382—531no—
215#340—534no—
216#336—532no—
217C53#2560911yescharacteristic, complemented, Sylow
218C22×Q8—532no—
219C22×Q8C22×D5312yes—
220C23×C4—531no—
221C2×Q16—1068no—
222C2×Q16—1068no—
223C2×Q16—2124no—
224C22×C8—532no—
225C22×C8C22×D5312yes—
226C4⋊Q8—22no—
227C4⋊Q8—22no—
228C4⋊Q8—534no—
229C4⋊Q8—1062no—
230#126—10616no—
231#125—22no—
232#125—10616no—
233#125C22⋊D5314yes—
234C22⋊Q8—1064no—
235C22⋊Q8—1064no—
236C22⋊Q8—2124no—
237C4×Q8—1064no—
238C4×Q8—1064no—
239#117—1062no—
240#117—1062no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—532no—
244C2×C4⋊C4—532no—
245C2×C4⋊C4—1062no—
246C2×C4⋊C4—1062no—
247C2×C4⋊C4—1062no—
248C2×C4⋊C4—1062no—
249C2×C4⋊C4C22×D5312yes—
250C2×C22⋊C4—532no—
251C2×C22⋊C4—1062no—
252C2×C22⋊C4—1062no—
253C2×C42—531no—
254C2×C42—1062no—
255C2×C42C22×D5311yescharacteristic
256#103—28no—
257#103—2124no—
258C22⋊C8—1064no—
259#97—22no—
260C4×C8—1062no—
261#95—1064no—
262#95—1062no—
263C24—531no—
264C2×Q8—22no—
265C2×Q8—22no—
266C2×Q8—24no—
267C2×Q8—24no—
268C2×Q8—534no—
269C2×Q8—1062no—
270C2×Q8—1064no—
271C2×Q8—1064no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—532no—
275C22×C4—531no—
276C22×C4—1062no—
277C22×C4—1061no—
278C22×C4—1061no—
279C22×C4—1062no—
280C22×C4—1062no—
281C22×C4—1062no—
282C22×C4D4×D5312yes—
283C22×C4C23×D5311yescharacteristic
284Q16—21216no—
285C2×C8—22no—
286C2×C8—22no—
287C2×C8—22no—
288C2×C8—534no—
289C2×C8—1062no—
290C2×C8—1062no—
291C4⋊C4—28no—
292C4⋊C4—22no—
293C4⋊C4—44no—
294C4⋊C4—1068no—
295C4⋊C4—1068no—
296C4⋊C4—1064no—
297C4⋊C4—1068no—
298C4⋊C4—1064no—
299C4⋊C4—2124no—
300C4⋊C4C2×C22⋊D5314yes—
301C22⋊C4—1068no—
302C22⋊C4—1064no—
303C22⋊C4—2124no—
304C42—1068no—
305C42—1061no—
306C42—1061no—
307C42D4×D5312yes—
308C42C2×C22⋊D5312yes—
309C23—1061no—
310C23—1061no—
311C23—1064no—
312C23—1061no—
313C23C2×D4×D5311yescharacteristic
314Q8—44no—
315Q8—44no—
316Q8—1068no—
317Q8—2124no—
318C2×C4—24no—
319C2×C4—24no—
320C2×C4—44no—
321C2×C4—42no—
322C2×C4—42no—
323C2×C4—42no—
324C2×C4—1062no—
325C2×C4—1064no—
326C2×C4—1061no—
327C2×C4—1061no—
328C2×C4—1064no—
329C2×C4—1061no—
330C2×C4—2124no—
331C2×C4—2122no—
332C2×C4—2122no—
333C2×C4—2122no—
334C2×C4—2122no—
335C2×C4—2122no—
336C2×C4#1313318514yes—
337C2×C4C2×D4×D5311yescharacteristic
338C2×C4C22×C22⋊D5311yescharacteristic
339C2×C4#41×D5312yes—
340C2×C4#1313326312yes—
341C8—42no—
342C8—1064no—
343C22—2121no—
344C22—2122no—
345C22—2122no—
346C22—2122no—
347C22C2×#1313318511yescharacteristic
348C22#1681117211yescharacteristic
349C22#1681119111yescharacteristic
350C22#1681143914yes—
351C4—24no—
352C4—44no—
353C4—82no—
354C4—4242no—
355C4—2121no—
356C4—2124no—
357C4—2122no—
358C4#1681126612yes—
359C4#1681129212yes—
360C2—4241no—
361C2#2027833711yescharacteristic
362C2#2028400912yes—
363C2#2028537312yes—
364C2#2028579912yes—
subgroupG/H
centerC23C2×D4×D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC2×C53×#491C2
socleC23×C53C22×D4
prime radicalC23×D53C2×D4
hypercenterC2×#491D53
hypocenterC53#25609
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#491

Complements

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

NH ≅ G/Ntype
2. C2×#16811002C2semidirect
5. #20275681C2semidirect
6. #20275535C2semidirect
7. #20275053C2semidirect
9. C2×C53×#491C2semidirect
180. C2×#491D53semidirect
217. C53#25609semidirect

Maximal subgroups

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

subgroupindex
C2×#168112362
C2×#168110022
#202804822
#202757242
#202756812
#202755352
#202750532
#202744842
C2×C53×#4912
#2560953