Finite Group Databasebeta

← #27393544

Subgroups — #27393544

Order 13568.

1130 conjugacy classes of subgroups; only the 235 normal subgroups are shown.

normal subgroups only

This group has 1130 conjugacy classes of subgroups — too many to draw. Only its 235 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×#16811249C211yesmaximal, characteristic
2#20277311C212yesmaximal
3#20275698C218yesmaximal, complemented
4#20274789C211yesmaximal, characteristic
5#20274787C212yesmaximal
6C53×#1819C211yesmaximal, characteristic
7#16811249—24no—
8#16811249—24no—
9C2×#13133196C414yescomplemented
10C2×#13133194C414yescomplemented
11C2×#13133194C2214yes—
12C22×C22⋊Dic53C2211yescharacteristic
13C22×C22⋊Dic53C2211yescharacteristic
14#16810517—24no—
15#16810511—24no—
16#16810481C2214yes—
17#16810479C2212yes—
18C22⋊C4×Dic53C2214yes—
19#16810390C2214yes—
20#16810389C2218yes—
21C22×C22⋊C4×C53C2211yescharacteristic
22C53×#379C2212yes—
23C53×#342C2214yes—
24C22×Dic106C2×C414yes—
25C23×Dic53C2×C412yes—
26C23×Dic53C2312yes—
27#13133196—28no—
28#13133196—28no—
29#13133194—28no—
30#13133194—28no—
31#13133194—28no—
32#13133194—28no—
33C2×C4⋊Dic53C2×C414yes—
34C2×#11719919—24no—
35C2×#11719919C2×C418yes—
36C2×C22⋊Dic53—22no—
37C2×C22⋊Dic53—22no—
38C2×C22⋊Dic53—22no—
39C2×C22⋊Dic53—22no—
40C2×C22⋊Dic53C2×C412yes—
41C2×C22⋊Dic53C2×C412yes—
42C2×C22⋊Dic53C2×C412yes—
43C2×C22⋊Dic53D414yes—
44C2×C22⋊Dic53D414yes—
45C2×C22⋊Dic53C2314yes—
46C2×C22⋊Dic53C2312yes—
47C2×C4×Dic53—24no—
48#13133124—24no—
49#13133124—24no—
50C25×C53C2311yescharacteristic
51C23×C4×C53C2×C412yes—
52C2×C22⋊C4×C53—22no—
53C2×C22⋊C4×C53—22no—
54C2×C22⋊C4×C53—22no—
55C2×C22⋊C4×C53—22no—
56C2×C22⋊C4×C53C2×C412yes—
57C2×C22⋊C4×C53C2314yes—
58C2×C22⋊C4×C53C2312yes—
59C24×C53—24no—
60C24×C53—82no—
61C24×C53C22×C411yescharacteristic
62C24×C53C22×C411yescharacteristic
63C24×C53C2×D412yes—
64C24×C53C2×D412yes—
65C24×C53C2411yescharacteristic
66C22×C4×C53—22no—
67C22×C4×C53—22no—
68C22×C4×C53—24no—
69C22×C4×C53—82no—
70C22×C4×C53C22×C414yes—
71C22×Dic53—22no—
72C22×Dic53—22no—
73C22×Dic53—24no—
74C22×Dic53—82no—
75C22×Dic53—82no—
76C22×Dic53C22×C414yes—
77C22×Dic53C22×C414yes—
78C22×Dic53C2×D414yes—
79C22×Dic53C2×D414yes—
80C2×Dic106—28no—
81C2×Dic106—84no—
82C22⋊C4×C53—22no—
83C22⋊C4×C53—22no—
84C22⋊C4×C53—24no—
85C22⋊C4×C53—28no—
86C22⋊C4×C53—82no—
87C22⋊C4×C53—82no—
88C22⋊Dic53—24no—
89C22⋊Dic53—24no—
90C22⋊Dic53—28no—
91C22⋊Dic53—82no—
92C22⋊Dic53—82no—
93C22⋊Dic53C22⋊C414yes—
94C22⋊Dic53C22⋊C414yes—
95C22⋊Dic53C22⋊C418yes—
96C4⋊Dic53—28no—
97#11719919—216no—
98#11719919—84no—
99C4×Dic53—84no—
100C23×C53—22no—
101C23×C53—22no—
102C23×C53—24no—
103C23×C53—24no—
104C23×C53—42no—
105C23×C53—42no—
106C23×C53—42no—
107C23×C53—42no—
108C23×C53—42no—
109C23×C53—42no—
110C23×C53—42no—
111C23×C53—88no—
112C23×C53C2×C22⋊C411yescharacteristic
113C23×C53C2×C22⋊C411yescharacteristic
114C23×C53C2×C22⋊C412yes—
115C23×C53C23×C411yescharacteristic
116C23×C53C22×D412yes—
117C23×C532−1+414yes—
118C2×C4×C53—24no—
119C2×C4×C53—44no—
120C2×C4×C53—44no—
121C2×C4×C53—44no—
122C2×C4×C53—44no—
123C2×C4×C53—82no—
124C2×C4×C53—82no—
125C2×Dic53—24no—
126C2×Dic53—44no—
127C2×Dic53—44no—
128C2×Dic53—44no—
129C2×Dic53—44no—
130C2×Dic53—44no—
131C2×Dic53—44no—
132C2×Dic53—44no—
133C2×Dic53—82no—
134C2×Dic53—82no—
135C2×Dic53C2×C22⋊C418yes—
136Dic106—88no—
137#21827—531nomaximal, Sylow
138C22×C53—22no—
139C22×C53—22no—
140C22×C53—22no—
141C22×C53—21no—
142C22×C53—21no—
143C22×C53—22no—
144C22×C53—22no—
145C22×C53—22no—
146C22×C53—22no—
147C22×C53—22no—
148C22×C53—44no—
149C22×C53—44no—
150C22×C53—44no—
151C22×C53—84no—
152C22×C53—84no—
153C22×C53C22×C22⋊C411yescharacteristic
154C22×C53#51912yes—
155C22×C53#53614yes—
156C4×C53—44no—
157C4×C53—84no—
158Dic53—28no—
159Dic53—44no—
160Dic53—84no—
161C2×#543—531no—
162#2312—532no—
163#1996—538no—
164#1819—532no—
165#1819—531no—
166#1819D5311yescharacteristic
167C2×C53—24no—
168C2×C53—22no—
169C2×C53—41no—
170C2×C53—41no—
171C2×C53—42no—
172C2×C53#231011yescharacteristic
173C2×C53#288312yes—
174#543—1064no—
175#543—1064no—
176C2×C22⋊Q8—534no—
177C2×C22⋊Q8—534no—
178C2×C22⋊Q8—534no—
179C22×C22⋊C4—531no—
180C22×C22⋊C4—531no—
181C22×C22⋊C4C2×D5311yescharacteristic
182#388—1064no—
183#388—1064no—
184#380—534no—
185#379—532no—
186#379C2×D5312yes—
187C4×C22⋊C4—534no—
188#342—538no—
189#342—534no—
190#342C2×D5314yes—
191C53#2182711yescharacteristic, complemented, Sylow
192C25C22×D5311yescharacteristic
193C22×Q8—534no—
194C23×C4—532no—
195C23×C4—532no—
196C23×C4C4×D5312yes—
197C22⋊Q8—1068no—
198C22⋊Q8—1068no—
199C22⋊Q8—1068no—
200C22⋊Q8—1068no—
201C22⋊Q8—1068no—
202C22⋊Q8—1068no—
203C2×C4⋊C4—538no—
204C2×C4⋊C4—534no—
205C2×C4⋊C4—1064no—
206C2×C22⋊C4—22no—
207C2×C22⋊C4—22no—
208C2×C22⋊C4—22no—
209C2×C22⋊C4—22no—
210C2×C22⋊C4—532no—
211C2×C22⋊C4—532no—
212C2×C22⋊C4—534no—
213C2×C22⋊C4—534no—
214C2×C22⋊C4—532no—
215C2×C22⋊C4—532no—
216C2×C22⋊C4—534no—
217C2×C22⋊C4—1062no—
218C2×C22⋊C4—1062no—
219C2×C22⋊C4—1062no—
220C2×C22⋊C4—1062no—
221C2×C22⋊C4C4×D5312yes—
222C2×C22⋊C4C22×D5314yes—
223C2×C22⋊C4C22×D5312yes—
224C2×C42—1064no—
225#95—1064no—
226#95—1064no—
227C24—24no—
228C24—82no—
229C24C2×C4×D5311yescharacteristic
230C24C2×C4×D5311yescharacteristic
231C24D4×D5312yes—
232C24D4×D5312yes—
233C24C23×D5311yescharacteristic
234C2×Q8—1068no—
235C2×Q8—4244no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—24no—
239C22×C4—82no—
240C22×C4—534no—
241C22×C4—534no—
242C22×C4—534no—
243C22×C4—534no—
244C22×C4—1062no—
245C22×C4—1062no—
246C22×C4—1064no—
247C22×C4—4242no—
248C22×C4—4242no—
249C22×C4C2×C4×D5314yes—
250C4⋊C4—10616no—
251C4⋊C4—1068no—
252C4⋊C4—4244no—
253C22⋊C4—22no—
254C22⋊C4—22no—
255C22⋊C4—24no—
256C22⋊C4—28no—
257C22⋊C4—82no—
258C22⋊C4—82no—
259C22⋊C4—534no—
260C22⋊C4—534no—
261C22⋊C4—538no—
262C22⋊C4—1064no—
263C22⋊C4—1064no—
264C22⋊C4—1068no—
265C22⋊C4—4242no—
266C22⋊C4—4242no—
267C42—4244no—
268C23—22no—
269C23—22no—
270C23—24no—
271C23—24no—
272C23—42no—
273C23—42no—
274C23—42no—
275C23—42no—
276C23—42no—
277C23—42no—
278C23—42no—
279C23—88no—
280C23C22⋊C4×D5311yescharacteristic
281C23C22⋊C4×D5311yescharacteristic
282C23C22⋊C4×D5312yes—
283C23C22×C4×D5311yescharacteristic
284C23C2×D4×D5312yes—
285C23#1313326414yes—
286Q8—4248no—
287C2×C4—24no—
288C2×C4—44no—
289C2×C4—44no—
290C2×C4—44no—
291C2×C4—44no—
292C2×C4—82no—
293C2×C4—82no—
294C2×C4—538no—
295C2×C4—1064no—
296C2×C4—4242no—
297C2×C4—4242no—
298C2×C4—2124no—
299C2×C4—2124no—
300C2×C4—2124no—
301C2×C4—2124no—
302C2×C4—2124no—
303C2×C4—2124no—
304C2×C4—2124no—
305C22—22no—
306C22—22no—
307C22—22no—
308C22—22no—
309C22—22no—
310C22—22no—
311C22—22no—
312C22—21no—
313C22—21no—
314C22—22no—
315C22—44no—
316C22—44no—
317C22—44no—
318C22—84no—
319C22—84no—
320C22C2×C22⋊C4×D5311yescharacteristic
321C22#1681110012yes—
322C22#1681119514yes—
323C4—44no—
324C4—84no—
325C4—1068no—
326C4—4244no—
327C4—2124no—
328C2—24no—
329C2—22no—
330C2—42no—
331C2—41no—
332C2—41no—
333C2#2027730011yescharacteristic
334C2#2028317312yes—
subgroupG/H
centerC22C2×C22⋊C4×D53
commutatorC23×C53C23×C4
FrattiniC24C23×D53
FittingC53×#1819C2
socleC22×C53C22×C22⋊C4
prime radicalC25×C53C23
cosocleC24×C53C24
hypercenter#1819D53
hypocenterC53#21827
LCSG ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C25 ◁ #1819

Complements

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

NH ≅ G/Ntype
3. #20275698C2semidirect
9. C2×#13133196C4semidirect
10. C2×#13133194C4semidirect
191. C53#21827semidirect

Maximal subgroups

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

subgroupindex
C2×#168112492
#202773112
#202756982
#202747892
#202747872
C53×#18192
#2182753