Finite Group Databasebeta

← #28310387

Subgroups — #28310387

Order 13568.

826 conjugacy classes of subgroups; only the 152 normal subgroups are shown.

normal subgroups only

This group has 826 conjugacy classes of subgroups — too many to draw. Only its 152 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×#16811602C212yesmaximal
2#20289810C214yesmaximal, complemented
3#20277308C211yesmaximal, characteristic
4C2×#13133296—22no—
5#16811602—24no—
6#16811602C418yescomplemented
7C2×#13133276C2214yes—
8C2×#13133275—22no—
9C22×#11719945C2211yescharacteristic
10C2×#13133176C411yescharacteristic, complemented
11#16810528C412yescomplemented
12#16810524C2212yes—
13#16810502—22no—
14#16810480—22no—
15C22×C4⋊C4×C53C411yescharacteristic, complemented
16#13133296—28no—
17#13133296—44no—
18C2×C4⋊#6157—42no—
19C2×C22⋊#6157—24no—
20C2×C4×#6157—42no—
21#13133276—24no—
22#13133276C2×C418yescomplemented
23#13133275—28no—
24C2×#11719945—42no—
25C2×#11719945—42no—
26C2×#11719945C2×C414yes—
27C2×#11719945D412yes—
28C2×#11719945Q812yes—
29C22×Dic106C2×C411yescharacteristic
30C22×D212C2311yescharacteristic
31C22×C22⋊D53D412yes—
32C22×C4×D53D412yes—
33#13133176—42no—
34C2×C4⋊Dic53—21no—
35C2×C22⋊Dic53—21no—
36C2×#11719921—24no—
37C2×C4×Dic53C2×C412yes—
38#13133124—24no—
39C23×C4×C53—21no—
40C23×C4×C53C2×C411yescharacteristic
41C2×C4⋊C4×C53—21no—
42C2×C4⋊C4×C53—42no—
43C2×C4⋊C4×C53C2×C412yes—
44C24×C53C22⋊C411yescharacteristic
45C23×D53C2×D412yes—
46C22×#6157—44no—
47C22×C4×C53—22no—
48C22×C4×C53—22no—
49C22×C4×C53—22no—
50C22×C4×C53—42no—
51C22×C4×C53—82no—
52C22×C4×C53C4212yes—
53C22×C4×C53C22⋊C411yescharacteristic
54C22×C4×C53C4⋊C412yes—
55C22×C4×C53C22×C411yescharacteristic
56C2×C22⋊D53—42no—
57C2×C22⋊D53—42no—
58C2×C22⋊D53C22⋊C414yes—
59C2×C22⋊D53C4⋊C414yes—
60C22×Dic53—22no—
61C22×Dic53C22⋊C412yes—
62#11719945—44no—
63#11719945—44no—
64#11719945—44no—
65#11719945—44no—
66C2×D212—41no—
67C2×D212—41no—
68C2×D212C22×C412yes—
69C2×D212C2×D411yescharacteristic
70C2×D212C2×Q811yescharacteristic
71C2×C4×D53—22no—
72C2×C4×D53—22no—
73C2×C4×D53—24no—
74C2×C4×D53C22⋊C414yes—
75C2×C4×D53C4⋊C414yes—
76C2×Dic106—41no—
77C2×Dic106—41no—
78C2×Dic106C4212yes—
79C2×Dic106C4⋊C412yes—
80C22⋊#6157—28no—
81C22⋊#6157—44no—
82C4⋊#6157—48no—
83C4×#6157—48no—
84C4⋊C4×C53—42no—
85C4⋊C4×C53—42no—
86C22⋊Dic53—42no—
87#11719921—84no—
88C4⋊Dic53—42no—
89C4×Dic53—42no—
90C23×C53—21no—
91C23×C53—21no—
92C23×C53—41no—
93C23×C53—41no—
94C23×C53#9512yes—
95C23×C53C2×C22⋊C411yescharacteristic
96C22×D53—22no—
97C22×D53—22no—
98C22×D53—42no—
99C22×D53—42no—
100C22×D53C2×C22⋊C412yes—
101C22×D53C2×C4⋊C412yes—
102C2×#6157—48no—
103C2×#6157—84no—
104C2×#6157—84no—
105C2×C4×C53—21no—
106C2×C4×C53—21no—
107C2×C4×C53—24no—
108C2×C4×C53—24no—
109C2×C4×C53—44no—
110C2×C4×C53—42no—
111C2×C4×C53—84no—
112C2×C4×C53#9512yes—
113C2×C4×C53#9912yes—
114C2×C4×C53#9912yes—
115C2×C4×C53C2×C4211yescharacteristic
116C2×C4×C53C2×C4⋊C411yescharacteristic
117C22⋊D53—44no—
118C22⋊D53—44no—
119C2×Dic53—22no—
120C2×Dic53—22no—
121C2×Dic53—42no—
122C2×Dic53—42no—
123C2×Dic53—42no—
124C2×Dic53#9514yes—
125D212—42no—
126D212—41no—
127D212—41no—
128C4×D53—24no—
129C4×D53—24no—
130C4×D53—24no—
131C4×D53—24no—
132Dic106—42no—
133Dic106—42no—
134#12583—531nomaximal, Sylow
135C22×C53—21no—
136C22×C53—21no—
137C22×C53—22no—
138C22×C53—42no—
139C22×C53—42no—
140C22×C53—41no—
141C22×C53—41no—
142C22×C53C2×#9511yescharacteristic
143C22×C53C2×#9911yescharacteristic
144C22×C53C2×#9911yescharacteristic
145C2×D53—22no—
146C2×D53—22no—
147C2×D53—42no—
148C2×D53—42no—
149C2×D53—42no—
150C2×D53—42no—
151C2×D53—82no—
152#6157—88no—
153C4×C53—21no—
154C4×C53—21no—
155C4×C53—82no—
156C4×C53#34214yes—
157Dic53—24no—
158Dic53—82no—
159C2×#410—532no—
160#2311—531no—
161#1393—534no—
162C2×C53—21no—
163C2×C53—21no—
164C2×C53—42no—
165C2×C53C2×#34211yescharacteristic
166C2×C53#212811yescharacteristic
167C2×C53#212911yescharacteristic
168D53—42no—
169D53—42no—
170C22×#41—531no—
171C2×#117—531no—
172C2×#117—1062no—
173C22×C4⋊C4#615711yescharacteristic, complemented
174#410—538no—
175#410—1064no—
176C2×#99—534no—
177C2×#99—1062no—
178#391—532no—
179#390—532no—
180#386—1062no—
181#380—1062no—
182C53#1258311yescharacteristic, complemented, Sylow
183C2×#41—534no—
184C2×#41—532no—
185C2×#41—532no—
186C2×#41—2122no—
187C2×#41—2122no—
188C22×Q8—531no—
189C22×D4—532no—
190C22×D4—531no—
191C23×C4—21no—
192C23×C4—532no—
193C23×C4C2×#615711yescharacteristic
194#117—1068no—
195#117—2124no—
196#117—2122no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—42no—
199C2×C4⋊C4—1061no—
200C2×C4⋊C4—2122no—
201C2×C4⋊C4C2×#615712yes—
202C2×C22⋊C4—1064no—
203C2×C22⋊C4—1064no—
204C2×C22⋊C4—1061no—
205C2×C42—532no—
206C2×C42—2122no—
207#99—538no—
208#99—1064no—
209#99—1068no—
210#95—1064no—
211C24—532no—
212C24C22⋊#615711yescharacteristic
213#41—2124no—
214#41—2124no—
215#41—2124no—
216#41—2124no—
217C2×Q8—532no—
218C2×Q8—532no—
219C2×Q8—2121no—
220C2×Q8—2121no—
221C2×D4—534no—
222C2×D4—534no—
223C2×D4—532no—
224C2×D4—531no—
225C2×D4—531no—
226C2×D4—2121no—
227C2×D4—2121no—
228C2×D4—2122no—
229C2×D4—2122no—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—42no—
234C22×C4—82no—
235C22×C4—532no—
236C22×C4—534no—
237C22×C4—534no—
238C22×C4—1062no—
239C22×C4—1062no—
240C22×C4—1064no—
241C22×C4—1062no—
242C22×C4—2124no—
243C22×C4C4×#615712yes—
244C22×C4C4⋊#615712yes—
245C22×C4C22⋊#615711yescharacteristic
246C22×C4C22×#615711yescharacteristic
247C4⋊C4—42no—
248C4⋊C4—42no—
249C4⋊C4—2128no—
250C4⋊C4—2122no—
251C22⋊C4—1068no—
252C22⋊C4—4244no—
253C22⋊C4—2122no—
254C22⋊C4—2124no—
255C42—2122no—
256C42—2128no—
257C23—21no—
258C23—21no—
259C23—41no—
260C23—41no—
261C23—532no—
262C23—532no—
263C23—1062no—
264C23—1062no—
265C23—2122no—
266C23—2122no—
267C23(C2×#6157)⋊C412yes—
268C23C2×C22⋊#615711yescharacteristic
269Q8—2122no—
270Q8—2122no—
271D4—2122no—
272D4—2121no—
273D4—2121no—
274D4—2124no—
275D4—2124no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—24no—
279C2×C4—24no—
280C2×C4—44no—
281C2×C4—42no—
282C2×C4—84no—
283C2×C4—534no—
284C2×C4—1062no—
285C2×C4—1062no—
286C2×C4—1064no—
287C2×C4—1064no—
288C2×C4—1064no—
289C2×C4—1064no—
290C2×C4—4244no—
291C2×C4—4244no—
292C2×C4—2122no—
293C2×C4—2122no—
294C2×C4—2128no—
295C2×C4—2122no—
296C2×C4(C2×#6157)⋊C412yes—
297C2×C4#1313327512yes—
298C2×C4#1313327612yes—
299C2×C4C2×C4×#615711yescharacteristic
300C2×C4C2×C4⋊#615711yescharacteristic
301C22—21no—
302C22—21no—
303C22—22no—
304C22—42no—
305C22—42no—
306C22—41no—
307C22—41no—
308C22—1062no—
309C22—1062no—
310C22—4242no—
311C22—2122no—
312C22—2122no—
313C22—2122no—
314C22—2122no—
315C22C2×(C2×#6157)⋊C411yescharacteristic
316C22C2×#1313327511yescharacteristic
317C22C2×#1313327611yescharacteristic
318C4—21no—
319C4—21no—
320C4—82no—
321C4—1064no—
322C4—4242no—
323C4—4248no—
324C4#1681150114yes—
325C2—21no—
326C2—21no—
327C2—42no—
328C2—2122no—
329C2—2122no—
330C2C2×#1681150111yescharacteristic
331C2#2029099411yescharacteristic
332C2#2029099711yescharacteristic
subgroupG/H
centerC22C2×#13133275
commutatorC2×C4×C53C2×C42
FrattiniC22×C4C22×#6157
FittingC22×C4⋊C4×C53C4
socleC22×C53C2×#99
prime radicalC22×#11719945C22
cosocleC22×D212C23
hypercenterC22×C4⋊C4#6157
hypocenterC53#12583
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C23×C4 ◁ C22×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. #20289810C2semidirect
6. #16811602C4semidirect
10. C2×#13133176C4semidirect
11. #16810528C4semidirect
15. C22×C4⋊C4×C53C4semidirect
22. #13133276C2×C4semidirect
173. C22×C4⋊C4#6157semidirect
182. C53#12583semidirect

Maximal subgroups

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

subgroupindex
C2×#168116022
#202898102
#202773082
#1258353