Finite Group Databasebeta

← #28312265

Subgroups — #28312265

Order 13568.

484 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 484 conjugacy classes of subgroups — too many to draw. Only its 84 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#20290575C211yesmaximal, characteristic
2C4×#13133277C211yesmaximal, characteristic
3#20289799C214yesmaximal
4#20279026C211yesmaximal, characteristic
5C2×#13133277C2212yes—
6C2×#13133275C2212yes—
7C4×#11719940—21no—
8#16811565—21no—
9#16811503—21no—
10#16811501—21no—
11C2×C4×C22⋊D53C2211yescharacteristic
12#16810561—21no—
13#16810558—21no—
14#16810519—21no—
15#16810502C2212yes—
16#16810496C412yescomplemented
17#16810483C411yescharacteristic, complemented
18#16810480—21no—
19C22×C4⋊C4×C53C411yescharacteristic, complemented
20C2×#11719940—22no—
21C2×C4⋊#6157—41no—
22C2×C22⋊#6157—22no—
23#13133277—22no—
24#13133277Q814yes—
25#13133275—24no—
26#13133272—41no—
27C4×#88324—41no—
28(C2×#6157)⋊C4—41no—
29C22×C22⋊D53C2311yescharacteristic
30C22×C4×D53D411yescharacteristic
31C4×C22⋊D53—41no—
32C4×C22⋊D53—41no—
33C2×C4⋊Dic53—21no—
34C2×#11719919—22no—
35C2×#11719919D411yescharacteristic
36C2×C22⋊Dic53C2×C412yes—
37C2×C22⋊Dic53C2×C411yescharacteristic
38C2×#11719921—22no—
39C2×#11719921D411yescharacteristic
40C2×C4×Dic53D411yescharacteristic
41#13133124—21no—
42#13133124—22no—
43#13133124—22no—
44C23×C4×C53C2×C411yescharacteristic
45C23×C4×C53C2×C412yes—
46C2×C4⋊C4×C53—21no—
47C2×C4⋊C4×C53—21no—
48C2×C4⋊C4×C53—41no—
49C2×C4⋊C4×C53—41no—
50C24×C53C22×C411yescharacteristic
51C23×D53C2×D411yescharacteristic
52C22×#6157—42no—
53C22×C4×C53—21no—
54C22×C4×C53—21no—
55C22×C4×C53—22no—
56C22×C4×C53—42no—
57C22×C4×C53—42no—
58C22×C4×C53—82no—
59C22×C4×C53C22⋊C411yescharacteristic
60C22×C4×C53C22⋊C411yescharacteristic
61C2×C22⋊D53—41no—
62C2×C22⋊D53—41no—
63C2×C22⋊D53C2×Q812yes—
64C2×C22⋊D53#4112yes—
65C22×Dic53—22no—
66C22×Dic53C22⋊C411yescharacteristic
67C22×Dic53C22⋊C411yescharacteristic
68C22×Dic53C2×D411yescharacteristic
69C2×C4×D53—21no—
70C2×C4×D53—21no—
71C2×C4×D53—42no—
72C22⋊#6157—42no—
73C22⋊#6157—42no—
74#11719940—24no—
75#11719940—42no—
76C2×#88324—42no—
77C4⋊#6157—82no—
78C4⋊C4×C53—42no—
79C4⋊C4×C53—41no—
80C4⋊C4×C53—41no—
81C22⋊Dic53—41no—
82C22⋊Dic53—42no—
83#11719921—41no—
84#11719921—82no—
85C4⋊Dic53—42no—
86#11719919—41no—
87#11719919—82no—
88C4×Dic53—21no—
89C4×Dic53—21no—
90C23×C53—21no—
91C23×C53—21no—
92C23×C53—41no—
93C23×C53—41no—
94C23×C53C2×C22⋊C411yescharacteristic
95C23×C53C4×Q812yes—
96C22×D53—21no—
97C22×D53—21no—
98C22×D53—41no—
99C22×D53—41no—
100C22×D53#12411yescharacteristic
101C22×D53C4⋊Q811yescharacteristic
102C2×#6157—82no—
103C2×#6157—82no—
104C2×#6157—82no—
105C2×C4×C53—22no—
106C2×C4×C53—21no—
107C2×C4×C53—42no—
108C2×C4×C53—42no—
109C2×C4×C53—41no—
110C2×C4×C53—42no—
111C2×C4×C53—42no—
112C2×C4×C53—42no—
113C2×C4×C53—84no—
114C2×C4×C53#9911yescharacteristic
115C2×C4×C53#9911yescharacteristic
116C22⋊D53—42no—
117C22⋊D53—42no—
118C2×Dic53—21no—
119C2×Dic53—21no—
120C2×Dic53—21no—
121C2×Dic53—21no—
122C2×Dic53—41no—
123C2×Dic53—41no—
124C2×Dic53—42no—
125C2×Dic53—42no—
126C2×Dic53—42no—
127C2×Dic53C22⋊Q812yes—
128C4×D53—42no—
129C4×D53—81no—
130#88324—44no—
131#13292—531nomaximal, Sylow
132C22×C53—21no—
133C22×C53—21no—
134C22×C53—22no—
135C22×C53—42no—
136C22×C53—42no—
137C22×C53—41no—
138C22×C53—41no—
139C22×C53#39111yescharacteristic
140C22×C53C2×#9911yescharacteristic
141C22×C53#41011yescharacteristic
142C2×D53—21no—
143C2×D53—21no—
144C2×D53—41no—
145C2×D53—41no—
146C2×D53—41no—
147C2×D53—41no—
148C2×D53—81no—
149#6157—162no—
150C4×C53—21no—
151C4×C53—41no—
152C4×C53—82no—
153C4×C53#35212yes—
154Dic53—22no—
155Dic53—41no—
156Dic53—41no—
157Dic53—82no—
158#2487—531no—
159#1799—531no—
160C4×#100—531no—
161#1390—534no—
162C2×C53—21no—
163C2×C53—21no—
164C2×C53—42no—
165C2×C53#199611yescharacteristic
166C2×C53C2×#35211yescharacteristic
167C2×C53#212211yescharacteristic
168D53—41no—
169D53—41no—
170C2×C4×D4—531no—
171C22×C4⋊C4#615711yescharacteristic, complemented
172C2×#100—532no—
173C2×#99—532no—
174C4×M16—1061no—
175#400—1061no—
176#399—1061no—
177#389—1061no—
178#386—532no—
179#385—532no—
180#380—531no—
181#380—1061no—
182#380—1061no—
183#343—1061no—
184#342—1061no—
185C53#1329211yescharacteristic, complemented, Sylow
186C22×D4—531no—
187C23×C4—531no—
188C23×C4C2×#615712yes—
189C23×C4C2×#615711yescharacteristic
190C2×M16—1062no—
191C4×D4—2121no—
192C4×D4—2121no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—41no—
196C2×C4⋊C4—41no—
197C2×C4⋊C4—531no—
198C2×C4⋊C4—1062no—
199C2×C4⋊C4—1061no—
200C2×C4⋊C4—2121no—
201C2×C22⋊C4—531no—
202C2×C22⋊C4—531no—
203C2×C22⋊C4—532no—
204C2×C22⋊C4—1062no—
205C2×C22⋊C4—1062no—
206C2×C42—531no—
207#100—534no—
208#100—1062no—
209#99—1064no—
210#97—2121no—
211C4×C8—2121no—
212#95—1062no—
213#95—1062no—
214#95—1061no—
215#95—2121no—
216C24—531no—
217C24C22×#615711yescharacteristic
218C2×D4—532no—
219C2×D4—532no—
220C2×D4—2121no—
221C2×D4—2121no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—22no—
225C22×C4—42no—
226C22×C4—42no—
227C22×C4—82no—
228C22×C4—531no—
229C22×C4—531no—
230C22×C4—531no—
231C22×C4—1061no—
232C22×C4—1061no—
233C22×C4—1062no—
234C22×C4—2122no—
235C22×C4—2122no—
236C22×C4C22⋊#615711yescharacteristic
237C22×C4C22⋊#615711yescharacteristic
238M16—1064no—
239M16—2122no—
240C2×C8—2122no—
241C4⋊C4—42no—
242C4⋊C4—41no—
243C4⋊C4—41no—
244C4⋊C4—4242no—
245C4⋊C4—4242no—
246C4⋊C4—2121no—
247C4⋊C4—2122no—
248C22⋊C4—4242no—
249C22⋊C4—2121no—
250C22⋊C4—2121no—
251C22⋊C4—2122no—
252C22⋊C4—2122no—
253C22⋊C4—2122no—
254C42—1061no—
255C42—1061no—
256C23—21no—
257C23—21no—
258C23—41no—
259C23—41no—
260C23—531no—
261C23—531no—
262C23—1061no—
263C23—1061no—
264C23—2121no—
265C23—2121no—
266C23C2×C22⋊#615711yescharacteristic
267C23Q8×#615712yes—
268D4—2122no—
269D4—2122no—
270C2×C4—22no—
271C2×C4—21no—
272C2×C4—42no—
273C2×C4—42no—
274C2×C4—41no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—84no—
279C2×C4—532no—
280C2×C4—1061no—
281C2×C4—1061no—
282C2×C4—1061no—
283C2×C4—1061no—
284C2×C4—4241no—
285C2×C4—4242no—
286C2×C4—4242no—
287C2×C4—4242no—
288C2×C4—2122no—
289C2×C4—2122no—
290C2×C4—2122no—
291C2×C4—2122no—
292C2×C4—2121no—
293C2×C4—2121no—
294C2×C4#1313327511yescharacteristic
295C2×C4#1313327511yescharacteristic
296C8—2124no—
297C22—21no—
298C22—21no—
299C22—22no—
300C22—42no—
301C22—42no—
302C22—41no—
303C22—41no—
304C22—1061no—
305C22—1061no—
306C22—4241no—
307C22—2121no—
308C22—2121no—
309C22—2121no—
310C22—2121no—
311C22(C2×#6157)⋊Q811yescharacteristic
312C22C2×#1313327511yescharacteristic
313C22#1681160411yescharacteristic
314C4—21no—
315C4—41no—
316C4—82no—
317C4—8482no—
318C4—1062no—
319C4—4242no—
320C4—2121no—
321C4—2121no—
322C4#1681152512yes—
323C2—21no—
324C2—21no—
325C2—42no—
326C2—2121no—
327C2—2121no—
328C2#2029090411yescharacteristic
329C2C2×#1681152511yescharacteristic
330C2#2029097411yescharacteristic
subgroupG/H
centerC22C2×#13133275
commutatorC24×C53C22×C4
FrattiniC24C22×#6157
FittingC22×C4⋊C4×C53C4
socleC22×C53C2×#99
prime radical, cosocleC22×C22⋊D53C23
hypercenterC22×C4⋊C4#6157
hypocenterC53#13292
LCSG ▷ C24×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
16. #16810496C4semidirect
17. #16810483C4semidirect
19. C22×C4⋊C4×C53C4semidirect
171. C22×C4⋊C4#6157semidirect
185. C53#13292semidirect

Maximal subgroups

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

subgroupindex
#202905752
C4×#131332772
#202897992
#202790262
#1329253