Finite Group Databasebeta

← #28310611

Subgroups — #28310611

Order 13568.

720 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 720 conjugacy classes of subgroups — too many to draw. Only its 102 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×#16811527C212yesmaximal
2#20290460C212yesmaximal, complemented
3#20289809C212yesmaximal, complemented
4#20277279C211yesmaximal, characteristic
5C2×#13133276C2214yes—
6C4×C22⋊#6157—22no—
7#16811527C418yescomplemented
8#16811501—22no—
9C2×#13133213C2211yescharacteristic
10C2×D4×Dic53C411yescharacteristic, complemented
11C2×#13133179C2211yescharacteristic
12C2×#13133177—22no—
13#16810525C2211yescharacteristic
14#16810523—22no—
15#16810492C411yescharacteristic, complemented
16C42×Dic53C411yescharacteristic, complemented
17C53×#390C411yescharacteristic, complemented
18C2×C22⋊#6157—24no—
19C2×C4×#6157—42no—
20#13133276—24no—
21#13133276C2×C418yescomplemented
22(C2×#6157)⋊C4—42no—
23C22×D212C2311yescharacteristic
24C22×C22⋊D53—22no—
25C23×Dic53—21no—
26C22×C4×D53D412yes—
27#13133213—22no—
28#13133213C2×C414yes—
29D4×Dic53—41no—
30D4×Dic53—41no—
31#13133179—41no—
32#13133179—41no—
33#13133177—48no—
34C2×C4⋊Dic53C2×C411yescharacteristic
35C2×#11719919—22no—
36C2×C22⋊Dic53—21no—
37C2×#11719921—22no—
38C2×#11719921D412yes—
39C2×C4×Dic53—21no—
40C2×C4×Dic53—21no—
41C2×C4×Dic53C2×C411yescharacteristic
42C2×C4×Dic53C2×C411yescharacteristic
43C22×D4×C53C2×C411yescharacteristic
44C2×C4⋊C4×C53C2×C411yescharacteristic
45C2×C22⋊C4×C53—22no—
46C2×C42×C53C2×C411yescharacteristic
47C24×C53—21no—
48C23×D53C2×D412yes—
49C22×#6157—44no—
50C2×D4×C53—21no—
51C2×D4×C53—41no—
52C2×D4×C53—41no—
53C2×D4×C53C4212yescomplemented
54C22×C4×C53—21no—
55C22×C4×C53C22⋊C412yes—
56C22×C4×C53C22×C411yescharacteristic
57C2×C22⋊D53—28no—
58C2×C22⋊D53—44no—
59C2×C22⋊D53—44no—
60C22×Dic53—21no—
61C22×Dic53—21no—
62C22×Dic53—44no—
63C22×Dic53—41no—
64C22×Dic53—41no—
65C22×Dic53C22⋊C411yescharacteristic
66C22×Dic53C22⋊C411yescharacteristic
67C2×D212—41no—
68C2×D212—41no—
69C2×D212C22×C412yes—
70C2×D212#4112yes—
71C2×C4×D53—22no—
72C2×C4×D53—22no—
73C2×C4×D53—44no—
74C22⋊#6157—28no—
75C22⋊#6157—44no—
76C4×#6157—84no—
77C4⋊C4×C53—41no—
78C22⋊C4×C53—44no—
79C42×C53—41no—
80C22⋊Dic53—42no—
81#11719921—42no—
82#11719921—44no—
83C4⋊Dic53—41no—
84#11719919—44no—
85C4×Dic53—24no—
86C4×Dic53—21no—
87C4×Dic53—21no—
88C4×Dic53—21no—
89C4×Dic53—42no—
90C4×Dic53C4212yescomplemented
91C23×C53—22no—
92C23×C53—42no—
93C23×C53—42no—
94C23×C53—41no—
95C23×C53—41no—
96C23×C53C2×C22⋊C411yescharacteristic
97C22×D53—22no—
98C22×D53—22no—
99C22×D53—42no—
100C22×D53—42no—
101C22×D53C4×D414yes—
102C2×#6157—48no—
103C2×#6157—84no—
104C2×#6157—84no—
105D4×C53—42no—
106D4×C53—42no—
107C2×C4×C53—21no—
108C2×C4×C53—21no—
109C2×C4×C53—22no—
110C2×C4×C53—22no—
111C2×C4×C53—41no—
112C2×C4×C53—41no—
113C2×C4×C53—42no—
114C2×C4×C53C2×C4211yescharacteristic
115C2×C4×C53#11711yescharacteristic
116C22⋊D53—416no—
117C2×Dic53—22no—
118C2×Dic53—22no—
119C2×Dic53—22no—
120C2×Dic53—22no—
121C2×Dic53—21no—
122C2×Dic53—41no—
123C2×Dic53—42no—
124C2×Dic53—41no—
125C2×Dic53—82no—
126C2×Dic53—82no—
127C2×Dic53#9911yescharacteristic
128C2×Dic53#9911yescharacteristic
129D212—42no—
130D212—42no—
131C4×D53—44no—
132C4×D53—82no—
133C4×#353—531nomaximal, Sylow
134C22×C53—21no—
135C22×C53—21no—
136C22×C53—44no—
137C22×C53—42no—
138C22×C53—82no—
139C22×C53—81no—
140C22×C53—81no—
141C22×C53C4×C22⋊C411yescharacteristic
142C22×C53C2×#9911yescharacteristic
143C22×C53#41011yescharacteristic
144C2×D53—24no—
145C2×D53—42no—
146C2×D53—42no—
147C2×D53—44no—
148C2×D53—82no—
149#6157—88no—
150C4×C53—21no—
151C4×C53—21no—
152C4×C53—42no—
153C4×C53—42no—
154Dic53—24no—
155Dic53—21no—
156Dic53—41no—
157Dic53—81no—
158Dic53#35312yescomplemented
159C4×#127—531no—
160C2×#353—532no—
161C4×#99—532no—
162#1393—532no—
163C2×C53—21no—
164C2×C53—21no—
165C2×C53—82no—
166C2×C53C4×#9911yescharacteristic
167C2×C53C2×#35311yescharacteristic
168C2×C53#212611yescharacteristic
169D53—44no—
170C2×#127—531no—
171C2×C4×D4—531no—
172C2×C4×D4—531no—
173C2×C4×D4—1062no—
174C2×#99—534no—
175#390—531no—
176#390—1062no—
177#390#615711yescharacteristic, complemented
178#384—531no—
179C4×C22⋊C4—1062no—
180C43—531no—
181#353—538no—
182#342—1062no—
183C53C4×#35311yescharacteristic, complemented, Sylow
184C22×D4—531no—
185C22×D4—1062no—
186C22×D4C2×#615711yescharacteristic
187C23×C4—532no—
188C23×C4—1061no—
189#127—534no—
190#127—1062no—
191C4×D4—2128no—
192C4×D4—2121no—
193C4×D4—2121no—
194C4×D4—2121no—
195C4×D4—2121no—
196C2×C4⋊C4—531no—
197C2×C4⋊C4—1062no—
198C2×C4⋊C4C2×#615711yescharacteristic
199C2×C22⋊C4—22no—
200C2×C22⋊C4—532no—
201C2×C22⋊C4—1064no—
202C2×C22⋊C4—1062no—
203C2×C22⋊C4—1061no—
204C2×C42—531no—
205C2×C42—531no—
206C2×C42—1061no—
207C2×C42—1061no—
208C2×C42—2122no—
209C2×C42C2×#615711yescharacteristic
210#99—538no—
211#99—1064no—
212#95—2122no—
213C24—21no—
214C24—532no—
215C2×D4—21no—
216C2×D4—41no—
217C2×D4—41no—
218C2×D4—532no—
219C2×D4—532no—
220C2×D4—1068no—
221C2×D4—2121no—
222C2×D4—2121no—
223C2×D4—2124no—
224C2×D4—2124no—
225C2×D4C4×#615712yescomplemented
226C22×C4—21no—
227C22×C4—531no—
228C22×C4—531no—
229C22×C4—1062no—
230C22×C4—1062no—
231C22×C4—1061no—
232C22×C4—1061no—
233C22×C4—2124no—
234C22×C4—2121no—
235C22×C4—2121no—
236C22×C4—2124no—
237C22×C4—2124no—
238C22×C4C22⋊#615712yes—
239C22×C4C22×#615711yescharacteristic
240C4⋊C4—41no—
241C4⋊C4—2121no—
242C4⋊C4—2124no—
243C22⋊C4—44no—
244C22⋊C4—1068no—
245C22⋊C4—2122no—
246C22⋊C4—2124no—
247C22⋊C4—2122no—
248C22⋊C4—2124no—
249C42—41no—
250C42—532no—
251C42—1061no—
252C42—1061no—
253C42—1064no—
254C42—1061no—
255C42—4244no—
256C42—2122no—
257C23—22no—
258C23—42no—
259C23—42no—
260C23—41no—
261C23—41no—
262C23—534no—
263C23—1062no—
264C23—1062no—
265C23—2122no—
266C23—2122no—
267C23C2×C22⋊#615711yescharacteristic
268D4—42no—
269D4—42no—
270D4—2122no—
271D4—2122no—
272D4—21216no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—22no—
276C2×C4—22no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—42no—
280C2×C4—531no—
281C2×C4—531no—
282C2×C4—1062no—
283C2×C4—1062no—
284C2×C4—1062no—
285C2×C4—1062no—
286C2×C4—1061no—
287C2×C4—4242no—
288C2×C4—4242no—
289C2×C4—4242no—
290C2×C4—4244no—
291C2×C4—4244no—
292C2×C4—2124no—
293C2×C4—2122no—
294C2×C4—2128no—
295C2×C4—2121no—
296C2×C4—2121no—
297C2×C4C2×C4×#615711yescharacteristic
298C2×C4#1313329611yescharacteristic
299C22—21no—
300C22—21no—
301C22—44no—
302C22—42no—
303C22—82no—
304C22—81no—
305C22—81no—
306C22—1064no—
307C22—4242no—
308C22—2124no—
309C22—2122no—
310C22—2122no—
311C22C4×C22⋊#615711yescharacteristic
312C22D53⋊#9911yescharacteristic
313C22#1681160311yescharacteristic
314C4—21no—
315C4—21no—
316C4—42no—
317C4—42no—
318C4—532no—
319C4—1064no—
320C4—1061no—
321C4—4248no—
322C4—4241no—
323C4—2121no—
324C2—21no—
325C2—21no—
326C2—82no—
327C2—2124no—
328C2#2029046211yescharacteristic
329C2D53⋊#35311yescharacteristic
330C2#2029098711yescharacteristic
subgroupG/H
centerC22D53⋊#99
commutatorC2×C4×C53C2×C42
FrattiniC22×C4C22×#6157
FittingC53×#390C4
socleC22×C53C2×#99
prime radicalC2×#13133213C22
cosocleC22×D212C23
hypercenter#390#6157
hypocenterC53C4×#353
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #390

Complements

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

NH ≅ G/Ntype
2. #20290460C2semidirect
3. #20289809C2semidirect
7. #16811527C4semidirect
10. C2×D4×Dic53C4semidirect
15. #16810492C4semidirect
16. C42×Dic53C4semidirect
17. C53×#390C4semidirect
21. #13133276C2×C4semidirect
53. C2×D4×C53C42semidirect
90. C4×Dic53C42semidirect
158. Dic53#353semidirect
177. #390#6157semidirect
183. C53C4×#353semidirect
225. C2×D4C4×#6157semidirect

Maximal subgroups

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

subgroupindex
C2×#168115272
#202904602
#202898092
#202772792
C4×#35353