Finite Group Databasebeta

← #28311186

Subgroups — #28311186

Order 13568.

588 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 588 conjugacy classes of subgroups — too many to draw. Only its 86 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#20290576C212yesmaximal, complemented
3#20289810C212yesmaximal, complemented
4#20277437C211yesmaximal, characteristic
5C2×#13133276C2214yes—
6#16811565—22no—
7#16811527—24no—
8#16811501—22no—
9C2×#13133213C2211yescharacteristic
10C2×C4×D212C2211yescharacteristic
11#16810526C411yescharacteristic, complemented
12#16810524C2211yescharacteristic
13#16810502—22no—
14#16810499—22no—
15#16810488C411yescharacteristic, complemented
16#16810469C411yescharacteristic, complemented
17C2×C4×D4×C53C411yescharacteristic, complemented
18C2×C4⋊#6157—42no—
19C2×C22⋊#6157—24no—
20#13133276—24no—
21#13133276—24no—
22(C2×#6157)⋊C4—42no—
23C22×D212C2311yescharacteristic
24C22×C22⋊D53—22no—
25C22×C4×D53D412yes—
26#13133213—22no—
27#13133213D412yes—
28#13133213Q812yes—
29C4×D212—42no—
30C2×C4⋊Dic53C2×C411yescharacteristic
31C2×C22⋊Dic53—22no—
32C2×#11719921—22no—
33C2×#11719921D412yes—
34C2×C4×Dic53C2×C411yescharacteristic
35C2×C4×Dic53C2×C411yescharacteristic
36#13133124—21no—
37#13133124—21no—
38#13133124—22no—
39C22×D4×C53C2×C411yescharacteristic
40C23×C4×C53—21no—
41C4×D4×C53—42no—
42C2×C4⋊C4×C53C2×C411yescharacteristic
43C2×C22⋊C4×C53—21no—
44C2×C42×C53C2×C411yescharacteristic
45C24×C53—21no—
46C23×D53C2×D412yes—
47C22×#6157—44no—
48C2×D4×C53—21no—
49C2×D4×C53—41no—
50C2×D4×C53—41no—
51C2×D4×C53C4⋊C412yescomplemented
52C22×C4×C53—21no—
53C22×C4×C53—42no—
54C22×C4×C53—82no—
55C22×C4×C53C22⋊C411yescharacteristic
56C22×C4×C53C22⋊C411yescharacteristic
57C22×C4×C53C22×C411yescharacteristic
58C2×C22⋊D53—24no—
59C2×C22⋊D53—24no—
60C2×C22⋊D53—82no—
61C2×C22⋊D53—82no—
62C22×Dic53—21no—
63C22×Dic53—21no—
64C22×Dic53C22⋊C412yes—
65C2×D212—41no—
66C2×D212—41no—
67C2×D212C2×D411yescharacteristic
68C2×D212C2×Q811yescharacteristic
69C2×D212#4112yes—
70C2×C4×D53—22no—
71C2×C4×D53—22no—
72C2×C4×D53—82no—
73C22⋊#6157—44no—
74C22⋊#6157—44no—
75C4⋊#6157—84no—
76C4⋊C4×C53—41no—
77C22⋊C4×C53—42no—
78C42×C53—22no—
79C22⋊Dic53—82no—
80#11719921—42no—
81#11719921—82no—
82C4⋊Dic53—41no—
83C4×Dic53—21no—
84C4×Dic53—41no—
85C4×Dic53C4⋊C412yescomplemented
86C23×C53—22no—
87C23×C53—42no—
88C23×C53—42no—
89C23×C53—41no—
90C23×C53—41no—
91C23×C53C2×C22⋊C411yescharacteristic
92C22×D53—22no—
93C22×D53—22no—
94C22×D53—42no—
95C22×D53—42no—
96C22×D53#12112yes—
97C22×D53C22⋊Q812yes—
98C2×#6157—84no—
99C2×#6157—84no—
100C2×#6157—84no—
101D4×C53—42no—
102D4×C53—42no—
103C2×C4×C53—21no—
104C2×C4×C53—21no—
105C2×C4×C53—21no—
106C2×C4×C53—41no—
107C2×C4×C53—41no—
108C2×C4×C53—42no—
109C2×C4×C53—41no—
110C2×C4×C53—84no—
111C2×C4×C53#9912yes—
112C2×C4×C53C2×C4⋊C411yescharacteristic
113C2×C4×C53#11711yescharacteristic
114C22⋊D53—84no—
115C22⋊D53—84no—
116C2×Dic53—22no—
117C2×Dic53—22no—
118C2×Dic53—41no—
119C2×Dic53—41no—
120C2×Dic53—42no—
121C2×Dic53—41no—
122C2×Dic53—41no—
123C2×Dic53—41no—
124D212—41no—
125D212—41no—
126D212—42no—
127C4×D53—44no—
128C4×D53—82no—
129#12799—531nomaximal, Sylow
130C22×C53—21no—
131C22×C53—21no—
132C22×C53—44no—
133C22×C53—42no—
134C22×C53—82no—
135C22×C53—81no—
136C22×C53—81no—
137C22×C53#38011yescharacteristic
138C22×C53C2×#9911yescharacteristic
139C22×C53#41011yescharacteristic
140C2×D53—22no—
141C2×D53—22no—
142C2×D53—42no—
143C2×D53—42no—
144C2×D53—42no—
145C2×D53—42no—
146C2×D53—82no—
147#6157—164no—
148C4×C53—21no—
149C4×C53—21no—
150C4×C53—22no—
151C4×C53—41no—
152C4×C53—81no—
153Dic53—42no—
154Dic53—42no—
155Dic53—81no—
156#2328—531no—
157C2×#353—532no—
158#1799—532no—
159#1393—532no—
160C2×C53—21no—
161C2×C53—21no—
162C2×C53—82no—
163C2×C53#179911yescharacteristic
164C2×C53#212711yescharacteristic
165C2×C53#212811yescharacteristic
166D53—42no—
167D53—42no—
168C2×#127—531no—
169C2×C4×D4—531no—
170C2×C4×D4#615711yescharacteristic, complemented
171C2×#99—534no—
172#390—531no—
173#390—531no—
174#386—1062no—
175#386—1062no—
176#382—531no—
177#380—1062no—
178#376—531no—
179#353—1064no—
180#342—1062no—
181C53#1279911yescharacteristic, complemented, Sylow
182C22×D4—531no—
183C22×D4—1062no—
184C22×D4C2×#615711yescharacteristic
185C23×C4—21no—
186C23×C4—532no—
187#127—532no—
188#127—532no—
189#127—1062no—
190C4×D4—42no—
191C4×D4—2122no—
192C2×C4⋊C4—531no—
193C2×C4⋊C4—2122no—
194C2×C4⋊C4C2×#615711yescharacteristic
195C2×C22⋊C4—21no—
196C2×C22⋊C4—532no—
197C2×C22⋊C4—1062no—
198C2×C22⋊C4—1062no—
199C2×C22⋊C4—1064no—
200C2×C42—531no—
201C2×C42—531no—
202C2×C42C2×#615711yescharacteristic
203#99—1064no—
204#99—1064no—
205#95—1062no—
206#95—1061no—
207#95—1061no—
208#95—2122no—
209C24—21no—
210C24—532no—
211C2×D4—21no—
212C2×D4—41no—
213C2×D4—41no—
214C2×D4—531no—
215C2×D4—531no—
216C2×D4—532no—
217C2×D4—1064no—
218C2×D4—1064no—
219C2×D4—4242no—
220C2×D4—4242no—
221C2×D4—2121no—
222C2×D4—2121no—
223C2×D4C4⋊#615712yescomplemented
224C22×C4—21no—
225C22×C4—42no—
226C22×C4—82no—
227C22×C4—532no—
228C22×C4—1061no—
229C22×C4—1062no—
230C22×C4—1062no—
231C22×C4—1061no—
232C22×C4—4242no—
233C22×C4—2124no—
234C22×C4C22⋊#615711yescharacteristic
235C22×C4C22⋊#615711yescharacteristic
236C22×C4C22×#615711yescharacteristic
237C4⋊C4—41no—
238C4⋊C4—4244no—
239C4⋊C4—2121no—
240C22⋊C4—42no—
241C22⋊C4—4242no—
242C22⋊C4—4242no—
243C22⋊C4—2122no—
244C22⋊C4—2124no—
245C22⋊C4—2124no—
246C42—22no—
247C42—532no—
248C42—1061no—
249C42—2121no—
250C23—22no—
251C23—42no—
252C23—42no—
253C23—41no—
254C23—41no—
255C23—532no—
256C23—532no—
257C23—1062no—
258C23—1062no—
259C23—2122no—
260C23—2122no—
261C23C2×C22⋊#615711yescharacteristic
262D4—42no—
263D4—42no—
264D4—4244no—
265D4—4244no—
266D4—2121no—
267D4—2121no—
268D4—2122no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—42no—
275C2×C4—41no—
276C2×C4—84no—
277C2×C4—1062no—
278C2×C4—1062no—
279C2×C4—4242no—
280C2×C4—4244no—
281C2×C4—4244no—
282C2×C4—4244no—
283C2×C4—2124no—
284C2×C4—2121no—
285C2×C4—2121no—
286C2×C4—2121no—
287C2×C4—2121no—
288C2×C4—2121no—
289C2×C4—2122no—
290C2×C4#1313327512yes—
291C2×C4C2×C4⋊#615711yescharacteristic
292C2×C4#1313329611yescharacteristic
293C22—21no—
294C22—21no—
295C22—44no—
296C22—42no—
297C22—82no—
298C22—81no—
299C22—81no—
300C22—1062no—
301C22—1062no—
302C22—4242no—
303C22—2122no—
304C22—2122no—
305C22—2122no—
306C22—2122no—
307C22#1681156511yescharacteristic
308C22C2×#1313327511yescharacteristic
309C22#1681160211yescharacteristic
310C4—21no—
311C4—21no—
312C4—22no—
313C4—41no—
314C4—81no—
315C4—8484no—
316C4—4241no—
317C4—2122no—
318C4—2122no—
319C2—21no—
320C2—21no—
321C2—82no—
322C2—2122no—
323C2—2122no—
324C2#2029057511yescharacteristic
325C2#2029099011yescharacteristic
326C2#2029099411yescharacteristic
subgroupG/H
centerC22C2×#13133275
commutatorC22×C4×C53C22×C4
FrattiniC22×C4C22×#6157
FittingC2×C4×D4×C53C4
socleC22×C53C2×#99
prime radicalC2×#13133213C22
cosocleC22×D212C23
hypercenterC2×C4×D4#6157
hypocenterC53#12799
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C42 ◁ C2×C4×D4

Complements

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

NH ≅ G/Ntype
2. #20290576C2semidirect
3. #20289810C2semidirect
11. #16810526C4semidirect
15. #16810488C4semidirect
16. #16810469C4semidirect
17. C2×C4×D4×C53C4semidirect
51. C2×D4×C53C4⋊C4semidirect
85. C4×Dic53C4⋊C4semidirect
170. C2×C4×D4#6157semidirect
181. C53#12799semidirect
223. 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
#202905762
#202898102
#202774372
#1279953