Finite Group Databasebeta

← #28311182

Subgroups — #28311182

Order 13568.

544 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 544 conjugacy classes of subgroups — too many to draw. Only its 90 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×#16811529C212yesmaximal
2#20290576C212yesmaximal
3#20289810C212yesmaximal
4C4×#13133206C211yesmaximal, characteristic
5C2×#13133276C2214yes—
6#16811565—22no—
7#16811529—24no—
8#16811501—22no—
9C2×#13133206C2211yescharacteristic
10C2×C4×D212C2211yescharacteristic
11#16810530C411yescharacteristic, complemented
12#16810524C2211yescharacteristic
13#16810513—22no—
14#16810488C411yescharacteristic, complemented
15C4×#11719921—22no—
16C42×Dic53C411yescharacteristic, complemented
17C2×C4×Q8×C53C411yescharacteristic, complemented
18C2×C4⋊#6157—42no—
19C2×C22⋊#6157—24no—
20#13133276—24no—
21#13133276—24no—
22(C2×#6157)⋊C4—42no—
23C22×D212C2311yescharacteristic
24C22×C4×D53D412yes—
25#13133206—22no—
26#13133206D412yes—
27#13133206Q812yes—
28C4×D212—41no—
29C4×D212—41no—
30C2×C4⋊Dic53C2×C411yescharacteristic
31C2×#11719921—22no—
32C2×#11719921—22no—
33C2×#11719921D412yes—
34C2×C4×Dic53—21no—
35C2×C4×Dic53—21no—
36C2×C4×Dic53C2×C411yescharacteristic
37C2×C4×Dic53C2×C411yescharacteristic
38#13133124—22no—
39#13133124—22no—
40C22×Q8×C53C2×C411yescharacteristic
41C4×Q8×C53—41no—
42C4×Q8×C53—41no—
43C2×C4⋊C4×C53—21no—
44C2×C4⋊C4×C53C2×C411yescharacteristic
45C2×C42×C53—21no—
46C2×C42×C53C2×C411yescharacteristic
47C23×D53C2×D412yes—
48C22×#6157—44no—
49C2×Q8×C53—21no—
50C2×Q8×C53—41no—
51C2×Q8×C53—41no—
52C2×Q8×C53C4⋊C412yescomplemented
53C22×C4×C53—21no—
54C22×C4×C53—21no—
55C22×C4×C53C22⋊C411yescharacteristic
56C22×C4×C53C22⋊C411yescharacteristic
57C22×C4×C53C22×C411yescharacteristic
58C22×Dic53—21no—
59C22×Dic53—21no—
60C22×Dic53C22⋊C412yes—
61C2×D212—41no—
62C2×D212—41no—
63C2×D212C2×D411yescharacteristic
64C2×D212C2×Q811yescharacteristic
65C2×D212#4112yes—
66C2×C4×D53—22no—
67C2×C4×D53—22no—
68C2×C4×D53—44no—
69C22⋊#6157—44no—
70C22⋊#6157—44no—
71C4⋊#6157—84no—
72C4⋊C4×C53—42no—
73C4⋊C4×C53—41no—
74C42×C53—21no—
75C42×C53—21no—
76C42×C53—41no—
77C42×C53—41no—
78#11719921—24no—
79#11719921—24no—
80#11719921—42no—
81#11719921—44no—
82C4⋊Dic53—41no—
83C4×Dic53—21no—
84C4×Dic53—42no—
85C4×Dic53—42no—
86C4×Dic53—41no—
87C4×Dic53C4⋊C412yescomplemented
88C23×C53C2×C22⋊C411yescharacteristic
89C22×D53—22no—
90C22×D53—22no—
91C22×D53—42no—
92C22×D53—42no—
93C22×D53#12112yes—
94C22×D53C22⋊Q812yes—
95C2×#6157—84no—
96C2×#6157—84no—
97C2×#6157—84no—
98Q8×C53—42no—
99Q8×C53—42no—
100C2×C4×C53—21no—
101C2×C4×C53—21no—
102C2×C4×C53—22no—
103C2×C4×C53—22no—
104C2×C4×C53—21no—
105C2×C4×C53—41no—
106C2×C4×C53—41no—
107C2×C4×C53—41no—
108C2×C4×C53—42no—
109C2×C4×C53—41no—
110C2×C4×C53#9911yescharacteristic
111C2×C4×C53#9911yescharacteristic
112C2×C4×C53C2×C4⋊C411yescharacteristic
113C2×C4×C53#11711yescharacteristic
114C2×Dic53—22no—
115C2×Dic53—22no—
116C2×Dic53—22no—
117C2×Dic53—41no—
118C2×Dic53—41no—
119C2×Dic53—41no—
120C2×Dic53—41no—
121C2×Dic53—42no—
122D212—41no—
123D212—41no—
124D212—42no—
125C4×D53—44no—
126C4×D53—82no—
127#12798—531nomaximal, Sylow
128C22×C53—21no—
129C22×C53—21no—
130C22×C53#38011yescharacteristic
131C22×C53C2×#9911yescharacteristic
132C22×C53#41011yescharacteristic
133C2×D53—22no—
134C2×D53—22no—
135C2×D53—42no—
136C2×D53—42no—
137C2×D53—42no—
138C2×D53—42no—
139C2×D53—82no—
140#6157—164no—
141C4×C53—21no—
142C4×C53—21no—
143C4×C53—21no—
144C4×C53—42no—
145C4×C53—41no—
146C4×C53—81no—
147C4×C53#35412yescomplemented
148Dic53—42no—
149Dic53—42no—
150Dic53—42no—
151C4×#124—531no—
152C2×#354—532no—
153#1799—532no—
154#1393—532no—
155C2×C53—21no—
156C2×C53—21no—
157C2×C53#179911yescharacteristic
158C2×C53C2×#35411yescharacteristic
159C2×C53#212611yescharacteristic
160D53—42no—
161D53—42no—
162C2×#124—531no—
163C2×C4×Q8#615711yescharacteristic, complemented
164C2×C4×D4—531no—
165C2×#99—534no—
166#391—531no—
167#390—531no—
168#388—1062no—
169#382—531no—
170#380—1062no—
171C4×C22⋊C4—1062no—
172C43—531no—
173#354—1064no—
174#342—1062no—
175C53#1279811yescharacteristic, complemented, Sylow
176C22×Q8C2×#615711yescharacteristic
177C22×D4—531no—
178C23×C4—532no—
179#124—532no—
180#124—532no—
181#124—1062no—
182C4×Q8—41no—
183C4×Q8—41no—
184C4×D4—2121no—
185C4×D4—2121no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—531no—
188C2×C4⋊C4—2122no—
189C2×C4⋊C4C2×#615711yescharacteristic
190C2×C22⋊C4—532no—
191C2×C22⋊C4—1062no—
192C2×C22⋊C4—1062no—
193C2×C22⋊C4—1064no—
194C2×C42—21no—
195C2×C42—531no—
196C2×C42—531no—
197C2×C42—1061no—
198C2×C42—1061no—
199C2×C42C2×#615711yescharacteristic
200#99—1064no—
201#99—1064no—
202#95—1062no—
203#95—1062no—
204#95—2122no—
205C24—532no—
206C2×Q8—21no—
207C2×Q8—41no—
208C2×Q8—41no—
209C2×Q8C4⋊#615712yescomplemented
210C2×D4—531no—
211C2×D4—531no—
212C2×D4—532no—
213C2×D4—2121no—
214C2×D4—2121no—
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—532no—
218C22×C4—1061no—
219C22×C4—1061no—
220C22×C4—1062no—
221C22×C4—1062no—
222C22×C4—2124no—
223C22×C4—2124no—
224C22×C4C22⋊#615711yescharacteristic
225C22×C4C22⋊#615711yescharacteristic
226C22×C4C22×#615711yescharacteristic
227C4⋊C4—41no—
228C4⋊C4—42no—
229C4⋊C4—4244no—
230C4⋊C4—2121no—
231C22⋊C4—1064no—
232C22⋊C4—1064no—
233C22⋊C4—2122no—
234C22⋊C4—2124no—
235C22⋊C4—2124no—
236C22⋊C4—2124no—
237C42—21no—
238C42—21no—
239C42—41no—
240C42—41no—
241C42—532no—
242C42—1061no—
243C42—2121no—
244C42—2122no—
245C42—2122no—
246C23—532no—
247C23—532no—
248C23—1062no—
249C23—1062no—
250C23—2122no—
251C23—2122no—
252C23C2×C22⋊#615711yescharacteristic
253Q8—42no—
254Q8—42no—
255D4—2121no—
256D4—2121no—
257D4—2122no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—22no—
261C2×C4—22no—
262C2×C4—21no—
263C2×C4—41no—
264C2×C4—41no—
265C2×C4—41no—
266C2×C4—42no—
267C2×C4—41no—
268C2×C4—1062no—
269C2×C4—1062no—
270C2×C4—1062no—
271C2×C4—4242no—
272C2×C4—4244no—
273C2×C4—4244no—
274C2×C4—4244no—
275C2×C4—2121no—
276C2×C4—2121no—
277C2×C4—2121no—
278C2×C4—2121no—
279C2×C4—2124no—
280C2×C4—2122no—
281C2×C4#1313327511yescharacteristic
282C2×C4#1313327511yescharacteristic
283C2×C4C2×C4⋊#615711yescharacteristic
284C2×C4#1313329611yescharacteristic
285C22—21no—
286C22—21no—
287C22—1062no—
288C22—1062no—
289C22—4242no—
290C22—2122no—
291C22—2122no—
292C22—2122no—
293C22—2122no—
294C22#1681156511yescharacteristic
295C22C2×#1313327511yescharacteristic
296C22#1681160211yescharacteristic
297C4—21no—
298C4—21no—
299C4—21no—
300C4—42no—
301C4—41no—
302C4—81no—
303C4—8484no—
304C4—2122no—
305C4—2122no—
306C4—2122no—
307C4#1681152912yescomplemented
308C2—21no—
309C2—21no—
310C2—2122no—
311C2—2122no—
312C2#2029057511yescharacteristic
313C2C2×#1681152911yescharacteristic
314C2#2029098611yescharacteristic
subgroupG/H
centerC22C2×#13133275
commutatorC22×C4×C53C22×C4
FrattiniC22×C4C22×#6157
FittingC2×C4×Q8×C53C4
socleC22×C53C2×#99
prime radical, cosocleC22×D212C23
hypercenterC2×C4×Q8#6157
hypocenterC53#12798
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C42 ◁ C2×C4×Q8

Complements

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

NH ≅ G/Ntype
11. #16810530C4semidirect
14. #16810488C4semidirect
16. C42×Dic53C4semidirect
17. C2×C4×Q8×C53C4semidirect
52. C2×Q8×C53C4⋊C4semidirect
87. C4×Dic53C4⋊C4semidirect
147. C4×C53#354semidirect
163. C2×C4×Q8#6157semidirect
175. C53#12798semidirect
209. C2×Q8C4⋊#6157semidirect
307. C4#16811529semidirect

Maximal subgroups

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

subgroupindex
C2×#168115292
#202905762
#202898102
C4×#131332062
#1279853