Finite Group Databasebeta

← #28311193

Subgroups — #28311193

Order 13568.

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

normal subgroups only

This group has 524 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
1C2×#16811530C212yesmaximal
2#20290578C212yesmaximal
3#20289810C212yesmaximal
4#20277423C211yesmaximal, characteristic
5C2×#13133276C2214yes—
6#16811565—22no—
7#16811530—24no—
8#16811501—22no—
9C2×#13133205C2211yescharacteristic
10C2×#13133182C411yescharacteristic, complemented
11C2×#13133179C2211yescharacteristic
12#16810528C411yescharacteristic, complemented
13#16810524C2211yescharacteristic
14#16810513—22no—
15#16810494—22no—
16#16810469C411yescharacteristic, complemented
17C53×#384C411yescharacteristic, complemented
18C2×C4⋊#6157—42no—
19C2×C22⋊#6157—24no—
20#13133276—24no—
21#13133276—24no—
22(C2×#6157)⋊C4—42no—
23C22×Dic106C2×C411yescharacteristic
24C22×D212C2311yescharacteristic
25C22×C4×D53D412yes—
26#13133205—22no—
27#13133205D412yes—
28#13133205Q812yes—
29#13133182—42no—
30#13133179—42no—
31C2×#11719919—21no—
32C2×#11719921—22no—
33C2×#11719921—22no—
34C2×#11719921D412yes—
35C2×C4×Dic53—21no—
36C2×C4×Dic53C2×C411yescharacteristic
37C2×C4×Dic53C2×C411yescharacteristic
38#13133124—22no—
39#13133124—21no—
40#13133124—21no—
41C2×C4⋊C4×C53—22no—
42C2×C4⋊C4×C53C2×C411yescharacteristic
43C2×C4⋊C4×C53C2×C411yescharacteristic
44C2×C42×C53C2×C411yescharacteristic
45C23×D53C2×D412yes—
46C22×#6157—44no—
47C22×C4×C53—21no—
48C22×C4×C53—21no—
49C22×C4×C53C22⋊C412yes—
50C22×C4×C53C22×C411yescharacteristic
51C22×Dic53—21no—
52C22×Dic53—21no—
53C22×Dic53C22⋊C411yescharacteristic
54C22×Dic53C22⋊C411yescharacteristic
55C2×D212—41no—
56C2×D212—41no—
57C2×D212C2×D411yescharacteristic
58C2×D212C2×Q811yescharacteristic
59C2×D212#4112yes—
60C2×C4×D53—22no—
61C2×C4×D53—22no—
62C2×C4×D53—82no—
63C2×Dic106—21no—
64C2×Dic106—41no—
65C2×Dic106—41no—
66C2×Dic106C4⋊C412yescomplemented
67C22⋊#6157—44no—
68C22⋊#6157—44no—
69C4⋊#6157—84no—
70C4⋊C4×C53—41no—
71C4⋊C4×C53—41no—
72C4⋊C4×C53—82no—
73C42×C53—21no—
74C42×C53C4⋊C412yescomplemented
75#11719921—24no—
76#11719921—24no—
77#11719921—42no—
78#11719921—82no—
79#11719919—42no—
80C4×Dic53—22no—
81C4×Dic53—42no—
82C4×Dic53—41no—
83C23×C53C2×C22⋊C411yescharacteristic
84C22×D53—22no—
85C22×D53—22no—
86C22×D53—42no—
87C22×D53—42no—
88C22×D53#12112yes—
89C22×D53C22⋊Q812yes—
90C2×#6157—84no—
91C2×#6157—84no—
92C2×#6157—84no—
93C2×C4×C53—21no—
94C2×C4×C53—21no—
95C2×C4×C53—22no—
96C2×C4×C53—22no—
97C2×C4×C53—41no—
98C2×C4×C53—41no—
99C2×C4×C53—42no—
100C2×C4×C53—42no—
101C2×C4×C53—41no—
102C2×C4×C53C2×C4⋊C411yescharacteristic
103C2×C4×C53#11711yescharacteristic
104C2×Dic53—22no—
105C2×Dic53—21no—
106C2×Dic53—41no—
107C2×Dic53—41no—
108C2×Dic53—41no—
109C2×Dic53—41no—
110C2×Dic53—41no—
111C2×Dic53—41no—
112C2×Dic53—41no—
113C2×Dic53#9912yes—
114D212—41no—
115D212—41no—
116D212—42no—
117C4×D53—44no—
118C4×D53—82no—
119Dic106—42no—
120Dic106—42no—
121#12800—531nomaximal, Sylow
122C22×C53—21no—
123C22×C53—21no—
124C22×C53#38011yescharacteristic
125C22×C53C2×#9911yescharacteristic
126C22×C53#41011yescharacteristic
127C2×D53—22no—
128C2×D53—22no—
129C2×D53—42no—
130C2×D53—42no—
131C2×D53—42no—
132C2×D53—42no—
133C2×D53—82no—
134#6157—164no—
135C4×C53—21no—
136C4×C53—21no—
137C4×C53—42no—
138C4×C53—42no—
139C4×C53—81no—
140Dic53—22no—
141Dic53—42no—
142Dic53—41no—
143Dic53—81no—
144#2326—531no—
145C2×#354—532no—
146#1799—532no—
147#1393—532no—
148C2×C53—21no—
149C2×C53—21no—
150C2×C53#179911yescharacteristic
151C2×C53#212711yescharacteristic
152C2×C53#212911yescharacteristic
153D53—42no—
154D53—42no—
155C2×#124—531no—
156C2×C4×Q8—531no—
157C2×C4×D4—531no—
158C2×#99—534no—
159#391—531no—
160#390—531no—
161#388—1062no—
162#385—1062no—
163#384#615711yescharacteristic, complemented
164#380—1062no—
165#376—531no—
166#354—1064no—
167#342—1062no—
168C53#1280011yescharacteristic, complemented, Sylow
169C22×Q8—531no—
170C22×D4—531no—
171C23×C4—532no—
172#124—532no—
173#124—532no—
174#124—1062no—
175C4×Q8—2122no—
176C4×D4—2122no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—1061no—
179C2×C4⋊C4—2122no—
180C2×C4⋊C4C2×#615711yescharacteristic
181C2×C4⋊C4C2×#615711yescharacteristic
182C2×C22⋊C4—532no—
183C2×C22⋊C4—1062no—
184C2×C22⋊C4—1062no—
185C2×C22⋊C4—1064no—
186C2×C42—531no—
187C2×C42—531no—
188C2×C42—1061no—
189C2×C42C2×#615711yescharacteristic
190#99—1064no—
191#99—1064no—
192#95—1061no—
193#95—1061no—
194#95—1062no—
195#95—2122no—
196C24—532no—
197C2×Q8—532no—
198C2×Q8—1061no—
199C2×Q8—2121no—
200C2×Q8—2121no—
201C2×D4—531no—
202C2×D4—531no—
203C2×D4—532no—
204C2×D4—2121no—
205C2×D4—2121no—
206C22×C4—21no—
207C22×C4—21no—
208C22×C4—531no—
209C22×C4—531no—
210C22×C4—1062no—
211C22×C4—1062no—
212C22×C4—1061no—
213C22×C4—1061no—
214C22×C4—4242no—
215C22×C4—2124no—
216C22×C4C22⋊#615712yes—
217C22×C4C22×#615711yescharacteristic
218C4⋊C4—41no—
219C4⋊C4—41no—
220C4⋊C4—82no—
221C4⋊C4—4244no—
222C4⋊C4—2122no—
223C22⋊C4—1064no—
224C22⋊C4—1064no—
225C22⋊C4—4242no—
226C22⋊C4—2122no—
227C22⋊C4—2124no—
228C22⋊C4—2124no—
229C42—21no—
230C42—1062no—
231C42—2121no—
232C42—2122no—
233C42C4⋊#615712yescomplemented
234C23—532no—
235C23—532no—
236C23—1062no—
237C23—1062no—
238C23—2122no—
239C23—2122no—
240C23C2×C22⋊#615711yescharacteristic
241Q8—2122no—
242Q8—2122no—
243D4—2121no—
244D4—2121no—
245D4—2122no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—41no—
251C2×C4—41no—
252C2×C4—42no—
253C2×C4—42no—
254C2×C4—41no—
255C2×C4—532no—
256C2×C4—1062no—
257C2×C4—1061no—
258C2×C4—4242no—
259C2×C4—4244no—
260C2×C4—4244no—
261C2×C4—4244no—
262C2×C4—2124no—
263C2×C4—2121no—
264C2×C4—2121no—
265C2×C4—2121no—
266C2×C4—2121no—
267C2×C4—2121no—
268C2×C4—2121no—
269C2×C4—2121no—
270C2×C4C2×C4⋊#615711yescharacteristic
271C2×C4#1313329611yescharacteristic
272C22—21no—
273C22—21no—
274C22—1062no—
275C22—1062no—
276C22—4242no—
277C22—2122no—
278C22—2122no—
279C22—2122no—
280C22—2122no—
281C22#1681156511yescharacteristic
282C22D53⋊#9911yescharacteristic
283C22#1681160411yescharacteristic
284C4—21no—
285C4—21no—
286C4—42no—
287C4—42no—
288C4—81no—
289C4—8484no—
290C4—1062no—
291C4—4241no—
292C4—2122no—
293C4—2121no—
294C2—21no—
295C2—21no—
296C2—2122no—
297C2—2122no—
298C2#2029057811yescharacteristic
299C2#2029099111yescharacteristic
300C2#2029099811yescharacteristic
subgroupG/H
centerC22D53⋊#99
commutatorC22×C4×C53C22×C4
FrattiniC22×C4C22×#6157
FittingC53×#384C4
socleC22×C53C2×#99
prime radical, cosocleC22×D212C23
hypercenter#384#6157
hypocenterC53#12800
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #384

Complements

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

NH ≅ G/Ntype
10. C2×#13133182C4semidirect
12. #16810528C4semidirect
16. #16810469C4semidirect
17. C53×#384C4semidirect
66. C2×Dic106C4⋊C4semidirect
74. C42×C53C4⋊C4semidirect
163. #384#6157semidirect
168. C53#12800semidirect
233. C42C4⋊#6157semidirect

Maximal subgroups

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

subgroupindex
C2×#168115302
#202905782
#202898102
#202774232
#1280053