Finite Group Databasebeta

← #28308422

Subgroups — #28308422

Order 13568.

358 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 358 conjugacy classes of subgroups — too many to draw. Only its 72 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#20289978C211yesmaximal, characteristic
2#20289958C211yesmaximal, characteristic
3#20289656C212yesmaximal
4#20289655C212yesmaximal
5#20279932C211yesmaximal, characteristic
6#16811643—21no—
7#16811639—21no—
8#16811468—21no—
9#16811468—21no—
10#16811468C2212yes—
11#16811468C2212yes—
12C2×#13133216C411yescharacteristic
13C2×Q8×Dic53C2211yescharacteristic
14#16810565—21no—
15#16810565—21no—
16#16810530C411yescharacteristic
17#16810527—21no—
18#16810527—21no—
19#16810489C2211yescharacteristic
20#16810488C2211yescharacteristic
21C4×#11719919C411yescharacteristic
22C53×#391C411yescharacteristic
23#13133284—41no—
24C4⋊#88324—41no—
25C22⋊#88324—22no—
26C22⋊#88324—22no—
27#13133272—41no—
28C4×#88324—41no—
29C22×Dic106—21no—
30#13133216—41no—
31#13133216—41no—
32Q8×Dic53—41no—
33Q8×Dic53—41no—
34C2×C4⋊Dic53C2×C411yescharacteristic
35C2×C4⋊Dic53D411yescharacteristic
36C2×C4⋊Dic53D411yescharacteristic
37C2×C4⋊Dic53C2311yescharacteristic
38C2×#11719919—21no—
39C2×#11719919—21no—
40C2×C4×Dic53C2×C411yescharacteristic
41C2×C4×Dic53C2×C411yescharacteristic
42C2×C4×Dic53D411yescharacteristic
43C2×C4×Dic53D411yescharacteristic
44#13133124—21no—
45#13133124—21no—
46#13133124—21no—
47#13133124—21no—
48#13133124—21no—
49#13133124—21no—
50#13133124—21no—
51C22×Q8×C53C2×C411yescharacteristic
52C2×C4⋊C4×C53C2×C411yescharacteristic
53C2×C42×C53C2×C411yescharacteristic
54#95×C53—21no—
55#95×C53—21no—
56C2×Q8×C53—22no—
57C2×Q8×C53—22no—
58C2×Q8×C53—41no—
59C22×C4×C53—21no—
60C22×C4×C53—21no—
61C22×C4×C53C22⋊C411yescharacteristic
62C22×C4×C53C22⋊C411yescharacteristic
63C22×C4×C53C22×C411yescharacteristic
64C22×Dic53—21no—
65C22×Dic53—21no—
66C22×Dic53C22⋊C411yescharacteristic
67C22×Dic53C22⋊C411yescharacteristic
68C22×Dic53C2×D411yescharacteristic
69C22×Dic53C2×D411yescharacteristic
70C2×Dic106—44no—
71C2×Dic106—41no—
72C2×Dic106—41no—
73C2×#88324—42no—
74C2×#88324—42no—
75C4⋊C4×C53—41no—
76C42×C53—22no—
77C4⋊Dic53—41no—
78C4⋊Dic53—41no—
79C4⋊Dic53—41no—
80C4⋊Dic53—41no—
81#11719919—42no—
82#11719919—42no—
83C4×Dic53—21no—
84C4×Dic53—21no—
85C4×Dic53—21no—
86C4×Dic53—21no—
87C4×Dic53—21no—
88C4×Dic53—21no—
89C4×Dic53—22no—
90C23×C53C2×C22⋊C411yescharacteristic
91Q8×C53—44no—
92C2×C4×C53—21no—
93C2×C4×C53—21no—
94C2×C4×C53—21no—
95C2×C4×C53—21no—
96C2×C4×C53—21no—
97C2×C4×C53—21no—
98C2×C4×C53—21no—
99C2×C4×C53—42no—
100C2×C4×C53—41no—
101C2×C4×C53—41no—
102C2×C4×C53—41no—
103C2×C4×C53—41no—
104C2×C4×C53#10112yes—
105C2×C4×C53#13112yes—
106C2×Dic53—22no—
107C2×Dic53—21no—
108C2×Dic53—21no—
109C2×Dic53—21no—
110C2×Dic53—41no—
111C2×Dic53—41no—
112C2×Dic53—41no—
113C2×Dic53—42no—
114C2×Dic53—41no—
115C2×Dic53—41no—
116C2×Dic53—41no—
117C2×Dic53—41no—
118C2×Dic53#9911yescharacteristic
119C2×Dic53#9911yescharacteristic
120Dic106—82no—
121Dic106—82no—
122#88324—82no—
123#88324—82no—
124#11573—531nomaximal, Sylow
125C22×C53—21no—
126C22×C53—21no—
127C22×C53#40811yescharacteristic
128C22×C53C2×#9911yescharacteristic
129C22×C53C2×#10111yescharacteristic
130C4×C53—22no—
131C4×C53—22no—
132C4×C53—41no—
133C4×C53—81no—
134C4×C53—81no—
135Dic53—21no—
136Dic53—41no—
137Dic53—41no—
138Dic53—41no—
139Dic53—81no—
140Dic53—81no—
141Dic53#35512yes—
142#2568—531no—
143#1467—531no—
144#1461—531no—
145#1324—532no—
146#1324—532no—
147C2×C53—21no—
148C2×C53—21no—
149C2×C53#147211yescharacteristic
150C2×C53#212911yescharacteristic
151C2×C53C2×#35511yescharacteristic
152C2×C4⋊Q8—531no—
153C2×C4×Q8—531no—
154#432—1061no—
155#431—1061no—
156#400—1061no—
157#400—1061no—
158#391—531no—
159#391—1061no—
160#391—1061no—
161#391#615711yescharacteristic
162#382—531no—
163#382—531no—
164C4×C4⋊C4—531no—
165#324—532no—
166#324—532no—
167#324—1061no—
168#324—1061no—
169C53#1157311yescharacteristic, complemented, Sylow
170C22×Q8—1061no—
171C22×Q8C2×#615711yescharacteristic
172C4⋊Q8—2121no—
173C4⋊Q8—2121no—
174C4×Q8—2121no—
175C4×Q8—2121no—
176C2×C4⋊C4—531no—
177C2×C4⋊C4—531no—
178C2×C4⋊C4—531no—
179C2×C4⋊C4—531no—
180C2×C4⋊C4—1061no—
181C2×C4⋊C4—1061no—
182C2×C4⋊C4C2×#615711yescharacteristic
183C2×C42—531no—
184C2×C42—531no—
185C2×C42—531no—
186C2×C42—531no—
187C2×C42C2×#615711yescharacteristic
188C4⋊C8—2121no—
189C4⋊C8—2121no—
190C22⋊C8—1062no—
191C22⋊C8—1062no—
192#97—2121no—
193C4×C8—2121no—
194#95—21no—
195#95—21no—
196#95—1061no—
197#95—1061no—
198#95—1061no—
199#95—1061no—
200#95—1061no—
201#95—1061no—
202#95—1061no—
203C2×Q8—22no—
204C2×Q8—22no—
205C2×Q8—41no—
206C2×Q8—2121no—
207C2×Q8—2121no—
208C2×Q8—2124no—
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—531no—
212C22×C4—531no—
213C22×C4—531no—
214C22×C4—531no—
215C22×C4—1061no—
216C22×C4—1061no—
217C22×C4C22⋊#615711yescharacteristic
218C22×C4C22⋊#615711yescharacteristic
219C22×C4C22×#615711yescharacteristic
220C2×C8—2122no—
221C2×C8—2122no—
222C4⋊C4—41no—
223C4⋊C4—2121no—
224C4⋊C4—2121no—
225C4⋊C4—2121no—
226C4⋊C4—2121no—
227C4⋊C4—2122no—
228C4⋊C4—2122no—
229C42—22no—
230C42—1061no—
231C42—1061no—
232C42—1061no—
233C42—1061no—
234C42—1061no—
235C42—1061no—
236C42—1062no—
237C23C2×C22⋊#615711yescharacteristic
238Q8—44no—
239Q8—4242no—
240Q8—4242no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—41no—
249C2×C4—41no—
250C2×C4—42no—
251C2×C4—41no—
252C2×C4—41no—
253C2×C4—531no—
254C2×C4—531no—
255C2×C4—1062no—
256C2×C4—1061no—
257C2×C4—1061no—
258C2×C4—1061no—
259C2×C4—2121no—
260C2×C4—2121no—
261C2×C4—2122no—
262C2×C4—2121no—
263C2×C4—2121no—
264C2×C4—2121no—
265C2×C4—2121no—
266C2×C4—2121no—
267C2×C4#1313327812yes—
268C2×C4#1313330412yes—
269C8—4242no—
270C8—4242no—
271C22—21no—
272C22—21no—
273C22#1681159611yescharacteristic
274C22D53⋊#9911yescharacteristic
275C22C2×#1313327811yescharacteristic
276C4—22no—
277C4—22no—
278C4—41no—
279C4—81no—
280C4—81no—
281C4—532no—
282C4—1061no—
283C4—4241no—
284C4—4241no—
285C4—2121no—
286C4—2121no—
287C4—2121no—
288C2—21no—
289C2—21no—
290C2#2028999411yescharacteristic
291C2#2029099811yescharacteristic
292C2D53⋊#35511yescharacteristic
subgroupG/H
centerC22D53⋊#99
commutatorC22×C4×C53C22×C4
FrattiniC22×C4C22×#6157
FittingC53×#391C4
socleC22×C53C2×#99
prime radicalC23×C53C2×C22⋊C4
cosocleC2×C4⋊Dic53C23
hypercenter#391#6157
hypocenterC53#11573
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23 ◁ C22×Q8 ◁ #391

Complements

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

NH ≅ G/Ntype
169. C53#11573semidirect

Maximal subgroups

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

subgroupindex
#202899782
#202899582
#202896562
#202896552
#202799322
#1157353