Finite Group Databasebeta

← #28298518

Subgroups — #28298518

Order 13568.

422 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 422 conjugacy classes of subgroups — too many to draw. Only its 60 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#20290614C211yesmaximal, characteristic
2#20289965C211yesmaximal, characteristic
3#20272948C211yesmaximal, characteristic
4#16811593—21no—
5#16811593—21no—
6#16811503—21no—
7#16811503C412yescomplemented
8#16811469—21no—
9#16811469C412yescomplemented
10C22×#11719919C2211yescharacteristic
11C2×#13133124—21no—
12C2×#13133124—21no—
13C2×#13133124C411yescharacteristic
14C22×C42×C53C411yescharacteristic
15C2×#11719940—22no—
16C2×#11719940—41no—
17C22⋊#88324—22no—
18C22⋊#88324—41no—
19C22⋊#88324—41no—
20C22⋊#88324—41no—
21C23×Dic53—21no—
22C23×Dic53D411yescharacteristic
23C23×Dic53D411yescharacteristic
24C2×#11719919—41no—
25C2×#11719919—41no—
26C2×#11719919C2×C411yescharacteristic
27C2×#11719919C2×C411yescharacteristic
28C2×#11719919D411yescharacteristic
29C2×#11719919Q811yescharacteristic
30#13133124—24no—
31#13133124—42no—
32#13133124—41no—
33C23×C4×C53—21no—
34C23×C4×C53C2×C411yescharacteristic
35C2×C42×C53—42no—
36C2×C42×C53—41no—
37C24×C53C22⋊C411yescharacteristic
38C22×C4×C53—22no—
39C22×C4×C53—22no—
40C22×C4×C53—22no—
41C22×C4×C53—21no—
42C22×C4×C53—21no—
43C22×C4×C53—41no—
44C22×C4×C53—41no—
45C22×C4×C53—41no—
46C22×C4×C53—41no—
47C22×C4×C53—41no—
48C22×C4×C53—41no—
49C22×C4×C53C4211yescharacteristic
50C22×C4×C53C4⋊C411yescharacteristic
51C22×Dic53—21no—
52C22×Dic53—21no—
53C22×Dic53—21no—
54C22×Dic53—21no—
55C22×Dic53—22no—
56C22×Dic53—22no—
57C22×Dic53—41no—
58C22×Dic53—41no—
59C22×Dic53—41no—
60C22×Dic53—41no—
61C22×Dic53—41no—
62C22×Dic53—41no—
63C22×Dic53—41no—
64C22×Dic53—41no—
65C22×Dic53—41no—
66C22×Dic53C22⋊C411yescharacteristic
67C22×Dic53C22⋊C411yescharacteristic
68C22×Dic53C4⋊C411yescharacteristic
69C22×Dic53C4⋊C411yescharacteristic
70#11719940—42no—
71#11719940—42no—
72#11719940—81no—
73#11719940—81no—
74C2×#88324—42no—
75C2×#88324—42no—
76C42×C53—42no—
77C42×C53—42no—
78#11719919—41no—
79#11719919—41no—
80#11719919—41no—
81#11719919—41no—
82C23×C53—21no—
83C23×C53—21no—
84C23×C53—21no—
85C23×C53—21no—
86C23×C53—21no—
87C23×C53—21no—
88C23×C53#9511yescharacteristic
89C23×C53#9911yescharacteristic
90C23×C53#9911yescharacteristic
91C2×C4×C53—24no—
92C2×C4×C53—21no—
93C2×C4×C53—21no—
94C2×C4×C53—42no—
95C2×C4×C53—42no—
96C2×C4×C53—42no—
97C2×C4×C53—42no—
98C2×C4×C53—42no—
99C2×C4×C53—42no—
100C2×C4×C53—42no—
101C2×C4×C53—41no—
102C2×C4×C53—41no—
103C2×C4×C53—41no—
104C2×C4×C53—41no—
105C2×Dic53—21no—
106C2×Dic53—21no—
107C2×Dic53—21no—
108C2×Dic53—21no—
109C2×Dic53—21no—
110C2×Dic53—21no—
111C2×Dic53—21no—
112C2×Dic53—42no—
113C2×Dic53—42no—
114C2×Dic53—41no—
115C2×Dic53—41no—
116C2×Dic53—41no—
117C2×Dic53—41no—
118C2×Dic53—42no—
119C2×Dic53—42no—
120C2×Dic53—42no—
121C2×Dic53—81no—
122C2×Dic53—81no—
123C2×Dic53—81no—
124C2×Dic53—81no—
125C2×Dic53—81no—
126C2×Dic53#10412yes—
127#88324—82no—
128#88324—82no—
129#7215—531nomaximal, Sylow
130C22×C53—21no—
131C22×C53—21no—
132C22×C53—21no—
133C22×C53—21no—
134C22×C53—21no—
135C22×C53—21no—
136C22×C53—21no—
137C22×C53—21no—
138C22×C53—21no—
139C22×C53—21no—
140C22×C53—21no—
141C22×C53—21no—
142C22×C53—21no—
143C22×C53—21no—
144C22×C53H(3,ℤ/4)11yescharacteristic
145C22×C53#33911yescharacteristic
146C22×C53#34211yescharacteristic
147C22×C53AΣL(1,𝔽16)11yescharacteristic
148C22×C53#35211yescharacteristic
149C22×C53#35511yescharacteristic
150C22×C53#35611yescharacteristic
151C4×C53—44no—
152C4×C53—41no—
153C4×C53—41no—
154Dic53—22no—
155Dic53—41no—
156Dic53—41no—
157Dic53—41no—
158Dic53—82no—
159#1821—531no—
160#1463—531no—
161#1436—531no—
162C2×C53—21no—
163C2×C53—21no—
164C2×C53—21no—
165C2×C53—21no—
166C2×C53—21no—
167C2×C53—21no—
168C2×C53#128411yescharacteristic
169C2×C53#139011yescharacteristic
170C2×C53#139411yescharacteristic
171C22×C4⋊C4—531no—
172C22×C42#615711yescharacteristic
173#407—1061no—
174#407—1061no—
175C2×#95—531no—
176C2×#95—1061no—
177C2×#95—1061no—
178#343—532no—
179#343—1061no—
180#324—532no—
181#324—1061no—
182C53#721511yescharacteristic, complemented, Sylow
183C23×C4—21no—
184C23×C4—531no—
185C23×C4—531no—
186C23×C4—1061no—
187C23×C4C2×#615711yescharacteristic
188C2×M16—1062no—
189C2×M16—2121no—
190C2×C4⋊C4—531no—
191C2×C4⋊C4—531no—
192C2×C4⋊C4—531no—
193C2×C4⋊C4—531no—
194C2×C4⋊C4—2121no—
195C2×C4⋊C4—2121no—
196C2×C42—41no—
197C2×C42—42no—
198C22⋊C8—1062no—
199C22⋊C8—2121no—
200C22⋊C8—2121no—
201C22⋊C8—2121no—
202#95—1064no—
203#95—2121no—
204#95—2122no—
205C24C22⋊#615711yescharacteristic
206C22×C4—22no—
207C22×C4—22no—
208C22×C4—22no—
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—41no—
212C22×C4—41no—
213C22×C4—41no—
214C22×C4—41no—
215C22×C4—41no—
216C22×C4—41no—
217C22×C4—531no—
218C22×C4—531no—
219C22×C4—531no—
220C22×C4—531no—
221C22×C4—1062no—
222C22×C4—1061no—
223C22×C4—1061no—
224C22×C4—1061no—
225C22×C4—1061no—
226C22×C4—1062no—
227C22×C4—2121no—
228C22×C4—2121no—
229C22×C4—2121no—
230C22×C4—2121no—
231C22×C4—2121no—
232C22×C4—2121no—
233C22×C4—2121no—
234C22×C4—2121no—
235C22×C4—2121no—
236C22×C4C4×#615711yescharacteristic
237C22×C4C4⋊#615711yescharacteristic
238M16—4241no—
239M16—4241no—
240M16—2122no—
241M16—2122no—
242C2×C8—2122no—
243C2×C8—2122no—
244C4⋊C4—2121no—
245C4⋊C4—2121no—
246C4⋊C4—2121no—
247C4⋊C4—2121no—
248C42—42no—
249C42—42no—
250C23—21no—
251C23—21no—
252C23—21no—
253C23—21no—
254C23—21no—
255C23—21no—
256C23(C2×#6157)⋊C411yescharacteristic
257C23#1313327511yescharacteristic
258C23#1313327611yescharacteristic
259C2×C4—24no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—42no—
263C2×C4—42no—
264C2×C4—42no—
265C2×C4—42no—
266C2×C4—42no—
267C2×C4—42no—
268C2×C4—41no—
269C2×C4—42no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—532no—
274C2×C4—1061no—
275C2×C4—1061no—
276C2×C4—1061no—
277C2×C4—1061no—
278C2×C4—1061no—
279C2×C4—1061no—
280C2×C4—1061no—
281C2×C4—4241no—
282C2×C4—4241no—
283C2×C4—4241no—
284C2×C4—4241no—
285C2×C4—4241no—
286C2×C4—2122no—
287C2×C4—2122no—
288C2×C4—2122no—
289C2×C4—2122no—
290C2×C4—2122no—
291C2×C4—2121no—
292C2×C4—2121no—
293C2×C4—2121no—
294C2×C4—2121no—
295C8—4242no—
296C8—4242no—
297C22—21no—
298C22—21no—
299C22—21no—
300C22—21no—
301C22—21no—
302C22—21no—
303C22—21no—
304C22—21no—
305C22—21no—
306C22—21no—
307C22—21no—
308C22—21no—
309C22—21no—
310C22—21no—
311C22#1681149011yescharacteristic
312C22#1681149411yescharacteristic
313C22#1681150111yescharacteristic
314C22#1681152311yescharacteristic
315C22#1681152511yescharacteristic
316C22#1681153211yescharacteristic
317C22#1681153411yescharacteristic
318C4—44no—
319C4—41no—
320C4—41no—
321C4—1062no—
322C4—4242no—
323C4—2121no—
324C4—2121no—
325C4—2121no—
326C2—21no—
327C2—21no—
328C2—21no—
329C2—21no—
330C2—21no—
331C2—21no—
332C2#2028956311yescharacteristic
333C2#2028979911yescharacteristic
334C2#2028981211yescharacteristic
subgroupG/H
centerC22#16811501
commutatorC22×C4×C53C42
FrattiniC23×C4C2×#6157
FittingC22×C42×C53C4
socleC22×C53#342
prime radicalC24×C53C22⋊C4
cosocleC22×#11719919C22
hypercenterC22×C42#6157
hypocenterC53#7215
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C24 ◁ C23×C4 ◁ C22×C42

Complements

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

NH ≅ G/Ntype
7. #16811503C4semidirect
9. #16811469C4semidirect
182. C53#7215semidirect

Maximal subgroups

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

subgroupindex
#202906142
#202899652
#202729482
#721553