Finite Group Databasebeta

← #27293786

Subgroups — #27293786

Order 13568.

552 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 552 conjugacy classes of subgroups — too many to draw. Only its 87 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#20278080C211yesmaximal, characteristic
2#20273053C211yesmaximal, characteristic
3#20272664C214yesmaximal, complemented
4C53×#1460C211yesmaximal, characteristic
5C23×Dic106C411yescharacteristic
6C2×#13133196—21no—
7#16810587—21no—
8#16810543—21no—
9#16810535—21no—
10#16810529—21no—
11#16810503C412yes—
12#16810495C2212yes—
13#16810483C411yescharacteristic
14#16810349—21no—
15#16810347C2212yes—
16C2×C4×D4×C53C2211yescharacteristic
17C53×#431—21no—
18C53×#323—21no—
19C53×#323C2212yes—
20C22×Dic106—21no—
21C22×Dic106—21no—
22C22×Dic106—44no—
23C22×Dic106—41no—
24C22×Dic106—41no—
25C23×Dic53C2×C412yes—
26C2×#11719917—41no—
27#13133196—44no—
28C2×C4⋊Dic53—21no—
29C2×#11719919—22no—
30C2×C22⋊Dic53—22no—
31C2×C22⋊Dic53C2×C412yes—
32C22⋊#88322—22no—
33C22⋊#88322—41no—
34#13133124—22no—
35#13133124—21no—
36#13133124—22no—
37C22×D4×C53C2×C411yescharacteristic
38C23×C4×C53C2×C411yescharacteristic
39C23×C4×C53D411yescharacteristic
40C4×D4×C53—41no—
41C4×D4×C53—41no—
42C2×C4⋊C4×C53D411yescharacteristic
43C2×C22⋊C4×C53D411yescharacteristic
44C2×C22⋊C4×C53C2311yescharacteristic
45C2×C42×C53D411yescharacteristic
46C4⋊C8×C53—41no—
47C22⋊C8×C53—22no—
48C4×C8×C53—41no—
49C24×C53C22⋊C411yescharacteristic
50C24×C53C22×C411yescharacteristic
51C2×D4×C53—22no—
52C2×D4×C53—22no—
53C2×D4×C53—41no—
54C22×C4×C53—21no—
55C22×C4×C53—21no—
56C22×C4×C53—21no—
57C22×C4×C53—21no—
58C22×C4×C53—42no—
59C22×C4×C53—42no—
60C22×C4×C53C22⋊C411yescharacteristic
61C22×C4×C53C22⋊C411yescharacteristic
62C22×C4×C53C22⋊C411yescharacteristic
63C22×C4×C53C2×D411yescharacteristic
64C22×C4×C53C2×D411yescharacteristic
65C22×Dic53—22no—
66C22×Dic53—22no—
67C22×Dic53—44no—
68C22×Dic53—42no—
69C2×Dic106—24no—
70C2×Dic106—44no—
71C2×Dic106—42no—
72C2×Dic106—42no—
73C2×Dic106—48no—
74C2×Dic106—44no—
75C2×Dic106—41no—
76C2×Dic106—41no—
77C2×Dic106—44no—
78C2×C8×C53—42no—
79C4⋊C4×C53—41no—
80C22⋊C4×C53—41no—
81C22⋊C4×C53—41no—
82C42×C53—21no—
83C42×C53—21no—
84C22⋊Dic53—24no—
85C22⋊Dic53—44no—
86C4⋊Dic53—42no—
87#11719919—44no—
88#11719917—82no—
89C2×#88322—42no—
90C23×C53—21no—
91C23×C53—21no—
92C23×C53—21no—
93C23×C53—21no—
94C23×C53—21no—
95C23×C53—41no—
96C23×C53—41no—
97C23×C53—41no—
98C23×C53—41no—
99C23×C53C2×C22⋊C411yescharacteristic
100C23×C53#13112yes—
101D4×C53—44no—
102C2×C4×C53—21no—
103C2×C4×C53—21no—
104C2×C4×C53—22no—
105C2×C4×C53—21no—
106C2×C4×C53—21no—
107C2×C4×C53—21no—
108C2×C4×C53—42no—
109C2×C4×C53—42no—
110C2×C4×C53—42no—
111C2×C4×C53—41no—
112C2×C4×C53—41no—
113C2×C4×C53—41no—
114C2×C4×C53—81no—
115C2×C4×C53#9911yescharacteristic
116C2×C4×C53#9911yescharacteristic
117C2×C4×C53#10112yes—
118C2×Dic53—24no—
119C2×Dic53—42no—
120C2×Dic53—44no—
121C2×Dic53—42no—
122C2×Dic53—44no—
123C2×Dic53—42no—
124C2×Dic53—42no—
125C2×Dic53—42no—
126Dic106—44no—
127Dic106—48no—
128Dic106—44no—
129C8×C53—82no—
130#88322—82no—
131#11580—531nomaximal, Sylow
132C22×C53—21no—
133C22×C53—21no—
134C22×C53—22no—
135C22×C53—42no—
136C22×C53—42no—
137C22×C53—42no—
138C22×C53—41no—
139C22×C53—41no—
140C22×C53—41no—
141C22×C53—41no—
142C22×C53—41no—
143C22×C53—81no—
144C22×C53#40811yescharacteristic
145C22×C53C2×#9911yescharacteristic
146C22×C53C2×#10111yescharacteristic
147C4×C53—21no—
148C4×C53—22no—
149C4×C53—41no—
150C4×C53—41no—
151C4×C53—41no—
152C4×C53AΣL(1,𝔽16)12yes—
153Dic53—44no—
154Dic53—82no—
155#2409—531no—
156#1465—531no—
157#1460D5311yescharacteristic
158#1326—534no—
159C2×C53—21no—
160C2×C53—21no—
161C2×C53—42no—
162C2×C53—81no—
163C2×C53#147211yescharacteristic
164C2×C53C2×AΣL(1,𝔽16)11yescharacteristic
165C2×C53#212311yescharacteristic
166C23×Q8—531no—
167C2×C22⋊Q8—1061no—
168C2×C4×D4C2×D5311yescharacteristic
169#431—21no—
170#407—1061no—
171#395—1061no—
172#393—1061no—
173#391—1061no—
174#386—532no—
175#385—532no—
176#380—531no—
177#324—1061no—
178#323—21no—
179#323—532no—
180#323C2×D5312yes—
181C53#1158011yescharacteristic, complemented, Sylow
182C22×Q8—1061no—
183C22×Q8—1061no—
184C22×Q8—2121no—
185C22×Q8—2121no—
186C22×Q8—2124no—
187C22×D4C4×D5311yescharacteristic
188C23×C4—532no—
189C23×C4C4×D5311yescharacteristic
190C23×C4D21211yescharacteristic
191C2×M16—2121no—
192C22⋊Q8—2124no—
193C4×D4—41no—
194C4×D4—41no—
195C2×C4⋊C4—1062no—
196C2×C4⋊C4—1061no—
197C2×C4⋊C4D21211yescharacteristic
198C2×C22⋊C4—532no—
199C2×C22⋊C4—1062no—
200C2×C22⋊C4C22⋊D5311yescharacteristic
201C2×C22⋊C4C22×D5311yescharacteristic
202C2×C42C22⋊D5311yescharacteristic
203C4⋊C8—41no—
204C22⋊C8—22no—
205C22⋊C8—1062no—
206C22⋊C8—2121no—
207C4×C8—41no—
208#95—1062no—
209#95—1061no—
210#95—1062no—
211C24#1171992111yescharacteristic
212C24C2×C4×D5311yescharacteristic
213C2×Q8—1064no—
214C2×Q8—2122no—
215C2×Q8—2122no—
216C2×Q8—2124no—
217C2×Q8—2124no—
218C2×Q8—2128no—
219C2×Q8—2124no—
220C2×Q8—2121no—
221C2×Q8—2121no—
222C2×D4—22no—
223C2×D4—22no—
224C2×D4—41no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—42no—
230C22×C4—42no—
231C22×C4—1062no—
232C22×C4—1062no—
233C22×C4—2124no—
234C22×C4—2122no—
235C22×C4#1171992111yescharacteristic
236C22×C4#1171992111yescharacteristic
237C22×C4#1171992111yescharacteristic
238C22×C4C2×D21211yescharacteristic
239C22×C4C2×C22⋊D5311yescharacteristic
240M16—4242no—
241C2×C8—42no—
242C2×C8—2122no—
243C4⋊C4—41no—
244C4⋊C4—2122no—
245C4⋊C4—2124no—
246C22⋊C4—41no—
247C22⋊C4—41no—
248C22⋊C4—1064no—
249C22⋊C4—2124no—
250C42—21no—
251C42—21no—
252C23—21no—
253C23—21no—
254C23—21no—
255C23—21no—
256C23—21no—
257C23—41no—
258C23—41no—
259C23—41no—
260C23—41no—
261C23C2×#1171992111yescharacteristic
262C23#1313322312yes—
263Q8—2124no—
264Q8—2128no—
265Q8—2124no—
266D4—44no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—22no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—42no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—81no—
280C2×C4—1064no—
281C2×C4—2122no—
282C2×C4—2122no—
283C2×C4—2122no—
284C2×C4—2122no—
285C2×C4—2124no—
286C2×C4—2122no—
287C2×C4—2124no—
288C2×C4#1313313111yescharacteristic
289C2×C4#1313313111yescharacteristic
290C2×C4#1313313512yes—
291C8—82no—
292C8—4242no—
293C22—21no—
294C22—21no—
295C22—22no—
296C22—42no—
297C22—42no—
298C22—42no—
299C22—41no—
300C22—41no—
301C22—41no—
302C22—41no—
303C22—41no—
304C22—81no—
305C22#1681058911yescharacteristic
306C22C2×#1313313111yescharacteristic
307C22C2×#1313313511yescharacteristic
308C4—21no—
309C4—22no—
310C4—41no—
311C4—41no—
312C4—41no—
313C4—4242no—
314C4—2124no—
315C4#1681041112yes—
316C2—21no—
317C2—21no—
318C2—42no—
319C2—81no—
320C2#2027307811yescharacteristic
321C2C2×#1681041111yescharacteristic
322C2#2027636811yescharacteristic
subgroupG/H
centerC22C2×#13133131
commutatorC24×C53C22×C4
FrattiniC2×C22⋊C4C22×D53
FittingC53×#1460C2
socleC22×C53C2×#99
prime radicalC22×D4×C53C2×C4
cosocleC2×C22⋊C4×C53C23
hypercenter#1460D53
hypocenterC53#11580
LCSG ▷ C24×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #1460

Complements

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

NH ≅ G/Ntype
3. #20272664C2semidirect
181. C53#11580semidirect

Maximal subgroups

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

subgroupindex
#202780802
#202730532
#202726642
C53×#14602
#1158053