Finite Group Databasebeta

← #27302736

Subgroups — #27302736

Order 13568.

484 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 484 conjugacy classes of subgroups — too many to draw. Only its 107 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#20279032C211yesmaximal, characteristic
2#20274703C211yesmaximal, characteristic
3#20272834C214yesmaximal
4C4×#100×C53C211yesmaximal, characteristic
5C22×C4⋊Dic53C411yescharacteristic, complemented
6C2×#13133132C2212yes—
7#16810562—21no—
8#16810559—21no—
9#16810521—21no—
10#16810503C2212yes—
11#16810495C412yescomplemented
12#16810483—21no—
13#16810483—21no—
14#16810483C411yescharacteristic, complemented
15#16810390—21no—
16C2×C4×D4×C53C2211yescharacteristic
17C2×#100×C53C2212yes—
18C4×M16×C53—21no—
19C53×#343—21no—
20C23×Dic53C2×C412yes—
21C2×C4⋊Dic53—21no—
22C2×C4⋊Dic53—21no—
23C2×C4⋊Dic53—21no—
24C2×C4⋊Dic53—41no—
25C2×C4⋊Dic53—41no—
26C2×C4⋊Dic53—41no—
27C2×#11719919—22no—
28C2×C22⋊Dic53—22no—
29C2×C22⋊Dic53—22no—
30C2×C22⋊Dic53C2×C412yes—
31#13133132—24no—
32#13133124—21no—
33#13133124—22no—
34#13133124—22no—
35#13133124—41no—
36C22×D4×C53C2311yescharacteristic
37C23×C4×C53C2×C411yescharacteristic
38C23×C4×C53D411yescharacteristic
39C2×M16×C53—22no—
40C4×D4×C53—41no—
41C4×D4×C53—41no—
42C2×C4⋊C4×C53D411yescharacteristic
43C2×C22⋊C4×C53C2×C411yescharacteristic
44C2×C22⋊C4×C53D411yescharacteristic
45C2×C42×C53D411yescharacteristic
46#100×C53—22no—
47#100×C53Q814yes—
48#97×C53—41no—
49C4×C8×C53—41no—
50C24×C53C22×C411yescharacteristic
51C24×C53C2×D411yescharacteristic
52C2×D4×C53—41no—
53C2×D4×C53—41no—
54C2×D4×C53C2×Q812yes—
55C2×D4×C53#4112yes—
56C22×C4×C53—21no—
57C22×C4×C53—21no—
58C22×C4×C53—21no—
59C22×C4×C53—21no—
60C22×C4×C53—42no—
61C22×C4×C53—42no—
62C22×C4×C53C22⋊C411yescharacteristic
63C22×C4×C53C22⋊C411yescharacteristic
64C22×C4×C53C22⋊C411yescharacteristic
65C22×C4×C53C22⋊C411yescharacteristic
66C22×C4×C53C2×D411yescharacteristic
67C22×Dic53—22no—
68C22×Dic53—22no—
69C22×Dic53—42no—
70C22×Dic53—42no—
71C22×Dic53—82no—
72M16×C53—24no—
73M16×C53—42no—
74C2×C8×C53—42no—
75C4⋊C4×C53—41no—
76C22⋊C4×C53—41no—
77C22⋊C4×C53—41no—
78C42×C53—21no—
79C42×C53—21no—
80C22⋊Dic53—42no—
81C22⋊Dic53—42no—
82C22⋊Dic53—42no—
83C22⋊Dic53—82no—
84C4⋊Dic53—42no—
85C4⋊Dic53—42no—
86C4⋊Dic53—41no—
87C4⋊Dic53—41no—
88C4⋊Dic53—82no—
89#11719919—82no—
90C23×C53—21no—
91C23×C53—21no—
92C23×C53—21no—
93C23×C53—21no—
94C23×C53—41no—
95C23×C53—41no—
96C23×C53—41no—
97C23×C53—41no—
98C23×C53C2×C22⋊C411yescharacteristic
99C23×C53C4×Q812yes—
100C23×C53#12411yescharacteristic
101C23×C53C4⋊Q811yescharacteristic
102D4×C53—42no—
103D4×C53—42no—
104C2×C4×C53—21no—
105C2×C4×C53—21no—
106C2×C4×C53—22no—
107C2×C4×C53—21no—
108C2×C4×C53—21no—
109C2×C4×C53—21no—
110C2×C4×C53—42no—
111C2×C4×C53—42no—
112C2×C4×C53—42no—
113C2×C4×C53—41no—
114C2×C4×C53—41no—
115C2×C4×C53—41no—
116C2×C4×C53—81no—
117C2×C4×C53#9911yescharacteristic
118C2×C4×C53#9911yescharacteristic
119C2×C4×C53C22⋊Q812yes—
120C2×Dic53—42no—
121C2×Dic53—42no—
122C2×Dic53—42no—
123C2×Dic53—42no—
124C2×Dic53—42no—
125C2×Dic53—42no—
126C2×Dic53—84no—
127C2×Dic53—82no—
128C2×Dic53—82no—
129C2×Dic53—82no—
130C8×C53—44no—
131#13292—531nomaximal, Sylow
132C22×C53—21no—
133C22×C53—21no—
134C22×C53—21no—
135C22×C53—22no—
136C22×C53—21no—
137C22×C53—42no—
138C22×C53—41no—
139C22×C53—41no—
140C22×C53—42no—
141C22×C53—41no—
142C22×C53—41no—
143C22×C53—41no—
144C22×C53—41no—
145C22×C53—81no—
146C22×C53#39111yescharacteristic
147C22×C53C2×#9911yescharacteristic
148C22×C53#41011yescharacteristic
149C4×C53—21no—
150C4×C53—22no—
151C4×C53—41no—
152C4×C53—41no—
153C4×C53—41no—
154C4×C53#35212yes—
155Dic53—82no—
156Dic53—82no—
157Dic53—162no—
158#2487—531no—
159#1799—531no—
160C4×#100D5311yescharacteristic
161#1390—534no—
162C2×C53—21no—
163C2×C53—21no—
164C2×C53—41no—
165C2×C53—42no—
166C2×C53—41no—
167C2×C53#199611yescharacteristic
168C2×C53C2×#35211yescharacteristic
169C2×C53#212211yescharacteristic
170C2×C4×D4C2×D5311yescharacteristic
171C22×C4⋊C4—531no—
172C2×#100C2×D5312yes—
173C2×#99—532no—
174C4×M16—21no—
175#400—1061no—
176#399—1061no—
177#389—1061no—
178#386—532no—
179#385—532no—
180#380—531no—
181#380—1061no—
182#380—1061no—
183#343—21no—
184#342—1061no—
185C53#1329211yescharacteristic, complemented, Sylow
186C22×D4C22×D5311yescharacteristic
187C23×C4—532no—
188C23×C4C4×D5311yescharacteristic
189C23×C4D21211yescharacteristic
190C2×M16—22no—
191C4×D4—41no—
192C4×D4—41no—
193C2×C4⋊C4—1062no—
194C2×C4⋊C4—1061no—
195C2×C4⋊C4—1061no—
196C2×C4⋊C4—1061no—
197C2×C4⋊C4—2121no—
198C2×C4⋊C4—2121no—
199C2×C4⋊C4—2121no—
200C2×C4⋊C4C22⋊D5311yescharacteristic
201C2×C22⋊C4—532no—
202C2×C22⋊C4—1062no—
203C2×C22⋊C4—1062no—
204C2×C22⋊C4C4×D5311yescharacteristic
205C2×C22⋊C4D21211yescharacteristic
206C2×C42C22⋊D5311yescharacteristic
207#100—22no—
208#100Dic10614yes—
209#99—1064no—
210#97—41no—
211C4×C8—41no—
212#95—1062no—
213#95—1061no—
214#95—1062no—
215#95—2121no—
216C24C2×C4×D5311yescharacteristic
217C24C2×D21211yescharacteristic
218C2×D4—41no—
219C2×D4—41no—
220C2×D4C2×Dic10612yes—
221C2×D4#1171994512yes—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—42no—
227C22×C4—42no—
228C22×C4—1062no—
229C22×C4—1062no—
230C22×C4—4242no—
231C22×C4—2122no—
232C22×C4—2122no—
233C22×C4#1171992111yescharacteristic
234C22×C4#1171992111yescharacteristic
235C22×C4#1171992111yescharacteristic
236C22×C4#1171992111yescharacteristic
237C22×C4C2×C22⋊D5311yescharacteristic
238M16—24no—
239M16—42no—
240C2×C8—42no—
241C4⋊C4—41no—
242C4⋊C4—4242no—
243C4⋊C4—4242no—
244C4⋊C4—2122no—
245C4⋊C4—2122no—
246C4⋊C4—2121no—
247C4⋊C4—2121no—
248C22⋊C4—41no—
249C22⋊C4—41no—
250C22⋊C4—4242no—
251C22⋊C4—2122no—
252C22⋊C4—2122no—
253C22⋊C4—2122no—
254C42—21no—
255C42—21no—
256C23—21no—
257C23—21no—
258C23—21no—
259C23—21no—
260C23—41no—
261C23—41no—
262C23—41no—
263C23—41no—
264C23C2×#1171992111yescharacteristic
265C23C4×Dic10612yes—
266C23#1313320511yescharacteristic
267C23#1313321711yescharacteristic
268D4—42no—
269D4—42no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—22no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—42no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—81no—
283C2×C4—4242no—
284C2×C4—4242no—
285C2×C4—4242no—
286C2×C4—4244no—
287C2×C4—2122no—
288C2×C4—2122no—
289C2×C4—2122no—
290C2×C4—2122no—
291C2×C4—2122no—
292C2×C4—2122no—
293C2×C4#1313313111yescharacteristic
294C2×C4#1313313111yescharacteristic
295C2×C4#1313319612yes—
296C8—44no—
297C22—21no—
298C22—21no—
299C22—21no—
300C22—22no—
301C22—21no—
302C22—42no—
303C22—41no—
304C22—41no—
305C22—42no—
306C22—41no—
307C22—41no—
308C22—41no—
309C22—41no—
310C22—81no—
311C22#1681052911yescharacteristic
312C22C2×#1313313111yescharacteristic
313C22#1681059611yescharacteristic
314C4—21no—
315C4—22no—
316C4—41no—
317C4—41no—
318C4—41no—
319C4—8482no—
320C4—4242no—
321C4—4242no—
322C4#1681041412yes—
323C2—21no—
324C2—21no—
325C2—41no—
326C2—42no—
327C2—41no—
328C2#2027570011yescharacteristic
329C2C2×#1681041411yescharacteristic
330C2#2027636611yescharacteristic
subgroupG/H
centerC22C2×#13133131
commutatorC24×C53C22×C4
FrattiniC22×D4C22×D53
FittingC4×#100×C53C2
socleC22×C53C2×#99
prime radical, cosocleC22×D4×C53C23
hypercenterC4×#100D53
hypocenterC53#13292
LCSG ▷ C24×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ C4×#100

Complements

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

NH ≅ G/Ntype
5. C22×C4⋊Dic53C4semidirect
11. #16810495C4semidirect
14. #16810483C4semidirect
185. C53#13292semidirect

Maximal subgroups

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

subgroupindex
#202790322
#202747032
#202728342
C4×#100×C532
#1329253