Finite Group Databasebeta

← #27295143

Subgroups — #27295143

Order 13568.

460 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 99 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#20279023C211yesmaximal, characteristic
2#20274707C211yesmaximal, characteristic
3#20272722C214yesmaximal, complemented
4C53×#1459C211yesmaximal, characteristic
5C2×#13133198—21no—
6C22×C4⋊Dic53C411yescharacteristic
7#16810647—21no—
8C2×#13133134—21no—
9#16810553C2212yes—
10#16810548C412yes—
11#16810521—21no—
12#16810484—21no—
13#16810483—21no—
14#16810483C411yescharacteristic
15#16810392C2212yes—
16C2×C4×D4×C53C2211yescharacteristic
17C2×C22⋊C8×C53—21no—
18C53×#324C2212yes—
19C53×#323—21no—
20C23×Dic53—21no—
21C2×#11719917—22no—
22#13133198—42no—
23#13133198—42no—
24C2×C4⋊Dic53—21no—
25C2×C4⋊Dic53—21no—
26C2×C4⋊Dic53—41no—
27C2×C4⋊Dic53—41no—
28C2×C4⋊Dic53C2×C412yes—
29C2×#11719919—22no—
30C2×C22⋊Dic53—22no—
31C2×C22⋊Dic53—21no—
32C4⋊#88322—41no—
33C4⋊#88322—41no—
34#13133134—42no—
35#13133124—22no—
36#13133124—22no—
37#13133124C2×C412yes—
38C22×D4×C53D411yescharacteristic
39C23×C4×C53C2×C411yescharacteristic
40C23×C4×C53D411yescharacteristic
41C22×C8×C53—41no—
42C4×D4×C53—41no—
43C4×D4×C53—41no—
44C2×C4⋊C4×C53C2311yescharacteristic
45C2×C22⋊C4×C53C2×C411yescharacteristic
46C2×C22⋊C4×C53D411yescharacteristic
47C2×C42×C53D411yescharacteristic
48C22⋊C8×C53—22no—
49C22⋊C8×C53—41no—
50C24×C53C22⋊C411yescharacteristic
51C24×C53C22⋊C411yescharacteristic
52C2×D4×C53—41no—
53C2×D4×C53—41no—
54C2×D4×C53D812yes—
55C2×D4×C53SD1612yes—
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×C53C2×D411yescharacteristic
66C22×C4×C53C2×D411yescharacteristic
67C22×Dic53—22no—
68C22×Dic53—22no—
69C22×Dic53—42no—
70C22×Dic53—42no—
71C22×Dic53—42no—
72C2×C8×C53—42no—
73C2×C8×C53—82no—
74C4⋊C4×C53—41no—
75C22⋊C4×C53—41no—
76C22⋊C4×C53—41no—
77C42×C53—21no—
78C42×C53—21no—
79C22⋊Dic53—44no—
80C22⋊Dic53—42no—
81C4⋊Dic53—42no—
82C4⋊Dic53—41no—
83C4⋊Dic53—41no—
84C4⋊Dic53—42no—
85#11719919—42no—
86#11719919—42no—
87#11719917—42no—
88#11719917—42no—
89C2×#88322—42no—
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×C53#10212yes—
99C23×C53#10212yes—
100C23×C53C2×C22⋊C411yescharacteristic
101D4×C53—42no—
102D4×C53—42no—
103C2×C4×C53—21no—
104C2×C4×C53—21no—
105C2×C4×C53—22no—
106C2×C4×C53—21no—
107C2×C4×C53—21no—
108C2×C4×C53—21no—
109C2×C4×C53—42no—
110C2×C4×C53—42no—
111C2×C4×C53—42no—
112C2×C4×C53—41no—
113C2×C4×C53—41no—
114C2×C4×C53—41no—
115C2×C4×C53—81no—
116C2×C4×C53#9911yescharacteristic
117C2×C4×C53#9911yescharacteristic
118C2×C4×C53C2×D811yescharacteristic
119C2×C4×C53C2×SD1611yescharacteristic
120C2×Dic53—42no—
121C2×Dic53—42no—
122C2×Dic53—42no—
123C2×Dic53—42no—
124C2×Dic53—42no—
125C2×Dic53—42no—
126C2×Dic53—84no—
127C8×C53—82no—
128#88322—82no—
129#11881—531nomaximal, Sylow
130C22×C53—21no—
131C22×C53—21no—
132C22×C53—22no—
133C22×C53—22no—
134C22×C53—42no—
135C22×C53—42no—
136C22×C53—42no—
137C22×C53—41no—
138C22×C53—41no—
139C22×C53—41no—
140C22×C53—41no—
141C22×C53—81no—
142C22×C53C2×#9911yescharacteristic
143C22×C53C2×#10211yescharacteristic
144C22×C53#42111yescharacteristic
145C4×C53—21no—
146C4×C53—21no—
147C4×C53—21no—
148C4×C53—41no—
149C4×C53—41no—
150C4×C53—41no—
151C4×C53#35212yes—
152Dic53—82no—
153Dic53—82no—
154#2486—531no—
155#1800—531no—
156#1459D5311yescharacteristic
157#1346—534no—
158C2×C53—21no—
159C2×C53—21no—
160C2×C53—42no—
161C2×C53—42no—
162C2×C53#151911yescharacteristic
163C2×C53C2×#35211yescharacteristic
164C2×C53#213511yescharacteristic
165C2×#123—1061no—
166C2×C4×D4C2×D5311yescharacteristic
167C22×C4⋊C4—531no—
168#423—1061no—
169C2×#100—1061no—
170C2×C22⋊C8—21no—
171#398—532no—
172#397—532no—
173#389—1061no—
174#381—1061no—
175#380—531no—
176#380—1061no—
177#343—532no—
178#324C2×D5312yes—
179#323—21no—
180C53#1188111yescharacteristic, complemented, Sylow
181C22×D4D21211yescharacteristic
182C23×C4—1061no—
183C23×C4C4×D5311yescharacteristic
184C23×C4C22⋊D5311yescharacteristic
185C2×M16—1062no—
186C22×C8—41no—
187#123—2122no—
188#123—2122no—
189C4×D4—41no—
190C4×D4—41no—
191C2×C4⋊C4—532no—
192C2×C4⋊C4—1062no—
193C2×C4⋊C4—1061no—
194C2×C4⋊C4—1061no—
195C2×C4⋊C4—2121no—
196C2×C4⋊C4—2121no—
197C2×C4⋊C4C22×D5311yescharacteristic
198C2×C22⋊C4—1061no—
199C2×C22⋊C4—1062no—
200C2×C22⋊C4C4×D5311yescharacteristic
201C2×C22⋊C4C22⋊D5311yescharacteristic
202C2×C42D21211yescharacteristic
203C4⋊C8—2121no—
204C4⋊C8—2121no—
205#100—2122no—
206C22⋊C8—22no—
207C22⋊C8—41no—
208#95—532no—
209#95—1062no—
210#95—1062no—
211C24#1171992111yescharacteristic
212C24#1171992111yescharacteristic
213C2×D4—41no—
214C2×D4—41no—
215C2×D4#1171991312yes—
216C2×D4D42412yes—
217C22×C4—21no—
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—42no—
222C22×C4—42no—
223C22×C4—1062no—
224C22×C4—1062no—
225C22×C4—2122no—
226C22×C4—2122no—
227C22×C4—2122no—
228C22×C4#1171992111yescharacteristic
229C22×C4#1171992111yescharacteristic
230C22×C4C2×C4×D5311yescharacteristic
231C22×C4C2×D21211yescharacteristic
232C22×C4C2×C22⋊D5311yescharacteristic
233M16—2122no—
234M16—2122no—
235C2×C8—42no—
236C2×C8—82no—
237C2×C8—2122no—
238C4⋊C4—41no—
239C4⋊C4—2122no—
240C4⋊C4—2122no—
241C4⋊C4—2121no—
242C4⋊C4—2121no—
243C4⋊C4—2122no—
244C4⋊C4—2122no—
245C22⋊C4—41no—
246C22⋊C4—41no—
247C22⋊C4—2122no—
248C22⋊C4—2124no—
249C42—21no—
250C42—21no—
251C23—21no—
252C23—21no—
253C23—21no—
254C23—21no—
255C23—41no—
256C23—41no—
257C23—41no—
258C23—41no—
259C23#1313313912yes—
260C23#1313313912yes—
261C23C2×#1171992111yescharacteristic
262D4—42no—
263D4—42no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—22no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—42no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—81no—
277C2×C4—4244no—
278C2×C4—2122no—
279C2×C4—2122no—
280C2×C4—2122no—
281C2×C4—2122no—
282C2×C4—2122no—
283C2×C4—2122no—
284C2×C4#1313313111yescharacteristic
285C2×C4#1313313111yescharacteristic
286C2×C4C2×D42411yescharacteristic
287C2×C4C2×#1171991311yescharacteristic
288C8—82no—
289C8—4242no—
290C22—21no—
291C22—21no—
292C22—22no—
293C22—22no—
294C22—42no—
295C22—42no—
296C22—42no—
297C22—41no—
298C22—41no—
299C22—41no—
300C22—41no—
301C22—81no—
302C22C2×#1313313111yescharacteristic
303C22C2×#1313313911yescharacteristic
304C22#1681063911yescharacteristic
305C4—21no—
306C4—21no—
307C4—21no—
308C4—41no—
309C4—41no—
310C4—41no—
311C4—4242no—
312C4—4242no—
313C4#1681041212yes—
314C2—21no—
315C2—21no—
316C2—42no—
317C2—42no—
318C2#2027328411yescharacteristic
319C2C2×#1681041211yescharacteristic
320C2#2027640911yescharacteristic
subgroupG/H
centerC22C2×#13133131
commutatorC22×C4×C53C22×C4
FrattiniC2×C4⋊C4C22×D53
FittingC53×#1459C2
socleC22×C53C2×#99
prime radicalC22×D4×C53D4
cosocleC2×C4⋊C4×C53C23
hypercenter#1459D53
hypocenterC53#11881
LCSG ▷ C22×C4×C53 ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #1459

Complements

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

NH ≅ G/Ntype
3. #20272722C2semidirect
180. C53#11881semidirect

Maximal subgroups

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

subgroupindex
#202790232
#202747072
#202727222
C53×#14592
#1188153