Finite Group Databasebeta

← #28309489

Subgroups — #28309489

Order 13568.

460 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 80 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#20290579C211yesmaximal, characteristic
2#20289950C211yesmaximal, characteristic
3#20289699C214yesmaximal, complemented
4#20279017C211yesmaximal, characteristic
5#16811626—21no—
6C2×#13133277—21no—
7D53⋊(C22⋊C8)—21no—
8#16811503C2212yes—
9#16811469C2212yes—
10#16811466—21no—
11C2×#13133200—21no—
12C2×C4×C22⋊D53C2211yescharacteristic
13#16810554C2212yes—
14#16810549C412yes—
15#16810519—21no—
16#16810486—21no—
17#16810483C411yescharacteristic
18#16810480—21no—
19C22×C4⋊C4×C53C411yescharacteristic
20C2×#11719940—22no—
21C2×D53⋊C8—41no—
22#13133284—41no—
23C4⋊#88324—41no—
24#13133277—42no—
25#13133274—41no—
26C22⋊#88324—22no—
27C22×C22⋊D53D411yescharacteristic
28C22×C4×D53D411yescharacteristic
29#13133200—42no—
30#13133200—42no—
31C4×C22⋊D53—41no—
32C4×C22⋊D53—41no—
33C2×C4⋊Dic53—21no—
34C2×#11719919—22no—
35C2×#11719919C2311yescharacteristic
36C2×C22⋊Dic53—21no—
37C2×C22⋊Dic53D411yescharacteristic
38C2×#11719921—22no—
39C2×#11719921C2×C411yescharacteristic
40C2×C4×Dic53D411yescharacteristic
41#13133124—22no—
42#13133124—22no—
43#13133124C2×C412yes—
44C23×C4×C53—21no—
45C23×C4×C53C2×C411yescharacteristic
46C2×C4⋊C4×C53—21no—
47C2×C4⋊C4×C53—41no—
48C2×C4⋊C4×C53—41no—
49C2×C4⋊C4×C53C2×C412yes—
50C24×C53C22⋊C411yescharacteristic
51C23×D53C22⋊C411yescharacteristic
52C22×C4×C53—21no—
53C22×C4×C53—21no—
54C22×C4×C53—22no—
55C22×C4×C53—42no—
56C22×C4×C53—42no—
57C22×C4×C53—42no—
58C22×C4×C53—42no—
59C22×C4×C53C22⋊C411yescharacteristic
60C22×C4×C53C22×C411yescharacteristic
61C2×C22⋊D53—41no—
62C2×C22⋊D53—41no—
63C2×C22⋊D53D812yes—
64C2×C22⋊D53SD1612yes—
65C22×Dic53—22no—
66C22×Dic53C22⋊C411yescharacteristic
67C22×Dic53C2×D411yescharacteristic
68C22×Dic53C2×D411yescharacteristic
69C2×C4×D53—21no—
70C2×C4×D53—21no—
71C2×C4×D53—42no—
72#11719940—42no—
73#11719940—42no—
74C2×#88324—42no—
75C2×#88324—42no—
76D53⋊C8—82no—
77C4⋊C4×C53—42no—
78C4⋊C4×C53—41no—
79C4⋊C4×C53—41no—
80C22⋊Dic53—41no—
81C22⋊Dic53—42no—
82#11719921—41no—
83#11719921—44no—
84C4⋊Dic53—42no—
85#11719919—41no—
86#11719919—42no—
87#11719919—42no—
88C4×Dic53—21no—
89C4×Dic53—21no—
90C23×C53—21no—
91C23×C53—21no—
92C23×C53—41no—
93C23×C53—41no—
94C23×C53#10212yes—
95C23×C53C2×C22⋊C411yescharacteristic
96C22×D53—21no—
97C22×D53—21no—
98C22×D53—41no—
99C22×D53—41no—
100C22×D53#10212yes—
101C2×C4×C53—22no—
102C2×C4×C53—21no—
103C2×C4×C53—42no—
104C2×C4×C53—42no—
105C2×C4×C53—41no—
106C2×C4×C53—42no—
107C2×C4×C53—42no—
108C2×C4×C53—42no—
109C2×C4×C53—84no—
110C2×C4×C53#9911yescharacteristic
111C2×C4×C53#9911yescharacteristic
112C22⋊D53—42no—
113C22⋊D53—42no—
114C2×Dic53—21no—
115C2×Dic53—21no—
116C2×Dic53—21no—
117C2×Dic53—21no—
118C2×Dic53—41no—
119C2×Dic53—41no—
120C2×Dic53—42no—
121C2×Dic53—42no—
122C2×Dic53—42no—
123C2×Dic53C2×D811yescharacteristic
124C2×Dic53C2×SD1611yescharacteristic
125C4×D53—42no—
126C4×D53—81no—
127#88324—82no—
128#88324—82no—
129#11881—531nomaximal, Sylow
130C22×C53—21no—
131C22×C53—21no—
132C22×C53—22no—
133C22×C53—42no—
134C22×C53—42no—
135C22×C53—41no—
136C22×C53—41no—
137C22×C53C2×#9911yescharacteristic
138C22×C53C2×#10211yescharacteristic
139C22×C53#42111yescharacteristic
140C2×D53—22no—
141C2×D53—41no—
142C2×D53—41no—
143C2×D53—42no—
144C2×D53—81no—
145C4×C53—21no—
146C4×C53—41no—
147C4×C53—82no—
148C4×C53#35212yes—
149Dic53—21no—
150Dic53—21no—
151Dic53—41no—
152Dic53—41no—
153Dic53—82no—
154#2486—531no—
155#1800—531no—
156#1459—531no—
157#1346—534no—
158C2×C53—21no—
159C2×C53—21no—
160C2×C53—42no—
161C2×C53#151911yescharacteristic
162C2×C53C2×#35211yescharacteristic
163C2×C53#213511yescharacteristic
164D53—42no—
165C2×#123—1061no—
166C2×C4×D4—531no—
167C22×C4⋊C4#615711yescharacteristic
168#423—1061no—
169C2×#100—1061no—
170C2×C22⋊C8—1061no—
171#398—532no—
172#397—532no—
173#389—1061no—
174#381—1061no—
175#380—531no—
176#380—1061no—
177#343—532no—
178#324—532no—
179#323—1061no—
180C53#1188111yescharacteristic, complemented, Sylow
181C22×D4—531no—
182C23×C4—21no—
183C23×C4—531no—
184C23×C4C2×#615711yescharacteristic
185C2×M16—1062no—
186C22×C8—2121no—
187#123—2122no—
188#123—2122no—
189C4×D4—2121no—
190C4×D4—2121no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—41no—
193C2×C4⋊C4—41no—
194C2×C4⋊C4—531no—
195C2×C4⋊C4—1062no—
196C2×C4⋊C4—1061no—
197C2×C4⋊C4C2×#615712yes—
198C2×C22⋊C4—531no—
199C2×C22⋊C4—531no—
200C2×C22⋊C4—1062no—
201C2×C22⋊C4—1061no—
202C2×C42—531no—
203C4⋊C8—2121no—
204C4⋊C8—2121no—
205#100—2122no—
206C22⋊C8—1062no—
207C22⋊C8—2121no—
208#95—532no—
209#95—1062no—
210#95—1062no—
211C24—531no—
212C24C22⋊#615711yescharacteristic
213C2×D4—532no—
214C2×D4—532no—
215C2×D4—2121no—
216C2×D4—2121no—
217C22×C4—21no—
218C22×C4—21no—
219C22×C4—22no—
220C22×C4—42no—
221C22×C4—42no—
222C22×C4—42no—
223C22×C4—42no—
224C22×C4—531no—
225C22×C4—531no—
226C22×C4—531no—
227C22×C4—1061no—
228C22×C4—1061no—
229C22×C4—1062no—
230C22×C4—2122no—
231C22×C4C22⋊#615711yescharacteristic
232C22×C4C22×#615711yescharacteristic
233M16—2122no—
234M16—2122no—
235C2×C8—4242no—
236C2×C8—2122no—
237C2×C8—2122no—
238C4⋊C4—42no—
239C4⋊C4—41no—
240C4⋊C4—41no—
241C4⋊C4—2121no—
242C4⋊C4—2122no—
243C4⋊C4—2122no—
244C4⋊C4—2122no—
245C22⋊C4—2121no—
246C22⋊C4—2124no—
247C22⋊C4—2121no—
248C22⋊C4—2122no—
249C42—1061no—
250C42—1061no—
251C23—21no—
252C23—21no—
253C23—41no—
254C23—41no—
255C23—532no—
256C23—1061no—
257C23—1061no—
258C23—2121no—
259C23—2121no—
260C23#1313327912yes—
261C23C2×C22⋊#615711yescharacteristic
262D4—2122no—
263D4—2122no—
264C2×C4—22no—
265C2×C4—21no—
266C2×C4—42no—
267C2×C4—42no—
268C2×C4—41no—
269C2×C4—42no—
270C2×C4—42no—
271C2×C4—42no—
272C2×C4—84no—
273C2×C4—531no—
274C2×C4—531no—
275C2×C4—1061no—
276C2×C4—1061no—
277C2×C4—1061no—
278C2×C4—1061no—
279C2×C4—4241no—
280C2×C4—2122no—
281C2×C4—2122no—
282C2×C4—2122no—
283C2×C4—2122no—
284C2×C4—2121no—
285C2×C4—2121no—
286C2×C4#1313327511yescharacteristic
287C2×C4#1313327511yescharacteristic
288C8—4242no—
289C8—4242no—
290C22—21no—
291C22—21no—
292C22—22no—
293C22—42no—
294C22—42no—
295C22—41no—
296C22—41no—
297C22—1062no—
298C22—4241no—
299C22—2122no—
300C22—2121no—
301C22—2121no—
302C22C2×#1313327511yescharacteristic
303C22C2×#1313327911yescharacteristic
304C22#1681162111yescharacteristic
305C4—21no—
306C4—41no—
307C4—82no—
308C4—1061no—
309C4—1061no—
310C4—4242no—
311C4—2121no—
312C4—2121no—
313C4#1681152512yes—
314C2—21no—
315C2—21no—
316C2—42no—
317C2—2122no—
318C2#2029015111yescharacteristic
319C2C2×#1681152511yescharacteristic
320C2#2029101711yescharacteristic
subgroupG/H
centerC22C2×#13133275
commutatorC22×C4×C53C22×C4
FrattiniC22×C4C22×#6157
FittingC22×C4⋊C4×C53C4
socleC22×C53C2×#99
prime radicalC22×C22⋊D53D4
cosocleC2×#11719919C23
hypercenterC22×C4⋊C4#6157
hypocenterC53#11881
LCSG ▷ C22×C4×C53 ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C23×C4 ◁ C22×C4⋊C4

Complements

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

NH ≅ G/Ntype
3. #20289699C2semidirect
180. C53#11881semidirect

Maximal subgroups

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

subgroupindex
#202905792
#202899502
#202896992
#202790172
#1188153