Finite Group Databasebeta

← #27293259

Subgroups — #27293259

Order 13568.

434 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 434 conjugacy classes of subgroups — too many to draw. Only its 115 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#20277335C211yesmaximal, characteristic
2#20273029C211yesmaximal, characteristic
3#20272660C214yesmaximal, complemented
4C53×#1458C211yesmaximal, characteristic
5C2×#13133176C411yescharacteristic
6C22×C4⋊Dic53C411yescharacteristic
7#16810676—21no—
8#16810521—21no—
9#16810521—21no—
10#16810495—21no—
11#16810495—21no—
12#16810492C2212yes—
13#16810488C412yes—
14#16810349C2212yes—
15#16810347—21no—
16C2×#117×C53C2211yescharacteristic
17C53×#431—21no—
18C53×#324C2212yes—
19C53×#323—21no—
20C23×Dic53—21no—
21#13133176—22no—
22#13133176C814yescomplemented
23C2×C4⋊Dic53—21no—
24C2×C4⋊Dic53—21no—
25C2×C4⋊Dic53—41no—
26C2×C4⋊Dic53—41no—
27C2×C4⋊Dic53C2×C412yes—
28C2×#11719919—22no—
29C2×#11719919—22no—
30C2×C22⋊Dic53—21no—
31C2×C4×Dic53C2×C412yes—
32C4⋊#88322—41no—
33C22⋊#88322—22no—
34C4×#88322—41no—
35#13133124—22no—
36#13133124—22no—
37C23×C4×C53C2×C411yescharacteristic
38#117×C53—41no—
39#117×C53—41no—
40C2×C4⋊C4×C53C2×C411yescharacteristic
41C2×C4⋊C4×C53C2311yescharacteristic
42C2×C22⋊C4×C53D411yescharacteristic
43C2×C22⋊C4×C53D411yescharacteristic
44C2×C42×C53D411yescharacteristic
45C2×C42×C53D411yescharacteristic
46C4⋊C8×C53—41no—
47C22⋊C8×C53—22no—
48C4×C8×C53—41no—
49C24×C53C22⋊C411yescharacteristic
50C22×C4×C53—21no—
51C22×C4×C53—21no—
52C22×C4×C53—22no—
53C22×C4×C53C22⋊C411yescharacteristic
54C22×C4×C53C22⋊C411yescharacteristic
55C22×C4×C53C22⋊C411yescharacteristic
56C22×C4×C53C2×C812yes—
57C22×C4×C53M1612yes—
58C22×C4×C53C22×C411yescharacteristic
59C22×C4×C53C2×D411yescharacteristic
60C22×C4×C53C2×D411yescharacteristic
61C22×Dic53—22no—
62C22×Dic53—22no—
63C22×Dic53—42no—
64C22×Dic53—82no—
65C2×C8×C53—42no—
66C4⋊C4×C53—41no—
67C4⋊C4×C53—41no—
68C22⋊C4×C53—41no—
69C22⋊C4×C53—41no—
70C42×C53—21no—
71C42×C53—21no—
72C42×C53—21no—
73C42×C53—21no—
74C22⋊Dic53—24no—
75C4⋊Dic53—24no—
76C4⋊Dic53—41no—
77C4⋊Dic53—41no—
78C4⋊Dic53—42no—
79#11719919—82no—
80#11719919—82no—
81C4×Dic53—22no—
82C4×Dic53C2×C814yes—
83C2×#88322—42no—
84C23×C53—21no—
85C23×C53—21no—
86C23×C53—41no—
87C23×C53—41no—
88C23×C53C22⋊C812yes—
89C23×C53C2×C22⋊C411yescharacteristic
90C2×C4×C53—21no—
91C2×C4×C53—21no—
92C2×C4×C53—22no—
93C2×C4×C53—22no—
94C2×C4×C53—22no—
95C2×C4×C53—22no—
96C2×C4×C53—21no—
97C2×C4×C53—21no—
98C2×C4×C53—21no—
99C2×C4×C53—21no—
100C2×C4×C53—41no—
101C2×C4×C53—41no—
102C2×C4×C53—41no—
103C2×C4×C53—41no—
104C2×C4×C53C22⋊C812yes—
105C2×C4×C53#9911yescharacteristic
106C2×C4×C53#9911yescharacteristic
107C2×C4×C53#10112yes—
108C2×C4×C53C22×C811yescharacteristic
109C2×C4×C53C2×M1611yescharacteristic
110C2×Dic53—24no—
111C2×Dic53—42no—
112C2×Dic53—42no—
113C2×Dic53—42no—
114C2×Dic53—42no—
115C2×Dic53—42no—
116C2×Dic53—84no—
117C8×C53—82no—
118#88322—82no—
119#11450—531nomaximal, Sylow
120C22×C53—21no—
121C22×C53—21no—
122C22×C53—22no—
123C22×C53—42no—
124C22×C53—42no—
125C22×C53—41no—
126C22×C53—41no—
127C22×C53C2×C22⋊C811yescharacteristic
128C22×C53C2×#9911yescharacteristic
129C22×C53C2×#10111yescharacteristic
130C4×C53—21no—
131C4×C53—21no—
132C4×C53—41no—
133C4×C53—41no—
134C4×C53—41no—
135C4×C53—41no—
136C4×C53#32412yes—
137C4×C53#32412yes—
138Dic53—44no—
139Dic53—82no—
140#2317—531no—
141#1458—531no—
142#1458D5311yescharacteristic
143#1325—534no—
144C2×C53—21no—
145C2×C53—21no—
146C2×C53—42no—
147C2×C53C2×#32411yescharacteristic
148C2×C53#212811yescharacteristic
149C2×C53#213511yescharacteristic
150C2×#117—531no—
151C2×#117C2×D5311yescharacteristic
152C22×C4⋊C4—531no—
153#431—21no—
154#431—1061no—
155#389—1061no—
156#389—1061no—
157#385—1061no—
158#385—1061no—
159#384—532no—
160#382—532no—
161#324—532no—
162#324C2×D5312yes—
163#323—21no—
164#323—1061no—
165C53#1145011yescharacteristic, complemented, Sylow
166C23×C4—1061no—
167C23×C4C4×D5311yescharacteristic
168#117—41no—
169#117—41no—
170#117—534no—
171#117—1062no—
172C2×C4⋊C4—532no—
173C2×C4⋊C4—1061no—
174C2×C4⋊C4—1062no—
175C2×C4⋊C4—1062no—
176C2×C4⋊C4—1061no—
177C2×C4⋊C4—2121no—
178C2×C4⋊C4—2121no—
179C2×C4⋊C4C4×D5311yescharacteristic
180C2×C4⋊C4C22×D5311yescharacteristic
181C2×C22⋊C4—1061no—
182C2×C22⋊C4D21211yescharacteristic
183C2×C22⋊C4C22⋊D5311yescharacteristic
184C2×C42—532no—
185C2×C42D21211yescharacteristic
186C2×C42C22⋊D5311yescharacteristic
187C4⋊C8—41no—
188C4⋊C8—2121no—
189C22⋊C8—22no—
190C22⋊C8—1062no—
191C4×C8—41no—
192C4×C8—2121no—
193#95—1062no—
194#95—1062no—
195C24#1171992111yescharacteristic
196C22×C4—21no—
197C22×C4—21no—
198C22×C4—22no—
199C22×C4—1062no—
200C22×C4—1062no—
201C22×C4—4242no—
202C22×C4—2122no—
203C22×C4C8×D5312yes—
204C22×C4#1171991212yes—
205C22×C4#1171992111yescharacteristic
206C22×C4#1171992111yescharacteristic
207C22×C4#1171992111yescharacteristic
208C22×C4C2×C4×D5311yescharacteristic
209C22×C4C2×D21211yescharacteristic
210C22×C4C2×C22⋊D5311yescharacteristic
211C2×C8—42no—
212C2×C8—2122no—
213C4⋊C4—41no—
214C4⋊C4—41no—
215C4⋊C4—1064no—
216C4⋊C4—4242no—
217C4⋊C4—4242no—
218C4⋊C4—2122no—
219C4⋊C4—2121no—
220C4⋊C4—2121no—
221C22⋊C4—41no—
222C22⋊C4—41no—
223C22⋊C4—1064no—
224C42—21no—
225C42—21no—
226C42—21no—
227C42—21no—
228C42—534no—
229C42—1062no—
230C23—21no—
231C23—21no—
232C23—41no—
233C23—41no—
234C23#1313312912yes—
235C23C2×#1171992111yescharacteristic
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—22no—
239C2×C4—22no—
240C2×C4—22no—
241C2×C4—22no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—41no—
247C2×C4—41no—
248C2×C4—41no—
249C2×C4—41no—
250C2×C4—1064no—
251C2×C4—4244no—
252C2×C4—2122no—
253C2×C4—2122no—
254C2×C4—2122no—
255C2×C4—2122no—
256C2×C4—2122no—
257C2×C4#1313312912yes—
258C2×C4#1313313111yescharacteristic
259C2×C4#1313313111yescharacteristic
260C2×C4#1313313512yes—
261C2×C4C2×C8×D5311yescharacteristic
262C2×C4C2×#1171991211yescharacteristic
263C8—82no—
264C8—4242no—
265C22—21no—
266C22—21no—
267C22—22no—
268C22—42no—
269C22—42no—
270C22—41no—
271C22—41no—
272C22C2×#1313312911yescharacteristic
273C22C2×#1313313111yescharacteristic
274C22C2×#1313313511yescharacteristic
275C4—21no—
276C4—21no—
277C4—41no—
278C4—41no—
279C4—41no—
280C4—41no—
281C4—4242no—
282C4—2124no—
283C4#1681034812yes—
284C4#1681034812yes—
285C2—21no—
286C2—21no—
287C2—42no—
288C2C2×#1681034811yescharacteristic
289C2#2027638611yescharacteristic
290C2#2027640911yescharacteristic
subgroupG/H
centerC22C2×#13133131
commutatorC2×C4×C53C22×C8
FrattiniC2×C4⋊C4C22×D53
FittingC53×#1458C2
socleC22×C53C2×#99
prime radicalC24×C53C22⋊C4
cosocleC2×C4⋊C4×C53C23
hypercenter#1458D53
hypocenterC53#11450
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #1458

Complements

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

NH ≅ G/Ntype
3. #20272660C2semidirect
22. #13133176C8semidirect
165. C53#11450semidirect

Maximal subgroups

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

subgroupindex
#202773352
#202730292
#202726602
C53×#14582
#1145053