Finite Group Databasebeta

← #27294818

Subgroups — #27294818

Order 13568.

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

normal subgroups only

This group has 382 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#20274792C211yesmaximal, characteristic
2#20273049C211yesmaximal, characteristic
3#20272736C214yesmaximal
4C53×#2628C211yesmaximal, characteristic
5#16810680—21no—
6#16810486—21no—
7#16810390—21no—
8#16810390C2212yes—
9#16810349—21no—
10#16810347C2212yes—
11C2×C4⋊Q8×C53C411yescharacteristic, complemented
12C2×#117×C53C2211yescharacteristic
13C53×#400—21no—
14C53×#399—21no—
15C53×#395C412yescomplemented
16C53×#393C2212yes—
17C53×#389—21no—
18C53×#382C411yescharacteristic, complemented
19C53×#380—21no—
20C2×C22⋊Dic53—22no—
21C4⋊#88322—41no—
22C22⋊#88322—22no—
23#13133128—41no—
24#13133124—41no—
25#13133124—41no—
26C22×Q8×C53C2×C412yes—
27C23×C4×C53D411yescharacteristic
28C4⋊Q8×C53—41no—
29C4⋊Q8×C53—41no—
30#117×C53—41no—
31#117×C53—41no—
32C2×C4⋊C4×C53—21no—
33C2×C4⋊C4×C53—21no—
34C2×C4⋊C4×C53—21no—
35C2×C4⋊C4×C53—22no—
36C2×C4⋊C4×C53C2×C411yescharacteristic
37C2×C4⋊C4×C53D411yescharacteristic
38C2×C22⋊C4×C53—22no—
39C2×C22⋊C4×C53D411yescharacteristic
40C2×C22⋊C4×C53C2311yescharacteristic
41C2×C42×C53C2×C411yescharacteristic
42C2×C42×C53D411yescharacteristic
43#95×C53—22no—
44#95×C53—21no—
45#95×C53—22no—
46#95×C53C2×C412yes—
47C24×C53C2×D411yescharacteristic
48C2×Q8×C53—42no—
49C2×Q8×C53—82no—
50C22×C4×C53—22no—
51C22×C4×C53—21no—
52C22×C4×C53—21no—
53C22×C4×C53—22no—
54C22×C4×C53—22no—
55C22×C4×C53—41no—
56C22×C4×C53C22⋊C411yescharacteristic
57C22×C4×C53C22⋊C411yescharacteristic
58C22×C4×C53C22⋊C411yescharacteristic
59C22×C4×C53C22⋊C411yescharacteristic
60C22×C4×C53C22×C411yescharacteristic
61C22×C4×C53C2×D411yescharacteristic
62C22×Dic53—42no—
63C4⋊C4×C53—41no—
64C4⋊C4×C53—41no—
65C4⋊C4×C53—42no—
66C4⋊C4×C53—42no—
67C4⋊C4×C53—41no—
68C4⋊C4×C53—41no—
69C4⋊C4×C53—82no—
70C22⋊C4×C53—41no—
71C22⋊C4×C53—41no—
72C22⋊C4×C53—82no—
73C42×C53—21no—
74C42×C53—21no—
75C42×C53—21no—
76C42×C53—21no—
77C22⋊Dic53—42no—
78C22⋊Dic53—42no—
79C2×#88322—42no—
80C23×C53—21no—
81C23×C53—21no—
82C23×C53—41no—
83C23×C53—41no—
84C23×C53C2×C22⋊C411yescharacteristic
85C23×C53#13712yes—
86Q8×C53—84no—
87C2×C4×C53—21no—
88C2×C4×C53—21no—
89C2×C4×C53—21no—
90C2×C4×C53—21no—
91C2×C4×C53—21no—
92C2×C4×C53—21no—
93C2×C4×C53—21no—
94C2×C4×C53—21no—
95C2×C4×C53—21no—
96C2×C4×C53—42no—
97C2×C4×C53—42no—
98C2×C4×C53—41no—
99C2×C4×C53—41no—
100C2×C4×C53—41no—
101C2×C4×C53—41no—
102C2×C4×C53—42no—
103C2×C4×C53—42no—
104C2×C4×C53—42no—
105C2×C4×C53—42no—
106C2×C4×C53—42no—
107C2×C4×C53—42no—
108C2×C4×C53#9911yescharacteristic
109C2×C4×C53#9911yescharacteristic
110C2×Dic53—82no—
111C2×Dic53—82no—
112C2×Dic53—82no—
113#88322—82no—
114#11812—531nomaximal, Sylow
115C22×C53—21no—
116C22×C53—21no—
117C22×C53—22no—
118C22×C53—42no—
119C22×C53—41no—
120C22×C53—41no—
121C22×C53—81no—
122C22×C53C2×#9911yescharacteristic
123C22×C53#41911yescharacteristic
124C22×C53#42111yescharacteristic
125C4×C53—21no—
126C4×C53—21no—
127C4×C53—41no—
128C4×C53—41no—
129C4×C53—41no—
130C4×C53—41no—
131C4×C53—41no—
132C4×C53—82no—
133C4×C53—82no—
134Dic53—162no—
135#2628D5311yescharacteristic
136#1820—531no—
137#1464—531no—
138#1351—534no—
139C2×C53—21no—
140C2×C53—21no—
141C2×C53—42no—
142C2×C53#151111yescharacteristic
143C2×C53#212211yescharacteristic
144C2×C53#212911yescharacteristic
145C2×C4⋊Q8Dic5311yescharacteristic, complemented
146C2×#117C2×D5311yescharacteristic
147#432—1061no—
148#400—21no—
149#399—21no—
150#395Dic5312yescomplemented
151#393C2×D5312yes—
152#389—21no—
153#382Dic5311yescharacteristic, complemented
154#381—1061no—
155#380—21no—
156#342—532no—
157#342—1061no—
158#324—1061no—
159#323—532no—
160C53#1181211yescharacteristic, complemented, Sylow
161C22×Q8C2×Dic5312yes—
162C23×C4C22⋊D5311yescharacteristic
163C4⋊Q8—41no—
164C4⋊Q8—41no—
165#117—41no—
166#117—41no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4C2×Dic5311yescharacteristic
172C2×C4⋊C4C22⋊D5311yescharacteristic
173C2×C22⋊C4—22no—
174C2×C22⋊C4—1062no—
175C2×C22⋊C4C22⋊D5311yescharacteristic
176C2×C22⋊C4C22×D5311yescharacteristic
177C2×C42C2×Dic5311yescharacteristic
178C2×C42C22⋊D5311yescharacteristic
179C4⋊C8—2121no—
180C22⋊C8—1062no—
181#97—2121no—
182#95—22no—
183#95—22no—
184#95—21no—
185#95—2121no—
186#95—2121no—
187#95C2×Dic5312yes—
188C24C2×C22⋊D5311yescharacteristic
189C2×Q8—42no—
190C2×Q8—82no—
191C22×C4—22no—
192C22×C4—21no—
193C22×C4—21no—
194C22×C4—22no—
195C22×C4—22no—
196C22×C4—41no—
197C22×C4—2122no—
198C22×C4C22⋊Dic5311yescharacteristic
199C22×C4C22⋊Dic5311yescharacteristic
200C22×C4C22⋊Dic5311yescharacteristic
201C22×C4C22⋊Dic5311yescharacteristic
202C22×C4C22×Dic5311yescharacteristic
203C22×C4C2×C22⋊D5311yescharacteristic
204C2×C8—2122no—
205C4⋊C4—41no—
206C4⋊C4—41no—
207C4⋊C4—42no—
208C4⋊C4—42no—
209C4⋊C4—41no—
210C4⋊C4—41no—
211C4⋊C4—82no—
212C22⋊C4—41no—
213C22⋊C4—41no—
214C22⋊C4—82no—
215C22⋊C4—2122no—
216C22⋊C4—2122no—
217C42—21no—
218C42—21no—
219C42—21no—
220C42—21no—
221C23—21no—
222C23—21no—
223C23—41no—
224C23—41no—
225C23C2×C22⋊Dic5311yescharacteristic
226C23#1313324812yes—
227Q8—84no—
228C2×C4—21no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—42no—
238C2×C4—42no—
239C2×C4—41no—
240C2×C4—41no—
241C2×C4—41no—
242C2×C4—41no—
243C2×C4—42no—
244C2×C4—42no—
245C2×C4—42no—
246C2×C4—42no—
247C2×C4—42no—
248C2×C4—42no—
249C2×C4—4242no—
250C2×C4—4242no—
251C2×C4—4242no—
252C2×C4#1313313211yescharacteristic
253C2×C4#1313313211yescharacteristic
254C8—4242no—
255C22—21no—
256C22—21no—
257C22—22no—
258C22—42no—
259C22—41no—
260C22—41no—
261C22—81no—
262C22C2×#1313313211yescharacteristic
263C22#1681063211yescharacteristic
264C22#1681064011yescharacteristic
265C4—21no—
266C4—21no—
267C4—41no—
268C4—41no—
269C4—41no—
270C4—41no—
271C4—41no—
272C4—82no—
273C4—82no—
274C4—8482no—
275C2—21no—
276C2—21no—
277C2—42no—
278C2#2027324811yescharacteristic
279C2#2027636511yescharacteristic
280C2#2027639011yescharacteristic
subgroupG/H
centerC22C2×#13133132
commutatorC22×C4×C53C22×C4
FrattiniC2×C22⋊C4C22×D53
FittingC53×#2628C2
socleC22×C53C2×#99
prime radicalC24×C53C2×D4
cosocleC2×C22⋊C4×C53C23
hypercenter#2628D53
hypocenterC53#11812
LCSG ▷ C22×C4×C53 ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #2628

Complements

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

NH ≅ G/Ntype
11. C2×C4⋊Q8×C53C4semidirect
15. C53×#395C4semidirect
18. C53×#382C4semidirect
145. C2×C4⋊Q8Dic53semidirect
150. #395Dic53semidirect
153. #382Dic53semidirect
160. C53#11812semidirect

Maximal subgroups

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

subgroupindex
#202747922
#202730492
#202727362
C53×#26282
#1181253