Finite Group Databasebeta

← #27293273

Subgroups — #27293273

Order 13568.

434 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 434 conjugacy classes of subgroups — too many to draw. Only its 131 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#20273029C211yesmaximal, characteristic
2#20273029C211yesmaximal, characteristic
3#20272663C214yesmaximal
4C53×#2317C211yesmaximal, characteristic
5#16810676—21no—
6#16810676—21no—
7#16810349—21no—
8#16810349—21no—
9#16810347C2212yes—
10#16810347C2212yes—
11C2×#117×C53C411yescharacteristic
12C2×#117×C53C2211yescharacteristic
13C22×C4⋊C4×C53C411yescharacteristic
14C53×#389—21no—
15C53×#389—21no—
16C53×#385C412yes—
17C53×#385C2212yes—
18C53×#384—21no—
19C53×#382—21no—
20C4⋊#88322—41no—
21C4⋊#88322—41no—
22C22⋊#88322—22no—
23C22⋊#88322—22no—
24C4×#88322—41no—
25C4×#88322—41no—
26C23×C4×C53C2×C411yescharacteristic
27C23×C4×C53C2×C412yes—
28#117×C53—22no—
29#117×C53—41no—
30#117×C53—41no—
31#117×C53C814yescomplemented
32C2×C4⋊C4×C53—21no—
33C2×C4⋊C4×C53—21no—
34C2×C4⋊C4×C53—21no—
35C2×C4⋊C4×C53—22no—
36C2×C4⋊C4×C53—22no—
37C2×C4⋊C4×C53—41no—
38C2×C4⋊C4×C53—41no—
39C2×C4⋊C4×C53D411yescharacteristic
40C2×C4⋊C4×C53D411yescharacteristic
41C2×C22⋊C4×C53C2×C412yes—
42C2×C22⋊C4×C53C2×C411yescharacteristic
43C2×C22⋊C4×C53C2311yescharacteristic
44C2×C42×C53—21no—
45C2×C42×C53D411yescharacteristic
46C2×C42×C53D411yescharacteristic
47#95×C53—22no—
48#95×C53—22no—
49C24×C53C22×C411yescharacteristic
50C22×C4×C53—21no—
51C22×C4×C53—21no—
52C22×C4×C53—22no—
53C22×C4×C53—22no—
54C22×C4×C53—22no—
55C22×C4×C53—42no—
56C22×C4×C53—82no—
57C22×C4×C53C22⋊C411yescharacteristic
58C22×C4×C53C22⋊C411yescharacteristic
59C22×C4×C53C22⋊C411yescharacteristic
60C22×C4×C53C22⋊C411yescharacteristic
61C22×C4×C53C2×C812yes—
62C22×C4×C53M1612yes—
63C22×C4×C53C2×D411yescharacteristic
64C22×C4×C53C2×D411yescharacteristic
65C4⋊C4×C53—24no—
66C4⋊C4×C53—41no—
67C4⋊C4×C53—41no—
68C4⋊C4×C53—42no—
69C4⋊C4×C53—41no—
70C4⋊C4×C53—41no—
71C4⋊C4×C53—82no—
72C4⋊C4×C53—82no—
73C22⋊C4×C53—22no—
74C22⋊C4×C53—41no—
75C22⋊C4×C53—41no—
76C22⋊C4×C53C2×C814yes—
77C42×C53—24no—
78C42×C53—21no—
79C42×C53—21no—
80C42×C53—21no—
81C42×C53—21no—
82C2×#88322—42no—
83C2×#88322—42no—
84C23×C53—21no—
85C23×C53—21no—
86C23×C53—41no—
87C23×C53—41no—
88C23×C53C2×C22⋊C411yescharacteristic
89C23×C53C22×C811yescharacteristic
90C23×C53C2×M1611yescharacteristic
91C2×C4×C53—21no—
92C2×C4×C53—21no—
93C2×C4×C53—22no—
94C2×C4×C53—22no—
95C2×C4×C53—22no—
96C2×C4×C53—22no—
97C2×C4×C53—24no—
98C2×C4×C53—21no—
99C2×C4×C53—21no—
100C2×C4×C53—21no—
101C2×C4×C53—21no—
102C2×C4×C53—42no—
103C2×C4×C53—42no—
104C2×C4×C53—41no—
105C2×C4×C53—41no—
106C2×C4×C53—41no—
107C2×C4×C53—41no—
108C2×C4×C53—42no—
109C2×C4×C53—42no—
110C2×C4×C53—42no—
111C2×C4×C53—84no—
112C2×C4×C53C22⋊C812yes—
113C2×C4×C53C22⋊C812yes—
114C2×C4×C53#9911yescharacteristic
115C2×C4×C53#9911yescharacteristic
116C2×C4×C53#10112yes—
117#88322—82no—
118#88322—82no—
119#11454—531nomaximal, Sylow
120C22×C53—21no—
121C22×C53—21no—
122C22×C53—21no—
123C22×C53—21no—
124C22×C53—41no—
125C22×C53—41no—
126C22×C53—42no—
127C22×C53—41no—
128C22×C53—41no—
129C22×C53C2×C22⋊C811yescharacteristic
130C22×C53C2×#9911yescharacteristic
131C22×C53C2×#10111yescharacteristic
132C4×C53—22no—
133C4×C53—44no—
134C4×C53—41no—
135C4×C53—41no—
136C4×C53—41no—
137C4×C53—41no—
138C4×C53—82no—
139C4×C53#32412yes—
140C4×C53#32412yes—
141#2317D5311yescharacteristic
142#1458—531no—
143#1458—531no—
144#1326—534no—
145C2×C53—21no—
146C2×C53—21no—
147C2×C53—41no—
148C2×C53—41no—
149C2×C53C2×#32411yescharacteristic
150C2×C53#212211yescharacteristic
151C2×C53#212311yescharacteristic
152C2×#117Dic5311yescharacteristic
153C2×#117C2×D5311yescharacteristic
154C22×C4⋊C4Dic5311yescharacteristic
155#431—1061no—
156#431—1061no—
157#389—21no—
158#389—21no—
159#385Dic5312yes—
160#385C2×D5312yes—
161#384—21no—
162#382—21no—
163#324—1061no—
164#324—1061no—
165#323—532no—
166#323—532no—
167C53#1145411yescharacteristic, complemented, Sylow
168C23×C4C2×Dic5311yescharacteristic
169C23×C4C2×Dic5312yes—
170#117—22no—
171#117—41no—
172#117—41no—
173#117#8832214yescomplemented
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—41no—
180C2×C4⋊C4—41no—
181C2×C4⋊C4C22⋊D5311yescharacteristic
182C2×C4⋊C4C22⋊D5311yescharacteristic
183C2×C22⋊C4C2×Dic5312yes—
184C2×C22⋊C4C2×Dic5311yescharacteristic
185C2×C22⋊C4C22×D5311yescharacteristic
186C2×C42—21no—
187C2×C42C22⋊D5311yescharacteristic
188C2×C42C22⋊D5311yescharacteristic
189C4⋊C8—2121no—
190C4⋊C8—2121no—
191C22⋊C8—1062no—
192C22⋊C8—1062no—
193C4×C8—2121no—
194C4×C8—2121no—
195#95—22no—
196#95—22no—
197C24C22×Dic5311yescharacteristic
198C22×C4—21no—
199C22×C4—21no—
200C22×C4—22no—
201C22×C4—22no—
202C22×C4—22no—
203C22×C4—42no—
204C22×C4—82no—
205C22×C4C2×#8832212yes—
206C22×C4#1171991712yes—
207C22×C4C22⋊Dic5311yescharacteristic
208C22×C4C22⋊Dic5311yescharacteristic
209C22×C4C22⋊Dic5311yescharacteristic
210C22×C4C22⋊Dic5311yescharacteristic
211C22×C4C2×C22⋊D5311yescharacteristic
212C22×C4C2×C22⋊D5311yescharacteristic
213C2×C8—2122no—
214C2×C8—2122no—
215C4⋊C4—24no—
216C4⋊C4—41no—
217C4⋊C4—41no—
218C4⋊C4—42no—
219C4⋊C4—41no—
220C4⋊C4—41no—
221C4⋊C4—82no—
222C4⋊C4—82no—
223C22⋊C4—22no—
224C22⋊C4—41no—
225C22⋊C4—41no—
226C22⋊C4C2×#8832214yes—
227C42—24no—
228C42—21no—
229C42—21no—
230C42—21no—
231C42—21no—
232C23—21no—
233C23—21no—
234C23—41no—
235C23—41no—
236C23C2×C22⋊Dic5311yescharacteristic
237C23C22×#8832211yescharacteristic
238C23C2×#1171991711yescharacteristic
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—24no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—42no—
251C2×C4—42no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—41no—
255C2×C4—41no—
256C2×C4—42no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—84no—
260C2×C4C22⋊#8832212yes—
261C2×C4C22⋊#8832212yes—
262C2×C4#1313313211yescharacteristic
263C2×C4#1313313211yescharacteristic
264C2×C4#1313313612yes—
265C8—4242no—
266C8—4242no—
267C22—21no—
268C22—21no—
269C22—21no—
270C22—21no—
271C22—41no—
272C22—41no—
273C22—42no—
274C22—41no—
275C22—41no—
276C22C2×C22⋊#8832211yescharacteristic
277C22C2×#1313313211yescharacteristic
278C22C2×#1313313611yescharacteristic
279C4—22no—
280C4—44no—
281C4—41no—
282C4—41no—
283C4—41no—
284C4—41no—
285C4—82no—
286C4#1681034912yes—
287C4#1681034912yes—
288C2—21no—
289C2—21no—
290C2—41no—
291C2—41no—
292C2C2×#1681034911yescharacteristic
293C2#2027636511yescharacteristic
294C2#2027636911yescharacteristic
subgroupG/H
centerC22C2×#13133132
commutatorC23×C53C22×C8
FrattiniC2×C22⋊C4C22×D53
FittingC53×#2317C2
socleC22×C53C2×#99
prime radicalC24×C53C22×C4
cosocleC2×C22⋊C4×C53C23
hypercenter#2317D53
hypocenterC53#11454
LCSG ▷ C23×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #2317

Complements

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

NH ≅ G/Ntype
31. #117×C53C8semidirect
167. C53#11454semidirect
173. #117#88322semidirect

Maximal subgroups

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

subgroupindex
#202730292
#202726632
C53×#23172
#1145453