Finite Group Databasebeta

← #28300001

Subgroups — #28300001

Order 13568.

400 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 400 conjugacy classes of subgroups — too many to draw. Only its 84 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#20290643C211yesmaximal, characteristic, complemented
2#20290642C211yesmaximal, characteristic, complemented
3#20289533C212yesmaximal, complemented
4#20289532C212yesmaximal
5#20283083C211yesmaximal, characteristic
6#16811489—21no—
7#16811489—21no—
8#16811489—21no—
9#16811489—21no—
10#16811489C2212yes—
11#16811489C2212yes—
12C2×#13133212—21no—
13#16810525C411yescharacteristic
14#16810524C411yescharacteristic
15#16810524C2211yescharacteristic
16#16810522C2211yescharacteristic
17#16810519C411yescharacteristic
18#16810515—21no—
19#16810513—21no—
20#16810512—21no—
21C42×Dic53C2211yescharacteristic
22C53×#389C411yescharacteristic
23C22×#88324—22no—
24C22×#88324—22no—
25C22×D212C2×C411yescharacteristic
26#13133212—44no—
27C2×C4⋊Dic53—21no—
28C2×C4⋊Dic53C2×C411yescharacteristic
29C2×#11719919—21no—
30C2×#11719921—21no—
31C2×#11719921—21no—
32C2×#11719921—21no—
33C2×#11719921—21no—
34C2×#11719921—21no—
35C2×#11719921—21no—
36C2×C4×Dic53C2×C411yescharacteristic
37C2×C4×Dic53D411yescharacteristic
38C2×C4×Dic53D411yescharacteristic
39C2×C4×Dic53D411yescharacteristic
40C2×C4×Dic53D411yescharacteristic
41C2×C4×Dic53C2311yescharacteristic
42#13133124—21no—
43#13133124—21no—
44C2×C4⋊C4×C53—21no—
45C2×C4⋊C4×C53C2×C411yescharacteristic
46C2×C4⋊C4×C53C2×C411yescharacteristic
47C2×C42×C53C2×C411yescharacteristic
48#95×C53—21no—
49C23×D53—21no—
50C22×C4×C53—21no—
51C22×C4×C53—21no—
52C22×C4×C53C22⋊C411yescharacteristic
53C22×C4×C53C22⋊C411yescharacteristic
54C22×C4×C53C22×C411yescharacteristic
55C22×Dic53—21no—
56C22×Dic53C22⋊C411yescharacteristic
57C22×Dic53C22⋊C411yescharacteristic
58C22×Dic53C2×D411yescharacteristic
59C22×Dic53C2×D411yescharacteristic
60C2×D212—22no—
61C2×D212—41no—
62C2×D212—41no—
63C2×#88324—42no—
64C2×#88324—42no—
65C2×#88324—42no—
66C2×#88324—42no—
67C2×#88324—42no—
68C2×#88324—42no—
69C4⋊C4×C53—22no—
70C4⋊C4×C53—42no—
71C4⋊C4×C53—41no—
72C42×C53—22no—
73#11719921—42no—
74#11719921—42no—
75#11719921—42no—
76#11719921—42no—
77#11719921—42no—
78#11719921—42no—
79C4⋊Dic53—42no—
80C4⋊Dic53—41no—
81#11719919—42no—
82C4×Dic53—22no—
83C4×Dic53—22no—
84C4×Dic53—41no—
85C4×Dic53—41no—
86C4×Dic53—41no—
87C4×Dic53—41no—
88C23×C53C2×C22⋊C411yescharacteristic
89C22×D53—41no—
90C22×D53—41no—
91C22×D53—41no—
92C22×D53—41no—
93C22×D53—41no—
94C22×D53—41no—
95C22×D53—41no—
96C2×C4×C53—21no—
97C2×C4×C53—21no—
98C2×C4×C53—21no—
99C2×C4×C53—21no—
100C2×C4×C53—21no—
101C2×C4×C53—21no—
102C2×C4×C53—21no—
103C2×C4×C53—21no—
104C2×C4×C53—41no—
105C2×C4×C53—41no—
106C2×C4×C53—41no—
107C2×C4×C53—41no—
108C2×C4×C53—41no—
109C2×C4×C53—41no—
110C2×C4×C53#13112yes—
111C2×Dic53—21no—
112C2×Dic53—21no—
113C2×Dic53—21no—
114C2×Dic53—21no—
115C2×Dic53—21no—
116C2×Dic53—21no—
117C2×Dic53—21no—
118C2×Dic53—21no—
119C2×Dic53—21no—
120C2×Dic53—21no—
121C2×Dic53—41no—
122C2×Dic53—41no—
123C2×Dic53—41no—
124C2×Dic53#13512yes—
125C2×Dic53AGL(1,ℤ/8)12yes—
126D212—82no—
127#88324—82no—
128#88324—82no—
129#7726—531nomaximal, Sylow
130C22×C53#40811yescharacteristic
131C22×C53#41011yescharacteristic
132C22×C53#41311yescharacteristic
133C22×C53#41611yescharacteristic
134C22×C53#41811yescharacteristic
135C22×C53#42111yescharacteristic
136C22×C53#42111yescharacteristic
137C2×D53—81no—
138C2×D53—81no—
139C2×D53—81no—
140C2×D53—81no—
141C2×D53—81no—
142C2×D53—81no—
143C2×D53—81no—
144C4×C53—22no—
145C4×C53—41no—
146C4×C53—41no—
147C4×C53—81no—
148C4×C53—81no—
149Dic53—22no—
150Dic53—22no—
151Dic53—41no—
152Dic53—41no—
153Dic53—81no—
154#2860—531no—
155#1839—531no—
156#1839—531no—
157#1271—532no—
158#1270—532no—
159C2×C53#147311yescharacteristic
160C2×C53#148211yescharacteristic
161C2×C53#149611yescharacteristic
162C2×C53#151411yescharacteristic
163C2×C53#152111yescharacteristic
164C2×C53#153511yescharacteristic
165C2×C53#155311yescharacteristic
166D53—161no—
167C2×#126—1061no—
168#390—531no—
169#390—531no—
170#390—531no—
171#389—531no—
172#389—531no—
173#389#615711yescharacteristic
174#388—1061no—
175#388—1061no—
176#388—1061no—
177C43—531no—
178#336—532no—
179#336—532no—
180#336—1061no—
181#336—1061no—
182#336—1061no—
183#336—1061no—
184C53#772611yescharacteristic, complemented, Sylow
185C22×D4—531no—
186C22×C8—1062no—
187C22×C8—1062no—
188#126—2124no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—531no—
191C2×C4⋊C4—1061no—
192C2×C4⋊C4—1061no—
193C2×C4⋊C4C2×#615711yescharacteristic
194C2×C4⋊C4C2×#615711yescharacteristic
195C2×C22⋊C4—1061no—
196C2×C22⋊C4—1061no—
197C2×C22⋊C4—1061no—
198C2×C22⋊C4—1061no—
199C2×C22⋊C4—1061no—
200C2×C22⋊C4—1061no—
201C2×C42—531no—
202C2×C42—531no—
203C2×C42—531no—
204C2×C42—531no—
205C2×C42—531no—
206C2×C42—531no—
207C2×C42C2×#615711yescharacteristic
208#95—21no—
209#95—1061no—
210#95—1061no—
211C24—1061no—
212C2×D4—1062no—
213C2×D4—2121no—
214C2×D4—2121no—
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—531no—
218C22×C4—531no—
219C22×C4—531no—
220C22×C4—531no—
221C22×C4—1061no—
222C22×C4C22⋊#615711yescharacteristic
223C22×C4C22⋊#615711yescharacteristic
224C22×C4C22×#615711yescharacteristic
225C2×C8—2122no—
226C2×C8—2122no—
227C2×C8—2122no—
228C2×C8—2122no—
229C2×C8—2122no—
230C2×C8—2122no—
231C4⋊C4—22no—
232C4⋊C4—41no—
233C4⋊C4—42no—
234C4⋊C4—2121no—
235C4⋊C4—2122no—
236C4⋊C4—2122no—
237C22⋊C4—2122no—
238C22⋊C4—2122no—
239C22⋊C4—2122no—
240C22⋊C4—2122no—
241C22⋊C4—2122no—
242C22⋊C4—2122no—
243C42—22no—
244C42—1062no—
245C42—1062no—
246C42—2121no—
247C42—2121no—
248C42—2121no—
249C42—2121no—
250C23—2121no—
251C23—2121no—
252C23—2121no—
253C23—2121no—
254C23—2121no—
255C23—2121no—
256C23—2121no—
257C23C2×C22⋊#615711yescharacteristic
258D4—4242no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—532no—
274C2×C4—532no—
275C2×C4—1061no—
276C2×C4—1061no—
277C2×C4—1061no—
278C2×C4—1061no—
279C2×C4—1061no—
280C2×C4—1061no—
281C2×C4—1061no—
282C2×C4—1061no—
283C2×C4—1061no—
284C2×C4—1061no—
285C2×C4—2121no—
286C2×C4—2121no—
287C2×C4—2121no—
288C2×C4#1313330412yes—
289C8—4242no—
290C8—4242no—
291C22—4241no—
292C22—4241no—
293C22—4241no—
294C22—4241no—
295C22—4241no—
296C22—4241no—
297C22—4241no—
298C22#1681159611yescharacteristic
299C22#1681160411yescharacteristic
300C22#1681160911yescharacteristic
301C22#1681161311yescharacteristic
302C22#1681161511yescharacteristic
303C22#1681162011yescharacteristic
304C22#1681162211yescharacteristic
305C4—22no—
306C4—41no—
307C4—41no—
308C4—81no—
309C4—81no—
310C4—1062no—
311C4—1062no—
312C4—4241no—
313C4—2121no—
314C4—2121no—
315C2—8481no—
316C2#2028999911yescharacteristic
317C2#2029002811yescharacteristic
318C2#2029007411yescharacteristic
319C2#2029013611yescharacteristic
320C2#2029015811yescharacteristic
321C2#2029020111yescharacteristic
322C2#2029023811yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6157
commutatorC22×C4×C53C22×C4
FrattiniC22×C4C22×#6157
FittingC53×#389C4
socleC23×C53C2×C22⋊C4
prime radicalC22×D212C2×C4
cosocleC2×C4×Dic53C23
hypercenter#389#6157
hypocenterC53#7726
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #389

Complements

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

NH ≅ G/Ntype
1. #20290643C2semidirect
2. #20290642C2semidirect
3. #20289533C2semidirect
184. C53#7726semidirect

Maximal subgroups

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

subgroupindex
#202906432
#202906422
#202895332
#202895322
#202830832
#772653