Finite Group Databasebeta

← #28309492

Subgroups — #28309492

Order 13568.

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

normal subgroups only

This group has 476 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#20290581C211yesmaximal, characteristic, complemented
2#20289957C211yesmaximal, characteristic, complemented
3#20289697C214yesmaximal, complemented
4#20279718C211yesmaximal, characteristic
5#16811640—21no—
6#16811626—21no—
7C2×#13133278—21no—
8#16811503C2212yes—
9#16811469C2212yes—
10#16811468—21no—
11C2×#13133204—21no—
12C2×C4×Dic106C2211yescharacteristic
13#16810537C412yes—
14#16810537C2212yes—
15#16810526—21no—
16#16810526—21no—
17#16810526—21no—
18C4×C4⋊Dic53C411yescharacteristic
19C2×#127×C53C411yescharacteristic
20C2×#11719940—22no—
21#13133284—41no—
22#13133284—41no—
23C4⋊#88324—41no—
24#13133278—42no—
25C22⋊#88324—22no—
26C4×#88324—41no—
27C22×Dic106D411yescharacteristic
28#13133204—42no—
29#13133204—42no—
30C4×Dic106—41no—
31C4×Dic106—41no—
32C2×C4⋊Dic53—21no—
33C2×C4⋊Dic53D411yescharacteristic
34C2×#11719919C2×C411yescharacteristic
35C2×#11719919C2311yescharacteristic
36C2×C22⋊Dic53—22no—
37C2×C22⋊Dic53—22no—
38C2×C22⋊Dic53—22no—
39C2×C22⋊Dic53—22no—
40C2×C4×Dic53—21no—
41C2×C4×Dic53D411yescharacteristic
42C2×C4×Dic53D411yescharacteristic
43#13133124C2×C412yes—
44C22×D4×C53—21no—
45C22×D4×C53—21no—
46C22×D4×C53C2×C412yes—
47#127×C53—41no—
48#127×C53—41no—
49C2×C42×C53C2×C411yescharacteristic
50C24×C53—22no—
51C2×D4×C53—42no—
52C2×D4×C53—42no—
53C2×D4×C53—42no—
54C2×D4×C53—42no—
55C2×D4×C53—42no—
56C2×D4×C53—41no—
57C2×D4×C53—41no—
58C2×D4×C53—42no—
59C2×D4×C53—82no—
60C22×C4×C53C22⋊C411yescharacteristic
61C22×C4×C53C22⋊C411yescharacteristic
62C22×C4×C53C22×C411yescharacteristic
63C22×Dic53—22no—
64C22×Dic53C22⋊C411yescharacteristic
65C22×Dic53C22⋊C411yescharacteristic
66C22×Dic53C2×D411yescharacteristic
67C22×Dic53C2×D411yescharacteristic
68C2×Dic106—41no—
69C2×Dic106—41no—
70C2×Dic106D812yes—
71C2×Dic106SD1612yes—
72#11719940—42no—
73#11719940—42no—
74C2×#88324—42no—
75C2×#88324—42no—
76C42×C53—21no—
77C42×C53—21no—
78C22⋊Dic53—44no—
79C22⋊Dic53—42no—
80C22⋊Dic53—42no—
81C22⋊Dic53—82no—
82C22⋊Dic53—82no—
83C4⋊Dic53—41no—
84C4⋊Dic53—42no—
85#11719919—41no—
86#11719919—41no—
87C4×Dic53—21no—
88C4×Dic53—21no—
89C4×Dic53—21no—
90C4×Dic53—21no—
91C4×Dic53—42no—
92C23×C53—42no—
93C23×C53—42no—
94C23×C53—42no—
95C23×C53—42no—
96C23×C53—42no—
97C23×C53—42no—
98C23×C53—42no—
99C23×C53C2×C22⋊C411yescharacteristic
100D4×C53—82no—
101D4×C53—84no—
102D4×C53—84no—
103D4×C53—82no—
104C2×C4×C53—21no—
105C2×C4×C53—21no—
106C2×C4×C53—22no—
107C2×C4×C53—21no—
108C2×C4×C53—41no—
109C2×C4×C53#9911yescharacteristic
110C2×C4×C53#9911yescharacteristic
111C2×C4×C53#10212yes—
112C2×Dic53—21no—
113C2×Dic53—21no—
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×Dic53#10212yes—
124C2×Dic53C2×D811yescharacteristic
125C2×Dic53C2×SD1611yescharacteristic
126Dic106—42no—
127Dic106—42no—
128#88324—82no—
129#88324—82no—
130#11882—531nomaximal, Sylow
131C22×C53—21no—
132C22×C53—21no—
133C22×C53—82no—
134C22×C53—82no—
135C22×C53—82no—
136C22×C53—82no—
137C22×C53—82no—
138C22×C53—82no—
139C22×C53—82no—
140C22×C53C2×#9911yescharacteristic
141C22×C53C2×#10211yescharacteristic
142C22×C53#42111yescharacteristic
143C4×C53—21no—
144C4×C53—22no—
145C4×C53—41no—
146C4×C53#35512yes—
147Dic53—21no—
148Dic53—22no—
149Dic53—21no—
150Dic53—41no—
151Dic53—41no—
152Dic53—82no—
153#2548—531no—
154#1801—531no—
155#1461—531no—
156#1345—534no—
157C2×C53—21no—
158C2×C53—21no—
159C2×C53—162no—
160C2×C53#151911yescharacteristic
161C2×C53#212211yescharacteristic
162C2×C53C2×#35511yescharacteristic
163C2×#127#615711yescharacteristic
164C2×#124—1061no—
165C2×C4×Q8—531no—
166#431—1061no—
167#423—1061no—
168C2×#101—1061no—
169#394—532no—
170#394—532no—
171#390—1061no—
172#390—1061no—
173#390—1061no—
174C4×C4⋊C4—531no—
175#343—532no—
176#324—532no—
177#324—1061no—
178C53#1188211yescharacteristic, complemented, Sylow
179C22×Q8—531no—
180C22×D4—21no—
181C22×D4—21no—
182C22×D4C2×#615712yes—
183C2×M16—1062no—
184#127—41no—
185#127—41no—
186#124—2122no—
187#124—2122no—
188C4×Q8—2121no—
189C4×Q8—2121no—
190C2×C4⋊C4—531no—
191C2×C4⋊C4—531no—
192C2×C4⋊C4—531no—
193C2×C4⋊C4—1061no—
194C2×C22⋊C4—1062no—
195C2×C22⋊C4—1062no—
196C2×C22⋊C4—1062no—
197C2×C22⋊C4—1062no—
198C2×C42—531no—
199C2×C42—531no—
200C2×C42—1061no—
201C2×C42C2×#615711yescharacteristic
202C4⋊C8—2121no—
203C4⋊C8—2121no—
204C4⋊C8—2121no—
205#101—2122no—
206C22⋊C8—1062no—
207C4×C8—2121no—
208#95—532no—
209C24—22no—
210C2×Q8—532no—
211C2×Q8—532no—
212C2×Q8—2121no—
213C2×Q8—2121no—
214C2×D4—42no—
215C2×D4—42no—
216C2×D4—42no—
217C2×D4—42no—
218C2×D4—42no—
219C2×D4—41no—
220C2×D4—41no—
221C2×D4—42no—
222C2×D4—82no—
223C22×C4—531no—
224C22×C4—531no—
225C22×C4—531no—
226C22×C4—531no—
227C22×C4—1062no—
228C22×C4C22⋊#615711yescharacteristic
229C22×C4C22⋊#615711yescharacteristic
230C22×C4C22×#615711yescharacteristic
231M16—2122no—
232M16—2122no—
233C2×C8—2122no—
234C2×C8—2122no—
235C4⋊C4—2121no—
236C4⋊C4—2121no—
237C4⋊C4—2121no—
238C4⋊C4—2122no—
239C22⋊C4—4242no—
240C22⋊C4—4242no—
241C22⋊C4—2124no—
242C22⋊C4—2122no—
243C22⋊C4—2122no—
244C42—21no—
245C42—21no—
246C42—1061no—
247C42—1061no—
248C42—1061no—
249C42—1061no—
250C42—2122no—
251C23—42no—
252C23—42no—
253C23—42no—
254C23—42no—
255C23—42no—
256C23—42no—
257C23—42no—
258C23C2×C22⋊#615711yescharacteristic
259Q8—2122no—
260Q8—2122no—
261D4—84no—
262D4—82no—
263D4—84no—
264D4—82no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—22no—
268C2×C4—21no—
269C2×C4—41no—
270C2×C4—532no—
271C2×C4—531no—
272C2×C4—531no—
273C2×C4—1061no—
274C2×C4—1061no—
275C2×C4—1061no—
276C2×C4—1061no—
277C2×C4—1061no—
278C2×C4—1061no—
279C2×C4—2122no—
280C2×C4—2122no—
281C2×C4—2122no—
282C2×C4—2121no—
283C2×C4—2121no—
284C2×C4#1313327511yescharacteristic
285C2×C4#1313327511yescharacteristic
286C2×C4#1313327912yes—
287C8—4242no—
288C8—4242no—
289C22—21no—
290C22—21no—
291C22—82no—
292C22—82no—
293C22—82no—
294C22—82no—
295C22—82no—
296C22—82no—
297C22—82no—
298C22C2×#1313327511yescharacteristic
299C22C2×#1313327911yescharacteristic
300C22#1681162111yescharacteristic
301C4—22no—
302C4—21no—
303C4—41no—
304C4—1061no—
305C4—1062no—
306C4—1061no—
307C4—4242no—
308C4—2121no—
309C4—2121no—
310C4#1681153112yes—
311C2—21no—
312C2—21no—
313C2—162no—
314C2#2029015111yescharacteristic
315C2#2029097411yescharacteristic
316C2C2×#1681153111yescharacteristic
subgroupG/H
centerC22C2×#13133275
commutatorC22×C4×C53C22×C4
FrattiniC22×C4C22×#6157
Fitting, prime radicalC2×#127×C53C4
socleC22×C53C2×#99
cosocleC2×#11719919C23
hypercenterC2×#127#6157
hypocenterC53#11882
LCSG ▷ C22×C4×C53 ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C42 ◁ C2×#127

Complements

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

NH ≅ G/Ntype
1. #20290581C2semidirect
2. #20289957C2semidirect
3. #20289697C2semidirect
178. C53#11882semidirect

Maximal subgroups

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

subgroupindex
#202905812
#202899572
#202896972
#202797182
#1188253