Finite Group Databasebeta

← #27295241

Subgroups — #27295241

Order 13568.

568 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 568 conjugacy classes of subgroups — too many to draw. Only its 107 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#20274707C211yesmaximal, characteristic
2#20273033C211yesmaximal, characteristic
3#20272745C214yesmaximal, complemented
4C53×#2409C211yesmaximal, characteristic
5#16810647—21no—
6C2×#13133134—21no—
7C2×C22⋊#88322—21no—
8#16810392C2212yes—
9#16810349C2212yes—
10#16810347—21no—
11C23×Q8×C53C411yescharacteristic
12C2×C22⋊Q8×C53—21no—
13C2×C4×D4×C53C2211yescharacteristic
14C53×#395C2212yes—
15C53×#393C412yes—
16C53×#391—21no—
17C53×#386—21no—
18C53×#385—21no—
19C53×#380C411yescharacteristic
20C2×#11719917—22no—
21C22×#88322—41no—
22C4⋊#88322—41no—
23C4⋊#88322—41no—
24#13133134—42no—
25C22⋊#88322—22no—
26C22⋊#88322—41no—
27C22×Q8×C53—21no—
28C22×Q8×C53—41no—
29C22×Q8×C53—41no—
30C22×Q8×C53—44no—
31C22×Q8×C53C2×C412yes—
32C22×D4×C53D411yescharacteristic
33C23×C4×C53—21no—
34C23×C4×C53C2×C411yescharacteristic
35C23×C4×C53D411yescharacteristic
36C22⋊Q8×C53—44no—
37C4×D4×C53—41no—
38C4×D4×C53—41no—
39C2×C4⋊C4×C53—21no—
40C2×C4⋊C4×C53—22no—
41C2×C4⋊C4×C53C2311yescharacteristic
42C2×C22⋊C4×C53—22no—
43C2×C22⋊C4×C53—21no—
44C2×C22⋊C4×C53C2×C411yescharacteristic
45C2×C22⋊C4×C53D411yescharacteristic
46C2×C42×C53D411yescharacteristic
47#95×C53—22no—
48#95×C53—22no—
49#95×C53C2×C412yes—
50C24×C53C22⋊C411yescharacteristic
51C24×C53C22⋊C411yescharacteristic
52C2×Q8×C53—24no—
53C2×Q8×C53—42no—
54C2×Q8×C53—44no—
55C2×Q8×C53—42no—
56C2×Q8×C53—48no—
57C2×Q8×C53—44no—
58C2×Q8×C53—44no—
59C2×Q8×C53—41no—
60C2×Q8×C53—41no—
61C2×D4×C53—41no—
62C2×D4×C53—41no—
63C2×D4×C53SD1612yes—
64C2×D4×C53Q1612yes—
65C22×C4×C53—22no—
66C22×C4×C53—21no—
67C22×C4×C53—21no—
68C22×C4×C53—21no—
69C22×C4×C53—21no—
70C22×C4×C53—22no—
71C22×C4×C53—44no—
72C22×C4×C53—42no—
73C22×C4×C53—42no—
74C22×C4×C53—42no—
75C22×C4×C53C22⋊C411yescharacteristic
76C22×C4×C53C22⋊C411yescharacteristic
77C22×C4×C53C22×C411yescharacteristic
78C22×C4×C53C2×D411yescharacteristic
79C22×C4×C53C2×D411yescharacteristic
80C4⋊C4×C53—41no—
81C4⋊C4×C53—42no—
82C4⋊C4×C53—44no—
83C22⋊C4×C53—41no—
84C22⋊C4×C53—41no—
85C22⋊C4×C53—44no—
86C22⋊C4×C53—42no—
87C42×C53—21no—
88C42×C53—21no—
89#11719917—42no—
90#11719917—42no—
91C2×#88322—42no—
92C2×#88322—42no—
93C2×#88322—82no—
94C23×C53—21no—
95C23×C53—21no—
96C23×C53—21no—
97C23×C53—21no—
98C23×C53—41no—
99C23×C53—41no—
100C23×C53—41no—
101C23×C53—41no—
102C23×C53#10312yes—
103C23×C53#10312yes—
104C23×C53C2×C22⋊C411yescharacteristic
105Q8×C53—44no—
106Q8×C53—44no—
107Q8×C53—44no—
108Q8×C53—44no—
109D4×C53—42no—
110D4×C53—42no—
111C2×C4×C53—21no—
112C2×C4×C53—21no—
113C2×C4×C53—22no—
114C2×C4×C53—24no—
115C2×C4×C53—21no—
116C2×C4×C53—21no—
117C2×C4×C53—21no—
118C2×C4×C53—42no—
119C2×C4×C53—42no—
120C2×C4×C53—48no—
121C2×C4×C53—42no—
122C2×C4×C53—42no—
123C2×C4×C53—42no—
124C2×C4×C53—41no—
125C2×C4×C53—41no—
126C2×C4×C53—41no—
127C2×C4×C53—42no—
128C2×C4×C53—42no—
129C2×C4×C53—42no—
130C2×C4×C53—81no—
131C2×C4×C53#9911yescharacteristic
132C2×C4×C53#9911yescharacteristic
133C2×C4×C53C2×SD1611yescharacteristic
134C2×C4×C53C2×Q1611yescharacteristic
135#88322—82no—
136#88322—82no—
137#11904—531nomaximal, Sylow
138C22×C53—21no—
139C22×C53—21no—
140C22×C53—22no—
141C22×C53—22no—
142C22×C53—42no—
143C22×C53—42no—
144C22×C53—42no—
145C22×C53—41no—
146C22×C53—41no—
147C22×C53—41no—
148C22×C53—41no—
149C22×C53—81no—
150C22×C53C2×#9911yescharacteristic
151C22×C53C2×#10311yescharacteristic
152C22×C53#42111yescharacteristic
153C4×C53—21no—
154C4×C53—21no—
155C4×C53—21no—
156C4×C53—41no—
157C4×C53—44no—
158C4×C53—41no—
159C4×C53—41no—
160C4×C53—82no—
161C4×C53AΣL(1,𝔽16)12yes—
162#2409D5311yescharacteristic
163#1800—531no—
164#1459—531no—
165#1354—534no—
166C2×C53—21no—
167C2×C53—21no—
168C2×C53—42no—
169C2×C53—42no—
170C2×C53#152011yescharacteristic
171C2×C53C2×AΣL(1,𝔽16)11yescharacteristic
172C2×C53#213511yescharacteristic
173C23×Q8Dic5311yescharacteristic
174C2×C22⋊Q8—21no—
175C2×C4×D4C2×D5311yescharacteristic
176#423—1061no—
177C2×#100—1061no—
178C2×C22⋊C8—1061no—
179#395C2×D5312yes—
180#393Dic5312yes—
181#391—21no—
182#386—21no—
183#385—21no—
184#380Dic5311yescharacteristic
185#343—532no—
186#324—532no—
187#323—1061no—
188C53#1190411yescharacteristic, complemented, Sylow
189C22×Q8—21no—
190C22×Q8—41no—
191C22×Q8—41no—
192C22×Q8—44no—
193C22×Q8C2×Dic5312yes—
194C22×D4C22⋊D5311yescharacteristic
195C23×C4—21no—
196C23×C4C2×Dic5311yescharacteristic
197C23×C4C22⋊D5311yescharacteristic
198C2×M16—1062no—
199C22×C8—2121no—
200C22⋊Q8—44no—
201C4×D4—41no—
202C4×D4—41no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4C22×D5311yescharacteristic
206C2×C22⋊C4—22no—
207C2×C22⋊C4—21no—
208C2×C22⋊C4C2×Dic5311yescharacteristic
209C2×C22⋊C4C22⋊D5311yescharacteristic
210C2×C42C22⋊D5311yescharacteristic
211C4⋊C8—2121no—
212C4⋊C8—2121no—
213#100—2122no—
214C22⋊C8—1062no—
215C22⋊C8—2121no—
216#95—22no—
217#95—22no—
218#95C2×Dic5312yes—
219C24C22⋊Dic5311yescharacteristic
220C24C22⋊Dic5311yescharacteristic
221C2×Q8—24no—
222C2×Q8—44no—
223C2×Q8—42no—
224C2×Q8—42no—
225C2×Q8—44no—
226C2×Q8—48no—
227C2×Q8—44no—
228C2×Q8—41no—
229C2×Q8—41no—
230C2×D4—41no—
231C2×D4—41no—
232C2×D4#1171992412yes—
233C2×D4#1171992512yes—
234C22×C4—22no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—22no—
240C22×C4—44no—
241C22×C4—42no—
242C22×C4—42no—
243C22×C4—42no—
244C22×C4C22⋊Dic5311yescharacteristic
245C22×C4C22⋊Dic5311yescharacteristic
246C22×C4C22×Dic5311yescharacteristic
247C22×C4C2×C22⋊D5311yescharacteristic
248C22×C4C2×C22⋊D5311yescharacteristic
249M16—2122no—
250M16—2122no—
251C2×C8—4242no—
252C2×C8—2122no—
253C2×C8—2122no—
254C4⋊C4—41no—
255C4⋊C4—42no—
256C4⋊C4—44no—
257C22⋊C4—41no—
258C22⋊C4—41no—
259C22⋊C4—44no—
260C22⋊C4—42no—
261C42—21no—
262C42—21no—
263C23—21no—
264C23—21no—
265C23—21no—
266C23—21no—
267C23—41no—
268C23—41no—
269C23—41no—
270C23—41no—
271C23#1313314112yes—
272C23#1313314112yes—
273C23C2×C22⋊Dic5311yescharacteristic
274Q8—44no—
275Q8—44no—
276Q8—44no—
277Q8—44no—
278D4—42no—
279D4—42no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—22no—
283C2×C4—24no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—48no—
290C2×C4—42no—
291C2×C4—42no—
292C2×C4—42no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—42no—
297C2×C4—42no—
298C2×C4—42no—
299C2×C4—81no—
300C2×C4#1313313211yescharacteristic
301C2×C4#1313313211yescharacteristic
302C2×C4C2×#1171992411yescharacteristic
303C2×C4C2×#1171992511yescharacteristic
304C8—4242no—
305C8—4242no—
306C22—21no—
307C22—21no—
308C22—22no—
309C22—22no—
310C22—42no—
311C22—42no—
312C22—42no—
313C22—41no—
314C22—41no—
315C22—41no—
316C22—41no—
317C22—81no—
318C22C2×#1313313211yescharacteristic
319C22C2×#1313314111yescharacteristic
320C22#1681064011yescharacteristic
321C4—21no—
322C4—21no—
323C4—21no—
324C4—41no—
325C4—44no—
326C4—41no—
327C4—41no—
328C4—82no—
329C4#1681041012yes—
330C2—21no—
331C2—21no—
332C2—42no—
333C2—42no—
334C2#2027328711yescharacteristic
335C2C2×#1681041011yescharacteristic
336C2#2027641011yescharacteristic
subgroupG/H
centerC22C2×#13133132
commutatorC22×C4×C53C22×C4
FrattiniC2×C4⋊C4C22×D53
FittingC53×#2409C2
socleC22×C53C2×#99
prime radicalC22×D4×C53D4
cosocleC2×C4⋊C4×C53C23
hypercenter#2409D53
hypocenterC53#11904
LCSG ▷ C22×C4×C53 ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2409

Complements

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

NH ≅ G/Ntype
3. #20272745C2semidirect
188. C53#11904semidirect

Maximal subgroups

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

subgroupindex
#202747072
#202730332
#202727452
C53×#24092
#1190453