Finite Group Databasebeta

← #27275444

Subgroups — #27275444

Order 13568.

810 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 810 conjugacy classes of subgroups — too many to draw. Only its 143 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
1C22×#13133207C211yesmaximal, characteristic
2#20274781C212yesmaximal
3C2×#16810349C211yesmaximal, characteristic
4C2×#16810348C212yesmaximal
5C2×C53×#324C211yesmaximal, characteristic
6C2×#13133207—24no—
7C2×#13133207—24no—
8C2×#13133207—22no—
9C2×#13133207—22no—
10C22×C4⋊Dic53C411yescharacteristic
11C22×#11719919C411yescharacteristic
12C22×#11719919C411yescharacteristic
13C22×#11719919C411yescharacteristic
14C22×#11719919C2211yescharacteristic
15C22×C4×Dic53C2211yescharacteristic
16C2×C22⋊#88322C2212yes—
17#16810377—24no—
18#16810349—22no—
19#16810348—24no—
20C22×C4⋊C4×C53C2211yescharacteristic
21C2×C22⋊C8×C53C2212yes—
22C53×#324—22no—
23C23×Dic53C2×C411yescharacteristic
24C23×Dic53C2×C411yescharacteristic
25C23×Dic53C2×C411yescharacteristic
26C23×Dic53C2×C411yescharacteristic
27C22×#88322—22no—
28#13133207—216no—
29#13133207—216no—
30C2×C4⋊Dic53—22no—
31C2×C4⋊Dic53—22no—
32C2×C4⋊Dic53—22no—
33C2×#11719919—22no—
34C2×#11719919—22no—
35C2×#11719919—22no—
36C2×#11719919—22no—
37C2×#11719919—21no—
38C2×#11719919—21no—
39C2×#11719919—22no—
40C2×#11719919—22no—
41C2×#11719919—22no—
42C2×#11719919—22no—
43C2×#11719919—22no—
44C2×#11719919—22no—
45C2×#11719919D414yes—
46C2×C4×Dic53—21no—
47C2×C4×Dic53—22no—
48C2×C4×Dic53—21no—
49C2×C4×Dic53D414yes—
50C22⋊#88322—22no—
51C22⋊#88322—22no—
52C23×C4×C53C2×C411yescharacteristic
53C23×C4×C53C2×C411yescharacteristic
54C23×C4×C53C2311yescharacteristic
55C22×C8×C53—22no—
56C2×C4⋊C4×C53—22no—
57C2×C4⋊C4×C53—22no—
58C2×C4⋊C4×C53D412yes—
59C2×C4⋊C4×C53D412yes—
60C22⋊C8×C53—22no—
61C22⋊C8×C53—22no—
62C24×C53C22×C411yescharacteristic
63C22×C4×C53—21no—
64C22×C4×C53—21no—
65C22×C4×C53—21no—
66C22×C4×C53—21no—
67C22×C4×C53—22no—
68C22×C4×C53—22no—
69C22×C4×C53—42no—
70C22×C4×C53—42no—
71C22×C4×C53C22⋊C412yes—
72C22×C4×C53C22⋊C412yes—
73C22×C4×C53C2×D412yes—
74C22×C4×C53C2×D411yescharacteristic
75C22×C4×C53C2×D412yes—
76C22×C4×C53C2×D411yescharacteristic
77C22×Dic53—21no—
78C22×Dic53—21no—
79C22×Dic53—24no—
80C22×Dic53—21no—
81C22×Dic53—21no—
82C22×Dic53—21no—
83C22×Dic53—21no—
84C22×Dic53—21no—
85C22×Dic53—21no—
86C22×Dic53—22no—
87C22×Dic53—22no—
88C22×Dic53—22no—
89C22×Dic53—22no—
90C22×Dic53—42no—
91C22×Dic53C22⋊C412yes—
92C22×Dic53C22⋊C412yes—
93C22×Dic53C22⋊C412yes—
94C22×Dic53C22⋊C412yes—
95C2×C8×C53—44no—
96C2×C8×C53—42no—
97C4⋊C4×C53—24no—
98C4⋊C4×C53—24no—
99C4⋊Dic53—28no—
100#11719919—28no—
101#11719919—28no—
102#11719919—28no—
103#11719919—24no—
104#11719919—24no—
105C4×Dic53—24no—
106C4×Dic53—24no—
107C2×#88322—44no—
108C2×#88322—42no—
109C23×C53—22no—
110C23×C53—22no—
111C23×C53C2×C22⋊C411yescharacteristic
112C23×C53C2×C22⋊C411yescharacteristic
113C23×C53C2×C22⋊C411yescharacteristic
114C23×C53#13112yes—
115C23×C53#13112yes—
116C2×C4×C53—24no—
117C2×C4×C53—22no—
118C2×C4×C53—22no—
119C2×C4×C53—22no—
120C2×C4×C53—22no—
121C2×C4×C53—41no—
122C2×C4×C53—42no—
123C2×C4×C53—41no—
124C2×C4×C53—42no—
125C2×C4×C53—42no—
126C2×C4×C53—42no—
127C2×C4×C53—42no—
128C2×C4×C53—42no—
129C2×C4×C53#12012yes—
130C2×C4×C53#12012yes—
131C2×Dic53—24no—
132C2×Dic53—24no—
133C2×Dic53—22no—
134C2×Dic53—22no—
135C2×Dic53—22no—
136C2×Dic53—22no—
137C2×Dic53—22no—
138C2×Dic53—24no—
139C2×Dic53—24no—
140C2×Dic53—42no—
141C2×Dic53—42no—
142C2×Dic53—42no—
143C2×Dic53—41no—
144C2×Dic53—41no—
145C2×Dic53—41no—
146C2×Dic53—41no—
147C2×Dic53—42no—
148C2×Dic53—42no—
149C2×Dic53#10114yes—
150C8×C53—82no—
151#88322—82no—
152#8306—531nomaximal, Sylow
153C22×C53—24no—
154C22×C53—24no—
155C22×C53—21no—
156C22×C53—21no—
157C22×C53—21no—
158C22×C53—21no—
159C22×C53#37911yescharacteristic
160C22×C53#40812yes—
161C22×C53#40812yes—
162C22×C53#40811yescharacteristic
163C22×C53#40811yescharacteristic
164C22×C53#41012yes—
165C22×C53C2×#10112yes—
166C4×C53—22no—
167C4×C53—22no—
168C4×C53—42no—
169C4×C53—42no—
170Dic53—24no—
171Dic53—42no—
172Dic53—42no—
173Dic53—42no—
174C22×#125—531no—
175#1817—532no—
176C2×#324—532no—
177C2×#324—531no—
178C2×#324D5311yescharacteristic
179C2×C53—22no—
180C2×C53—21no—
181C2×C53—21no—
182C2×C53#147014yes—
183C2×C53#178011yescharacteristic
184C2×C53#178211yescharacteristic
185C2×C53#178411yescharacteristic
186C2×#125—1064no—
187C2×#125—1064no—
188C2×#125—1062no—
189C2×#125—1062no—
190C22×C4⋊C4—531no—
191C22×C4⋊C4—531no—
192C22×C4⋊C4—531no—
193C22×C4⋊C4—531no—
194C22×C4⋊C4—531no—
195C22×C4⋊C4C2×D5311yescharacteristic
196C22×C42—531no—
197C2×C22⋊C8—532no—
198C2×C22⋊C8C2×D5312yes—
199#336—1064no—
200#324—22no—
201#324—1064no—
202#324—1062no—
203C53#830611yescharacteristic, complemented, Sylow
204C23×C4—531no—
205C23×C4—531no—
206C23×C4—531no—
207C23×C4—531no—
208C23×C4C4×D5311yescharacteristic
209C23×C4C4×D5311yescharacteristic
210C23×C4C22×D5311yescharacteristic
211C22×C8—22no—
212C22×C8—1062no—
213#125—10616no—
214#125—10616no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—534no—
218C2×C4⋊C4—1062no—
219C2×C4⋊C4—1062no—
220C2×C4⋊C4—1062no—
221C2×C4⋊C4—1062no—
222C2×C4⋊C4—1062no—
223C2×C4⋊C4—1062no—
224C2×C4⋊C4—1062no—
225C2×C4⋊C4—1062no—
226C2×C4⋊C4—1061no—
227C2×C4⋊C4—1062no—
228C2×C4⋊C4—1061no—
229C2×C4⋊C4—1062no—
230C2×C4⋊C4—1062no—
231C2×C4⋊C4—1062no—
232C2×C4⋊C4—1062no—
233C2×C4⋊C4D21212yes—
234C2×C4⋊C4C22⋊D5312yes—
235C2×C42—534no—
236C2×C42—1061no—
237C2×C42—1062no—
238C2×C42—1061no—
239C22⋊C8—22no—
240C22⋊C8—22no—
241C22⋊C8—1062no—
242C22⋊C8—1062no—
243C24C2×C4×D5311yescharacteristic
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—42no—
251C22×C4—42no—
252C22×C4—532no—
253C22×C4—532no—
254C22×C4—532no—
255C22×C4—532no—
256C22×C4—1064no—
257C22×C4—1061no—
258C22×C4—1061no—
259C22×C4—1061no—
260C22×C4—1061no—
261C22×C4—1061no—
262C22×C4—1061no—
263C22×C4—1061no—
264C22×C4—1061no—
265C22×C4—1062no—
266C22×C4—1062no—
267C22×C4—1062no—
268C22×C4—1062no—
269C22×C4—2122no—
270C22×C4#1171992112yes—
271C22×C4#1171992112yes—
272C22×C4C2×D21211yescharacteristic
273C22×C4D4×D5312yes—
274C22×C4D4×D5312yes—
275C22×C4C2×C22⋊D5311yescharacteristic
276C2×C8—44no—
277C2×C8—42no—
278C2×C8—2124no—
279C2×C8—2122no—
280C4⋊C4—24no—
281C4⋊C4—24no—
282C4⋊C4—1068no—
283C4⋊C4—1068no—
284C4⋊C4—1068no—
285C4⋊C4—1068no—
286C4⋊C4—1064no—
287C4⋊C4—1064no—
288C42—1064no—
289C42—1064no—
290C23—22no—
291C23—22no—
292C23C22⋊C4×D5311yescharacteristic
293C23C22⋊C4×D5311yescharacteristic
294C23C2×#1171992111yescharacteristic
295C23#1313322312yes—
296C23#1313322512yes—
297C2×C4—24no—
298C2×C4—22no—
299C2×C4—22no—
300C2×C4—22no—
301C2×C4—22no—
302C2×C4—41no—
303C2×C4—42no—
304C2×C4—41no—
305C2×C4—42no—
306C2×C4—42no—
307C2×C4—42no—
308C2×C4—42no—
309C2×C4—42no—
310C2×C4—534no—
311C2×C4—1064no—
312C2×C4—1064no—
313C2×C4—1062no—
314C2×C4—1062no—
315C2×C4—1064no—
316C2×C4—1064no—
317C2×C4—1062no—
318C2×C4—1062no—
319C2×C4—1062no—
320C2×C4—2122no—
321C2×C4—2122no—
322C2×C4—2122no—
323C2×C4—2121no—
324C2×C4—2121no—
325C2×C4—2121no—
326C2×C4—2121no—
327C2×C4—2122no—
328C2×C4—2122no—
329C2×C4#1313318512yes—
330C2×C4#1313318712yes—
331C8—82no—
332C8—4242no—
333C22—24no—
334C22—24no—
335C22—21no—
336C22—21no—
337C22—21no—
338C22—21no—
339C22#1681047811yescharacteristic
340C22#1681058812yes—
341C22#1681058812yes—
342C22#1681058911yescharacteristic
343C22#1681059111yescharacteristic
344C22#1681059512yes—
345C22#101×D5312yes—
346C4—22no—
347C4—22no—
348C4—42no—
349C4—42no—
350C4—1064no—
351C4—2122no—
352C4—2122no—
353C4—2122no—
354C2—22no—
355C2—21no—
356C2—21no—
357C2#2027306914yes—
358C2#2027462611yescharacteristic
359C2#2027463111yescharacteristic
360C2#2027463711yescharacteristic
subgroupG/H
centerC23C22⋊C4×D53
commutator, prime radicalC24×C53C22×C4
FrattiniC23×C4C22×D53
FittingC2×C53×#324C2
socleC23×C53C2×C22⋊C4
cosocleC23×C4×C53C23
hypercenterC2×#324D53
hypocenterC53#8306
LCSG ▷ C24×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#324

Complements

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

NH ≅ G/Ntype
203. C53#8306semidirect

Maximal subgroups

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

subgroupindex
C22×#131332072
#202747812
C2×#168103492
C2×#168103482
C2×C53×#3242
#830653