Finite Group Databasebeta

← #28308244

Subgroups — #28308244

Order 13568.

470 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 470 conjugacy classes of subgroups — too many to draw. Only its 76 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#20289975C211yesmaximal, characteristic
2#20289971C211yesmaximal, characteristic
3#20289659C212yesmaximal, complemented
4#20289641C212yesmaximal, complemented
5#123×Dic53C211yesmaximal, characteristic
6#16811643—21no—
7#16811593—21no—
8#16811468—21no—
9#16811466—21no—
10#16811466C2212yes—
11#16811466C2212yes—
12C2×D4×Dic53C2211yescharacteristic
13C22×C4×Dic53C411yescharacteristic
14#16810516—21no—
15#16810509—21no—
16#16810503C411yescharacteristic
17#16810498C2211yescharacteristic
18#16810486C2211yescharacteristic
19C4⋊C4×Dic53—21no—
20C22⋊C4×Dic53—21no—
21C22⋊C4×Dic53C411yescharacteristic
22C2×#123×C53C411yescharacteristic
23C2×#11719940—41no—
24#13133284—41no—
25C22⋊#88324—22no—
26C22⋊#88324—22no—
27C22⋊#88324—41no—
28#13133272—41no—
29C23×Dic53C2×C411yescharacteristic
30C23×Dic53C2×C411yescharacteristic
31C23×Dic53D411yescharacteristic
32D4×Dic53—41no—
33D4×Dic53—41no—
34C2×C4⋊Dic53—21no—
35C2×C4⋊Dic53D411yescharacteristic
36C2×C22⋊Dic53—21no—
37C2×C22⋊Dic53C2×C411yescharacteristic
38C2×C22⋊Dic53D411yescharacteristic
39C2×C22⋊Dic53C2311yescharacteristic
40C2×C4×Dic53—21no—
41C2×C4×Dic53—21no—
42C2×C4×Dic53—21no—
43C2×C4×Dic53—41no—
44C2×C4×Dic53—41no—
45C2×C4×Dic53D411yescharacteristic
46#13133124—21no—
47#13133124—21no—
48#13133124—21no—
49#13133124—21no—
50#13133124—21no—
51C22×D4×C53C2×C411yescharacteristic
52C23×C4×C53C2×C411yescharacteristic
53#123×C53—22no—
54#123×C53—22no—
55C2×C4⋊C4×C53—21no—
56C2×C22⋊C4×C53—21no—
57C2×C22⋊C4×C53C2×C411yescharacteristic
58C24×C53C22⋊C411yescharacteristic
59C24×C53C22×C411yescharacteristic
60C2×D4×C53—21no—
61C2×D4×C53—21no—
62C2×D4×C53—41no—
63C2×D4×C53—41no—
64C22×C4×C53—21no—
65C22×C4×C53—21no—
66C22×C4×C53—21no—
67C22×C4×C53—41no—
68C22×C4×C53—41no—
69C22×C4×C53C22⋊C411yescharacteristic
70C22×C4×C53M1612yescomplemented
71C22×Dic53—21no—
72C22×Dic53—21no—
73C22×Dic53—21no—
74C22×Dic53—21no—
75C22×Dic53—21no—
76C22×Dic53—21no—
77C22×Dic53—42no—
78C22×Dic53—42no—
79C22×Dic53—42no—
80C22×Dic53—41no—
81C22×Dic53C22⋊C411yescharacteristic
82C22×Dic53C22⋊C411yescharacteristic
83C22×Dic53C2×D411yescharacteristic
84C22×Dic53C2×D411yescharacteristic
85#11719940—82no—
86C2×#88324—42no—
87C2×#88324—42no—
88C4⋊C4×C53—24no—
89C22⋊C4×C53—24no—
90C22⋊C4×C53—21no—
91C22⋊C4×C53M1612yescomplemented
92C22⋊Dic53—41no—
93C22⋊Dic53—41no—
94C22⋊Dic53—42no—
95C22⋊Dic53—41no—
96C4⋊Dic53—41no—
97C4⋊Dic53—42no—
98C4×Dic53—21no—
99C4×Dic53—21no—
100C4×Dic53—42no—
101C4×Dic53—42no—
102C4×Dic53—41no—
103C4×Dic53—41no—
104C23×C53—21no—
105C23×C53—21no—
106C23×C53—21no—
107C23×C53—21no—
108C23×C53—21no—
109C23×C53—41no—
110C23×C53—41no—
111C23×C53—41no—
112C23×C53—41no—
113C23×C53C2×C22⋊C411yescharacteristic
114C23×C53C2×M1611yescharacteristic
115C23×C53C2×M1611yescharacteristic
116D4×C53—42no—
117D4×C53—42no—
118C2×C4×C53—22no—
119C2×C4×C53—22no—
120C2×C4×C53—21no—
121C2×C4×C53—21no—
122C2×C4×C53—21no—
123C2×C4×C53—42no—
124C2×C4×C53—41no—
125C2×C4×C53—41no—
126C2×C4×C53—41no—
127C2×C4×C53—41no—
128C2×C4×C53—42no—
129C2×Dic53—22no—
130C2×Dic53—22no—
131C2×Dic53—21no—
132C2×Dic53—21no—
133C2×Dic53—21no—
134C2×Dic53—42no—
135C2×Dic53—41no—
136C2×Dic53—42no—
137C2×Dic53—41no—
138C2×Dic53—42no—
139C2×Dic53—42no—
140C2×Dic53—41no—
141C2×Dic53—41no—
142C2×Dic53—41no—
143C2×Dic53—42no—
144C2×Dic53—41no—
145C2×Dic53—41no—
146C2×Dic53—41no—
147C2×Dic53—81no—
148C2×Dic53#9911yescharacteristic
149C2×Dic53#9911yescharacteristic
150#88324—82no—
151#88324—82no—
152#11526—531nomaximal, Sylow
153C22×C53—21no—
154C22×C53—21no—
155C22×C53—21no—
156C22×C53—21no—
157C22×C53—41no—
158C22×C53—41no—
159C22×C53—42no—
160C22×C53—42no—
161C22×C53—41no—
162C22×C53—41no—
163C22×C53—41no—
164C22×C53—41no—
165C22×C53—41no—
166C22×C53—81no—
167C22×C53#40711yescharacteristic
168C22×C53C2×#9911yescharacteristic
169C22×C53#41311yescharacteristic
170C4×C53—42no—
171C4×C53—41no—
172C4×C53—42no—
173Dic53—21no—
174Dic53—41no—
175Dic53—41no—
176Dic53—41no—
177Dic53—42no—
178Dic53—81no—
179Dic53#35212yes—
180C4×#123—531no—
181#1466—531no—
182#1465—531no—
183#1326—532no—
184#1317—532no—
185C2×C53—21no—
186C2×C53—21no—
187C2×C53—41no—
188C2×C53—41no—
189C2×C53—81no—
190C2×C53#146511yescharacteristic
191C2×C53C2×#35211yescharacteristic
192C2×C53#212011yescharacteristic
193C2×#123#615711yescharacteristic
194C2×C4×D4—531no—
195C22×C42—531no—
196#432—1061no—
197#407—1061no—
198#388—1061no—
199#387—1061no—
200#386—531no—
201#385—531no—
202#381—531no—
203C4×C4⋊C4—1061no—
204C4×C22⋊C4—531no—
205C4×C22⋊C4—1061no—
206#324—1061no—
207#323—532no—
208#323—532no—
209#323—1061no—
210C53#1152611yescharacteristic, complemented, Sylow
211C22×D4C2×#615711yescharacteristic
212C23×C4—531no—
213C23×C4—531no—
214C23×C4—531no—
215C23×C4C2×#615711yescharacteristic
216C2×M16—2121no—
217#123—22no—
218#123—22no—
219C4×D4—2121no—
220C4×D4—2121no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—531no—
223C2×C4⋊C4—1061no—
224C2×C22⋊C4—21no—
225C2×C22⋊C4—531no—
226C2×C22⋊C4—531no—
227C2×C22⋊C4—531no—
228C2×C22⋊C4—1061no—
229C2×C22⋊C4C2×#615711yescharacteristic
230C2×C42—531no—
231C2×C42—1061no—
232C2×C42—1061no—
233C2×C42—1061no—
234C2×C42—2121no—
235C2×C42—2121no—
236C4⋊C8—2121no—
237C22⋊C8—1062no—
238C22⋊C8—1062no—
239C22⋊C8—2121no—
240#97—2121no—
241#95—1061no—
242#95—1061no—
243#95—1061no—
244#95—1061no—
245#95—1061no—
246C24C22⋊#615711yescharacteristic
247C24C22×#615711yescharacteristic
248C2×D4—21no—
249C2×D4—21no—
250C2×D4—41no—
251C2×D4—41no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—41no—
256C22×C4—41no—
257C22×C4—531no—
258C22×C4—531no—
259C22×C4—531no—
260C22×C4—531no—
261C22×C4—1061no—
262C22×C4—1061no—
263C22×C4—1061no—
264C22×C4—1061no—
265C22×C4—1061no—
266C22×C4—1061no—
267C22×C4—2122no—
268C22×C4—2122no—
269C22×C4—2122no—
270C22×C4—2121no—
271C22×C4#1171994012yescomplemented
272C22×C4C22⋊#615711yescharacteristic
273M16—4242no—
274C2×C8—2122no—
275C2×C8—2122no—
276C4⋊C4—24no—
277C4⋊C4—2121no—
278C4⋊C4—2122no—
279C22⋊C4—24no—
280C22⋊C4—21no—
281C22⋊C4—2121no—
282C22⋊C4—2121no—
283C22⋊C4—2122no—
284C22⋊C4—2121no—
285C22⋊C4#1171994012yescomplemented
286C42—1061no—
287C42—1061no—
288C42—2121no—
289C42—2121no—
290C42—2122no—
291C42—2122no—
292C23—21no—
293C23—21no—
294C23—21no—
295C23—21no—
296C23—21no—
297C23—41no—
298C23—41no—
299C23—41no—
300C23—41no—
301C23C2×C22⋊#615711yescharacteristic
302C23C2×#1171994011yescharacteristic
303C23C2×#1171994011yescharacteristic
304D4—42no—
305D4—42no—
306C2×C4—22no—
307C2×C4—22no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—42no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—42no—
317C2×C4—531no—
318C2×C4—531no—
319C2×C4—1062no—
320C2×C4—1062no—
321C2×C4—1061no—
322C2×C4—1061no—
323C2×C4—1061no—
324C2×C4—4241no—
325C2×C4—2122no—
326C2×C4—2122no—
327C2×C4—2122no—
328C2×C4—2121no—
329C2×C4—2121no—
330C2×C4—2121no—
331C2×C4—2121no—
332C2×C4—2122no—
333C2×C4—2121no—
334C2×C4—2121no—
335C2×C4—2121no—
336C2×C4—2121no—
337C2×C4—2122no—
338C8—4242no—
339C8—4242no—
340C22—21no—
341C22—21no—
342C22—21no—
343C22—21no—
344C22—41no—
345C22—41no—
346C22—42no—
347C22—42no—
348C22—41no—
349C22—41no—
350C22—41no—
351C22—41no—
352C22—41no—
353C22—81no—
354C22#1681159311yescharacteristic
355C22D53⋊#9911yescharacteristic
356C22#1681161011yescharacteristic
357C4—42no—
358C4—41no—
359C4—42no—
360C4—532no—
361C4—1061no—
362C4—4241no—
363C4—2121no—
364C4—2121no—
365C4—2121no—
366C4—2122no—
367C2—21no—
368C2—21no—
369C2—41no—
370C2—41no—
371C2—81no—
372C2#2028997111yescharacteristic
373C2D53⋊#35211yescharacteristic
374C2#2029096911yescharacteristic
subgroupG/H
centerC22D53⋊#99
commutatorC24×C53C22×C4
FrattiniC24C22×#6157
FittingC2×#123×C53C4
socleC22×C53C2×#99
prime radicalC22×D4×C53C2×C4
cosocleC2×C22⋊Dic53C23
hypercenterC2×#123#6157
hypocenterC53#11526
LCSG ▷ C24×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23 ◁ C22×D4 ◁ C2×#123

Complements

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

NH ≅ G/Ntype
3. #20289659C2semidirect
4. #20289641C2semidirect
70. C22×C4×C53M16semidirect
91. C22⋊C4×C53M16semidirect
210. C53#11526semidirect
271. C22×C4#11719940semidirect
285. C22⋊C4#11719940semidirect

Maximal subgroups

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

subgroupindex
#202899752
#202899712
#202896592
#202896412
#123×Dic532
#1152653