Finite Group Databasebeta

← #28299814

Subgroups — #28299814

Order 13568.

550 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 550 conjugacy classes of subgroups — too many to draw. Only its 94 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#20290441C211yesmaximal, characteristic, complemented
2C4×C22⋊#88324C211yesmaximal, characteristic, complemented
3#20289533C212yesmaximal, complemented
4#20289532C212yesmaximal, complemented
5#20283047C211yesmaximal, characteristic
6C2×C22⋊#88324—21no—
7C2×C22⋊#88324—21no—
8C2×#13133272—21no—
9C2×C4×#88324—21no—
10#16811489C2212yes—
11#16811489C2212yes—
12C2×#13133201—21no—
13C2×#13133191C411yescharacteristic
14C22×C4×Dic53C2211yescharacteristic
15#16810524C411yescharacteristic
16#16810524C2211yescharacteristic
17#16810522C411yescharacteristic
18#16810522C2211yescharacteristic
19#16810515—21no—
20#16810515—21no—
21#16810500—21no—
22C53×#380C411yescharacteristic
23C22×#88324—22no—
24C22×#88324—22no—
25C22⋊#88324—24no—
26C22⋊#88324—24no—
27#13133272—42no—
28C4×#88324—42no—
29C22×D212C2×C411yescharacteristic
30C22×C22⋊D53—21no—
31C23×Dic53D411yescharacteristic
32C23×Dic53D411yescharacteristic
33#13133201—44no—
34#13133191—42no—
35C2×C4⋊Dic53—21no—
36C2×C4⋊Dic53—21no—
37C2×C4⋊Dic53C2×C411yescharacteristic
38C2×#11719921—21no—
39C2×#11719921—21no—
40C2×#11719921—21no—
41C2×#11719921—21no—
42C2×#11719921—21no—
43C2×C4×Dic53—42no—
44C2×C4×Dic53C2×C411yescharacteristic
45C2×C4×Dic53D411yescharacteristic
46C2×C4×Dic53D411yescharacteristic
47C2×C4×Dic53C2311yescharacteristic
48#13133124—21no—
49#13133124—21no—
50C23×C4×C53C2×C411yescharacteristic
51C2×C4⋊C4×C53C2×C411yescharacteristic
52C2×C4⋊C4×C53C2×C411yescharacteristic
53C2×C22⋊C4×C53—21no—
54#95×C53—21no—
55C24×C53C22⋊C411yescharacteristic
56C23×D53—21no—
57C22×C4×C53—22no—
58C22×C4×C53—22no—
59C22×C4×C53—21no—
60C22×C4×C53—21no—
61C22×C4×C53—41no—
62C22×C4×C53C22⋊C411yescharacteristic
63C22×C4×C53C22×C411yescharacteristic
64C2×C22⋊D53—42no—
65C2×C22⋊D53—42no—
66C2×C22⋊D53—42no—
67C22×Dic53—22no—
68C22×Dic53—22no—
69C22×Dic53—22no—
70C22×Dic53—22no—
71C22×Dic53—21no—
72C22×Dic53—41no—
73C22×Dic53—41no—
74C22×Dic53C22⋊C411yescharacteristic
75C22×Dic53C22⋊C411yescharacteristic
76C22×Dic53C2×D411yescharacteristic
77C22×Dic53C2×D411yescharacteristic
78C2×D212—22no—
79C2×D212—41no—
80C2×D212—41no—
81C2×#88324—24no—
82C2×#88324—24no—
83C2×#88324—42no—
84C2×#88324—42no—
85C2×#88324—42no—
86C2×#88324—42no—
87C4⋊C4×C53—22no—
88C4⋊C4×C53—41no—
89C22⋊C4×C53—42no—
90#11719921—42no—
91#11719921—42no—
92#11719921—42no—
93#11719921—42no—
94#11719921—42no—
95C4⋊Dic53—42no—
96C4⋊Dic53—42no—
97C4⋊Dic53—41no—
98C4×Dic53—22no—
99C4×Dic53—22no—
100C4×Dic53—41no—
101C4×Dic53—41no—
102C23×C53—21no—
103C23×C53—21no—
104C23×C53—21no—
105C23×C53—21no—
106C23×C53—21no—
107C23×C53—21no—
108C23×C53#10412yes—
109C23×C53C2×C22⋊C411yescharacteristic
110C22×D53—41no—
111C22×D53—41no—
112C22×D53—41no—
113C22×D53—41no—
114C22×D53—41no—
115C22×D53—41no—
116C22×D53—41no—
117C2×C4×C53—21no—
118C2×C4×C53—21no—
119C2×C4×C53—21no—
120C2×C4×C53—21no—
121C2×C4×C53—44no—
122C2×C4×C53—41no—
123C2×C4×C53—41no—
124C2×C4×C53—41no—
125C2×C4×C53—41no—
126C2×C4×C53—41no—
127C2×C4×C53—41no—
128C2×C4×C53#9912yes—
129C2×C4×C53#13112yes—
130C22⋊D53—84no—
131C2×Dic53—21no—
132C2×Dic53—21no—
133C2×Dic53—21no—
134C2×Dic53—21no—
135C2×Dic53—21no—
136C2×Dic53—21no—
137C2×Dic53—21no—
138C2×Dic53—21no—
139C2×Dic53—21no—
140C2×Dic53—44no—
141C2×Dic53—44no—
142C2×Dic53—41no—
143C2×Dic53—41no—
144C2×Dic53—41no—
145C2×Dic53#10412yes—
146C2×Dic53#13512yes—
147C2×Dic53AGL(1,ℤ/8)12yes—
148D212—82no—
149#88324—44no—
150#88324—44no—
151#7674—531nomaximal, Sylow
152C22×C53—22no—
153C22×C53—22no—
154C22×C53—22no—
155C22×C53—41no—
156C22×C53—41no—
157C22×C53—41no—
158C22×C53—41no—
159C22×C53#40811yescharacteristic
160C22×C53C2×#9911yescharacteristic
161C22×C53#41311yescharacteristic
162C22×C53#41611yescharacteristic
163C22×C53#41811yescharacteristic
164C22×C53C2×#10411yescharacteristic
165C22×C53C2×#10411yescharacteristic
166C2×D53—81no—
167C2×D53—81no—
168C2×D53—81no—
169C2×D53—81no—
170C2×D53—81no—
171C2×D53—81no—
172C2×D53—81no—
173C4×C53—22no—
174C4×C53—22no—
175C4×C53—81no—
176C4×C53—81no—
177Dic53—22no—
178Dic53—22no—
179Dic53—22no—
180Dic53—41no—
181Dic53—81no—
182#2854—531no—
183#1749—531no—
184C4×C22⋊C8—531no—
185#1271—532no—
186#1270—532no—
187C2×C53—42no—
188C2×C53#146811yescharacteristic
189C2×C53#148411yescharacteristic
190C2×C53#149411yescharacteristic
191C2×C53#150111yescharacteristic
192C2×C53#150711yescharacteristic
193C2×C53#152411yescharacteristic
194C2×C53#155311yescharacteristic
195D53—161no—
196C2×#123—1061no—
197C2×#121—531no—
198C22×C42—531no—
199C2×C22⋊C8—1061no—
200C2×C22⋊C8—1061no—
201C2×#97—1061no—
202C2×C4×C8—1061no—
203#390—531no—
204#390—531no—
205#389—531no—
206#389—531no—
207#388—1061no—
208#388—1061no—
209#386—1061no—
210#380#615711yescharacteristic
211#336—532no—
212#336—532no—
213C53#767411yescharacteristic, complemented, Sylow
214C22×D4—531no—
215C22×D4—1061no—
216C23×C4—531no—
217C23×C4—531no—
218C23×C4C2×#615711yescharacteristic
219C22×C8—1062no—
220C22×C8—1062no—
221#123—2124no—
222#121—2122no—
223C2×C4⋊C4—531no—
224C2×C4⋊C4—1061no—
225C2×C4⋊C4—1061no—
226C2×C4⋊C4C2×#615711yescharacteristic
227C2×C4⋊C4C2×#615711yescharacteristic
228C2×C22⋊C4—21no—
229C2×C22⋊C4—1061no—
230C2×C22⋊C4—1061no—
231C2×C22⋊C4—1061no—
232C2×C22⋊C4—1061no—
233C2×C22⋊C4—1061no—
234C2×C42—531no—
235C2×C42—531no—
236C2×C42—531no—
237C2×C42—531no—
238C2×C42—2122no—
239C22⋊C8—1064no—
240C22⋊C8—1064no—
241#97—2122no—
242C4×C8—2122no—
243#95—21no—
244#95—1061no—
245#95—1061no—
246C24—1061no—
247C24C22⋊#615711yescharacteristic
248C2×D4—1062no—
249C2×D4—2121no—
250C2×D4—2121no—
251C2×D4—2122no—
252C2×D4—2122no—
253C2×D4—2122no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—41no—
259C22×C4—531no—
260C22×C4—531no—
261C22×C4—531no—
262C22×C4—531no—
263C22×C4—1062no—
264C22×C4—1062no—
265C22×C4—1062no—
266C22×C4—1062no—
267C22×C4—1061no—
268C22×C4—2121no—
269C22×C4—2121no—
270C22×C4C22⋊#615711yescharacteristic
271C22×C4C22×#615711yescharacteristic
272C2×C8—1064no—
273C2×C8—1064no—
274C2×C8—2122no—
275C2×C8—2122no—
276C2×C8—2122no—
277C2×C8—2122no—
278C4⋊C4—22no—
279C4⋊C4—41no—
280C4⋊C4—2121no—
281C4⋊C4—2122no—
282C4⋊C4—2122no—
283C22⋊C4—42no—
284C22⋊C4—2122no—
285C22⋊C4—2122no—
286C22⋊C4—2122no—
287C22⋊C4—2122no—
288C22⋊C4—2122no—
289C42—1062no—
290C42—1062no—
291C42—2121no—
292C42—2121no—
293C23—21no—
294C23—21no—
295C23—21no—
296C23—21no—
297C23—21no—
298C23—21no—
299C23—2121no—
300C23—2121no—
301C23—2121no—
302C23—2121no—
303C23—2121no—
304C23—2121no—
305C23—2121no—
306C23#1313328212yes—
307C23C2×C22⋊#615711yescharacteristic
308D4—4242no—
309D4—4244no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—44no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—532no—
322C2×C4—532no—
323C2×C4—532no—
324C2×C4—1061no—
325C2×C4—1061no—
326C2×C4—1061no—
327C2×C4—1061no—
328C2×C4—1061no—
329C2×C4—1061no—
330C2×C4—1061no—
331C2×C4—1061no—
332C2×C4—1061no—
333C2×C4—2124no—
334C2×C4—2124no—
335C2×C4—2121no—
336C2×C4—2121no—
337C2×C4—2121no—
338C2×C4#1313327612yes—
339C2×C4#1313330412yes—
340C8—2124no—
341C8—2124no—
342C22—22no—
343C22—22no—
344C22—22no—
345C22—41no—
346C22—41no—
347C22—41no—
348C22—41no—
349C22—4241no—
350C22—4241no—
351C22—4241no—
352C22—4241no—
353C22—4241no—
354C22—4241no—
355C22—4241no—
356C22#1681159611yescharacteristic
357C22C2×#1313327611yescharacteristic
358C22#1681161011yescharacteristic
359C22#1681161311yescharacteristic
360C22#1681161511yescharacteristic
361C22D53⋊#10411yescharacteristic
362C22C2×#1313328211yescharacteristic
363C4—22no—
364C4—22no—
365C4—81no—
366C4—81no—
367C4—1062no—
368C4—1062no—
369C4—1062no—
370C4—4241no—
371C4—2121no—
372C2—42no—
373C2—8481no—
374C2#2028998211yescharacteristic
375C2#2029003411yescharacteristic
376C2#2029006811yescharacteristic
377C2#2029009211yescharacteristic
378C2#2029011211yescharacteristic
379C2#2029016811yescharacteristic
380C2#2029023911yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6157
commutatorC22×C4×C53C22×C4
FrattiniC22×C4C22×#6157
FittingC53×#380C4
socleC23×C53C2×C22⋊C4
prime radicalC2×#13133191C4
cosocleC2×C4×Dic53C23
hypercenter#380#6157
hypocenterC53#7674
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ #380

Complements

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

NH ≅ G/Ntype
1. #20290441C2semidirect
2. C4×C22⋊#88324C2semidirect
3. #20289533C2semidirect
4. #20289532C2semidirect
213. C53#7674semidirect

Maximal subgroups

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

subgroupindex
#202904412
C4×C22⋊#883242
#202895332
#202895322
#202830472
#767453