Finite Group Databasebeta

← #28299812

Subgroups — #28299812

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#20289534C212yesmaximal, complemented
4#20289531C212yesmaximal, complemented
5#20283041C211yesmaximal, characteristic
6C2×C22⋊#88324—21no—
7C2×C22⋊#88324—21no—
8C2×#13133272—21no—
9C2×C4×#88324—21no—
10#16811489C2212yes—
11#16811489C2212yes—
12C2×#13133198—21no—
13C22×C4×Dic53C2211yescharacteristic
14#16810526C411yescharacteristic
15#16810526C2211yescharacteristic
16#16810519C411yescharacteristic
17#16810519C2211yescharacteristic
18#16810511—21no—
19#16810511—21no—
20#16810503—21no—
21#16810483C411yescharacteristic
22C2×#121×C53C411yescharacteristic
23C22×#88324—22no—
24C22×#88324—22no—
25C22⋊#88324—24no—
26C22⋊#88324—24no—
27#13133272—42no—
28C4×#88324—42no—
29C23×Dic53D411yescharacteristic
30C23×Dic53D411yescharacteristic
31#13133198—44no—
32C2×C4⋊Dic53C2×C411yescharacteristic
33C2×C4⋊Dic53C2×C411yescharacteristic
34C2×#11719919—21no—
35C2×#11719919—21no—
36C2×C22⋊Dic53—21no—
37C2×C22⋊Dic53—21no—
38C2×C22⋊Dic53—21no—
39C2×C22⋊Dic53—21no—
40C2×C22⋊Dic53—21no—
41C2×C4×Dic53—42no—
42C2×C4×Dic53C2×C411yescharacteristic
43C2×C4×Dic53D411yescharacteristic
44C2×C4×Dic53D411yescharacteristic
45C2×C4×Dic53C2311yescharacteristic
46#13133124—21no—
47#13133124—21no—
48#13133124—21no—
49C22×D4×C53—21no—
50C22×D4×C53C2×C411yescharacteristic
51C23×C4×C53C2×C411yescharacteristic
52#121×C53—42no—
53C2×C4⋊C4×C53C2×C411yescharacteristic
54C2×C22⋊C4×C53—21no—
55C24×C53—21no—
56C24×C53C22⋊C411yescharacteristic
57C2×D4×C53—22no—
58C2×D4×C53—41no—
59C2×D4×C53—41no—
60C2×D4×C53—42no—
61C2×D4×C53—42no—
62C2×D4×C53—42no—
63C22×C4×C53—22no—
64C22×C4×C53—22no—
65C22×C4×C53—21no—
66C22×C4×C53—41no—
67C22×C4×C53C22⋊C411yescharacteristic
68C22×C4×C53C22×C411yescharacteristic
69C22×Dic53—22no—
70C22×Dic53—22no—
71C22×Dic53—22no—
72C22×Dic53—22no—
73C22×Dic53—21no—
74C22×Dic53—21no—
75C22×Dic53—41no—
76C22×Dic53—41no—
77C22×Dic53C22⋊C411yescharacteristic
78C22×Dic53C22⋊C411yescharacteristic
79C22×Dic53C2×D411yescharacteristic
80C22×Dic53C2×D411yescharacteristic
81C2×#88324—24no—
82C2×#88324—24no—
83C2×#88324—42no—
84C2×#88324—42no—
85C2×#88324—42no—
86C2×#88324—42no—
87C4⋊C4×C53—41no—
88C22⋊C4×C53—42no—
89C22⋊Dic53—42no—
90C22⋊Dic53—42no—
91C22⋊Dic53—42no—
92C22⋊Dic53—42no—
93C22⋊Dic53—42no—
94C4⋊Dic53—22no—
95C4⋊Dic53—41no—
96#11719919—42no—
97#11719919—42no—
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—41no—
109C23×C53—41no—
110C23×C53—41no—
111C23×C53—41no—
112C23×C53—41no—
113C23×C53—41no—
114C23×C53—41no—
115C23×C53#10412yes—
116C23×C53C2×C22⋊C411yescharacteristic
117D4×C53—82no—
118D4×C53—84no—
119C2×C4×C53—21no—
120C2×C4×C53—21no—
121C2×C4×C53—21no—
122C2×C4×C53—21no—
123C2×C4×C53—44no—
124C2×C4×C53—41no—
125C2×C4×C53—41no—
126C2×C4×C53—41no—
127C2×C4×C53#9912yes—
128C2×C4×C53#13112yes—
129C2×Dic53—21no—
130C2×Dic53—21no—
131C2×Dic53—21no—
132C2×Dic53—21no—
133C2×Dic53—21no—
134C2×Dic53—21no—
135C2×Dic53—21no—
136C2×Dic53—21no—
137C2×Dic53—21no—
138C2×Dic53—44no—
139C2×Dic53—44no—
140C2×Dic53—41no—
141C2×Dic53—41no—
142C2×Dic53—41no—
143C2×Dic53—41no—
144C2×Dic53—41no—
145C2×Dic53—41no—
146C2×Dic53#10412yes—
147C2×Dic53#13512yes—
148C2×Dic53AGL(1,ℤ/8)12yes—
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—81no—
160C22×C53—81no—
161C22×C53—81no—
162C22×C53—81no—
163C22×C53—81no—
164C22×C53—81no—
165C22×C53—81no—
166C22×C53#40811yescharacteristic
167C22×C53C2×#9911yescharacteristic
168C22×C53#41311yescharacteristic
169C22×C53#41611yescharacteristic
170C22×C53#41811yescharacteristic
171C22×C53C2×#10411yescharacteristic
172C22×C53C2×#10411yescharacteristic
173C4×C53—22no—
174C4×C53—22no—
175C4×C53—81no—
176Dic53—22no—
177Dic53—22no—
178Dic53—22no—
179Dic53—41no—
180Dic53—81no—
181Dic53—81no—
182#2854—531no—
183#1749—531no—
184C4×C22⋊C8—531no—
185#1271—532no—
186#1270—532no—
187C2×C53—42no—
188C2×C53—161no—
189C2×C53#146811yescharacteristic
190C2×C53#148411yescharacteristic
191C2×C53#149411yescharacteristic
192C2×C53#150111yescharacteristic
193C2×C53#150711yescharacteristic
194C2×C53#152411yescharacteristic
195C2×C53#155311yescharacteristic
196C2×#123—1061no—
197C2×#121#615711yescharacteristic
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—531no—
211#336—532no—
212#336—532no—
213C53#767411yescharacteristic, complemented, Sylow
214C22×D4—21no—
215C22×D4C2×#615711yescharacteristic
216C23×C4—531no—
217C23×C4—531no—
218C23×C4C2×#615711yescharacteristic
219C22×C8—1062no—
220C22×C8—1062no—
221#123—2124no—
222#121—42no—
223C2×C4⋊C4—531no—
224C2×C4⋊C4—531no—
225C2×C4⋊C4—1061no—
226C2×C4⋊C4—1061no—
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—1061no—
244#95—1061no—
245#95—1061no—
246C24—21no—
247C24C22⋊#615711yescharacteristic
248C2×D4—22no—
249C2×D4—41no—
250C2×D4—41no—
251C2×D4—42no—
252C2×D4—42no—
253C2×D4—42no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—21no—
257C22×C4—41no—
258C22×C4—531no—
259C22×C4—531no—
260C22×C4—531no—
261C22×C4—531no—
262C22×C4—1062no—
263C22×C4—1062no—
264C22×C4—1062no—
265C22×C4—1062no—
266C22×C4—1061no—
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—41no—
279C4⋊C4—1062no—
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—41no—
300C23—41no—
301C23—41no—
302C23—41no—
303C23—41no—
304C23—41no—
305C23—41no—
306C23#1313328112yes—
307C23C2×C22⋊#615711yescharacteristic
308D4—82no—
309D4—84no—
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—532no—
319C2×C4—532no—
320C2×C4—532no—
321C2×C4—1061no—
322C2×C4—1061no—
323C2×C4—1061no—
324C2×C4—1061no—
325C2×C4—1061no—
326C2×C4—1061no—
327C2×C4—1061no—
328C2×C4—1061no—
329C2×C4—1061no—
330C2×C4—2124no—
331C2×C4—2124no—
332C2×C4—2121no—
333C2×C4—2121no—
334C2×C4—2121no—
335C2×C4—2121no—
336C2×C4—2121no—
337C2×C4—2121no—
338C2×C4#1313327512yes—
339C2×C4#1313330412yes—
340C8—2124no—
341C8—2124no—
342C22—22no—
343C22—22no—
344C22—22no—
345C22—41no—
346C22—41no—
347C22—41no—
348C22—41no—
349C22—81no—
350C22—81no—
351C22—81no—
352C22—81no—
353C22—81no—
354C22—81no—
355C22—81no—
356C22#1681159611yescharacteristic
357C22C2×#1313327511yescharacteristic
358C22#1681161011yescharacteristic
359C22#1681161311yescharacteristic
360C22#1681161511yescharacteristic
361C22C2×#1313328111yescharacteristic
362C22D53⋊#10411yescharacteristic
363C4—22no—
364C4—22no—
365C4—81no—
366C4—1062no—
367C4—1062no—
368C4—1062no—
369C4—4241no—
370C4—4241no—
371C4—2121no—
372C2—42no—
373C2—161no—
374C2#2028998011yescharacteristic
375C2#2029003411yescharacteristic
376C2#2029006611yescharacteristic
377C2#2029009011yescharacteristic
378C2#2029011111yescharacteristic
379C2#2029016611yescharacteristic
380C2#2029023911yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6157
commutatorC22×C4×C53C22×C4
FrattiniC22×C4C22×#6157
Fitting, prime radicalC2×#121×C53C4
socleC23×C53C2×C22⋊C4
cosocleC2×C4×Dic53C23
hypercenterC2×#121#6157
hypocenterC53#7674
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#121

Complements

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

NH ≅ G/Ntype
1. #20290441C2semidirect
2. C4×C22⋊#88324C2semidirect
3. #20289534C2semidirect
4. #20289531C2semidirect
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
#202895342
#202895312
#202830412
#767453