Finite Group Databasebeta

← #28299625

Subgroups — #28299625

Order 13568.

590 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 590 conjugacy classes of subgroups — too many to draw. Only its 98 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#20290606C211yesmaximal, characteristic, complemented
2#20290441C211yesmaximal, characteristic, complemented
3#20289533C212yesmaximal, complemented
4#20289531C212yesmaximal, complemented
5#121×Dic53C211yesmaximal, characteristic
6C2×C22⋊#88324—21no—
7C2×C22⋊#88324—21no—
8C2×#13133272—21no—
9#16811489—21no—
10#16811489C2212yes—
11#16811489C2212yes—
12C2×D4×Dic53—21no—
13C2×D4×Dic53C411yescharacteristic
14C22×C4×Dic53C2211yescharacteristic
15#16810526C2211yescharacteristic
16#16810522C2211yescharacteristic
17#16810516—21no—
18#16810503—21no—
19#16810483C411yescharacteristic
20C4⋊C4×Dic53C411yescharacteristic
21C22⋊C4×Dic53—21no—
22C2×#121×C53C411yescharacteristic
23C22×#88324—22no—
24C22×#88324—22no—
25C22⋊#88324—24no—
26C22⋊#88324—42no—
27#13133272—42no—
28C23×Dic53—21no—
29C23×Dic53D411yescharacteristic
30C23×Dic53D411yescharacteristic
31D4×Dic53—44no—
32D4×Dic53—42no—
33C2×C4⋊Dic53—21no—
34C2×C4⋊Dic53C2×C411yescharacteristic
35C2×C4⋊Dic53C2×C411yescharacteristic
36C2×C22⋊Dic53—21no—
37C2×C22⋊Dic53—21no—
38C2×C22⋊Dic53—21no—
39C2×C22⋊Dic53—21no—
40C2×C4×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—22no—
53#121×C53—22no—
54C2×C4⋊C4×C53C2×C411yescharacteristic
55C2×C22⋊C4×C53—21no—
56C24×C53—21no—
57C24×C53C22⋊C411yescharacteristic
58C2×D4×C53—24no—
59C2×D4×C53—21no—
60C2×D4×C53—41no—
61C2×D4×C53—41no—
62C2×D4×C53—42no—
63C2×D4×C53—42no—
64C2×D4×C53M1612yes—
65C22×C4×C53—21no—
66C22×C4×C53—21no—
67C22×C4×C53—21no—
68C22×C4×C53—41no—
69C22×C4×C53—41no—
70C22×C4×C53C22⋊C411yescharacteristic
71C22×C4×C53C22×C411yescharacteristic
72C22×Dic53—22no—
73C22×Dic53—22no—
74C22×Dic53—22no—
75C22×Dic53—22no—
76C22×Dic53—21no—
77C22×Dic53—21no—
78C22×Dic53—41no—
79C22×Dic53—41no—
80C22×Dic53—42no—
81C22×Dic53—42no—
82C22×Dic53—42no—
83C22×Dic53C22⋊C411yescharacteristic
84C22×Dic53C22⋊C411yescharacteristic
85C22×Dic53C2×D411yescharacteristic
86C22×Dic53C2×D411yescharacteristic
87C2×#88324—24no—
88C2×#88324—42no—
89C2×#88324—42no—
90C2×#88324—42no—
91C2×#88324—42no—
92C2×#88324—42no—
93C4⋊C4×C53—21no—
94C4⋊C4×C53M1612yes—
95C22⋊C4×C53—24no—
96C22⋊Dic53—42no—
97C22⋊Dic53—42no—
98C22⋊Dic53—42no—
99C22⋊Dic53—42no—
100C4⋊Dic53—42no—
101C4⋊Dic53—41no—
102C4⋊Dic53—41no—
103C4×Dic53—22no—
104C4×Dic53—22no—
105C4×Dic53—41no—
106C4×Dic53—41no—
107C4×Dic53—42no—
108C23×C53—21no—
109C23×C53—21no—
110C23×C53—21no—
111C23×C53—21no—
112C23×C53—21no—
113C23×C53—21no—
114C23×C53—22no—
115C23×C53—41no—
116C23×C53—41no—
117C23×C53—41no—
118C23×C53—41no—
119C23×C53—41no—
120C23×C53—41no—
121C23×C53#10412yes—
122C23×C53C2×C22⋊C411yescharacteristic
123D4×C53—42no—
124D4×C53—42no—
125D4×C53—48no—
126C2×C4×C53—21no—
127C2×C4×C53—21no—
128C2×C4×C53—21no—
129C2×C4×C53—22no—
130C2×C4×C53—21no—
131C2×C4×C53—21no—
132C2×C4×C53—42no—
133C2×C4×C53—42no—
134C2×C4×C53—41no—
135C2×C4×C53—41no—
136C2×C4×C53C2×M1611yescharacteristic
137C2×C4×C53C2×M1611yescharacteristic
138C2×Dic53—21no—
139C2×Dic53—21no—
140C2×Dic53—21no—
141C2×Dic53—21no—
142C2×Dic53—21no—
143C2×Dic53—21no—
144C2×Dic53—21no—
145C2×Dic53—21no—
146C2×Dic53—21no—
147C2×Dic53—44no—
148C2×Dic53—44no—
149C2×Dic53—41no—
150C2×Dic53—41no—
151C2×Dic53—41no—
152C2×Dic53—41no—
153C2×Dic53—41no—
154C2×Dic53—41no—
155C2×Dic53—84no—
156C2×Dic53#9912yes—
157C2×Dic53#13512yes—
158C2×Dic53AGL(1,ℤ/8)12yes—
159#88324—44no—
160#88324—82no—
161#7620—531nomaximal, Sylow
162C22×C53—22no—
163C22×C53—22no—
164C22×C53—22no—
165C22×C53—41no—
166C22×C53—41no—
167C22×C53—41no—
168C22×C53—41no—
169C22×C53—42no—
170C22×C53—42no—
171C22×C53—42no—
172C22×C53—81no—
173C22×C53—81no—
174C22×C53—81no—
175C22×C53—81no—
176C22×C53#40711yescharacteristic
177C22×C53C2×#9911yescharacteristic
178C22×C53#41311yescharacteristic
179C22×C53#41611yescharacteristic
180C22×C53#41811yescharacteristic
181C22×C53C2×#10411yescharacteristic
182C22×C53#42111yescharacteristic
183C4×C53—21no—
184C4×C53—21no—
185C4×C53—41no—
186C4×C53—42no—
187Dic53—22no—
188Dic53—22no—
189Dic53—22no—
190Dic53—41no—
191Dic53—81no—
192Dic53—81no—
193C4×#121—531no—
194#1817—531no—
195#1749—531no—
196#1271—532no—
197#1270—532no—
198C2×C53—42no—
199C2×C53—82no—
200C2×C53#146511yescharacteristic
201C2×C53#148811yescharacteristic
202C2×C53#149011yescharacteristic
203C2×C53#150111yescharacteristic
204C2×C53#151011yescharacteristic
205C2×C53#153211yescharacteristic
206C2×C53#155311yescharacteristic
207C2×#121#615711yescharacteristic
208C2×C4×D4—531no—
209C2×C4×D4—1061no—
210C22×C42—531no—
211C2×C22⋊C8—1061no—
212C2×C22⋊C8—1061no—
213C2×#97—1061no—
214#390—531no—
215#389—531no—
216#388—1061no—
217#386—1061no—
218#380—531no—
219C4×C4⋊C4—531no—
220C4×C22⋊C4—1061no—
221#336—532no—
222#336—532no—
223#336—1061no—
224C53#762011yescharacteristic, complemented, Sylow
225C22×D4—21no—
226C22×D4C2×#615711yescharacteristic
227C23×C4—531no—
228C23×C4—531no—
229C23×C4—1061no—
230C23×C4C2×#615711yescharacteristic
231C22×C8—1062no—
232C22×C8—1062no—
233#121—22no—
234#121—22no—
235C4×D4—2122no—
236C4×D4—2124no—
237C2×C4⋊C4—531no—
238C2×C4⋊C4—531no—
239C2×C4⋊C4—1061no—
240C2×C4⋊C4C2×#615711yescharacteristic
241C2×C22⋊C4—21no—
242C2×C22⋊C4—1061no—
243C2×C22⋊C4—1061no—
244C2×C22⋊C4—1061no—
245C2×C22⋊C4—1061no—
246C2×C42—531no—
247C2×C42—531no—
248C2×C42—531no—
249C2×C42—531no—
250C2×C42—1061no—
251C2×C42—2122no—
252C22⋊C8—1064no—
253C22⋊C8—2122no—
254#97—2122no—
255#95—1061no—
256#95—1061no—
257#95—1061no—
258C24—21no—
259C24C22⋊#615711yescharacteristic
260C2×D4—24no—
261C2×D4—21no—
262C2×D4—41no—
263C2×D4—41no—
264C2×D4—42no—
265C2×D4—42no—
266C2×D4#1171994012yes—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—41no—
271C22×C4—41no—
272C22×C4—531no—
273C22×C4—531no—
274C22×C4—531no—
275C22×C4—531no—
276C22×C4—1062no—
277C22×C4—1062no—
278C22×C4—1062no—
279C22×C4—1062no—
280C22×C4—1061no—
281C22×C4—1061no—
282C22×C4—2122no—
283C22×C4—2121no—
284C22×C4—2121no—
285C22×C4—2122no—
286C22×C4—2122no—
287C22×C4C22⋊#615711yescharacteristic
288C22×C4C22×#615711yescharacteristic
289C2×C8—1064no—
290C2×C8—2122no—
291C2×C8—2122no—
292C2×C8—2122no—
293C2×C8—2122no—
294C2×C8—2122no—
295C4⋊C4—21no—
296C4⋊C4—2121no—
297C4⋊C4—2121no—
298C4⋊C4—2122no—
299C4⋊C4#1171994012yes—
300C22⋊C4—24no—
301C22⋊C4—2122no—
302C22⋊C4—2122no—
303C22⋊C4—2122no—
304C22⋊C4—2122no—
305C42—1062no—
306C42—1062no—
307C42—2121no—
308C42—2121no—
309C42—2122no—
310C23—21no—
311C23—21no—
312C23—21no—
313C23—21no—
314C23—21no—
315C23—21no—
316C23—22no—
317C23—41no—
318C23—41no—
319C23—41no—
320C23—41no—
321C23—41no—
322C23—41no—
323C23#1313328112yes—
324C23C2×C22⋊#615711yescharacteristic
325D4—42no—
326D4—42no—
327D4—48no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—22no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—42no—
335C2×C4—42no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—532no—
339C2×C4—532no—
340C2×C4—532no—
341C2×C4—1061no—
342C2×C4—1061no—
343C2×C4—1061no—
344C2×C4—1061no—
345C2×C4—1061no—
346C2×C4—1061no—
347C2×C4—1061no—
348C2×C4—1061no—
349C2×C4—1061no—
350C2×C4—4244no—
351C2×C4—2124no—
352C2×C4—2124no—
353C2×C4—2121no—
354C2×C4—2121no—
355C2×C4—2121no—
356C2×C4—2121no—
357C2×C4—2121no—
358C2×C4—2121no—
359C2×C4C2×#1171994011yescharacteristic
360C2×C4C2×#1171994011yescharacteristic
361C8—4242no—
362C8—2124no—
363C22—22no—
364C22—22no—
365C22—22no—
366C22—41no—
367C22—41no—
368C22—41no—
369C22—41no—
370C22—42no—
371C22—42no—
372C22—42no—
373C22—81no—
374C22—81no—
375C22—81no—
376C22—81no—
377C22#1681159311yescharacteristic
378C22D53⋊#9911yescharacteristic
379C22#1681161011yescharacteristic
380C22#1681161311yescharacteristic
381C22#1681161511yescharacteristic
382C22C2×#1313328111yescharacteristic
383C22#1681162011yescharacteristic
384C4—21no—
385C4—21no—
386C4—42no—
387C4—41no—
388C4—1062no—
389C4—1062no—
390C4—1062no—
391C4—4241no—
392C4—4241no—
393C4—2121no—
394C2—42no—
395C2—82no—
396C2#2028997111yescharacteristic
397C2#2029004611yescharacteristic
398C2#2029005411yescharacteristic
399C2#2029009111yescharacteristic
400C2#2029012111yescharacteristic
401C2#2029019111yescharacteristic
402C2#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#7620
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. #20290606C2semidirect
2. #20290441C2semidirect
3. #20289533C2semidirect
4. #20289531C2semidirect
224. C53#7620semidirect

Maximal subgroups

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

subgroupindex
#202906062
#202904412
#202895332
#202895312
#121×Dic532
#762053