Finite Group Databasebeta

← #28299799

Subgroups — #28299799

Order 13568.

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

normal subgroups only

This group has 538 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#20290606C211yesmaximal, characteristic, complemented
2C4×C22⋊#88324C211yesmaximal, characteristic, complemented
3#20289535C212yesmaximal, complemented
4#20289532C212yesmaximal, complemented
5#20283018C211yesmaximal, characteristic
6C2×C22⋊#88324—21no—
7C2×C22⋊#88324—21no—
8C2×C4×#88324—21no—
9#16811489—21no—
10#16811489C2212yes—
11#16811489C2212yes—
12C2×#13133206C411yescharacteristic
13C2×#13133191C411yescharacteristic
14C22×C4×Dic53C2211yescharacteristic
15#16810530C2211yescharacteristic
16#16810524C2211yescharacteristic
17#16810513—21no—
18#16810513—21no—
19#16810500—21no—
20#16810500—21no—
21#16810488C411yescharacteristic
22C2×C22⋊Q8×C53C411yescharacteristic
23C22×#88324—22no—
24C22×#88324—22no—
25C22⋊#88324—24no—
26C22⋊#88324—42no—
27C4×#88324—42no—
28C22×D212C2×C411yescharacteristic
29C22×C22⋊D53—21no—
30C23×Dic53D411yescharacteristic
31C23×Dic53D411yescharacteristic
32#13133206—42no—
33#13133191—42no—
34C2×C4⋊Dic53C2×C411yescharacteristic
35C2×#11719921—21no—
36C2×#11719921—21no—
37C2×#11719921—21no—
38C2×#11719921—21no—
39C2×#11719921—21no—
40C2×C4×Dic53—42no—
41C2×C4×Dic53C2×C411yescharacteristic
42C2×C4×Dic53D411yescharacteristic
43C2×C4×Dic53D411yescharacteristic
44C2×C4×Dic53C2311yescharacteristic
45#13133124—21no—
46#13133124—21no—
47#13133124—21no—
48#13133124—21no—
49C22×Q8×C53C2×C411yescharacteristic
50C23×C4×C53C2×C411yescharacteristic
51C22⋊Q8×C53—42no—
52C2×C4⋊C4×C53—21no—
53C2×C4⋊C4×C53C2×C411yescharacteristic
54C2×C22⋊C4×C53—21no—
55C24×C53C22⋊C411yescharacteristic
56C23×D53—21no—
57C2×Q8×C53—22no—
58C2×Q8×C53—41no—
59C2×Q8×C53—41no—
60C22×C4×C53—22no—
61C22×C4×C53—22no—
62C22×C4×C53—21no—
63C22×C4×C53—21no—
64C22×C4×C53—41no—
65C22×C4×C53C22⋊C411yescharacteristic
66C22×C4×C53C22×C411yescharacteristic
67C2×C22⋊D53—42no—
68C2×C22⋊D53—42no—
69C2×C22⋊D53—42no—
70C22×Dic53—22no—
71C22×Dic53—22no—
72C22×Dic53—22no—
73C22×Dic53—22no—
74C22×Dic53—21no—
75C22×Dic53—41no—
76C22×Dic53—41no—
77C22×Dic53C22⋊C411yescharacteristic
78C22×Dic53C22⋊C411yescharacteristic
79C22×Dic53C2×D411yescharacteristic
80C22×Dic53C2×D411yescharacteristic
81C2×D212—22no—
82C2×D212—41no—
83C2×D212—41no—
84C2×#88324—24no—
85C2×#88324—42no—
86C2×#88324—42no—
87C2×#88324—42no—
88C2×#88324—42no—
89C2×#88324—42no—
90C4⋊C4×C53—41no—
91C4⋊C4×C53—42no—
92C22⋊C4×C53—42no—
93#11719921—42no—
94#11719921—42no—
95#11719921—42no—
96#11719921—42no—
97#11719921—42no—
98C4⋊Dic53—41no—
99C4×Dic53—22no—
100C4×Dic53—22no—
101C4×Dic53—41no—
102C4×Dic53—41no—
103C23×C53—21no—
104C23×C53—21no—
105C23×C53—21no—
106C23×C53—21no—
107C23×C53—21no—
108C23×C53—21no—
109C23×C53#10412yes—
110C23×C53C2×C22⋊C411yescharacteristic
111C22×D53—41no—
112C22×D53—41no—
113C22×D53—41no—
114C22×D53—41no—
115C22×D53—41no—
116C22×D53—41no—
117C22×D53—41no—
118Q8×C53—82no—
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—41no—
128C2×C4×C53—41no—
129C2×C4×C53—41no—
130C2×C4×C53#10012yes—
131C2×C4×C53#13112yes—
132C22⋊D53—84no—
133C2×Dic53—21no—
134C2×Dic53—21no—
135C2×Dic53—21no—
136C2×Dic53—21no—
137C2×Dic53—21no—
138C2×Dic53—21no—
139C2×Dic53—21no—
140C2×Dic53—21no—
141C2×Dic53—21no—
142C2×Dic53—44no—
143C2×Dic53—44no—
144C2×Dic53—41no—
145C2×Dic53—41no—
146C2×Dic53—41no—
147C2×Dic53#9912yes—
148C2×Dic53#13512yes—
149C2×Dic53#13512yes—
150D212—82no—
151#88324—44no—
152#88324—82no—
153#7670—531nomaximal, Sylow
154C22×C53—22no—
155C22×C53—22no—
156C22×C53—22no—
157C22×C53—41no—
158C22×C53—41no—
159C22×C53—41no—
160C22×C53—41no—
161C22×C53#40811yescharacteristic
162C22×C53C2×#9911yescharacteristic
163C22×C53C2×#10011yescharacteristic
164C22×C53#41611yescharacteristic
165C22×C53#41611yescharacteristic
166C22×C53C2×#10411yescharacteristic
167C22×C53#42111yescharacteristic
168C2×D53—81no—
169C2×D53—81no—
170C2×D53—81no—
171C2×D53—81no—
172C2×D53—81no—
173C2×D53—81no—
174C2×D53—81no—
175C4×C53—22no—
176C4×C53—22no—
177C4×C53—81no—
178C4×C53—81no—
179Dic53—22no—
180Dic53—22no—
181Dic53—22no—
182Dic53—41no—
183Dic53—81no—
184#2850—531no—
185#1817—531no—
186C4×C22⋊C8—531no—
187#1272—532no—
188#1270—532no—
189C2×C53—42no—
190C2×C53#147111yescharacteristic
191C2×C53#148211yescharacteristic
192C2×C53#149411yescharacteristic
193C2×C53#150111yescharacteristic
194C2×C53#151611yescharacteristic
195C2×C53#152711yescharacteristic
196C2×C53#154311yescharacteristic
197D53—161no—
198C2×#124—531no—
199C2×C22⋊Q8#615711yescharacteristic
200C2×#121—531no—
201C22×C42—531no—
202C2×C22⋊C8—1061no—
203C2×C22⋊C8—1061no—
204C2×C4×C8—1061no—
205#391—531no—
206#390—531no—
207#388—1061no—
208#388—1061no—
209#386—1061no—
210#386—1061no—
211#382—531no—
212#336—532no—
213#336—532no—
214#336—1061no—
215C53#767011yescharacteristic, complemented, Sylow
216C22×Q8C2×#615711yescharacteristic
217C22×D4—531no—
218C22×D4—1061no—
219C23×C4—531no—
220C23×C4—531no—
221C23×C4C2×#615711yescharacteristic
222C22×C8—1062no—
223C22×C8—1062no—
224#124—2122no—
225C22⋊Q8—42no—
226#121—2122no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—531no—
229C2×C4⋊C4C2×#615711yescharacteristic
230C2×C22⋊C4—21no—
231C2×C22⋊C4—1061no—
232C2×C22⋊C4—1061no—
233C2×C22⋊C4—1061no—
234C2×C22⋊C4—1061no—
235C2×C22⋊C4—1061no—
236C2×C42—531no—
237C2×C42—531no—
238C2×C42—531no—
239C2×C42—531no—
240C2×C42—2122no—
241C22⋊C8—1064no—
242C22⋊C8—2122no—
243C4×C8—2122no—
244#95—1061no—
245#95—1061no—
246#95—1061no—
247#95—1061no—
248C24—1061no—
249C24C22⋊#615711yescharacteristic
250C2×Q8—22no—
251C2×Q8—41no—
252C2×Q8—41no—
253C2×D4—1062no—
254C2×D4—2121no—
255C2×D4—2121no—
256C2×D4—2122no—
257C2×D4—2122no—
258C2×D4—2122no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—41no—
264C22×C4—531no—
265C22×C4—531no—
266C22×C4—531no—
267C22×C4—531no—
268C22×C4—1062no—
269C22×C4—1062no—
270C22×C4—1062no—
271C22×C4—1062no—
272C22×C4—1061no—
273C22×C4—2121no—
274C22×C4—2121no—
275C22×C4C22⋊#615711yescharacteristic
276C22×C4C22×#615711yescharacteristic
277C2×C8—1064no—
278C2×C8—2122no—
279C2×C8—2122no—
280C2×C8—2122no—
281C2×C8—2122no—
282C2×C8—2122no—
283C4⋊C4—41no—
284C4⋊C4—42no—
285C4⋊C4—2121no—
286C22⋊C4—42no—
287C22⋊C4—2122no—
288C22⋊C4—2122no—
289C22⋊C4—2122no—
290C22⋊C4—2122no—
291C22⋊C4—2122no—
292C42—1062no—
293C42—1062no—
294C42—2121no—
295C42—2121no—
296C23—21no—
297C23—21no—
298C23—21no—
299C23—21no—
300C23—21no—
301C23—21no—
302C23—2121no—
303C23—2121no—
304C23—2121no—
305C23—2121no—
306C23—2121no—
307C23—2121no—
308C23—2121no—
309C23#1313328212yes—
310C23C2×C22⋊#615711yescharacteristic
311Q8—82no—
312D4—4242no—
313D4—4244no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—44no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—532no—
326C2×C4—532no—
327C2×C4—532no—
328C2×C4—1061no—
329C2×C4—1061no—
330C2×C4—1061no—
331C2×C4—1061no—
332C2×C4—1061no—
333C2×C4—1061no—
334C2×C4—1061no—
335C2×C4—1061no—
336C2×C4—1061no—
337C2×C4—2124no—
338C2×C4—2124no—
339C2×C4—2121no—
340C2×C4—2121no—
341C2×C4—2121no—
342C2×C4#1313327712yes—
343C2×C4#1313330412yes—
344C8—4242no—
345C8—2124no—
346C22—22no—
347C22—22no—
348C22—22no—
349C22—41no—
350C22—41no—
351C22—41no—
352C22—41no—
353C22—4241no—
354C22—4241no—
355C22—4241no—
356C22—4241no—
357C22—4241no—
358C22—4241no—
359C22—4241no—
360C22#1681159611yescharacteristic
361C22D53⋊#9911yescharacteristic
362C22C2×#1313327711yescharacteristic
363C22#1681161311yescharacteristic
364C22#1681161311yescharacteristic
365C22C2×#1313328211yescharacteristic
366C22#1681162211yescharacteristic
367C4—22no—
368C4—22no—
369C4—81no—
370C4—81no—
371C4—1062no—
372C4—1062no—
373C4—1062no—
374C4—4241no—
375C4—2121no—
376C2—42no—
377C2—8481no—
378C2#2028999111yescharacteristic
379C2#2029002811yescharacteristic
380C2#2029006811yescharacteristic
381C2#2029009311yescharacteristic
382C2#2029014211yescharacteristic
383C2#2029017611yescharacteristic
384C2#2029021811yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6157
commutatorC22×C4×C53C22×C4
FrattiniC22×C4C22×#6157
FittingC2×C22⋊Q8×C53C4
socleC23×C53C2×C22⋊C4
prime radicalC2×#13133191C4
cosocleC2×C4×Dic53C23
hypercenterC2×C22⋊Q8#6157
hypocenterC53#7670
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. #20290606C2semidirect
2. C4×C22⋊#88324C2semidirect
3. #20289535C2semidirect
4. #20289532C2semidirect
215. C53#7670semidirect

Maximal subgroups

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

subgroupindex
#202906062
C4×C22⋊#883242
#202895352
#202895322
#202830182
#767053