Finite Group Databasebeta

← #28300015

Subgroups — #28300015

Order 13568.

520 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 520 conjugacy classes of subgroups — too many to draw. Only its 90 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#20290605C211yesmaximal, characteristic, complemented
2#20290571C211yesmaximal, characteristic, complemented
3#20289535C212yesmaximal, complemented
4#20289532C212yesmaximal, complemented
5#20283018C211yesmaximal, characteristic
6C2×#13133284—21no—
7C2×C22⋊#88324—21no—
8C2×C22⋊#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—
25#13133284—42no—
26C22⋊#88324—42no—
27C22⋊#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—21no—
73C22×Dic53—21no—
74C22×Dic53—21no—
75C22×Dic53—41no—
76C22×Dic53—41no—
77C22×Dic53—41no—
78C22×Dic53C22⋊C411yescharacteristic
79C22×Dic53C22⋊C411yescharacteristic
80C22×Dic53C2×D411yescharacteristic
81C22×Dic53C2×D411yescharacteristic
82C2×D212—22no—
83C2×D212—41no—
84C2×D212—41no—
85C2×#88324—42no—
86C2×#88324—42no—
87C2×#88324—42no—
88C2×#88324—42no—
89C2×#88324—42no—
90C2×#88324—42no—
91C4⋊C4×C53—42no—
92C4⋊C4×C53—41no—
93C22⋊C4×C53—42no—
94#11719921—42no—
95#11719921—42no—
96#11719921—42no—
97#11719921—42no—
98#11719921—41no—
99#11719921—41no—
100C4⋊Dic53—41no—
101C4×Dic53—22no—
102C4×Dic53—22no—
103C4×Dic53—41no—
104C4×Dic53—41no—
105C23×C53—21no—
106C23×C53—21no—
107C23×C53—21no—
108C23×C53—21no—
109C23×C53—21no—
110C23×C53—21no—
111C23×C53#10012yes—
112C23×C53C2×C22⋊C411yescharacteristic
113C22×D53—41no—
114C22×D53—41no—
115C22×D53—41no—
116C22×D53—41no—
117C22×D53—41no—
118C22×D53—41no—
119C22×D53—41no—
120Q8×C53—82no—
121C2×C4×C53—21no—
122C2×C4×C53—21no—
123C2×C4×C53—21no—
124C2×C4×C53—21no—
125C2×C4×C53—21no—
126C2×C4×C53—44no—
127C2×C4×C53—41no—
128C2×C4×C53—41no—
129C2×C4×C53—41no—
130C2×C4×C53—41no—
131C2×C4×C53—41no—
132C2×C4×C53—41no—
133C2×C4×C53#13112yes—
134C22⋊D53—84no—
135C2×Dic53—21no—
136C2×Dic53—21no—
137C2×Dic53—21no—
138C2×Dic53—21no—
139C2×Dic53—21no—
140C2×Dic53—21no—
141C2×Dic53—21no—
142C2×Dic53—21no—
143C2×Dic53—21no—
144C2×Dic53—44no—
145C2×Dic53—42no—
146C2×Dic53—42no—
147C2×Dic53—41no—
148C2×Dic53—41no—
149C2×Dic53—41no—
150C2×Dic53#9912yes—
151C2×Dic53#13512yes—
152C2×Dic53AGL(1,ℤ/8)11yescharacteristic
153C2×Dic53#13711yescharacteristic
154D212—82no—
155#88324—82no—
156#88324—82no—
157#7730—531nomaximal, Sylow
158C22×C53—22no—
159C22×C53—22no—
160C22×C53—22no—
161C22×C53—41no—
162C22×C53—41no—
163C22×C53—41no—
164C22×C53—41no—
165C22×C53#40811yescharacteristic
166C22×C53C2×#9911yescharacteristic
167C22×C53C2×#10011yescharacteristic
168C22×C53#41611yescharacteristic
169C22×C53#41711yescharacteristic
170C22×C53#42111yescharacteristic
171C22×C53#42111yescharacteristic
172C2×D53—81no—
173C2×D53—81no—
174C2×D53—81no—
175C2×D53—81no—
176C2×D53—81no—
177C2×D53—81no—
178C2×D53—81no—
179C4×C53—22no—
180C4×C53—41no—
181C4×C53—81no—
182C4×C53—81no—
183Dic53—21no—
184Dic53—21no—
185Dic53—22no—
186Dic53—22no—
187Dic53—41no—
188Dic53—81no—
189#2850—531no—
190#1817—531no—
191#1797—531no—
192#1272—532no—
193#1270—532no—
194C2×C53—42no—
195C2×C53#147111yescharacteristic
196C2×C53#148211yescharacteristic
197C2×C53#149511yescharacteristic
198C2×C53#151311yescharacteristic
199C2×C53#151611yescharacteristic
200C2×C53#153311yescharacteristic
201C2×C53#155111yescharacteristic
202D53—161no—
203C2×#124—531no—
204C2×C22⋊Q8#615711yescharacteristic
205C2×#121—531no—
206C22×C42—531no—
207C2×C4⋊C8—1061no—
208C2×C22⋊C8—1061no—
209C2×C22⋊C8—1061no—
210#391—531no—
211#390—531no—
212#388—1061no—
213#388—1061no—
214#386—1061no—
215#386—1061no—
216#382—531no—
217#336—532no—
218#336—532no—
219#336—1061no—
220C53#773011yescharacteristic, complemented, Sylow
221C22×Q8C2×#615711yescharacteristic
222C22×D4—531no—
223C22×D4—1061no—
224C23×C4—531no—
225C23×C4—531no—
226C23×C4C2×#615711yescharacteristic
227C22×C8—1062no—
228C22×C8—1062no—
229#124—2122no—
230C22⋊Q8—42no—
231#121—2122no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—531no—
234C2×C4⋊C4C2×#615711yescharacteristic
235C2×C22⋊C4—21no—
236C2×C22⋊C4—1061no—
237C2×C22⋊C4—1061no—
238C2×C22⋊C4—1061no—
239C2×C22⋊C4—1061no—
240C2×C22⋊C4—1061no—
241C2×C42—531no—
242C2×C42—531no—
243C2×C42—531no—
244C2×C42—531no—
245C2×C42—2122no—
246C4⋊C8—2122no—
247C22⋊C8—2122no—
248C22⋊C8—2122no—
249#95—1061no—
250#95—1061no—
251#95—1061no—
252#95—1061no—
253C24—1061no—
254C24C22⋊#615711yescharacteristic
255C2×Q8—22no—
256C2×Q8—41no—
257C2×Q8—41no—
258C2×D4—1062no—
259C2×D4—2121no—
260C2×D4—2121no—
261C2×D4—2122no—
262C2×D4—2122no—
263C2×D4—2122no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—41no—
269C22×C4—531no—
270C22×C4—531no—
271C22×C4—531no—
272C22×C4—531no—
273C22×C4—1061no—
274C22×C4—1062no—
275C22×C4—1062no—
276C22×C4—1061no—
277C22×C4—1061no—
278C22×C4—2121no—
279C22×C4—2121no—
280C22×C4—2121no—
281C22×C4C22⋊#615711yescharacteristic
282C22×C4C22×#615711yescharacteristic
283C2×C8—2122no—
284C2×C8—2122no—
285C2×C8—2122no—
286C2×C8—2122no—
287C2×C8—2122no—
288C2×C8—2122no—
289C4⋊C4—41no—
290C4⋊C4—42no—
291C4⋊C4—2121no—
292C22⋊C4—42no—
293C22⋊C4—2122no—
294C22⋊C4—2122no—
295C22⋊C4—2122no—
296C22⋊C4—2121no—
297C22⋊C4—2121no—
298C22⋊C4—2122no—
299C42—1062no—
300C42—1062no—
301C42—2121no—
302C42—2121no—
303C23—21no—
304C23—21no—
305C23—21no—
306C23—21no—
307C23—21no—
308C23—21no—
309C23—2121no—
310C23—2121no—
311C23—2121no—
312C23—2121no—
313C23—2121no—
314C23—2121no—
315C23—2121no—
316C23#1313327712yes—
317C23C2×C22⋊#615711yescharacteristic
318Q8—82no—
319D4—4242no—
320D4—4244no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—44no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—532no—
334C2×C4—532no—
335C2×C4—531no—
336C2×C4—531no—
337C2×C4—1061no—
338C2×C4—1061no—
339C2×C4—1061no—
340C2×C4—1061no—
341C2×C4—1061no—
342C2×C4—1061no—
343C2×C4—1061no—
344C2×C4—1061no—
345C2×C4—1061no—
346C2×C4—2124no—
347C2×C4—2122no—
348C2×C4—2122no—
349C2×C4—2121no—
350C2×C4—2121no—
351C2×C4—2121no—
352C2×C4#1313330412yes—
353C8—4242no—
354C8—4242no—
355C22—22no—
356C22—22no—
357C22—22no—
358C22—41no—
359C22—41no—
360C22—41no—
361C22—41no—
362C22—4241no—
363C22—4241no—
364C22—4241no—
365C22—4241no—
366C22—4241no—
367C22—4241no—
368C22—4241no—
369C22#1681159611yescharacteristic
370C22D53⋊#9911yescharacteristic
371C22C2×#1313327711yescharacteristic
372C22#1681161311yescharacteristic
373C22#1681161411yescharacteristic
374C22#1681162211yescharacteristic
375C22#1681162211yescharacteristic
376C4—22no—
377C4—41no—
378C4—81no—
379C4—81no—
380C4—1061no—
381C4—1061no—
382C4—1062no—
383C4—1062no—
384C4—4241no—
385C4—2121no—
386C2—42no—
387C2—8481no—
388C2#2028999111yescharacteristic
389C2#2029002811yescharacteristic
390C2#2029007211yescharacteristic
391C2#2029013311yescharacteristic
392C2#2029014211yescharacteristic
393C2#2029019611yescharacteristic
394C2#2029023411yescharacteristic
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#7730
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. #20290605C2semidirect
2. #20290571C2semidirect
3. #20289535C2semidirect
4. #20289532C2semidirect
220. C53#7730semidirect

Maximal subgroups

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

subgroupindex
#202906052
#202905712
#202895352
#202895322
#202830182
#773053