Finite Group Databasebeta

← #28299756

Subgroups — #28299756

Order 13568.

544 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 544 conjugacy classes of subgroups — too many to draw. Only its 100 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#20290571C211yesmaximal, characteristic, complemented
2C4×C22⋊#88324C211yesmaximal, characteristic, complemented
3#20289534C212yesmaximal, complemented
4#20289531C212yesmaximal, complemented
5#20283041C211yesmaximal, characteristic
6C2×#13133284—21no—
7C2×C22⋊#88324—21no—
8C2×C22⋊#88324—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—
25#13133284—42no—
26C22⋊#88324—24no—
27C22⋊#88324—42no—
28C4×#88324—42no—
29C23×Dic53D411yescharacteristic
30C23×Dic53D411yescharacteristic
31#13133198—42no—
32#13133198—42no—
33C2×C4⋊Dic53C2×C411yescharacteristic
34C2×C4⋊Dic53C2×C411yescharacteristic
35C2×#11719919—21no—
36C2×#11719919—21no—
37C2×C22⋊Dic53—21no—
38C2×C22⋊Dic53—21no—
39C2×C22⋊Dic53—21no—
40C2×C22⋊Dic53—21no—
41C2×C22⋊Dic53—21no—
42C2×C4×Dic53—42no—
43C2×C4×Dic53C2×C411yescharacteristic
44C2×C4×Dic53D411yescharacteristic
45C2×C4×Dic53D411yescharacteristic
46C2×C4×Dic53C2311yescharacteristic
47#13133124—21no—
48#13133124—21no—
49#13133124—21no—
50C22×D4×C53—21no—
51C22×D4×C53C2×C411yescharacteristic
52C23×C4×C53C2×C411yescharacteristic
53#121×C53—42no—
54C2×C4⋊C4×C53C2×C411yescharacteristic
55C2×C22⋊C4×C53—21no—
56C24×C53—21no—
57C24×C53C22⋊C411yescharacteristic
58C2×D4×C53—22no—
59C2×D4×C53—41no—
60C2×D4×C53—41no—
61C2×D4×C53—42no—
62C2×D4×C53—42no—
63C2×D4×C53—42no—
64C22×C4×C53—22no—
65C22×C4×C53—22no—
66C22×C4×C53—21no—
67C22×C4×C53—41no—
68C22×C4×C53C22⋊C411yescharacteristic
69C22×C4×C53C22×C411yescharacteristic
70C22×Dic53—22no—
71C22×Dic53—22no—
72C22×Dic53—21no—
73C22×Dic53—21no—
74C22×Dic53—41no—
75C22×Dic53—41no—
76C22×Dic53—41no—
77C22×Dic53C22⋊C411yescharacteristic
78C22×Dic53C22⋊C411yescharacteristic
79C22×Dic53D812yes—
80C22×Dic53SD1612yes—
81C22×Dic53C2×D411yescharacteristic
82C22×Dic53C2×D411yescharacteristic
83C2×#88324—24no—
84C2×#88324—42no—
85C2×#88324—42no—
86C2×#88324—42no—
87C2×#88324—42no—
88C2×#88324—42no—
89C4⋊C4×C53—41no—
90C22⋊C4×C53—42no—
91C22⋊Dic53—42no—
92C22⋊Dic53—42no—
93C22⋊Dic53—42no—
94C22⋊Dic53—41no—
95C22⋊Dic53—41no—
96C22⋊Dic53—42no—
97C4⋊Dic53—22no—
98C4⋊Dic53—41no—
99#11719919—42no—
100#11719919—41no—
101#11719919—41no—
102C4×Dic53—22no—
103C4×Dic53—22no—
104C4×Dic53—41no—
105C4×Dic53—41no—
106C23×C53—21no—
107C23×C53—21no—
108C23×C53—21no—
109C23×C53—21no—
110C23×C53—21no—
111C23×C53—21no—
112C23×C53—41no—
113C23×C53—41no—
114C23×C53—41no—
115C23×C53—41no—
116C23×C53—41no—
117C23×C53—41no—
118C23×C53—41no—
119C23×C53#10212yes—
120C23×C53C2×C22⋊C411yescharacteristic
121D4×C53—82no—
122D4×C53—84no—
123C2×C4×C53—21no—
124C2×C4×C53—21no—
125C2×C4×C53—21no—
126C2×C4×C53—21no—
127C2×C4×C53—44no—
128C2×C4×C53—41no—
129C2×C4×C53—41no—
130C2×C4×C53—41no—
131C2×C4×C53#9912yes—
132C2×C4×C53#13112yes—
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—42no—
142C2×Dic53—42no—
143C2×Dic53—44no—
144C2×Dic53—41no—
145C2×Dic53—41no—
146C2×Dic53—41no—
147C2×Dic53—41no—
148C2×Dic53—41no—
149C2×Dic53—41no—
150C2×Dic53#10212yes—
151C2×Dic53#10412yes—
152C2×Dic53C2×D811yescharacteristic
153C2×Dic53C2×SD1611yescharacteristic
154C2×Dic53#13512yes—
155#88324—44no—
156#88324—82no—
157#7658—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—81no—
166C22×C53—81no—
167C22×C53—81no—
168C22×C53—81no—
169C22×C53—81no—
170C22×C53—81no—
171C22×C53—81no—
172C22×C53#40811yescharacteristic
173C22×C53C2×#9911yescharacteristic
174C22×C53#41311yescharacteristic
175C22×C53C2×#10211yescharacteristic
176C22×C53#41611yescharacteristic
177C22×C53C2×#10411yescharacteristic
178C22×C53#42111yescharacteristic
179C4×C53—22no—
180C4×C53—22no—
181C4×C53—81no—
182Dic53—22no—
183Dic53—22no—
184Dic53—21no—
185Dic53—22no—
186Dic53—21no—
187Dic53—81no—
188Dic53—81no—
189#2854—531no—
190#1797—531no—
191C4×C22⋊C8—531no—
192#1271—532no—
193#1270—532no—
194C2×C53—42no—
195C2×C53—161no—
196C2×C53#146811yescharacteristic
197C2×C53#148211yescharacteristic
198C2×C53#149211yescharacteristic
199C2×C53#150111yescharacteristic
200C2×C53#151911yescharacteristic
201C2×C53#152911yescharacteristic
202C2×C53#154011yescharacteristic
203C2×#123—1061no—
204C2×#121#615711yescharacteristic
205C22×C42—531no—
206C2×C4⋊C8—1061no—
207C2×C22⋊C8—1061no—
208C2×C22⋊C8—1061no—
209C2×C4×C8—1061no—
210#390—531no—
211#390—531no—
212#389—531no—
213#389—531no—
214#388—1061no—
215#388—1061no—
216#386—1061no—
217#380—531no—
218#336—532no—
219#336—532no—
220C53#765811yescharacteristic, complemented, Sylow
221C22×D4—21no—
222C22×D4C2×#615711yescharacteristic
223C23×C4—531no—
224C23×C4—531no—
225C23×C4C2×#615711yescharacteristic
226C22×C8—1062no—
227C22×C8—1062no—
228#123—2122no—
229#123—2122no—
230#121—42no—
231C2×C4⋊C4—531no—
232C2×C4⋊C4—531no—
233C2×C4⋊C4—1061no—
234C2×C4⋊C4—1061no—
235C2×C4⋊C4C2×#615711yescharacteristic
236C2×C22⋊C4—21no—
237C2×C22⋊C4—1061no—
238C2×C22⋊C4—1061no—
239C2×C22⋊C4—1061no—
240C2×C22⋊C4—1061no—
241C2×C22⋊C4—1061no—
242C2×C42—531no—
243C2×C42—531no—
244C2×C42—531no—
245C2×C42—531no—
246C2×C42—2122no—
247C4⋊C8—2122no—
248C22⋊C8—1064no—
249C22⋊C8—2122no—
250C4×C8—2122no—
251#95—1061no—
252#95—1061no—
253#95—1061no—
254C24—21no—
255C24C22⋊#615711yescharacteristic
256C2×D4—22no—
257C2×D4—41no—
258C2×D4—41no—
259C2×D4—42no—
260C2×D4—42no—
261C2×D4—42no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—21no—
265C22×C4—41no—
266C22×C4—531no—
267C22×C4—531no—
268C22×C4—531no—
269C22×C4—531no—
270C22×C4—532no—
271C22×C4—532no—
272C22×C4—1062no—
273C22×C4—1062no—
274C22×C4—1061no—
275C22×C4—1061no—
276C22×C4—2121no—
277C22×C4—2121no—
278C22×C4—2121no—
279C22×C4C22⋊#615711yescharacteristic
280C22×C4C22×#615711yescharacteristic
281C2×C8—1064no—
282C2×C8—2122no—
283C2×C8—2122no—
284C2×C8—2122no—
285C2×C8—2122no—
286C2×C8—2122no—
287C4⋊C4—41no—
288C4⋊C4—1062no—
289C4⋊C4—2121no—
290C4⋊C4—2122no—
291C4⋊C4—2121no—
292C4⋊C4—2121no—
293C22⋊C4—42no—
294C22⋊C4—2122no—
295C22⋊C4—2121no—
296C22⋊C4—2121no—
297C22⋊C4—2122no—
298C22⋊C4—2122no—
299C22⋊C4—2122no—
300C42—1062no—
301C42—1062no—
302C42—2121no—
303C42—2121no—
304C23—21no—
305C23—21no—
306C23—21no—
307C23—21no—
308C23—21no—
309C23—21no—
310C23—41no—
311C23—41no—
312C23—41no—
313C23—41no—
314C23—41no—
315C23—41no—
316C23—41no—
317C23#1313327912yes—
318C23C2×C22⋊#615711yescharacteristic
319D4—82no—
320D4—84no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—44no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—532no—
330C2×C4—531no—
331C2×C4—531no—
332C2×C4—532no—
333C2×C4—532no—
334C2×C4—1061no—
335C2×C4—1061no—
336C2×C4—1061no—
337C2×C4—1061no—
338C2×C4—1061no—
339C2×C4—1061no—
340C2×C4—1061no—
341C2×C4—1061no—
342C2×C4—2122no—
343C2×C4—2122no—
344C2×C4—2124no—
345C2×C4—2121no—
346C2×C4—2121no—
347C2×C4—2121no—
348C2×C4—2121no—
349C2×C4—2121no—
350C2×C4—2121no—
351C2×C4#1313327512yes—
352C2×C4#1313330412yes—
353C8—4242no—
354C8—2124no—
355C22—22no—
356C22—22no—
357C22—22no—
358C22—41no—
359C22—41no—
360C22—41no—
361C22—41no—
362C22—81no—
363C22—81no—
364C22—81no—
365C22—81no—
366C22—81no—
367C22—81no—
368C22—81no—
369C22#1681159611yescharacteristic
370C22C2×#1313327511yescharacteristic
371C22#1681161011yescharacteristic
372C22C2×#1313327911yescharacteristic
373C22#1681161311yescharacteristic
374C22D53⋊#10411yescharacteristic
375C22#1681162111yescharacteristic
376C4—22no—
377C4—22no—
378C4—81no—
379C4—1062no—
380C4—1062no—
381C4—1061no—
382C4—1062no—
383C4—1061no—
384C4—4241no—
385C4—4241no—
386C2—42no—
387C2—161no—
388C2#2028998011yescharacteristic
389C2#2029002611yescharacteristic
390C2#2029006011yescharacteristic
391C2#2029009011yescharacteristic
392C2#2029015111yescharacteristic
393C2#2029018111yescharacteristic
394C2#2029021311yescharacteristic
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#7658
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. #20290571C2semidirect
2. C4×C22⋊#88324C2semidirect
3. #20289534C2semidirect
4. #20289531C2semidirect
220. C53#7658semidirect

Maximal subgroups

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

subgroupindex
#202905712
C4×C22⋊#883242
#202895342
#202895312
#202830412
#765853