Finite Group Databasebeta

← #28303319

Subgroups — #28303319

Order 13568.

540 conjugacy classes of subgroups; only the 124 normal subgroups are shown.

normal subgroups only

This group has 540 conjugacy classes of subgroups — too many to draw. Only its 124 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#20290569C211yesmaximal, characteristic
2C4×C22⋊#88324C211yesmaximal, characteristic
3#20289551C212yesmaximal, complemented
4#20289551C212yesmaximal, complemented
5#20277117C211yesmaximal, characteristic
6C2×C4⋊#88324C2212yes—
7C2×C22⋊#88324—21no—
8C2×C22⋊#88324—21no—
9C2×C4×#88324C2212yes—
10#16811489—21no—
11#16811489—21no—
12C2×#13133173—21no—
13C22×C4×Dic53C2211yescharacteristic
14#16810492C411yescharacteristic
15#16810492C411yescharacteristic
16#16810487—21no—
17#16810483C411yescharacteristic
18#16810481—21no—
19C4⋊C4×Dic53C2211yescharacteristic
20C4⋊C4×Dic53C2211yescharacteristic
21#16810474—21no—
22C53×#380C411yescharacteristic
23C22×#88324—22no—
24C22×#88324—22no—
25C4⋊#88324—24no—
26C22⋊#88324—24no—
27C22⋊#88324—42no—
28C4×#88324—24no—
29C23×Dic53D411yescharacteristic
30C23×Dic53D411yescharacteristic
31#13133173—44no—
32C2×C4⋊Dic53C2×C411yescharacteristic
33C2×C4⋊Dic53C2×C411yescharacteristic
34C2×#11719919—21no—
35C2×#11719919—21no—
36C2×#11719919—21no—
37C2×#11719919—21no—
38C2×C22⋊Dic53—21no—
39C2×C4×Dic53—24no—
40C2×C4×Dic53—21no—
41C2×C4×Dic53—21no—
42C2×C4×Dic53C2×C411yescharacteristic
43C2×C4×Dic53D411yescharacteristic
44C2×C4×Dic53D411yescharacteristic
45C2×C4×Dic53C2311yescharacteristic
46#13133124—21no—
47#13133124—21no—
48#13133124—21no—
49C23×C4×C53C2×C411yescharacteristic
50C2×C4⋊C4×C53C2×C411yescharacteristic
51C2×C4⋊C4×C53C2×C411yescharacteristic
52C2×C22⋊C4×C53—21no—
53#95×C53—21no—
54C24×C53C22⋊C411yescharacteristic
55C22×C4×C53—22no—
56C22×C4×C53—22no—
57C22×C4×C53—22no—
58C22×C4×C53—21no—
59C22×C4×C53—21no—
60C22×C4×C53C22⋊C411yescharacteristic
61C22×C4×C53C22×C411yescharacteristic
62C22×Dic53—22no—
63C22×Dic53—22no—
64C22×Dic53—22no—
65C22×Dic53—22no—
66C22×Dic53—21no—
67C22×Dic53—21no—
68C22×Dic53—41no—
69C22×Dic53C22⋊C411yescharacteristic
70C22×Dic53C22⋊C411yescharacteristic
71C22×Dic53SD1612yes—
72C22×Dic53Q1612yes—
73C22×Dic53C2×D411yescharacteristic
74C22×Dic53C2×D411yescharacteristic
75C2×#88324—24no—
76C2×#88324—42no—
77C2×#88324—42no—
78C2×#88324—42no—
79C2×#88324—42no—
80C2×#88324—42no—
81C4⋊C4×C53—22no—
82C4⋊C4×C53—22no—
83C22⋊C4×C53—42no—
84C22⋊Dic53—42no—
85C4⋊Dic53—22no—
86C4⋊Dic53—22no—
87#11719919—42no—
88#11719919—42no—
89#11719919—42no—
90#11719919—42no—
91C4×Dic53—22no—
92C4×Dic53—22no—
93C4×Dic53—22no—
94C4×Dic53—42no—
95C4×Dic53—42no—
96C4×Dic53#4114yes—
97C23×C53—21no—
98C23×C53—21no—
99C23×C53—21no—
100C23×C53—21no—
101C23×C53—21no—
102C23×C53—21no—
103C23×C53#10312yes—
104C23×C53C2×C22⋊C411yescharacteristic
105C2×C4×C53—21no—
106C2×C4×C53—21no—
107C2×C4×C53—21no—
108C2×C4×C53—44no—
109C2×C4×C53—41no—
110C2×C4×C53—41no—
111C2×C4×C53—41no—
112C2×C4×C53—41no—
113C2×C4×C53—41no—
114C2×C4×C53—41no—
115C2×C4×C53#11712yes—
116C2×C4×C53#13112yes—
117C2×C4×C53#13112yes—
118C2×Dic53—21no—
119C2×Dic53—21no—
120C2×Dic53—21no—
121C2×Dic53—21no—
122C2×Dic53—21no—
123C2×Dic53—21no—
124C2×Dic53—24no—
125C2×Dic53—24no—
126C2×Dic53—44no—
127C2×Dic53—41no—
128C2×Dic53—41no—
129C2×Dic53—41no—
130C2×Dic53—41no—
131C2×Dic53—41no—
132C2×Dic53—41no—
133C2×Dic53#10312yes—
134C2×Dic53#10412yes—
135C2×Dic53#12312yes—
136C2×Dic53#12312yes—
137C2×Dic53C2×SD1611yescharacteristic
138C2×Dic53C2×Q1611yescharacteristic
139C2×Dic53AGL(1,ℤ/8)12yes—
140#88324—44no—
141#88324—82no—
142#8710—531nomaximal, Sylow
143C22×C53—22no—
144C22×C53—22no—
145C22×C53—22no—
146C22×C53—41no—
147C22×C53—41no—
148C22×C53—41no—
149C22×C53—41no—
150C22×C53#38111yescharacteristic
151C22×C53#40811yescharacteristic
152C22×C53#40811yescharacteristic
153C22×C53C2×#10311yescharacteristic
154C22×C53#41811yescharacteristic
155C22×C53C2×#10411yescharacteristic
156C22×C53#42111yescharacteristic
157C4×C53—41no—
158C4×C53—81no—
159C4×C53—81no—
160C4×C53#43314yes—
161Dic53—22no—
162Dic53—22no—
163Dic53—22no—
164Dic53—81no—
165Dic53—81no—
166Dic53#48112yes—
167Dic53#48412yes—
168#2285—531no—
169#1797—531no—
170C4×C22⋊C8—531no—
171#1279—532no—
172#1279—532no—
173C2×C53—42no—
174C2×C53#148111yescharacteristic
175C2×C53#148411yescharacteristic
176C2×C53#149311yescharacteristic
177C2×C53#149611yescharacteristic
178C2×C53#181811yescharacteristic
179C2×C53#182411yescharacteristic
180C2×C53#182611yescharacteristic
181C2×#117—1061no—
182C22×C42—531no—
183C2×C4⋊C8—532no—
184C2×C22⋊C8—1061no—
185C2×C22⋊C8—1061no—
186C2×C4×C8—532no—
187#384—531no—
188#384—531no—
189#382—1061no—
190#380—531no—
191#380—1061no—
192#380#615711yescharacteristic
193C4×C4⋊C4—531no—
194C4×C4⋊C4—531no—
195C4×C4⋊C4—1061no—
196#336—1061no—
197#336—1061no—
198C53#871011yescharacteristic, complemented, Sylow
199C23×C4—531no—
200C23×C4—531no—
201C23×C4C2×#615711yescharacteristic
202C22×C8—1062no—
203C22×C8—1062no—
204#117—2124no—
205C2×C4⋊C4—531no—
206C2×C4⋊C4—531no—
207C2×C4⋊C4—1061no—
208C2×C4⋊C4—1061no—
209C2×C4⋊C4—1061no—
210C2×C4⋊C4—1061no—
211C2×C4⋊C4C2×#615711yescharacteristic
212C2×C4⋊C4C2×#615711yescharacteristic
213C2×C22⋊C4—21no—
214C2×C22⋊C4—1061no—
215C2×C42—531no—
216C2×C42—531no—
217C2×C42—531no—
218C2×C42—531no—
219C2×C42—1064no—
220C2×C42—1061no—
221C2×C42—1061no—
222C4⋊C8—1064no—
223C22⋊C8—1064no—
224C22⋊C8—2122no—
225C4×C8—1064no—
226#95—21no—
227#95—1061no—
228#95—1061no—
229#95—1061no—
230C24C22⋊#615711yescharacteristic
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—531no—
237C22×C4—532no—
238C22×C4—532no—
239C22×C4—531no—
240C22×C4—531no—
241C22×C4—531no—
242C22×C4—1062no—
243C22×C4—1062no—
244C22×C4—1062no—
245C22×C4—1062no—
246C22×C4—1061no—
247C22×C4—1061no—
248C22×C4—2121no—
249C22×C4C22⋊#615711yescharacteristic
250C22×C4C22×#615711yescharacteristic
251C2×C8—1064no—
252C2×C8—2122no—
253C2×C8—2122no—
254C2×C8—2122no—
255C2×C8—2122no—
256C2×C8—2122no—
257C4⋊C4—22no—
258C4⋊C4—22no—
259C4⋊C4—1062no—
260C4⋊C4—1062no—
261C4⋊C4—2122no—
262C4⋊C4—2122no—
263C4⋊C4—2122no—
264C4⋊C4—2122no—
265C22⋊C4—42no—
266C22⋊C4—2122no—
267C42—534no—
268C42—1062no—
269C42—1062no—
270C42—1062no—
271C42—2122no—
272C42—2122no—
273C23—21no—
274C23—21no—
275C23—21no—
276C23—21no—
277C23—21no—
278C23—21no—
279C23#1313328012yes—
280C23C2×C22⋊#615711yescharacteristic
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—44no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—532no—
292C2×C4—531no—
293C2×C4—531no—
294C2×C4—532no—
295C2×C4—532no—
296C2×C4—532no—
297C2×C4—532no—
298C2×C4—1064no—
299C2×C4—1064no—
300C2×C4—1061no—
301C2×C4—1061no—
302C2×C4—1061no—
303C2×C4—1061no—
304C2×C4—1061no—
305C2×C4—1061no—
306C2×C4—2124no—
307C2×C4—2121no—
308C2×C4—2121no—
309C2×C4—2121no—
310C2×C4—2121no—
311C2×C4—2121no—
312C2×C4—2121no—
313C2×C4#1313329612yes—
314C2×C4#1313330412yes—
315C2×C4#1313330412yes—
316C8—4242no—
317C8—2124no—
318C22—22no—
319C22—22no—
320C22—22no—
321C22—41no—
322C22—41no—
323C22—41no—
324C22—41no—
325C22#1681156611yescharacteristic
326C22#1681159611yescharacteristic
327C22#1681159611yescharacteristic
328C22C2×#1313328011yescharacteristic
329C22#1681161511yescharacteristic
330C22D53⋊#10411yescharacteristic
331C22#1681162011yescharacteristic
332C4—41no—
333C4—81no—
334C4—81no—
335C4—532no—
336C4—532no—
337C4—1062no—
338C4—1062no—
339C4—1062no—
340C4—4241no—
341C4—4241no—
342C4#1681164714yes—
343C2—42no—
344C2#2029002311yescharacteristic
345C2#2029003411yescharacteristic
346C2#2029006311yescharacteristic
347C2#2029007411yescharacteristic
348C2#2029060811yescharacteristic
349C2#2029061911yescharacteristic
350C2#2029062311yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6157
commutatorC22×C4×C53C22×C4
FrattiniC22×C4C22×#6157
FittingC53×#380C4
socleC23×C53C2×C22⋊C4
prime radicalC24×C53C22⋊C4
cosocleC2×C4×Dic53C23
hypercenter#380#6157
hypocenterC53#8710
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ #380

Complements

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

NH ≅ G/Ntype
3. #20289551C2semidirect
4. #20289551C2semidirect
198. C53#8710semidirect

Maximal subgroups

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

subgroupindex
#202905692
C4×C22⋊#883242
#202895512
#202771172
#871053