Finite Group Databasebeta

← #27294449

Subgroups — #27294449

Order 13568.

442 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 442 conjugacy classes of subgroups — too many to draw. Only its 83 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#20280583C211yesmaximal, characteristic
2#20273033C211yesmaximal, characteristic
3#20272738C212yesmaximal, complemented
4#20272714C212yesmaximal, complemented
5C4×#99×C53C211yesmaximal, characteristic
6C2×#13133209C411yescharacteristic, complemented
7C2×C22⋊#88322—21no—
8#16810559C411yescharacteristic, complemented
9#16810559C2211yescharacteristic
10#16810554—21no—
11#16810548C411yescharacteristic, complemented
12#16810548C2211yescharacteristic
13#16810537—21no—
14#16810496—21no—
15#16810496—21no—
16#16810491C411yescharacteristic, complemented
17#16810349—21no—
18#16810347C2212yes—
19C2×C4×D4×C53C2211yescharacteristic
20C2×#99×C53—21no—
21C4×C22⋊C4×C53—21no—
22C53×#342C2212yes—
23C22×#88322—41no—
24#13133209—41no—
25#13133209—41no—
26C2×C4⋊Dic53C2×C411yescharacteristic
27C2×C4⋊Dic53C2×C411yescharacteristic
28C2×#11719919—21no—
29C2×#11719919—21no—
30C2×#11719919—21no—
31C2×#11719919—21no—
32C2×C22⋊Dic53—21no—
33C2×C22⋊Dic53—21no—
34C2×C22⋊Dic53—21no—
35C2×C22⋊Dic53—21no—
36C22⋊#88322—22no—
37C22⋊#88322—41no—
38#13133124—21no—
39#13133124—21no—
40#13133124—21no—
41#13133124—21no—
42#13133124C2×C411yescharacteristic
43#13133124C2×C411yescharacteristic
44C22×D4×C53D411yescharacteristic
45C23×C4×C53D411yescharacteristic
46C23×C4×C53D411yescharacteristic
47C4×D4×C53—41no—
48C4×D4×C53—41no—
49C2×C4⋊C4×C53D411yescharacteristic
50C2×C22⋊C4×C53—22no—
51C2×C22⋊C4×C53C2×C411yescharacteristic
52C2×C22⋊C4×C53C2311yescharacteristic
53C2×C42×C53—41no—
54C2×C42×C53C2×C411yescharacteristic
55#99×C53—24no—
56#95×C53—41no—
57C24×C53C22⋊C411yescharacteristic
58C24×C53C2×D411yescharacteristic
59C2×D4×C53—22no—
60C2×D4×C53—22no—
61C2×D4×C53—41no—
62C22×C4×C53—21no—
63C22×C4×C53—21no—
64C22×C4×C53—21no—
65C22×C4×C53—21no—
66C22×C4×C53—42no—
67C22×C4×C53—81no—
68C22×C4×C53—81no—
69C22×C4×C53C22⋊C411yescharacteristic
70C22×C4×C53C22⋊C411yescharacteristic
71C22×C4×C53C22⋊C411yescharacteristic
72C22×C4×C53C22×C411yescharacteristic
73C22×C4×C53C2×D411yescharacteristic
74C22×Dic53—21no—
75C22×Dic53—21no—
76C22×Dic53—21no—
77C22×Dic53—21no—
78C4⋊C4×C53—41no—
79C22⋊C4×C53—24no—
80C22⋊C4×C53—41no—
81C22⋊C4×C53—41no—
82C22⋊C4×C53—42no—
83C42×C53—21no—
84C42×C53—21no—
85C42×C53—82no—
86C22⋊Dic53—42no—
87C22⋊Dic53—42no—
88C22⋊Dic53—81no—
89C22⋊Dic53—81no—
90C4⋊Dic53—41no—
91C4⋊Dic53—41no—
92#11719919—42no—
93#11719919—41no—
94#11719919—41no—
95#11719919—81no—
96#11719919—81no—
97C2×#88322—42no—
98C2×#88322—82no—
99C23×C53—21no—
100C23×C53—21no—
101C23×C53—21no—
102C23×C53—21no—
103C23×C53—41no—
104C23×C53—41no—
105C23×C53—41no—
106C23×C53—41no—
107C23×C53#10412yes—
108C23×C53C2×C22⋊C411yescharacteristic
109C23×C53#13512yes—
110D4×C53—44no—
111C2×C4×C53—21no—
112C2×C4×C53—21no—
113C2×C4×C53—21no—
114C2×C4×C53—21no—
115C2×C4×C53—21no—
116C2×C4×C53—21no—
117C2×C4×C53—41no—
118C2×C4×C53—42no—
119C2×C4×C53—42no—
120C2×C4×C53—41no—
121C2×C4×C53—41no—
122C2×C4×C53—41no—
123C2×C4×C53—44no—
124C2×C4×C53—81no—
125C2×C4×C53—81no—
126C2×C4×C53—82no—
127C2×C4×C53—82no—
128C2×C4×C53#9911yescharacteristic
129C2×C4×C53#9911yescharacteristic
130C2×Dic53—41no—
131C2×Dic53—41no—
132C2×Dic53—41no—
133C2×Dic53—41no—
134C2×Dic53—41no—
135C2×Dic53—41no—
136C2×Dic53—41no—
137C2×Dic53—41no—
138C2×Dic53—41no—
139C2×Dic53—41no—
140C2×Dic53—41no—
141C2×Dic53—41no—
142#88322—82no—
143#11732—531nomaximal, Sylow
144C22×C53—21no—
145C22×C53—21no—
146C22×C53—22no—
147C22×C53—22no—
148C22×C53—42no—
149C22×C53—42no—
150C22×C53—41no—
151C22×C53—41no—
152C22×C53—41no—
153C22×C53—41no—
154C22×C53—81no—
155C22×C53—81no—
156C22×C53C2×#9911yescharacteristic
157C22×C53#41611yescharacteristic
158C22×C53C2×#10411yescharacteristic
159C4×C53—21no—
160C4×C53—21no—
161C4×C53—41no—
162C4×C53—41no—
163C4×C53—41no—
164C4×C53—41no—
165C4×C53—84no—
166Dic53—81no—
167Dic53—81no—
168Dic53—81no—
169Dic53—81no—
170#2633—531no—
171C4×#99D5311yescharacteristic
172#1459—531no—
173#1352—532no—
174#1344—532no—
175C2×C53—21no—
176C2×C53—21no—
177C2×C53—42no—
178C2×C53—42no—
179C2×C53#150111yescharacteristic
180C2×C53#212311yescharacteristic
181C2×C53#212711yescharacteristic
182C2×#125—531no—
183C2×C4×D4C2×D5311yescharacteristic
184C2×#99—21no—
185C2×C22⋊C8—1061no—
186#399—531no—
187#399—531no—
188#398—1061no—
189#397—531no—
190#397—531no—
191#394—1061no—
192#385—1061no—
193#385—1061no—
194#383—531no—
195C4×C22⋊C4—21no—
196#342C2×D5312yes—
197#324—1061no—
198#323—532no—
199C53#1173211yescharacteristic, complemented, Sylow
200C22×D4C22⋊D5311yescharacteristic
201C23×C4D21211yescharacteristic
202C23×C4C22⋊D5311yescharacteristic
203C22×C8—2121no—
204#125—2121no—
205#125—2121no—
206C4×D4—41no—
207C4×D4—41no—
208C2×C4⋊C4—531no—
209C2×C4⋊C4—531no—
210C2×C4⋊C4—1061no—
211C2×C4⋊C4—1061no—
212C2×C4⋊C4—1061no—
213C2×C4⋊C4—1061no—
214C2×C4⋊C4D21211yescharacteristic
215C2×C22⋊C4—22no—
216C2×C22⋊C4—1061no—
217C2×C22⋊C4—1061no—
218C2×C22⋊C4—1061no—
219C2×C22⋊C4—1061no—
220C2×C22⋊C4C4×D5311yescharacteristic
221C2×C22⋊C4C22×D5311yescharacteristic
222C2×C42—41no—
223C2×C42C4×D5311yescharacteristic
224#99—24no—
225C22⋊C8—1062no—
226C22⋊C8—2121no—
227#95—41no—
228#95—531no—
229#95—531no—
230#95—1061no—
231#95—1061no—
232#95—1061no—
233#95—1061no—
234C24#1171992111yescharacteristic
235C24C2×C22⋊D5311yescharacteristic
236C2×D4—22no—
237C2×D4—22no—
238C2×D4—41no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—42no—
244C22×C4—81no—
245C22×C4—81no—
246C22×C4—1061no—
247C22×C4—1061no—
248C22×C4—1061no—
249C22×C4—1061no—
250C22×C4#1171992111yescharacteristic
251C22×C4#1171992111yescharacteristic
252C22×C4#1171992111yescharacteristic
253C22×C4C2×C4×D5311yescharacteristic
254C22×C4C2×D21211yescharacteristic
255C2×C8—4242no—
256C2×C8—2122no—
257C4⋊C4—41no—
258C4⋊C4—4241no—
259C4⋊C4—4241no—
260C4⋊C4—2121no—
261C4⋊C4—2121no—
262C4⋊C4—2122no—
263C4⋊C4—2121no—
264C4⋊C4—2121no—
265C22⋊C4—24no—
266C22⋊C4—41no—
267C22⋊C4—41no—
268C22⋊C4—42no—
269C22⋊C4—4241no—
270C22⋊C4—4241no—
271C22⋊C4—2122no—
272C22⋊C4—2122no—
273C42—21no—
274C42—21no—
275C42—82no—
276C23—21no—
277C23—21no—
278C23—21no—
279C23—21no—
280C23—41no—
281C23—41no—
282C23—41no—
283C23—41no—
284C23#1313314512yes—
285C23C2×#1171992111yescharacteristic
286C23#1313323712yes—
287D4—44no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—41no—
295C2×C4—42no—
296C2×C4—42no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—44no—
301C2×C4—81no—
302C2×C4—81no—
303C2×C4—82no—
304C2×C4—82no—
305C2×C4—2121no—
306C2×C4—2121no—
307C2×C4—2121no—
308C2×C4—2121no—
309C2×C4—2121no—
310C2×C4—2121no—
311C2×C4—2121no—
312C2×C4—2121no—
313C2×C4—2121no—
314C2×C4—2121no—
315C2×C4—2121no—
316C2×C4—2121no—
317C2×C4#1313313111yescharacteristic
318C2×C4#1313313111yescharacteristic
319C8—4242no—
320C22—21no—
321C22—21no—
322C22—22no—
323C22—22no—
324C22—42no—
325C22—42no—
326C22—41no—
327C22—41no—
328C22—41no—
329C22—41no—
330C22—81no—
331C22—81no—
332C22C2×#1313313111yescharacteristic
333C22#1681061711yescharacteristic
334C22C2×#1313314511yescharacteristic
335C4—21no—
336C4—21no—
337C4—41no—
338C4—41no—
339C4—41no—
340C4—41no—
341C4—84no—
342C4—4241no—
343C4—4241no—
344C4—4241no—
345C4—4241no—
346C2—21no—
347C2—21no—
348C2—42no—
349C2—42no—
350C2#2027320411yescharacteristic
351C2#2027636811yescharacteristic
352C2#2027638211yescharacteristic
subgroupG/H
centerC22C2×#13133131
commutatorC22×C4×C53C22×C4
FrattiniC2×C22⋊C4C22×D53
FittingC4×#99×C53C2
socleC22×C53C2×#99
prime radicalC22×D4×C53D4
cosocleC2×C22⋊C4×C53C23
hypercenterC4×#99D53
hypocenterC53#11732
LCSG ▷ C22×C4×C53 ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ C4×#99

Complements

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

NH ≅ G/Ntype
3. #20272738C2semidirect
4. #20272714C2semidirect
6. C2×#13133209C4semidirect
8. #16810559C4semidirect
11. #16810548C4semidirect
16. #16810491C4semidirect
199. C53#11732semidirect

Maximal subgroups

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

subgroupindex
#202805832
#202730332
#202727382
#202727142
C4×#99×C532
#1173253