Finite Group Databasebeta

← #27294117

Subgroups — #27294117

Order 13568.

430 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 430 conjugacy classes of subgroups — too many to draw. Only its 97 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
1C4×D8×D53C211yesmaximal, characteristic
2#20276512C211yesmaximal, characteristic, complemented
3#20273472C211yesmaximal, characteristic, complemented
4#20272677C211yesmaximal, characteristic, complemented
5#20272676C211yesmaximal, characteristic, complemented
6#20272675C211yesmaximal, characteristic, complemented
7C53×#1331C211yesmaximal, characteristic, complemented
8C2×D8×D53C411yescharacteristic
9C4×D4×D53—21no—
10C4×D424C2211yescharacteristic
11D8×Dic53C411yescharacteristic
12#16810705C411yescharacteristic
13C4×#11719922—21no—
14#16810703—21no—
15#107×D53C411yescharacteristic
16#16810645—21no—
17#102×D53—21no—
18C4×C8×D53C2211yescharacteristic
19#16810441—21no—
20#16810402—21no—
21C4×#11719908C2211yescharacteristic, complemented
22#16810376C2211yescharacteristic, complemented
23#16810375—21no—
24C4×D8×C53C2211yescharacteristic
25C4×C16×C53C2211yescharacteristic, complemented
26C53×#335C2211yescharacteristic, complemented
27C2×D4×D53—21no—
28C22×C4×D53—41no—
29C2×D424C2×C411yescharacteristic
30C2×#11719922—21no—
31D8×D53—22no—
32D8×D53—22no—
33C2×#11719912—41no—
34C2×C8×D53C2×C411yescharacteristic
35C2×C8×D53D411yescharacteristic
36C4×D212—21no—
37D4×Dic53—21no—
38#13133179—21no—
39C4×C22⋊D53—41no—
40#13133177—41no—
41C4⋊C4×D53—21no—
42C22⋊C4×D53—41no—
43C42×D53D411yescharacteristic
44C2×#11719908—21no—
45#13133151C2×C411yescharacteristic
46#13133150—21no—
47C4⋊#88322—21no—
48#13133146—41no—
49#13133139—21no—
50#13133138—21no—
51#13133137—21no—
52C4×#88322D411yescharacteristic
53C8×Dic53C2×C411yescharacteristic
54C8×Dic53D411yescharacteristic
55C2×D8×C53C2×C411yescharacteristic
56C4×D4×C53—21no—
57C2×C16×C53—21no—
58#107×C53C2×C411yescharacteristic
59C4⋊C8×C53—21no—
60#102×C53—21no—
61C4×C8×C53C2311yescharacteristic
62C23×D53—41no—
63C2×D4×C53—21no—
64C22×C4×C53—41no—
65C2×C22⋊D53—41no—
66C22×Dic53—41no—
67D4×D53—24no—
68D4×D53—42no—
69C2×D212—21no—
70C2×C4×D53—41no—
71C2×C4×D53—41no—
72C2×C4×D53—41no—
73C2×C4×D53—82no—
74C2×C4×D53—82no—
75C2×C4×D53C22⋊C411yescharacteristic
76C2×C4×D53D811yescharacteristic
77C2×C4×D53SD1611yescharacteristic
78D8×C53—21no—
79D8×C53M1612yescomplemented
80C2×C8×C53—41no—
81C2×C8×C53C22×C411yescharacteristic
82C2×C8×C53C2×D411yescharacteristic
83C4⋊C4×C53—21no—
84C22⋊C4×C53—41no—
85C42×C53C2×D411yescharacteristic
86C22⋊Dic53—41no—
87#11719922—24no—
88#11719921—41no—
89#11719921—41no—
90C4⋊Dic53—21no—
91#11719919—41no—
92C4×Dic53C22⋊C411yescharacteristic
93C4×Dic53D811yescharacteristic
94C4×Dic53SD1611yescharacteristic
95C2×#88322—41no—
96C2×#88322C22⋊C411yescharacteristic
97C2×#88322C22⋊C411yescharacteristic
98D424—21no—
99D424M1612yescomplemented
100#11719912—82no—
101C8×D53—21no—
102C8×D53—21no—
103C8×D53—41no—
104C16×C53—41no—
105#11719908—41no—
106C23×C53—41no—
107C22×D53—44no—
108C22×D53—41no—
109C22×D53—82no—
110C22×D53—82no—
111C22×D53#10211yescharacteristic
112D4×C53—22no—
113D4×C53—41no—
114C2×C4×C53—41no—
115C2×C4×C53—81no—
116C2×C4×C53—81no—
117C2×C4×C53C2×C22⋊C411yescharacteristic
118C2×C4×C53C2×D811yescharacteristic
119C2×C4×C53C2×SD1611yescharacteristic
120C22⋊D53—44no—
121C2×Dic53—41no—
122C2×Dic53—82no—
123C2×Dic53#10211yescharacteristic
124C2×Dic53#10211yescharacteristic
125C2×Dic53#10211yescharacteristic
126D212—22no—
127D212—41no—
128C4×D53—21no—
129C4×D53—21no—
130C4×D53—41no—
131C4×D53—82no—
132C4×D53—81no—
133C4×D53—81no—
134C4×D53—82no—
135C4×D53D1612yes—
136C4×D53SD3212yes—
137C8×C53—21no—
138C8×C53—81no—
139C8×C53C2×M1611yescharacteristic
140C8×C53C2×M1611yescharacteristic
141#88322—21no—
142#88322—21no—
143#88322—81no—
144#11657—531nomaximal, Sylow
145C22×C53—42no—
146C22×C53—81no—
147C22×C53—81no—
148C22×C53C2×#10211yescharacteristic
149C2×D53—21no—
150C2×D53—21no—
151C2×D53—42no—
152C2×D53—88no—
153C2×D53—81no—
154C2×D53—81no—
155C2×D53#35712yes—
156C4×C53—21no—
157C4×C53—81no—
158C4×C53#40711yescharacteristic
159C4×C53#42111yescharacteristic
160C4×C53C2×D1611yescharacteristic
161C4×C53C2×SD3211yescharacteristic
162Dic53—21no—
163Dic53—21no—
164Dic53—81no—
165Dic53#35712yes—
166C2×C4×D8—531no—
167#2162—531no—
168#1567—531no—
169#1331—531no—
170#1331—531no—
171#1331—531no—
172#1331D5311yescharacteristic, complemented
173C2×C53—82no—
174C2×C53#148611yescharacteristic
175C2×C53C2×#35711yescharacteristic
176C2×C53#214311yescharacteristic
177D53—22no—
178D53—82no—
179C22×D8—531no—
180C2×C4×D4—1061no—
181C4×D8—531no—
182C4×D8—531no—
183C4×D8—531no—
184C4×D8—1061no—
185C4×D8—1061no—
186C4×D8C2×D5311yescharacteristic
187C2×#107—531no—
188#423—1061no—
189C2×#102—1061no—
190C2×C4×C8—531no—
191C4⋊C16—1061no—
192C22⋊C16—1061no—
193C4×C16—531no—
194C4×C16C2×D5311yescharacteristic, complemented
195#335—531no—
196#335—1061no—
197#335C2×D5311yescharacteristic, complemented
198C53#1165711yescharacteristic, complemented, Sylow
199C22×D4—1061no—
200C23×C4—2121no—
201C2×D8—531no—
202C2×D8—1061no—
203C2×D8—1062no—
204C2×D8—1062no—
205C2×D8C4×D5311yescharacteristic
206C2×M16—2121no—
207C22×C8—531no—
208C22×C8—531no—
209C4×D4—21no—
210C4×D4—1061no—
211C4×D4—1061no—
212C4×D4—1061no—
213C4×D4—2121no—
214C4×D4—2121no—
215C2×C4⋊C4—1061no—
216C2×C22⋊C4—2121no—
217C2×C42—531no—
218C2×C16—21no—
219C2×C16—1061no—
220#107—531no—
221#107—1061no—
222#107C4×D5311yescharacteristic
223C4⋊C8—21no—
224C4⋊C8—1061no—
225C4⋊C8—2121no—
226#102—21no—
227#102—1061no—
228#102—1061no—
229#102—1061no—
230C4×C8—531no—
231C4×C8—531no—
232C4×C8—531no—
233C4×C8C22×D5311yescharacteristic
234C24—2121no—
235C2×D4—21no—
236C2×D4—1064no—
237C2×D4—1061no—
238C2×D4—2121no—
239C2×D4—2122no—
240C22×C4—41no—
241C22×C4—531no—
242C22×C4—531no—
243C22×C4—531no—
244C22×C4—4242no—
245C22×C4—4242no—
246C22×C4—2121no—
247C22×C4—2121no—
248C22×C4—2121no—
249C22×C4—2121no—
250D8—21no—
251D8—532no—
252D8—1064no—
253D8—1061no—
254D8#1171991212yescomplemented
255M16—4242no—
256C2×C8—41no—
257C2×C8—531no—
258C2×C8—531no—
259C2×C8—1061no—
260C2×C8—1061no—
261C2×C8—2121no—
262C2×C8—2121no—
263C2×C8C2×C4×D5311yescharacteristic
264C2×C8D4×D5311yescharacteristic
265C4⋊C4—21no—
266C4⋊C4—1061no—
267C4⋊C4—2121no—
268C22⋊C4—41no—
269C22⋊C4—2121no—
270C22⋊C4—2121no—
271C22⋊C4—2121no—
272C42—531no—
273C42—531no—
274C42—531no—
275C42D4×D5311yescharacteristic
276C16—41no—
277C16—2121no—
278C23—41no—
279C23—531no—
280C23—4242no—
281C23—4242no—
282C23—2124no—
283C23—2121no—
284D4—22no—
285D4—41no—
286D4—1062no—
287D4—2124no—
288D4—2121no—
289C2×C4—41no—
290C2×C4—81no—
291C2×C4—81no—
292C2×C4—531no—
293C2×C4—532no—
294C2×C4—532no—
295C2×C4—531no—
296C2×C4—531no—
297C2×C4—1061no—
298C2×C4—1061no—
299C2×C4—4242no—
300C2×C4—4242no—
301C2×C4—4242no—
302C2×C4—4241no—
303C2×C4—4241no—
304C2×C4—2121no—
305C2×C4—2121no—
306C2×C4C22⋊C4×D5311yescharacteristic
307C2×C4D8×D5311yescharacteristic
308C2×C4SD16×D5311yescharacteristic
309C8—21no—
310C8—81no—
311C8—1061no—
312C8—1061no—
313C8—4241no—
314C8M16×D5311yescharacteristic
315C8C2×#1171991211yescharacteristic
316C22—42no—
317C22—81no—
318C22—81no—
319C22—532no—
320C22—1061no—
321C22—1061no—
322C22—4248no—
323C22—4241no—
324C22—4241no—
325C22—2122no—
326C22#102×D5311yescharacteristic
327C4—21no—
328C4—81no—
329C4—532no—
330C4—1061no—
331C4—1061no—
332C4—4241no—
333C4#1681058511yescharacteristic
334C4#1681063811yescharacteristic
335C4D16×D5311yescharacteristic
336C4SD32×D5311yescharacteristic
337C2—82no—
338C2—1062no—
339C2—4242no—
340C2#2027313811yescharacteristic
341C2#357×D5311yescharacteristic
342C2#2027644111yescharacteristic
subgroupG/H
centerC2×C4D8×D53
commutatorC2×C8×C53C22×C4
FrattiniC4×C8C22×D53
FittingC53×#1331C2
socleC22×C53C2×#102
prime radicalC2×D8×D53C4
cosocleC4×C8×C53C23
hypercenter#1331D53
hypocenterC53#11657
LCSG ▷ C2×C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C42 ◁ C4×C8 ◁ #1331

Complements

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

NH ≅ G/Ntype
2. #20276512C2semidirect
3. #20273472C2semidirect
4. #20272677C2semidirect
5. #20272676C2semidirect
6. #20272675C2semidirect
7. C53×#1331C2semidirect
21. C4×#11719908C22semidirect
22. #16810376C22semidirect
25. C4×C16×C53C22semidirect
26. C53×#335C22semidirect
79. D8×C53M16semidirect
99. D424M16semidirect
172. #1331D53semidirect
194. C4×C16C2×D53semidirect
197. #335C2×D53semidirect
198. C53#11657semidirect
254. D8#11719912semidirect

Maximal subgroups

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

subgroupindex
C4×D8×D532
#202765122
#202734722
#202726772
#202726762
#202726752
C53×#13312
#1165753