Finite Group Databasebeta

← #27540680

Subgroups — #27540680

Order 13568.

712 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 712 conjugacy classes of subgroups — too many to draw. Only its 147 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
1C2×#16811057C212yesmaximal, complemented
2#20286401C212yesmaximal, complemented
3#20276976C211yesmaximal, characteristic, complemented
4#20276829C212yesmaximal, complemented
5#20276770C212yesmaximal, complemented
6#20276656C212yesmaximal, complemented
7#20276628C212yesmaximal, complemented
8#20276548C211yesmaximal, characteristic, complemented
9C4×D16×C53C211yesmaximal, characteristic, complemented
10C2×#13133236C2212yes—
11C2×D8×D53C2212yescomplemented
12#16811268—22no—
13#16811057—28no—
14C2×Dic424C2211yescharacteristic, complemented
15C2×#13133162C2212yescomplemented
16C2×#13133161C2212yescomplemented
17C2×D848C2211yescharacteristic
18C2×#13133158C2212yes—
19C2×#13133156C2212yescomplemented
20#16811007C2212yescomplemented
21#16810865C2212yescomplemented
22#16810860C2212yes—
23#16810838—22no—
24#16810775—22no—
25#16810769—22no—
26C8×D212C2211yescharacteristic
27C4⋊#11719908C2211yescharacteristic, complemented
28#16810426C2212yescomplemented
29#16810423C2212yescomplemented
30#16810402C2212yescomplemented
31C2×D16×C53C2211yescharacteristic
32C4×D8×C53C2212yescomplemented
33C53×#366C2211yescharacteristic, complemented
34C53×#357C2212yescomplemented
35C4×C16×C53C2211yescharacteristic, complemented
36C2×#11719947—22no—
37C2×D4×D53—22no—
38#13133236—24no—
39#13133236—42no—
40C2×Dic212C2312yes—
41C2×#11719913—22no—
42C2×#11719923—22no—
43C2×D424C2312yes—
44C2×#11719922—22no—
45D8×D53—24no—
46D8×D53—42no—
47C2×C8×D53C2312yes—
48#13133205—22no—
49#13133201—42no—
50#13133197—22no—
51#13133192—22no—
52#13133191—42no—
53#13133187—42no—
54C4×D212D411yescharacteristic
55C2×#11719921—42no—
56Dic424—41no—
57#13133162—24no—
58#13133161—24no—
59D848—41no—
60#13133158—24no—
61#13133156—24no—
62C2×#11719908C2312yes—
63C4⋊#88322D411yescharacteristic
64#13133140—22no—
65#13133137—22no—
66#13133129D412yes—
67C2×D8×C53C2312yes—
68C4×D4×C53—22no—
69D16×C53D414yescomplemented
70C2×C16×C53C2311yescharacteristic
71C2×C16×C53C2311yescharacteristic
72#107×C53C2312yes—
73#102×C53—22no—
74C4×C8×C53C2311yescharacteristic
75C23×D53—42no—
76C2×D4×C53—22no—
77C22×C4×C53—42no—
78C2×C22⋊D53—42no—
79C2×C22⋊D53—42no—
80C22×Dic53—42no—
81#11719947—44no—
82#11719947—82no—
83D4×D53—44no—
84D4×D53—82no—
85C2×D212—22no—
86C2×D212—42no—
87C2×D212D811yescharacteristic
88C2×C4×D53C2×D412yes—
89C2×Dic106—22no—
90D8×C53—22no—
91D8×C53C2×D414yes—
92C2×C8×C53C2×D411yescharacteristic
93C2×C8×C53C2411yescharacteristic
94C4⋊C4×C53—22no—
95C22⋊C4×C53—42no—
96C42×C53C2×D411yescharacteristic
97#11719923—44no—
98#11719922—44no—
99#11719921—42no—
100#11719921—42no—
101#11719921—42no—
102#11719921—42no—
103#11719921D812yes—
104C4⋊Dic53—42no—
105C4⋊Dic53D811yescharacteristic
106C2×#88322C2×D412yes—
107Dic212—22no—
108Dic212—22no—
109D424—22no—
110D424—22no—
111#11719913—82no—
112C8×D53—22no—
113C8×D53—22no—
114C16×C53—21no—
115C16×C53C2×D412yes—
116#11719908—22no—
117C23×C53—42no—
118C22×D53—42no—
119C22×D53—84no—
120C22×D53—84no—
121C22×D53—162no—
122C22×D53C2×D812yes—
123D4×C53—24no—
124D4×C53—42no—
125C2×C4×C53—42no—
126C2×C4×C53—162no—
127C2×C4×C53C2×D811yescharacteristic
128C2×C4×C53C2×D811yescharacteristic
129C2×C4×C53C22×D411yescharacteristic
130C22⋊D53—84no—
131C22⋊D53—84no—
132C2×Dic53—42no—
133C2×Dic53—84no—
134C2×Dic53C2×D812yes—
135D212—41no—
136D212—42no—
137D212—42no—
138D212—162no—
139C4×D53—22no—
140C4×D53—22no—
141Dic106—42no—
142Dic106—42no—
143C8×C53—21no—
144C8×C53C22×D411yescharacteristic
145C8×C53H(5,𝔽2)11yescharacteristic
146#88322—22no—
147#33839—531nomaximal, Sylow
148C22×C53—44no—
149C22×C53—82no—
150C22×C53—82no—
151C22×C53C22×D811yescharacteristic
152C2×D53—22no—
153C2×D53—22no—
154C2×D53—22no—
155C2×D53—82no—
156C2×D53—82no—
157C2×D53—82no—
158C2×D53—168no—
159C4×C53—21no—
160C4×C53—21no—
161C4×C53—82no—
162C4×C53D4211yescharacteristic
163C4×C53#57611yescharacteristic
164Dic53—22no—
165Dic53—82no—
166C2×#509—532no—
167#3280—532no—
168#2249—531no—
169#2220—532no—
170#2210—532no—
171#2189—532no—
172#2184—532no—
173C4×D16D5311yescharacteristic, complemented
174#2169—531no—
175C2×C53—84no—
176C2×C53D4×D811yescharacteristic
177C2×C53#341411yescharacteristic
178C2×C53#341511yescharacteristic
179D53—42no—
180D53—162no—
181C2×#135—532no—
182C22×D8—532no—
183#546—1062no—
184#509—1068no—
185C2×Q32—531no—
186C2×SD32—532no—
187C2×SD32—532no—
188C2×D16—532no—
189C2×D16—531no—
190C2×D16C2×D5311yescharacteristic
191C2×M32—532no—
192#495—532no—
193#467—532no—
194#466—532no—
195#463—1062no—
196#450—1062no—
197#449—1062no—
198C4×D8C2×D5312yescomplemented
199C8×D4—531no—
200#366C2×D5311yescharacteristic, complemented
201C4⋊C16—531no—
202#358—532no—
203#357—532no—
204#357C2×D5312yescomplemented
205C22⋊C16—532no—
206C4×C16C2×D5311yescharacteristic, complemented
207C53#3383911yescharacteristic, complemented, Sylow
208C2×#41—1062no—
209C22×D4—1062no—
210#135—1064no—
211#135—2122no—
212C2×Q16—532no—
213C2×SD16—1062no—
214C2×SD16—1062no—
215C2×D8—532no—
216C2×D8—1062no—
217C2×D8—1064no—
218C2×D8—2122no—
219C2×D8C22×D5312yes—
220C22×C8—532no—
221#124—1062no—
222#123—2122no—
223C22⋊Q8—1062no—
224#121—1062no—
225#121—2122no—
226#120—2122no—
227C4×D4—22no—
228C4×D4—531no—
229C2×C22⋊C4—2122no—
230Q32—2121no—
231SD32—1064no—
232SD32—1064no—
233D16—1064no—
234D16—2121no—
235D16D21214yescomplemented
236M32—1064no—
237C2×C16—532no—
238C2×C16C22×D5311yescharacteristic
239C2×C16C22×D5311yescharacteristic
240#107C22×D5312yes—
241C4⋊C8—531no—
242#103—1062no—
243#102—22no—
244#102—1062no—
245C22⋊C8—532no—
246C4×C8C22×D5311yescharacteristic
247C24—2122no—
248#41—4242no—
249#41—2124no—
250C2×Q8—1062no—
251C2×D4—22no—
252C2×D4—531no—
253C2×D4—1062no—
254C2×D4—4242no—
255C2×D4—2122no—
256C2×D4—2122no—
257C2×D4—2124no—
258C2×D4—2122no—
259C22×C4—42no—
260C22×C4—532no—
261C22×C4—2122no—
262Q16—1062no—
263Q16—1062no—
264SD16—4242no—
265SD16—2124no—
266D8—22no—
267D8—1062no—
268D8—1062no—
269D8—2124no—
270D8C2×D21214yes—
271C2×C8—532no—
272C2×C8—1062no—
273C2×C8—1062no—
274C2×C8D4×D5311yescharacteristic
275C2×C8C23×D5311yescharacteristic
276C4⋊C4—22no—
277C4⋊C4—531no—
278C4⋊C4—2122no—
279C22⋊C4—42no—
280C22⋊C4—532no—
281C22⋊C4—2122no—
282C22⋊C4—2122no—
283C22⋊C4—2122no—
284C22⋊C4—2122no—
285C42D4×D5311yescharacteristic
286C16—21no—
287C16—1062no—
288C16C2×D21212yes—
289C23—42no—
290C23—532no—
291C23—8482no—
292C23—4244no—
293C23—4244no—
294C23—2122no—
295Q8—2122no—
296Q8—2122no—
297D4—24no—
298D4—42no—
299D4—8482no—
300D4—4244no—
301D4—4244no—
302D4—2121no—
303D4—2122no—
304D4—2122no—
305C2×C4—42no—
306C2×C4—162no—
307C2×C4—532no—
308C2×C4—1062no—
309C2×C4—1062no—
310C2×C4—4244no—
311C2×C4—2122no—
312C2×C4D8×D5311yescharacteristic
313C2×C4D8×D5311yescharacteristic
314C2×C4C2×D4×D5311yescharacteristic
315C8—21no—
316C8—1062no—
317C8C22×D21211yescharacteristic
318C8#1313326311yescharacteristic
319C22—44no—
320C22—82no—
321C22—82no—
322C22—8488no—
323C22—1062no—
324C22—1062no—
325C22—1062no—
326C22—4242no—
327C22—4242no—
328C22—4242no—
329C22C2×D8×D5311yescharacteristic
330C4—21no—
331C4—21no—
332C4—82no—
333C4—1062no—
334C4—4242no—
335C4D4×D21211yescharacteristic
336C4#1681142511yescharacteristic
337C2—84no—
338C2—8482no—
339C2—2122no—
340C2D8×D21211yescharacteristic
341C2#2028784811yescharacteristic
342C2#2028785211yescharacteristic
subgroupG/H
centerC22C2×D8×D53
commutator, cosocleC2×C8×C53C24
FrattiniC2×C8C23×D53
FittingC4×D16×C53C2
socleC22×C53C22×D8
hypercenterC4×D16D53
hypocenterC53#33839
LCSG ▷ C2×C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ C4×D16

Complements

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

NH ≅ G/Ntype
1. C2×#16811057C2semidirect
2. #20286401C2semidirect
3. #20276976C2semidirect
4. #20276829C2semidirect
5. #20276770C2semidirect
6. #20276656C2semidirect
7. #20276628C2semidirect
8. #20276548C2semidirect
9. C4×D16×C53C2semidirect
11. C2×D8×D53C22semidirect
14. C2×Dic424C22semidirect
15. C2×#13133162C22semidirect
16. C2×#13133161C22semidirect
19. C2×#13133156C22semidirect
20. #16811007C22semidirect
21. #16810865C22semidirect
27. C4⋊#11719908C22semidirect
28. #16810426C22semidirect
29. #16810423C22semidirect
30. #16810402C22semidirect
32. C4×D8×C53C22semidirect
33. C53×#366C22semidirect
34. C53×#357C22semidirect
35. C4×C16×C53C22semidirect
69. D16×C53D4semidirect
173. C4×D16D53semidirect
198. C4×D8C2×D53semidirect
200. #366C2×D53semidirect
204. #357C2×D53semidirect
206. C4×C16C2×D53semidirect
207. C53#33839semidirect
235. D16D212semidirect

Maximal subgroups

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

subgroupindex
C2×#168110572
#202864012
#202769762
#202768292
#202767702
#202766562
#202766282
#202765482
C4×D16×C532
#3383953