Finite Group Databasebeta

← #27296194

Subgroups — #27296194

Order 13568.

476 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 476 conjugacy classes of subgroups — too many to draw. Only its 71 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#20276643C212yesmaximal, complemented
2#20275713C211yesmaximal, characteristic
3#20272727C212yesmaximal, complemented
4#20272639C211yesmaximal, characteristic, complemented
5C53×#3189C211yesmaximal, characteristic, complemented
6C22×D424C2211yescharacteristic
7C2×#13133191—21no—
8C2×#13133160—22no—
9C2×#13133139—21no—
10#16810423—22no—
11C22⋊#11719908C2212yes—
12#16810386C2211yescharacteristic
13C2×C22⋊Q8×C53—21no—
14C53×#484—22no—
15C53×#467—21no—
16C53×#467C2212yes—
17C53×#451—22no—
18C2×C22⋊C8×C53C2211yescharacteristic
19C22×D212—21no—
20C22×C22⋊D53—41no—
21C2×D424—22no—
22C2×D424—42no—
23#13133191—42no—
24#13133191—42no—
25C2×C4⋊Dic53—21no—
26C2×#11719921—41no—
27#13133160—44no—
28C2×#11719908—22no—
29#13133139—42no—
30C22×Q8×C53—21no—
31C23×C4×C53D411yescharacteristic
32C2×Q16×C53—21no—
33C2×Q16×C53D412yescomplemented
34C22×C8×C53D411yescharacteristic
35C22×C8×C53C2311yescharacteristic
36C22⋊Q8×C53—22no—
37C22⋊Q8×C53—22no—
38C22⋊Q8×C53—41no—
39C22⋊Q8×C53—41no—
40C2×C4⋊C4×C53—21no—
41C2×C4⋊C4×C53—41no—
42C2×C22⋊C4×C53—41no—
43#107×C53—21no—
44#107×C53D412yescomplemented
45#103×C53—22no—
46#103×C53—22no—
47C22⋊C8×C53—22no—
48C24×C53D811yescharacteristic
49C23×D53—41no—
50C2×Q8×C53—22no—
51C2×Q8×C53—41no—
52C2×Q8×C53—81no—
53C2×Q8×C53—81no—
54C22×C4×C53—21no—
55C22×C4×C53—21no—
56C22×C4×C53—41no—
57C22×C4×C53—41no—
58C22×C4×C53—41no—
59C22×C4×C53—41no—
60C22×C4×C53D811yescharacteristic
61C22×C4×C53C2×D411yescharacteristic
62C2×C22⋊D53—44no—
63C2×C22⋊D53—82no—
64C2×C22⋊D53—82no—
65C22×Dic53—41no—
66C2×D212—22no—
67C2×D212—41no—
68C2×D212—41no—
69C2×D212—41no—
70C2×D212—41no—
71C2×D212—41no—
72Q16×C53—42no—
73Q16×C53—81no—
74C2×C8×C53—21no—
75C2×C8×C53—41no—
76C2×C8×C53—41no—
77C2×C8×C53C2×D412yes—
78C4⋊C4×C53—22no—
79C4⋊C4×C53—41no—
80C4⋊C4×C53—42no—
81C4⋊C4×C53—42no—
82C22⋊C4×C53—42no—
83C22⋊C4×C53—42no—
84#11719921—44no—
85C4⋊Dic53—22no—
86C4⋊Dic53—41no—
87D424—24no—
88D424—42no—
89#11719908—42no—
90C23×C53—21no—
91C23×C53—21no—
92C23×C53—41no—
93C23×C53—41no—
94C23×C53SD3212yescomplemented
95C23×C53C2×D811yescharacteristic
96C22×D53—42no—
97C22×D53—81no—
98C22×D53—81no—
99C22×D53—81no—
100C22×D53—81no—
101C22×D53—81no—
102C22×D53—81no—
103Q8×C53—42no—
104Q8×C53—42no—
105Q8×C53—82no—
106C2×C4×C53—21no—
107C2×C4×C53—21no—
108C2×C4×C53—21no—
109C2×C4×C53—41no—
110C2×C4×C53—41no—
111C2×C4×C53—41no—
112C2×C4×C53—42no—
113C2×C4×C53—42no—
114C2×C4×C53—81no—
115C2×C4×C53—82no—
116C2×C4×C53—82no—
117C2×C4×C53#12011yescharacteristic
118C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
119C22⋊D53—88no—
120C2×Dic53—42no—
121C2×Dic53—81no—
122C2×Dic53—81no—
123D212—42no—
124D212—42no—
125D212—81no—
126D212—81no—
127C8×C53—22no—
128C8×C53—41no—
129#12103—531nomaximal, Sylow
130C22×C53—21no—
131C22×C53—21no—
132C22×C53—22no—
133C22×C53—42no—
134C22×C53—41no—
135C22×C53—41no—
136C22×C53—81no—
137C22×C53#44711yescharacteristic
138C22×C53C2×SD3211yescharacteristic
139C22×C53#51011yescharacteristic
140C2×D53—82no—
141C2×D53—82no—
142C2×D53—82no—
143C2×D53—161no—
144C2×D53—161no—
145C2×D53—161no—
146C2×D53—161no—
147C4×C53—21no—
148C4×C53—41no—
149C4×C53—82no—
150C4×C53—82no—
151C4×C53#45311yescharacteristic
152C4×C53B(4,𝔽2)11yescharacteristic
153Dic53—82no—
154#3189D5311yescharacteristic, complemented
155#2187—532no—
156#1999—531no—
157#1348—532no—
158#1314—531no—
159C2×C53—21no—
160C2×C53—21no—
161C2×C53—42no—
162C2×C53#159511yescharacteristic
163C2×C53#218511yescharacteristic
164C2×C53#219011yescharacteristic
165D53—162no—
166C22×D8—531no—
167C2×C22⋊Q8—21no—
168C2×#121—1061no—
169C2×SD32—1062no—
170#484—22no—
171#467—21no—
172#467C2×D5312yes—
173#451—22no—
174C2×#102—1061no—
175C2×C22⋊C8C2×D5311yescharacteristic
176#357—1062no—
177C22⋊C16—532no—
178#340—531no—
179C53#1210311yescharacteristic, complemented, Sylow
180C22×Q8—21no—
181C22×D4—1061no—
182C22×D4—2121no—
183C23×C4C22⋊D5311yescharacteristic
184C2×Q16—21no—
185C2×Q16C22⋊D5312yescomplemented
186C2×D8—1062no—
187C2×D8—2122no—
188C22×C8C22⋊D5311yescharacteristic
189C22×C8C22×D5311yescharacteristic
190C22⋊Q8—22no—
191C22⋊Q8—22no—
192C22⋊Q8—41no—
193C22⋊Q8—41no—
194#121—2122no—
195#121—2122no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—41no—
198C2×C4⋊C4—1061no—
199C2×C22⋊C4—41no—
200C2×C22⋊C4—2121no—
201SD32—2124no—
202C2×C16—1062no—
203#107—21no—
204#107C22⋊D5312yescomplemented
205#103—22no—
206#103—22no—
207#102—2122no—
208C22⋊C8—22no—
209C24—2121no—
210C24#1171992211yescharacteristic
211C2×Q8—22no—
212C2×Q8—41no—
213C2×Q8—81no—
214C2×Q8—81no—
215C2×D4—1062no—
216C2×D4—4242no—
217C2×D4—4242no—
218C2×D4—2121no—
219C2×D4—2121no—
220C2×D4—2121no—
221C2×D4—2124no—
222C2×D4—2121no—
223C2×D4—2121no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—41no—
227C22×C4—41no—
228C22×C4—41no—
229C22×C4—41no—
230C22×C4—2121no—
231C22×C4#1171992211yescharacteristic
232C22×C4C2×C22⋊D5311yescharacteristic
233Q16—42no—
234Q16—81no—
235D8—1064no—
236D8—2122no—
237C2×C8—21no—
238C2×C8—41no—
239C2×C8—41no—
240C2×C8C2×C22⋊D5312yes—
241C4⋊C4—22no—
242C4⋊C4—41no—
243C4⋊C4—42no—
244C4⋊C4—42no—
245C4⋊C4—1062no—
246C4⋊C4—2121no—
247C22⋊C4—42no—
248C22⋊C4—42no—
249C22⋊C4—2124no—
250C16—2122no—
251C23—21no—
252C23—21no—
253C23—41no—
254C23—41no—
255C23—4241no—
256C23—4241no—
257C23—4241no—
258C23—4241no—
259C23—4241no—
260C23—4241no—
261C23—2122no—
262C23#1313316012yescomplemented
263C23C2×#1171992211yescharacteristic
264Q8—42no—
265Q8—42no—
266Q8—82no—
267D4—4241no—
268D4—4248no—
269D4—4241no—
270D4—2122no—
271D4—2122no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—42no—
279C2×C4—42no—
280C2×C4—81no—
281C2×C4—82no—
282C2×C4—82no—
283C2×C4—4241no—
284C2×C4—4241no—
285C2×C4—2122no—
286C2×C4#1313318611yescharacteristic
287C2×C4#1313324311yescharacteristic
288C8—22no—
289C8—41no—
290C22—21no—
291C22—21no—
292C22—22no—
293C22—42no—
294C22—41no—
295C22—41no—
296C22—81no—
297C22—8481no—
298C22—8481no—
299C22—8481no—
300C22—8481no—
301C22—4242no—
302C22—4242no—
303C22—4242no—
304C22#1681075711yescharacteristic
305C22C2×#1313316011yescharacteristic
306C22#1681106411yescharacteristic
307C4—21no—
308C4—41no—
309C4—82no—
310C4—82no—
311C4—4242no—
312C4#1681079111yescharacteristic
313C4#1681081011yescharacteristic
314C2—21no—
315C2—21no—
316C2—42no—
317C2—8482no—
318C2#2027363311yescharacteristic
319C2#2027663211yescharacteristic
320C2#2027666111yescharacteristic
subgroupG/H
centerC22#16810757
commutator, cosocleC22×C8×C53C23
FrattiniC22×C8C22×D53
FittingC53×#3189C2
socleC22×C53#447
prime radical#20275713C2
hypercenter#3189D53
hypocenterC53#12103
LCSG ▷ C22×C8×C53 ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C22⋊C8 ◁ #3189

Complements

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

NH ≅ G/Ntype
1. #20276643C2semidirect
3. #20272727C2semidirect
4. #20272639C2semidirect
5. C53×#3189C2semidirect
33. C2×Q16×C53D4semidirect
44. #107×C53D4semidirect
94. C23×C53SD32semidirect
154. #3189D53semidirect
179. C53#12103semidirect
185. C2×Q16C22⋊D53semidirect
204. #107C22⋊D53semidirect
262. C23#13133160semidirect

Maximal subgroups

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

subgroupindex
#202766432
#202757132
#202727272
#202726392
C53×#31892
#1210353