Finite Group Databasebeta

← #27296210

Subgroups — #27296210

Order 13568.

424 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 424 conjugacy classes of subgroups — too many to draw. Only its 67 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#20285490C211yesmaximal, characteristic
2#20276649C212yesmaximal, complemented
3#20272726C212yesmaximal, complemented
4#20272639C211yesmaximal, characteristic
5C53×#2001C211yesmaximal, characteristic
6C2×#13133196—21no—
7C2×#13133163—22no—
8#16810959—22no—
9#16810864—21no—
10#16810864C2212yes—
11#16810781—22no—
12#16810427—22no—
13C22⋊#11719908C2212yes—
14C22×Q16×C53C2211yescharacteristic
15C2×C22⋊Q8×C53—21no—
16C2×#103×C53—21no—
17C2×C22⋊C8×C53C2211yescharacteristic
18C53×#340C2211yescharacteristic
19C22×Dic106—21no—
20C2×Dic212—21no—
21C2×Dic212D412yes—
22#13133196—22no—
23#13133196—22no—
24#13133196—41no—
25#13133196—41no—
26C2×C4⋊Dic53—21no—
27C2×#11719919—41no—
28C2×C22⋊Dic53—41no—
29#13133163—44no—
30C2×#11719908—22no—
31#13133151—21no—
32#13133151D412yes—
33#13133142—22no—
34#13133142—22no—
35C22×Q8×C53—21no—
36C23×C4×C53D411yescharacteristic
37C2×Q16×C53—22no—
38C2×Q16×C53—42no—
39C22×C8×C53D411yescharacteristic
40C22×C8×C53C2311yescharacteristic
41C22⋊Q8×C53—42no—
42C22⋊Q8×C53—42no—
43C2×C4⋊C4×C53—21no—
44C2×C4⋊C4×C53—41no—
45C2×C22⋊C4×C53—41no—
46#103×C53—42no—
47C22⋊C8×C53—22no—
48C24×C53D811yescharacteristic
49C2×Q8×C53—22no—
50C2×Q8×C53—41no—
51C2×Q8×C53—41no—
52C2×Q8×C53—41no—
53C2×Q8×C53—41no—
54C2×Q8×C53—41no—
55C22×C4×C53—21no—
56C22×C4×C53—21no—
57C22×C4×C53—41no—
58C22×C4×C53—41no—
59C22×C4×C53—41no—
60C22×C4×C53—41no—
61C22×C4×C53D811yescharacteristic
62C22×C4×C53C2×D411yescharacteristic
63C22×Dic53—41no—
64C22×Dic53—41no—
65C2×Dic106—22no—
66C2×Dic106—41no—
67C2×Dic106—81no—
68C2×Dic106—81no—
69Q16×C53—24no—
70Q16×C53—42no—
71C2×C8×C53—21no—
72C2×C8×C53—41no—
73C2×C8×C53—41no—
74C2×C8×C53C2×D412yes—
75C4⋊C4×C53—22no—
76C4⋊C4×C53—41no—
77C4⋊C4×C53—44no—
78C22⋊C4×C53—44no—
79C22⋊Dic53—42no—
80C22⋊Dic53—42no—
81C4⋊Dic53—22no—
82C4⋊Dic53—41no—
83#11719919—42no—
84#11719919—42no—
85Dic212—42no—
86Dic212—81no—
87#11719908—42no—
88C23×C53—21no—
89C23×C53—21no—
90C23×C53—41no—
91C23×C53—41no—
92C23×C53Q3212yescomplemented
93C23×C53C2×D811yescharacteristic
94Q8×C53—42no—
95Q8×C53—42no—
96Q8×C53—81no—
97Q8×C53—81no—
98C2×C4×C53—21no—
99C2×C4×C53—21no—
100C2×C4×C53—21no—
101C2×C4×C53—41no—
102C2×C4×C53—41no—
103C2×C4×C53—41no—
104C2×C4×C53—42no—
105C2×C4×C53—42no—
106C2×C4×C53—81no—
107C2×C4×C53—81no—
108C2×C4×C53—81no—
109C2×C4×C53—81no—
110C2×C4×C53—81no—
111C2×C4×C53#12011yescharacteristic
112C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
113C2×Dic53—42no—
114C2×Dic53—42no—
115C2×Dic53—82no—
116C2×Dic53—82no—
117Dic106—42no—
118Dic106—42no—
119Dic106—82no—
120C8×C53—22no—
121C8×C53—41no—
122#12106—531nomaximal, Sylow
123C22×C53—21no—
124C22×C53—21no—
125C22×C53—22no—
126C22×C53—42no—
127C22×C53—41no—
128C22×C53—41no—
129C22×C53—81no—
130C22×C53#44711yescharacteristic
131C22×C53C2×Q3211yescharacteristic
132C22×C53#51011yescharacteristic
133C4×C53—21no—
134C4×C53—41no—
135C4×C53—82no—
136C4×C53—82no—
137C4×C53#45311yescharacteristic
138C4×C53B(4,𝔽2)11yescharacteristic
139Dic53—82no—
140Dic53—82no—
141#3189—531no—
142#2188—532no—
143#2001D5311yescharacteristic
144#1348—532no—
145#1314—531no—
146C2×C53—21no—
147C2×C53—21no—
148C2×C53—42no—
149C2×C53#159511yescharacteristic
150C2×C53#218811yescharacteristic
151C2×C53#219111yescharacteristic
152C22×Q16C2×D5311yescharacteristic
153C2×C22⋊Q8—21no—
154C2×C22⋊Q8—1061no—
155C2×Q32—1062no—
156#484—1062no—
157#467—532no—
158#467—1061no—
159#451—1062no—
160C2×#103—21no—
161C2×C22⋊C8C2×D5311yescharacteristic
162#358—1062no—
163C22⋊C16—532no—
164#340C2×D5311yescharacteristic
165C53#1210611yescharacteristic, complemented, Sylow
166C22×Q8—21no—
167C22×Q8—1061no—
168C23×C4C22⋊D5311yescharacteristic
169C2×Q16—22no—
170C2×Q16—42no—
171C2×Q16—532no—
172C2×Q16—1061no—
173C22×C8C22⋊D5311yescharacteristic
174C22×C8C22×D5311yescharacteristic
175C22⋊Q8—42no—
176C22⋊Q8—42no—
177C22⋊Q8—1062no—
178C22⋊Q8—1062no—
179C22⋊Q8—2121no—
180C22⋊Q8—2121no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—41no—
183C2×C4⋊C4—1061no—
184C2×C4⋊C4—2121no—
185C2×C22⋊C4—41no—
186C2×C22⋊C4—2121no—
187Q32—2124no—
188C2×C16—1062no—
189#107—532no—
190#107—1061no—
191#103—42no—
192#103—1062no—
193#103—1062no—
194C22⋊C8—22no—
195C24#1171992211yescharacteristic
196C2×Q8—22no—
197C2×Q8—41no—
198C2×Q8—41no—
199C2×Q8—41no—
200C2×Q8—41no—
201C2×Q8—41no—
202C2×Q8—1062no—
203C2×Q8—4241no—
204C2×Q8—4241no—
205C2×Q8—2121no—
206C22×C4—21no—
207C22×C4—21no—
208C22×C4—41no—
209C22×C4—41no—
210C22×C4—41no—
211C22×C4—41no—
212C22×C4—2121no—
213C22×C4—2121no—
214C22×C4#1171992211yescharacteristic
215C22×C4C2×C22⋊D5311yescharacteristic
216Q16—24no—
217Q16—42no—
218Q16—4241no—
219Q16—2122no—
220C2×C8—21no—
221C2×C8—41no—
222C2×C8—41no—
223C2×C8D4×D5312yes—
224C4⋊C4—22no—
225C4⋊C4—41no—
226C4⋊C4—44no—
227C4⋊C4—1062no—
228C4⋊C4—2121no—
229C4⋊C4—2122no—
230C4⋊C4—2122no—
231C22⋊C4—44no—
232C22⋊C4—2122no—
233C22⋊C4—2122no—
234C16—2122no—
235C23—21no—
236C23—21no—
237C23—41no—
238C23—41no—
239C23#1313316312yescomplemented
240C23C2×#1171992211yescharacteristic
241Q8—42no—
242Q8—42no—
243Q8—81no—
244Q8—81no—
245Q8—4242no—
246Q8—2122no—
247Q8—2122no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—41no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—42no—
255C2×C4—42no—
256C2×C4—81no—
257C2×C4—81no—
258C2×C4—81no—
259C2×C4—81no—
260C2×C4—81no—
261C2×C4—4242no—
262C2×C4—4242no—
263C2×C4—2122no—
264C2×C4—2122no—
265C2×C4#1313318511yescharacteristic
266C2×C4#1313324211yescharacteristic
267C8—22no—
268C8—41no—
269C22—21no—
270C22—21no—
271C22—22no—
272C22—42no—
273C22—41no—
274C22—41no—
275C22—81no—
276C22#1681075611yescharacteristic
277C22C2×#1313316311yescharacteristic
278C22#1681106311yescharacteristic
279C4—21no—
280C4—41no—
281C4—82no—
282C4—82no—
283C4—4242no—
284C4—4242no—
285C4#1681079011yescharacteristic
286C4#1681080911yescharacteristic
287C2—21no—
288C2—21no—
289C2—42no—
290C2#2027363211yescharacteristic
291C2#2027664811yescharacteristic
292C2#2027666511yescharacteristic
subgroupG/H
centerC22#16810756
commutator, cosocleC22×C8×C53C23
FrattiniC22×C8C22×D53
FittingC53×#2001C2
socleC22×C53#447
prime radicalC24×C53D8
hypercenter#2001D53
hypocenterC53#12106
LCSG ▷ C22×C8×C53 ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C22⋊C8 ◁ #2001

Complements

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

NH ≅ G/Ntype
2. #20276649C2semidirect
3. #20272726C2semidirect
92. C23×C53Q32semidirect
165. C53#12106semidirect
239. C23#13133163semidirect

Maximal subgroups

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

subgroupindex
#202854902
#202766492
#202727262
#202726392
C53×#20012
#1210653