Finite Group Databasebeta

← #27281967

Subgroups — #27281967

Order 13568.

522 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 522 conjugacy classes of subgroups — too many to draw. Only its 119 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#20283800C211yesmaximal, characteristic, complemented
2#20274711C211yesmaximal, characteristic, complemented
3#20273140C212yesmaximal
4#20272573C212yesmaximal, complemented
5C53×#2536C211yesmaximal, characteristic
6#16810754—24no—
7#16810680—22no—
8#16810676C2212yes—
9#16810647C2212yes—
10C2×#13133136—21no—
11#16810392—21no—
12D4⋊#88322—24no—
13C2×C22⋊Q8×C53—21no—
14C2×C4×Q8×C53C2211yescharacteristic
15C2×C4×D4×C53C2211yescharacteristic
16C53×#398C412yes—
17C53×#394C412yes—
18C53×#388—22no—
19C53×#382C2211yescharacteristic
20C53×#380—21no—
21C2×#11719917—22no—
22C4⋊#88322—24no—
23C4⋊#88322—22no—
24C4⋊#88322—22no—
25C4⋊#88322—22no—
26#13133136—42no—
27C22⋊#88322—22no—
28#13133128—24no—
29C4×#88322D414yes—
30C22×Q8×C53D411yescharacteristic
31C22×D4×C53C2×C411yescharacteristic
32C23×C4×C53—21no—
33C22⋊Q8×C53—44no—
34C4×Q8×C53—22no—
35C4×Q8×C53—22no—
36C4×D4×C53—22no—
37C4×D4×C53—22no—
38C2×C4⋊C4×C53—22no—
39C2×C4⋊C4×C53—21no—
40C2×C4⋊C4×C53C2×C411yescharacteristic
41C2×C4⋊C4×C53C2×C412yes—
42C2×C4⋊C4×C53D411yescharacteristic
43C2×C22⋊C4×C53—22no—
44C2×C22⋊C4×C53—22no—
45C2×C22⋊C4×C53—21no—
46C2×C42×C53D412yes—
47C2×C42×C53C2311yescharacteristic
48#95×C53—22no—
49#95×C53—21no—
50#95×C53C2×C412yes—
51C24×C53—21no—
52C2×Q8×C53—22no—
53C2×Q8×C53—22no—
54C2×Q8×C53—21no—
55C2×Q8×C53—21no—
56C2×D4×C53—22no—
57C2×D4×C53—42no—
58C22×C4×C53—22no—
59C22×C4×C53—21no—
60C22×C4×C53—42no—
61C22×C4×C53—42no—
62C22×C4×C53—41no—
63C22×C4×C53—41no—
64C22×C4×C53C22⋊C412yes—
65C22×C4×C53C22⋊C412yes—
66C22×C4×C53C22×C411yescharacteristic
67C22×C4×C53C2×D411yescharacteristic
68C22×C4×C53C2×D411yescharacteristic
69C4⋊C4×C53—24no—
70C4⋊C4×C53—22no—
71C4⋊C4×C53—22no—
72C4⋊C4×C53—44no—
73C4⋊C4×C53—42no—
74C22⋊C4×C53—44no—
75C22⋊C4×C53—44no—
76C22⋊C4×C53—41no—
77C22⋊C4×C53—41no—
78C42×C53—22no—
79C42×C53—22no—
80C42×C53C2×D412yes—
81C42×C53#4112yes—
82#11719917—42no—
83#11719917—42no—
84C2×#88322—24no—
85C2×#88322—42no—
86C23×C53—22no—
87C23×C53—41no—
88C23×C53—41no—
89C23×C53—41no—
90C23×C53—41no—
91C23×C53—41no—
92C23×C53—41no—
93C23×C53C2×C22⋊C411yescharacteristic
94Q8×C53—42no—
95Q8×C53—42no—
96D4×C53—24no—
97D4×C53—42no—
98C2×C4×C53—22no—
99C2×C4×C53—22no—
100C2×C4×C53—22no—
101C2×C4×C53—22no—
102C2×C4×C53—41no—
103C2×C4×C53—41no—
104C2×C4×C53—42no—
105C2×C4×C53—42no—
106C2×C4×C53—42no—
107C2×C4×C53—41no—
108C2×C4×C53—41no—
109C2×C4×C53—41no—
110C2×C4×C53—42no—
111C2×C4×C53—82no—
112C2×C4×C53—82no—
113C2×C4×C53C4×D412yes—
114C2×C4×C53#12011yescharacteristic
115C2×C4×C53#12112yes—
116C2×C4×C53#12311yescharacteristic
117C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
118C2×C4×C53AGL(1,ℤ/8)12yes—
119C2×C4×C53#13711yescharacteristic
120#88322—44no—
121#88322—82no—
122#9490—531nomaximal, Sylow
123C22×C53—21no—
124C22×C53—21no—
125C22×C53—42no—
126C22×C53—42no—
127C22×C53—42no—
128C22×C53—81no—
129C22×C53—81no—
130C22×C53—81no—
131C22×C53—81no—
132C22×C53#38611yescharacteristic
133C22×C53#41711yescharacteristic
134C22×C53#41811yescharacteristic
135C4×C53—22no—
136C4×C53—21no—
137C4×C53—21no—
138C4×C53—42no—
139C4×C53—42no—
140C4×C53—82no—
141C4×C53—81no—
142C4×C53#46812yes—
143C4×C53#46912yes—
144#2983—531no—
145#2536D5311yescharacteristic
146#1801—531no—
147#1486—532no—
148#1286—532no—
149C2×C53—21no—
150C2×C53—21no—
151C2×C53—82no—
152C2×C53#189111yescharacteristic
153C2×C53#189511yescharacteristic
154C2×C53#191211yescharacteristic
155C2×C22⋊Q8—21no—
156C2×C4×Q8C2×D5311yescharacteristic
157C2×C4×D4C2×D5311yescharacteristic
158#446—1064no—
159#432—1062no—
160#431—532no—
161#423—532no—
162C2×#101—1061no—
163#398Dic5312yes—
164#394Dic5312yes—
165#388—22no—
166#382C2×D5311yescharacteristic
167#380—21no—
168#343—1061no—
169D4⋊C8—1064no—
170C53#949011yescharacteristic, complemented, Sylow
171C22×Q8C22⋊D5311yescharacteristic
172C22×D4C2×Dic5311yescharacteristic
173C23×C4—21no—
174C2×M16—1062no—
175C22⋊Q8—44no—
176C4×Q8—22no—
177C4×Q8—22no—
178C4×D4—22no—
179C4×D4—22no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4C2×Dic5311yescharacteristic
183C2×C4⋊C4C2×Dic5312yes—
184C2×C4⋊C4C22⋊D5311yescharacteristic
185C2×C22⋊C4—21no—
186C2×C22⋊C4—22no—
187C2×C22⋊C4—22no—
188C2×C42C22⋊D5312yes—
189C2×C42C22×D5311yescharacteristic
190C4⋊C8—1064no—
191C4⋊C8—1062no—
192C4⋊C8—1062no—
193C4⋊C8—1062no—
194#101—2122no—
195C22⋊C8—1062no—
196#97—1064no—
197C4×C8—534no—
198#95—21no—
199#95—22no—
200#95C2×Dic5312yes—
201C24—21no—
202C2×Q8—22no—
203C2×Q8—21no—
204C2×Q8—22no—
205C2×Q8—21no—
206C2×D4—22no—
207C2×D4—42no—
208C22×C4—22no—
209C22×C4—21no—
210C22×C4—42no—
211C22×C4—41no—
212C22×C4—42no—
213C22×C4—41no—
214C22×C4C22⋊Dic5312yes—
215C22×C4C22⋊Dic5312yes—
216C22×C4C22×Dic5311yescharacteristic
217C22×C4C2×C22⋊D5311yescharacteristic
218C22×C4C2×C22⋊D5311yescharacteristic
219M16—2122no—
220M16—2122no—
221C2×C8—1064no—
222C2×C8—2122no—
223C4⋊C4—24no—
224C4⋊C4—22no—
225C4⋊C4—22no—
226C4⋊C4—44no—
227C4⋊C4—42no—
228C22⋊C4—44no—
229C22⋊C4—44no—
230C22⋊C4—41no—
231C22⋊C4—41no—
232C42—22no—
233C42—22no—
234C42D4×D5312yes—
235C42#1171994712yes—
236C23—22no—
237C23—41no—
238C23—41no—
239C23—41no—
240C23—41no—
241C23—41no—
242C23—41no—
243C23C2×C22⋊Dic5311yescharacteristic
244Q8—42no—
245Q8—42no—
246D4—24no—
247D4—42no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—42no—
255C2×C4—42no—
256C2×C4—42no—
257C2×C4—41no—
258C2×C4—41no—
259C2×C4—42no—
260C2×C4—41no—
261C2×C4—82no—
262C2×C4—82no—
263C2×C4D4×Dic5312yes—
264C2×C4#1313318511yescharacteristic
265C2×C4#1313318912yes—
266C2×C4#1313319811yescharacteristic
267C2×C4#1313324211yescharacteristic
268C2×C4#1313324312yes—
269C2×C4#1313324711yescharacteristic
270C8—4242no—
271C8—2124no—
272C22—21no—
273C22—21no—
274C22—42no—
275C22—42no—
276C22—42no—
277C22—81no—
278C22—81no—
279C22—81no—
280C22—81no—
281C22#1681050311yescharacteristic
282C22#1681062411yescharacteristic
283C22#1681062811yescharacteristic
284C4—21no—
285C4—21no—
286C4—22no—
287C4—42no—
288C4—42no—
289C4—82no—
290C4—81no—
291C4#1681086712yes—
292C4#1681087412yes—
293C2—21no—
294C2—21no—
295C2—82no—
296C2#2027513411yescharacteristic
297C2#2027515411yescharacteristic
298C2#2027525311yescharacteristic
subgroupG/H
centerC22#16810503
commutatorC22×C4×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#2536C2
socleC22×C53#386
prime radicalC22×D4×C53C2×C4
cosocleC2×C42×C53C23
hypercenter#2536D53
hypocenterC53#9490
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2536

Complements

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

NH ≅ G/Ntype
1. #20283800C2semidirect
2. #20274711C2semidirect
4. #20272573C2semidirect
170. C53#9490semidirect

Maximal subgroups

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

subgroupindex
#202838002
#202747112
#202731402
#202725732
C53×#25362
#949053