Finite Group Databasebeta

← #27279413

Subgroups — #27279413

Order 13568.

530 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 530 conjugacy classes of subgroups — too many to draw. Only its 107 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#20285141C211yesmaximal, characteristic
2#20273307C211yesmaximal, characteristic, complemented
3#20273131C212yesmaximal, complemented
4#20272551C212yesmaximal, complemented
5C53×#2979C211yesmaximal, characteristic, complemented
6C2×#13133237—22no—
7#16811173C2211yescharacteristic
8#16811005—22no—
9#16811002C411yescharacteristic
10#16810999C411yescharacteristic
11#16810997C411yescharacteristic
12#16810995C411yescharacteristic
13C4×#11719917C2212yes—
14C2×C4×#88322C2211yescharacteristic
15#16810363—22no—
16#16810350—22no—
17#16810350—22no—
18C53×#532C2211yescharacteristic
19C53×#436—22no—
20C8×D4×C53—22no—
21C53×#431C2212yes—
22C53×#408—22no—
23C2×#11719945—21no—
24#13133237—44no—
25#13133237—44no—
26C2×#11719925—22no—
27C2×#11719923—22no—
28C2×#11719924—22no—
29C2×#11719922—22no—
30C2×#11719917—22no—
31C22×#88322—21no—
32#13133217C2×C411yescharacteristic
33#13133214C2×C411yescharacteristic
34#13133205—21no—
35#13133191—22no—
36C4×D212—22no—
37#13133128—22no—
38C4×#88322—21no—
39C4×#88322—22no—
40C4×#88322C2×C411yescharacteristic
41C4×#88322C2×C411yescharacteristic
42C2×#41×C53D412yes—
43C2×M16×C53—22no—
44C22×C8×C53—22no—
45C4⋊Q8×C53C2×C411yescharacteristic
46#127×C53C2×C411yescharacteristic
47#124×C53D412yes—
48#121×C53—22no—
49C4×D4×C53—22no—
50C2×C42×C53C2311yescharacteristic
51C4⋊C8×C53D412yes—
52C4⋊C8×C53D412yes—
53C22⋊C8×C53—24no—
54C22⋊C8×C53—22no—
55C4×C8×C53—22no—
56#41×C53—42no—
57#41×C53—42no—
58C2×Q8×C53C22⋊C412yes—
59C2×D4×C53—22no—
60C2×D4×C53—22no—
61C2×D4×C53C22⋊C412yes—
62C22×C4×C53—22no—
63C22×C4×C53C2×D411yescharacteristic
64C22×C4×C53#4112yes—
65C2×C22⋊D53—41no—
66#11719945—42no—
67#11719945—42no—
68C2×D212—21no—
69C2×D212—22no—
70C2×C4×D53—41no—
71C2×Dic106—21no—
72M16×C53—82no—
73C2×C8×C53—22no—
74C2×C8×C53—22no—
75C2×C8×C53—82no—
76C4⋊C4×C53—22no—
77C22⋊C4×C53—24no—
78C42×C53C22×C411yescharacteristic
79C42×C53C2×D411yescharacteristic
80C42×C53C2×D412yes—
81#11719925—44no—
82#11719924—44no—
83#11719923—44no—
84#11719922—44no—
85#11719921—42no—
86C4⋊Dic53—22no—
87#11719917—24no—
88#11719917—42no—
89C2×#88322—22no—
90C2×#88322—22no—
91C2×#88322—21no—
92C2×#88322—21no—
93C2×#88322—42no—
94C23×C53—22no—
95C23×C53#12311yescharacteristic
96C22×D53—41no—
97Q8×C53—42no—
98D4×C53—42no—
99D4×C53—42no—
100D4×C53—42no—
101D4×C53—82no—
102C2×C4×C53—22no—
103C2×C4×C53—22no—
104C2×C4×C53—21no—
105C2×C4×C53—21no—
106C2×C4×C53—22no—
107C2×C4×C53—42no—
108C2×C4×C53—42no—
109C2×C4×C53C2×C22⋊C411yescharacteristic
110C2×C4×C53C4×D412yes—
111C2×C4×C53#12011yescharacteristic
112C2×C4×C53#12112yes—
113C22⋊D53—82no—
114C2×Dic53—41no—
115D212—42no—
116D212—42no—
117D212—42no—
118C4×D53—81no—
119C4×D53—81no—
120Dic106—42no—
121C8×C53—42no—
122C8×C53—42no—
123#88322—22no—
124#88322—22no—
125#88322—44no—
126#9069—531nomaximal, Sylow
127C22×C53—22no—
128C22×C53—21no—
129C22×C53—44no—
130C22×C53—42no—
131C22×C53#38611yescharacteristic
132C2×D53—82no—
133C2×D53—81no—
134C4×C53—22no—
135C4×C53—21no—
136C4×C53—42no—
137C4×C53#44312yes—
138C4×C53#44412yes—
139C4×C53#45311yescharacteristic
140C4×C53#45311yescharacteristic
141Dic53—81no—
142#3147—531no—
143#2979D5311yescharacteristic, complemented
144#1525—531no—
145#1484—532no—
146#1278—532no—
147C2×C53—41no—
148C2×C53—82no—
149C2×C53#187612yes—
150C2×C53#188711yescharacteristic
151D53—161no—
152C2×#135—1062no—
153#532—531no—
154#532C2×D5311yescharacteristic
155#495—1062no—
156#494—531no—
157#493—531no—
158#492—531no—
159#492—531no—
160#436—22no—
161C8×D4—22no—
162#431C2×D5312yes—
163#408—22no—
164C4×M16—532no—
165C2×C4×C8—531no—
166#329—1062no—
167D4⋊C8—1062no—
168D4⋊C8—1062no—
169C53#906911yescharacteristic, complemented, Sylow
170C2×#41—1061no—
171C2×#41C22⋊D5312yes—
172#135—2124no—
173#135—2124no—
174C2×Q16—1062no—
175C2×SD16—1062no—
176C2×SD16—1062no—
177C2×D8—1062no—
178C2×M16—22no—
179C2×M16—1062no—
180C22×C8—22no—
181C22×C8—1061no—
182C4⋊Q8—531no—
183C4⋊Q8C4×D5311yescharacteristic
184#127—531no—
185#127C4×D5311yescharacteristic
186#124—1061no—
187#124D21212yes—
188#121—22no—
189#121—1062no—
190C4×D4—22no—
191C4×D4—1062no—
192C2×C42C22×D5311yescharacteristic
193C4⋊C8C22⋊D5312yes—
194C4⋊C8C22⋊D5312yes—
195C22⋊C8—24no—
196C22⋊C8—22no—
197#97—1062no—
198C4×C8—22no—
199C4×C8—531no—
200C4×C8—531no—
201C4×C8—1061no—
202C4×C8—1062no—
203#41—42no—
204#41—42no—
205#41—2122no—
206#41—2122no—
207C2×Q8—1061no—
208C2×Q8#1171992112yes—
209C2×D4—22no—
210C2×D4—22no—
211C2×D4—1061no—
212C2×D4—1062no—
213C2×D4—2121no—
214C2×D4#1171992112yes—
215C22×C4—22no—
216C22×C4—2121no—
217C22×C4#1171994512yes—
218C22×C4C2×C22⋊D5311yescharacteristic
219Q16—2124no—
220SD16—2124no—
221SD16—2124no—
222D8—2124no—
223M16—82no—
224M16—1064no—
225M16—2122no—
226C2×C8—22no—
227C2×C8—22no—
228C2×C8—82no—
229C2×C8—1062no—
230C2×C8—1062no—
231C2×C8—1061no—
232C2×C8—1061no—
233C2×C8—2122no—
234C4⋊C4—22no—
235C4⋊C4—1062no—
236C22⋊C4—24no—
237C22⋊C4—2122no—
238C42C2×C4×D5311yescharacteristic
239C42C2×D21211yescharacteristic
240C42C2×C22⋊D5312yes—
241C23—22no—
242C23—2121no—
243C23#1313320011yescharacteristic
244Q8—42no—
245Q8—2122no—
246D4—42no—
247D4—42no—
248D4—42no—
249D4—82no—
250D4—4242no—
251D4—2122no—
252D4—2122no—
253D4—2122no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—22no—
257C2×C4—22no—
258C2×C4—22no—
259C2×C4—42no—
260C2×C4—42no—
261C2×C4—4241no—
262C2×C4—4241no—
263C2×C4—2121no—
264C2×C4C2×#1171992111yescharacteristic
265C2×C4C4×C22⋊D5312yes—
266C2×C4#1313318611yescharacteristic
267C2×C4#1313319112yes—
268C8—42no—
269C8—42no—
270C8—1062no—
271C8—1062no—
272C8—2124no—
273C22—22no—
274C22—21no—
275C22—44no—
276C22—42no—
277C22—4242no—
278C22—4241no—
279C22#1681050211yescharacteristic
280C4—22no—
281C4—21no—
282C4—42no—
283C4—4241no—
284C4#1681073812yes—
285C4#1681074312yes—
286C4#1681079111yescharacteristic
287C4#1681079111yescharacteristic
288C2—41no—
289C2—82no—
290C2—8481no—
291C2#2027504212yes—
292C2#2027511311yescharacteristic
subgroupG/H
centerC2×C4#13133186
commutatorC42×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#2979C2
socleC22×C53#386
prime radical#20285141C2
cosocleC2×C42×C53C23
hypercenter#2979D53
hypocenterC53#9069
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #2979

Complements

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

NH ≅ G/Ntype
2. #20273307C2semidirect
3. #20273131C2semidirect
4. #20272551C2semidirect
5. C53×#2979C2semidirect
143. #2979D53semidirect
169. C53#9069semidirect

Maximal subgroups

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

subgroupindex
#202851412
#202733072
#202731312
#202725512
C53×#29792
#906953