Finite Group Databasebeta

← #27280757

Subgroups — #27280757

Order 13568.

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

normal subgroups only

This group has 422 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#20285665C211yesmaximal, characteristic
2#20283652C211yesmaximal, characteristic
3#20275687C211yesmaximal, characteristic
4#20273401C211yesmaximal, characteristic
5#20273278C211yesmaximal, characteristic
6#20272545C211yesmaximal, characteristic
7C53×#1286C211yesmaximal, characteristic
8#16811180C2211yescharacteristic
9C2×#13133217C2211yescharacteristic
10#16811003C412yescomplemented
11#16810974C412yescomplemented
12#16810834—22no—
13#16810781—22no—
14#16810752—22no—
15#16810752—22no—
16#16810676C2211yescharacteristic
17#16810651—21no—
18#16810581—21no—
19C2×#13133128C2211yescharacteristic
20#16810385—21no—
21#16810385—21no—
22#16810368—22no—
23#16810359—22no—
24C53×#431C2211yescharacteristic
25C53×#423C2211yescharacteristic
26C53×#382C2211yescharacteristic
27C22×Dic106—21no—
28C2×Dic212—24no—
29C22×#88322D411yescharacteristic
30C22×#88322D411yescharacteristic
31#13133217—21no—
32#13133217—21no—
33#13133217—21no—
34#13133217C2×C412yes—
35#13133217C2×C411yescharacteristic
36#13133217C2×C411yescharacteristic
37#13133209—21no—
38#13133196—21no—
39#13133196D412yes—
40C4×Dic106—21no—
41C4×Dic106D412yes—
42#13133176—21no—
43C2×C4⋊Dic53—21no—
44C2×C4⋊Dic53—21no—
45C4⋊#88322—21no—
46#13133146—24no—
47#13133142—24no—
48C22⋊#88322—21no—
49#13133128—21no—
50#13133128—21no—
51#13133127—22no—
52C4×#88322—21no—
53C8×Dic53—22no—
54C2×M16×C53—21no—
55C2×C4⋊C4×C53—21no—
56C2×C42×C53C2311yescharacteristic
57C4⋊C8×C53—21no—
58C4⋊C8×C53—21no—
59C4⋊C8×C53D412yes—
60C22⋊C8×C53—21no—
61C4×C8×C53C2×C412yes—
62#95×C53—21no—
63#95×C53—21no—
64C22×C4×C53—41no—
65C22×C4×C53C2×D411yescharacteristic
66C22×C4×C53C2×D411yescharacteristic
67C22×C4×C53#4111yescharacteristic
68C22×Dic53—41no—
69C2×Dic106—22no—
70C2×Dic106—21no—
71C2×Dic106—41no—
72C2×Dic106—82no—
73C2×Dic106C22⋊C412yes—
74M16×C53—41no—
75M16×C53—41no—
76C2×C8×C53—22no—
77C2×C8×C53—22no—
78C4⋊C4×C53—41no—
79C4⋊C4×C53—41no—
80C42×C53C22×C411yescharacteristic
81C42×C53C2×D411yescharacteristic
82C42×C53C2×D411yescharacteristic
83C42×C53#4111yescharacteristic
84C22⋊Dic53—22no—
85C4⋊Dic53—21no—
86C4⋊Dic53—21no—
87C4⋊Dic53—21no—
88C4⋊Dic53—22no—
89C4⋊Dic53—22no—
90C4⋊Dic53—41no—
91C4⋊Dic53—41no—
92C4⋊Dic53C22⋊C412yes—
93#11719919—22no—
94#11719919—22no—
95C4×Dic53—22no—
96C2×#88322—21no—
97C2×#88322—21no—
98C2×#88322—21no—
99C2×#88322—21no—
100C2×#88322—21no—
101C2×#88322—41no—
102C2×#88322Q1612yescomplemented
103Dic212—84no—
104C23×C53#12111yescharacteristic
105C2×C4×C53—41no—
106C2×C4×C53—41no—
107C2×C4×C53—81no—
108C2×C4×C53—81no—
109C2×C4×C53—81no—
110C2×C4×C53C2×C22⋊C411yescharacteristic
111C2×C4×C53C4×D411yescharacteristic
112C2×C4×C53C4×D411yescharacteristic
113C2×C4×C53#12011yescharacteristic
114C2×C4×C53#12111yescharacteristic
115C2×C4×C53#12311yescharacteristic
116C2×C4×C53C2×Q1611yescharacteristic
117C2×C4×C53#13511yescharacteristic
118C2×C4×C53#13711yescharacteristic
119C2×C4×C53#13711yescharacteristic
120C2×Dic53—22no—
121C2×Dic53—22no—
122C2×Dic53—42no—
123C2×Dic53—82no—
124Dic106—42no—
125Dic106—44no—
126Dic106—81no—
127Dic106—82no—
128C8×C53—42no—
129C8×C53—42no—
130#88322—22no—
131#88322—41no—
132#88322—81no—
133#9290—531nomaximal, Sylow
134C22×C53—21no—
135C22×C53—21no—
136C22×C53#38611yescharacteristic
137C22×C53#46211yescharacteristic
138C22×C53#46411yescharacteristic
139C4×C53—21no—
140C4×C53—21no—
141C4×C53—21no—
142C4×C53—21no—
143C4×C53—161no—
144C4×C53C4×Q1611yescharacteristic
145C4×C53#44111yescharacteristic
146C4×C53#45111yescharacteristic
147C4×C53#45211yescharacteristic
148Dic53—42no—
149Dic53—42no—
150Dic53—82no—
151#3209—531no—
152#2961—531no—
153#1993—531no—
154#1549—531no—
155#1518—531no—
156#1286D5311yescharacteristic
157#1275—531no—
158C2×C53—21no—
159C2×C53—21no—
160C2×C53#187811yescharacteristic
161C2×C53#190211yescharacteristic
162C2×C53#191011yescharacteristic
163#533—531no—
164C2×C4⋊Q8—531no—
165#494—532no—
166#487—532no—
167#462—1062no—
168#451—1062no—
169#446—1062no—
170#446—1062no—
171#431—531no—
172#431C2×D5311yescharacteristic
173#424—1061no—
174#423C2×D5311yescharacteristic
175#405—1061no—
176C2×#97—531no—
177#382C2×D5311yescharacteristic
178#340—1061no—
179#340—1061no—
180#332—1062no—
181#328—1062no—
182C53#929011yescharacteristic, complemented, Sylow
183C22×Q8—1061no—
184C2×Q16—1064no—
185C2×M16—21no—
186C22×C8—531no—
187C22×C8—531no—
188C4⋊Q8—531no—
189C4⋊Q8—531no—
190C4⋊Q8—532no—
191C4⋊Q8—1061no—
192C4⋊Q8—1061no—
193C4⋊Q8—1061no—
194#125—1061no—
195C22⋊Q8—532no—
196C22⋊Q8—1061no—
197C4×Q8—532no—
198C4×Q8—1061no—
199#117—1061no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—1061no—
202C2×C4⋊C4—1061no—
203C2×C42C22×D5311yescharacteristic
204C4⋊C8—21no—
205C4⋊C8—21no—
206C4⋊C8—1064no—
207C4⋊C8—1061no—
208C4⋊C8D21212yes—
209#103—1064no—
210C22⋊C8—21no—
211C22⋊C8—1061no—
212#97—1062no—
213#97—1061no—
214#97—1061no—
215C4×C8—1062no—
216C4×C8—1061no—
217C4×C8C4×D5312yes—
218#95—21no—
219#95—21no—
220C2×Q8—532no—
221C2×Q8—1061no—
222C2×Q8—1062no—
223C2×Q8—4242no—
224C2×Q8—2121no—
225C22×C4—41no—
226C22×C4—2121no—
227C22×C4#1171994511yescharacteristic
228C22×C4D4×D5311yescharacteristic
229C22×C4D4×D5311yescharacteristic
230Q16—4244no—
231M16—41no—
232M16—41no—
233C2×C8—22no—
234C2×C8—22no—
235C2×C8—532no—
236C2×C8—1061no—
237C2×C8—1061no—
238C2×C8—1061no—
239C2×C8—1061no—
240C2×C8—1061no—
241C2×C8—2121no—
242C4⋊C4—41no—
243C4⋊C4—41no—
244C4⋊C4—532no—
245C4⋊C4—1061no—
246C4⋊C4—1061no—
247C4⋊C4—1061no—
248C4⋊C4—1062no—
249C4⋊C4—1062no—
250C4⋊C4—1062no—
251C4⋊C4—1062no—
252C4⋊C4—2121no—
253C4⋊C4—2121no—
254C22⋊C4—1062no—
255C42—1062no—
256C42C2×C4×D5311yescharacteristic
257C42C2×D21211yescharacteristic
258C42D4×D5311yescharacteristic
259C42#1171994911yescharacteristic
260C23#1313319011yescharacteristic
261Q8—4242no—
262Q8—4241no—
263Q8—2124no—
264Q8—2122no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—81no—
268C2×C4—81no—
269C2×C4—81no—
270C2×C4—1062no—
271C2×C4—1062no—
272C2×C4—4242no—
273C2×C4—2122no—
274C2×C4C22⋊C4×D5311yescharacteristic
275C2×C4#1313317911yescharacteristic
276C2×C4C4×D21211yescharacteristic
277C2×C4#1313318711yescharacteristic
278C2×C4#1313319211yescharacteristic
279C2×C4#1313320211yescharacteristic
280C2×C4Q16×D5311yescharacteristic
281C2×C4#1313323611yescharacteristic
282C2×C4#1313324611yescharacteristic
283C2×C4#1313324611yescharacteristic
284C8—42no—
285C8—42no—
286C8—1062no—
287C8—4241no—
288C8—2121no—
289C22—21no—
290C22—21no—
291C22#1681050111yescharacteristic
292C22#1681083211yescharacteristic
293C22#1681084411yescharacteristic
294C4—21no—
295C4—21no—
296C4—21no—
297C4—21no—
298C4—161no—
299C4—4242no—
300C4—2122no—
301C4—2122no—
302C4#1681071711yescharacteristic
303C4#1681072911yescharacteristic
304C4#1681078011yescharacteristic
305C4#1681078611yescharacteristic
306C2—21no—
307C2—21no—
308C2#2027505711yescharacteristic
309C2#2027518711yescharacteristic
310C2#2027524111yescharacteristic
subgroupG/H
centerC22#16810501
commutatorC42×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#1286C2
socleC22×C53#386
prime radicalC23×C53#121
cosocleC2×C42×C53C23
hypercenter#1286D53
hypocenterC53#9290
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1286

Complements

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

NH ≅ G/Ntype
10. #16811003C4semidirect
11. #16810974C4semidirect
102. C2×#88322Q16semidirect
182. C53#9290semidirect

Maximal subgroups

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

subgroupindex
#202856652
#202836522
#202756872
#202734012
#202732782
#202725452
C53×#12862
#929053