Finite Group Databasebeta

← #27297670

Subgroups — #27297670

Order 13568.

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

normal subgroups only

This group has 320 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#20276987C211yesmaximal, characteristic, complemented
2#20276760C211yesmaximal, characteristic, complemented
3#20274209C211yesmaximal, characteristic, complemented
4#20272758C211yesmaximal, characteristic, complemented
5#20272697C211yesmaximal, characteristic, complemented
6#20272682C211yesmaximal, characteristic, complemented
7C53×#3403C211yesmaximal, characteristic
8C2×#13133163—21no—
9C2×#13133161—21no—
10C2×#13133161—21no—
11#16811003C2211yescharacteristic
12#16810908—21no—
13C4⋊#11719908C2211yescharacteristic
14#16810426—21no—
15#16810399C2211yescharacteristic
16#16810374C2211yescharacteristic
17#16810370—21no—
18C53×#559—21no—
19C53×#496—21no—
20C53×#495C2211yescharacteristic
21C53×#464—21no—
22C53×#463—21no—
23C53×#459—21no—
24C53×#446C2211yescharacteristic
25C53×#440—21no—
26C4×D8×C53C2211yescharacteristic
27C2×Dic212—21no—
28C2×Dic212—21no—
29#13133217—21no—
30#13133163—42no—
31#13133161—42no—
32#13133161—42no—
33C2×#11719908—21no—
34C2×#11719908—21no—
35#13133148—41no—
36C4⋊#88322—21no—
37#13133141—41no—
38C2×Q16×C53D411yescharacteristic
39C2×SD16×C53—21no—
40C2×SD16×C53—21no—
41C2×D8×C53—21no—
42C2×D8×C53D411yescharacteristic
43C2×D8×C53D411yescharacteristic
44#127×C53—21no—
45#124×C53—21no—
46#124×C53—21no—
47#124×C53—41no—
48#123×C53—41no—
49#123×C53—41no—
50C4×Q8×C53—21no—
51C4×Q8×C53D411yescharacteristic
52C4×D4×C53—21no—
53C4×D4×C53—21no—
54C4×D4×C53—41no—
55#107×C53D411yescharacteristic
56C4⋊C8×C53—21no—
57C4⋊C8×C53D411yescharacteristic
58#103×C53—21no—
59#102×C53—21no—
60#102×C53—21no—
61#97×C53—21no—
62C4×C8×C53C2311yescharacteristic
63C2×Q8×C53—21no—
64C2×Q8×C53D811yescharacteristic
65C2×D4×C53—21no—
66C2×D4×C53—21no—
67C2×D4×C53—41no—
68C2×D4×C53—41no—
69C22×C4×C53—41no—
70C22×C4×C53—41no—
71C2×Dic106—41no—
72Q16×C53—41no—
73SD16×C53—41no—
74SD16×C53—41no—
75SD16×C53—81no—
76D8×C53—21no—
77D8×C53—21no—
78D8×C53—41no—
79D8×C53—81no—
80C2×C8×C53—21no—
81C2×C8×C53C2×D411yescharacteristic
82C2×C8×C53C2×D411yescharacteristic
83C4⋊C4×C53—21no—
84C4⋊C4×C53—21no—
85C4⋊C4×C53—41no—
86C4⋊C4×C53D811yescharacteristic
87C22⋊C4×C53—41no—
88C22⋊C4×C53—41no—
89C22⋊C4×C53—41no—
90C22⋊C4×C53—41no—
91C22⋊C4×C53—41no—
92C22⋊C4×C53—41no—
93C42×C53—21no—
94C42×C53—41no—
95C42×C53C2×D411yescharacteristic
96C4⋊Dic53—41no—
97C4⋊Dic53—41no—
98C2×#88322—41no—
99Dic212—41no—
100Dic212—41no—
101Dic212—41no—
102Dic212—41no—
103#11719908—22no—
104#11719908—41no—
105C23×C53—41no—
106C23×C53—41no—
107Q8×C53—41no—
108Q8×C53—41no—
109Q8×C53—41no—
110D4×C53—41no—
111D4×C53—41no—
112D4×C53—41no—
113D4×C53—41no—
114D4×C53—82no—
115D4×C53—161no—
116C2×C4×C53—21no—
117C2×C4×C53—21no—
118C2×C4×C53—41no—
119C2×C4×C53—41no—
120C2×C4×C53—161no—
121C2×C4×C53—161no—
122C2×C4×C53#12011yescharacteristic
123C2×C4×C53C2×D811yescharacteristic
124C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
125C2×Dic53—81no—
126Dic106—81no—
127Dic106—81no—
128C8×C53—21no—
129C8×C53—41no—
130C8×C53#13511yescharacteristic
131C8×C53AGL(1,ℤ/8)11yescharacteristic
132#88322—81no—
133#12338—531nomaximal, Sylow
134C22×C53—81no—
135C22×C53—81no—
136C22×C53—81no—
137C22×C53—81no—
138C22×C53—81no—
139C22×C53—81no—
140C22×C53#44711yescharacteristic
141C4×C53—21no—
142C4×C53—21no—
143C4×C53—41no—
144C4×C53—41no—
145C4×C53—81no—
146C4×C53—81no—
147C4×C53#44911yescharacteristic
148C4×C53#45311yescharacteristic
149Dic53—161no—
150#3403D5311yescharacteristic
151#2251—531no—
152#2208—531no—
153#1690—531no—
154#1360—531no—
155#1338—531no—
156#1333—531no—
157C2×C53—161no—
158C2×C53—161no—
159C2×C53#162111yescharacteristic
160C2×C53#219111yescharacteristic
161C2×C53#219411yescharacteristic
162#559—21no—
163C2×Q32—1061no—
164C2×SD32—1061no—
165C2×SD32—1061no—
166#496—21no—
167#495C2×D5311yescharacteristic
168#494—531no—
169#475—1061no—
170#464—21no—
171#463—21no—
172#459—21no—
173#446C2×D5311yescharacteristic
174#440—21no—
175C4×D8C2×D5311yescharacteristic
176C4⋊C16—531no—
177#358—1061no—
178#346—531no—
179#334—531no—
180#332—1061no—
181C53#1233811yescharacteristic, complemented, Sylow
182C2×Q16—1061no—
183C2×Q16—1061no—
184C2×Q16C22⋊D5311yescharacteristic
185C2×SD16—21no—
186C2×SD16—21no—
187C2×D8—21no—
188C2×D8C22⋊D5311yescharacteristic
189C2×D8C22⋊D5311yescharacteristic
190C4⋊Q8—1061no—
191#127—21no—
192#124—21no—
193#124—21no—
194#124—41no—
195#123—41no—
196#123—41no—
197C4×Q8—21no—
198C4×Q8C22⋊D5311yescharacteristic
199C4×D4—21no—
200C4×D4—21no—
201C4×D4—41no—
202Q32—2122no—
203SD32—2122no—
204SD32—2122no—
205C2×C16—1061no—
206C2×C16—1061no—
207#107C22⋊D5311yescharacteristic
208#106—2121no—
209C4⋊C8—21no—
210C4⋊C8—1061no—
211C4⋊C8C22⋊D5311yescharacteristic
212#103—21no—
213#103—2121no—
214#102—21no—
215#102—21no—
216#97—21no—
217C4×C8C22×D5311yescharacteristic
218C2×Q8—21no—
219C2×Q8—2121no—
220C2×Q8#1171992211yescharacteristic
221C2×D4—21no—
222C2×D4—21no—
223C2×D4—41no—
224C2×D4—41no—
225C22×C4—41no—
226C22×C4—41no—
227Q16—41no—
228Q16—2121no—
229Q16—2121no—
230Q16—2121no—
231Q16—2121no—
232SD16—41no—
233SD16—41no—
234SD16—81no—
235D8—21no—
236D8—21no—
237D8—41no—
238D8—81no—
239C2×C8—21no—
240C2×C8—2121no—
241C2×C8C2×C22⋊D5311yescharacteristic
242C2×C8C2×C22⋊D5311yescharacteristic
243C4⋊C4—21no—
244C4⋊C4—21no—
245C4⋊C4—41no—
246C4⋊C4—2121no—
247C4⋊C4—2121no—
248C4⋊C4#1171992211yescharacteristic
249C22⋊C4—41no—
250C22⋊C4—41no—
251C22⋊C4—41no—
252C22⋊C4—41no—
253C22⋊C4—41no—
254C22⋊C4—41no—
255C42—21no—
256C42—41no—
257C42C2×C22⋊D5311yescharacteristic
258C16—1062no—
259C16—2121no—
260C23—41no—
261C23—41no—
262Q8—41no—
263Q8—41no—
264Q8—41no—
265Q8—4241no—
266Q8—4241no—
267D4—41no—
268D4—41no—
269D4—41no—
270D4—41no—
271D4—82no—
272D4—161no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—161no—
278C2×C4—161no—
279C2×C4—4241no—
280C2×C4#1313318611yescharacteristic
281C2×C4C2×#1171992211yescharacteristic
282C2×C4#1313324311yescharacteristic
283C8—21no—
284C8—41no—
285C8—4241no—
286C8#1313323711yescharacteristic
287C8#1313324211yescharacteristic
288C22—81no—
289C22—81no—
290C22—81no—
291C22—81no—
292C22—81no—
293C22—81no—
294C22#1681075711yescharacteristic
295C4—21no—
296C4—21no—
297C4—41no—
298C4—41no—
299C4—81no—
300C4—81no—
301C4—8481no—
302C4#1681076811yescharacteristic
303C4#1681079111yescharacteristic
304C2—161no—
305C2—161no—
306C2#2027378011yescharacteristic
307C2#2027666611yescharacteristic
308C2#2027668611yescharacteristic
subgroupG/H
centerC22#16810757
commutator, cosocleC4×C8×C53C23
FrattiniC4×C8C22×D53
Fitting, prime radicalC53×#3403C2
socleC22×C53#447
hypercenter#3403D53
hypocenterC53#12338
LCSG ▷ C4×C8×C53 ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ #446 ◁ #3403

Complements

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

NH ≅ G/Ntype
1. #20276987C2semidirect
2. #20276760C2semidirect
3. #20274209C2semidirect
4. #20272758C2semidirect
5. #20272697C2semidirect
6. #20272682C2semidirect
181. C53#12338semidirect

Maximal subgroups

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

subgroupindex
#202769872
#202767602
#202742092
#202727582
#202726972
#202726822
C53×#34032
#1233853