Finite Group Databasebeta

← #27298026

Subgroups — #27298026

Order 13568.

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

normal subgroups only

This group has 302 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#20276951C211yesmaximal, characteristic, complemented
2#20276862C211yesmaximal, characteristic, complemented
3#20274233C211yesmaximal, characteristic, complemented
4#20272758C211yesmaximal, characteristic, complemented
5#20272697C211yesmaximal, characteristic, complemented
6#20272682C211yesmaximal, characteristic, complemented
7C53×#3403C211yesmaximal, characteristic
8#16811028C2211yescharacteristic
9#16810833—21no—
10#16810450—21no—
11C4⋊#11719908C2211yescharacteristic
12#16810427—21no—
13#16810424—21no—
14#16810424—21no—
15#16810399C2211yescharacteristic
16#16810374C2211yescharacteristic
17#16810368—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×#11719925—41no—
28#13133217—21no—
29C2×#11719908—21no—
30C2×#11719908—21no—
31#13133151—21no—
32#13133151—21no—
33C4⋊#88322—21no—
34#13133140—41no—
35C2×Q16×C53D411yescharacteristic
36C2×SD16×C53—21no—
37C2×SD16×C53—21no—
38C2×D8×C53—21no—
39C2×D8×C53D411yescharacteristic
40C2×D8×C53D411yescharacteristic
41#127×C53—21no—
42#124×C53—21no—
43#124×C53—21no—
44#124×C53—41no—
45#123×C53—41no—
46#123×C53—41no—
47C4×Q8×C53—21no—
48C4×Q8×C53D411yescharacteristic
49C4×D4×C53—21no—
50C4×D4×C53—21no—
51C4×D4×C53—41no—
52#107×C53D411yescharacteristic
53C4⋊C8×C53—21no—
54C4⋊C8×C53D411yescharacteristic
55#103×C53—21no—
56#102×C53—21no—
57#102×C53—21no—
58#97×C53—21no—
59C4×C8×C53C2311yescharacteristic
60C2×Q8×C53—21no—
61C2×Q8×C53D811yescharacteristic
62C2×D4×C53—21no—
63C2×D4×C53—21no—
64C2×D4×C53—41no—
65C2×D4×C53—41no—
66C22×C4×C53—41no—
67C22×C4×C53—41no—
68C2×Dic106—41no—
69Q16×C53—41no—
70SD16×C53—41no—
71SD16×C53—41no—
72SD16×C53—81no—
73D8×C53—21no—
74D8×C53—21no—
75D8×C53—41no—
76D8×C53—81no—
77C2×C8×C53—21no—
78C2×C8×C53C2×D411yescharacteristic
79C2×C8×C53C2×D411yescharacteristic
80C4⋊C4×C53—21no—
81C4⋊C4×C53—21no—
82C4⋊C4×C53—41no—
83C4⋊C4×C53D811yescharacteristic
84C22⋊C4×C53—41no—
85C22⋊C4×C53—41no—
86C22⋊C4×C53—41no—
87C22⋊C4×C53—41no—
88C22⋊C4×C53—41no—
89C22⋊C4×C53—41no—
90C42×C53—21no—
91C42×C53—41no—
92C42×C53C2×D411yescharacteristic
93#11719925—82no—
94C4⋊Dic53—41no—
95C4⋊Dic53—41no—
96C2×#88322—41no—
97#11719908—41no—
98#11719908—41no—
99C23×C53—41no—
100C23×C53—41no—
101Q8×C53—41no—
102Q8×C53—41no—
103Q8×C53—41no—
104D4×C53—41no—
105D4×C53—41no—
106D4×C53—41no—
107D4×C53—41no—
108D4×C53—82no—
109D4×C53—161no—
110C2×C4×C53—21no—
111C2×C4×C53—21no—
112C2×C4×C53—41no—
113C2×C4×C53—41no—
114C2×C4×C53—161no—
115C2×C4×C53—161no—
116C2×C4×C53#12011yescharacteristic
117C2×C4×C53C2×D811yescharacteristic
118C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
119C2×Dic53—81no—
120Dic106—81no—
121Dic106—81no—
122C8×C53—21no—
123C8×C53—41no—
124C8×C53#13511yescharacteristic
125C8×C53#13711yescharacteristic
126#88322—81no—
127#12390—531nomaximal, Sylow
128C22×C53—81no—
129C22×C53—81no—
130C22×C53—81no—
131C22×C53—81no—
132C22×C53—81no—
133C22×C53—81no—
134C22×C53#44711yescharacteristic
135C4×C53—21no—
136C4×C53—21no—
137C4×C53—41no—
138C4×C53—41no—
139C4×C53—81no—
140C4×C53—81no—
141C4×C53#45211yescharacteristic
142C4×C53#45511yescharacteristic
143Dic53—161no—
144#3403D5311yescharacteristic
145#2243—531no—
146#2226—531no—
147#1694—531no—
148#1360—531no—
149#1338—531no—
150#1333—531no—
151C2×C53—161no—
152C2×C53—161no—
153C2×C53#164111yescharacteristic
154C2×C53#219011yescharacteristic
155C2×C53#219411yescharacteristic
156#559—21no—
157#500—531no—
158#496—21no—
159#495C2×D5311yescharacteristic
160#464—21no—
161#463—21no—
162#462—1061no—
163#459—21no—
164#446C2×D5311yescharacteristic
165#440—21no—
166C4×D8C2×D5311yescharacteristic
167#367—1061no—
168C4⋊C16—531no—
169#358—1061no—
170#357—1061no—
171#357—1061no—
172#346—531no—
173#334—531no—
174#332—1061no—
175C53#1239011yescharacteristic, complemented, Sylow
176C2×Q16—2121no—
177C2×Q16C22⋊D5311yescharacteristic
178C2×SD16—21no—
179C2×SD16—21no—
180C2×D8—21no—
181C2×D8C22⋊D5311yescharacteristic
182C2×D8C22⋊D5311yescharacteristic
183C4⋊Q8—1061no—
184#127—21no—
185#124—21no—
186#124—21no—
187#124—41no—
188#123—41no—
189#123—41no—
190C4×Q8—21no—
191C4×Q8C22⋊D5311yescharacteristic
192C4×D4—21no—
193C4×D4—21no—
194C4×D4—41no—
195C2×C16—1061no—
196C2×C16—1061no—
197#107—1061no—
198#107—1061no—
199#107C22⋊D5311yescharacteristic
200C4⋊C8—21no—
201C4⋊C8—1061no—
202C4⋊C8C22⋊D5311yescharacteristic
203#103—21no—
204#103—2121no—
205#102—21no—
206#102—21no—
207#97—21no—
208C4×C8C22×D5311yescharacteristic
209C2×Q8—21no—
210C2×Q8—2121no—
211C2×Q8#1171992211yescharacteristic
212C2×D4—21no—
213C2×D4—21no—
214C2×D4—41no—
215C2×D4—41no—
216C22×C4—41no—
217C22×C4—41no—
218Q16—41no—
219Q16—4242no—
220SD16—41no—
221SD16—41no—
222SD16—81no—
223D8—21no—
224D8—21no—
225D8—41no—
226D8—81no—
227C2×C8—21no—
228C2×C8—2121no—
229C2×C8C2×C22⋊D5311yescharacteristic
230C2×C8C2×C22⋊D5311yescharacteristic
231C4⋊C4—21no—
232C4⋊C4—21no—
233C4⋊C4—41no—
234C4⋊C4—2121no—
235C4⋊C4—2121no—
236C4⋊C4#1171992211yescharacteristic
237C22⋊C4—41no—
238C22⋊C4—41no—
239C22⋊C4—41no—
240C22⋊C4—41no—
241C22⋊C4—41no—
242C22⋊C4—41no—
243C42—21no—
244C42—41no—
245C42C2×C22⋊D5311yescharacteristic
246C16—2121no—
247C16—2121no—
248C23—41no—
249C23—41no—
250Q8—41no—
251Q8—41no—
252Q8—41no—
253Q8—4241no—
254Q8—4241no—
255D4—41no—
256D4—41no—
257D4—41no—
258D4—41no—
259D4—82no—
260D4—161no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—41no—
264C2×C4—41no—
265C2×C4—161no—
266C2×C4—161no—
267C2×C4—4241no—
268C2×C4#1313318611yescharacteristic
269C2×C4C2×#1171992211yescharacteristic
270C2×C4#1313324311yescharacteristic
271C8—21no—
272C8—41no—
273C8—4241no—
274C8#1313323711yescharacteristic
275C8#1313324711yescharacteristic
276C22—81no—
277C22—81no—
278C22—81no—
279C22—81no—
280C22—81no—
281C22—81no—
282C22#1681075711yescharacteristic
283C4—21no—
284C4—21no—
285C4—41no—
286C4—41no—
287C4—81no—
288C4—81no—
289C4—8481no—
290C4#1681078511yescharacteristic
291C4#1681080111yescharacteristic
292C2—161no—
293C2—161no—
294C2#2027390811yescharacteristic
295C2#2027666111yescharacteristic
296C2#2027668611yescharacteristic
subgroupG/H
centerC22#16810757
commutator, cosocleC4×C8×C53C23
FrattiniC4×C8C22×D53
Fitting, prime radicalC53×#3403C2
socleC22×C53#447
hypercenter#3403D53
hypocenterC53#12390
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. #20276951C2semidirect
2. #20276862C2semidirect
3. #20274233C2semidirect
4. #20272758C2semidirect
5. #20272697C2semidirect
6. #20272682C2semidirect
175. C53#12390semidirect

Maximal subgroups

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

subgroupindex
#202769512
#202768622
#202742332
#202727582
#202726972
#202726822
C53×#34032
#1239053