Finite Group Databasebeta

← #27539332

Subgroups — #27539332

Order 13568.

520 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 520 conjugacy classes of subgroups — too many to draw. Only its 135 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#20285460C211yesmaximal, characteristic, complemented
2#20276934C212yesmaximal, complemented
3#20276886C212yesmaximal, complemented
4#20276796C212yesmaximal, complemented
5#20276666C212yesmaximal
6#20276606C214yesmaximal, complemented
7#20276527C211yesmaximal, characteristic, complemented
8C53×#2956C211yesmaximal, characteristic
9#16811180—21no—
10#16811063C2214yes—
11#16811021C2211yescharacteristic, complemented
12#16811003C2211yescharacteristic, complemented
13#16810974—21no—
14#16810881—21no—
15#16810881C2212yes—
16#16810731—21no—
17#16810731C2212yescomplemented
18#16810447C2214yescomplemented
19#16810444C2212yescomplemented
20#16810438C2214yescomplemented
21#16810433C2214yes—
22#16810405C2212yes—
23#16810401C2212yescomplemented
24C53×#575C2211yescharacteristic
25C53×#568—21no—
26C53×#444—21no—
27C53×#444C2212yes—
28C53×#443—21no—
29C53×#443C2212yes—
30C53×#430C2211yescharacteristic
31C2×#104×C53—21no—
32C4×M16×C53C2211yescharacteristic
33C2×Dic212—21no—
34C2×Dic212C2312yes—
35#13133217—21no—
36#13133217—21no—
37#13133209—41no—
38#13133196—22no—
39#13133196—41no—
40C4×Dic106—22no—
41C4×Dic106—41no—
42#13133176—41no—
43#13133163—24no—
44#13133161—24no—
45#13133157C2314yes—
46#13133149—21no—
47#13133149C2312yes—
48#13133142—22no—
49#13133142—22no—
50C2×#41×C53—21no—
51#137×C53—22no—
52AGL(1,ℤ/8)×C53—22no—
53#135×C53—21no—
54#135×C53C2312yes—
55#131×C53—22no—
56#131×C53—22no—
57#131×C53—42no—
58C2×M16×C53—21no—
59C2×M16×C53—42no—
60C2×M16×C53D411yescharacteristic
61C2×M16×C53C2311yescharacteristic
62C2×C42×C53D411yescharacteristic
63#108×C53—21no—
64#108×C53C2312yes—
65#104×C53—22no—
66#104×C53—22no—
67#97×C53D411yescharacteristic
68#97×C53C2311yescharacteristic
69C4×C8×C53D411yescharacteristic
70C4×C8×C53C2311yescharacteristic
71#41×C53—22no—
72#41×C53—22no—
73#41×C53—41no—
74#41×C53—41no—
75C2×Q8×C53—21no—
76C2×D4×C53—21no—
77C2×D4×C53—41no—
78C22×C4×C53—21no—
79C22×C4×C53—41no—
80C22×C4×C53C2×D411yescharacteristic
81C2×Dic106—22no—
82C2×Dic106—41no—
83Q16×C53—21no—
84Q16×C53#4112yes—
85SD16×C53—22no—
86SD16×C53—22no—
87D8×C53—21no—
88D8×C53#4112yes—
89M16×C53—22no—
90M16×C53—22no—
91M16×C53—22no—
92M16×C53—42no—
93M16×C53—41no—
94M16×C53—42no—
95C2×C8×C53—42no—
96C2×C8×C53—41no—
97C2×C8×C53—42no—
98C2×C8×C53C2×D411yescharacteristic
99C2×C8×C53C2×D411yescharacteristic
100C2×C8×C53C2×D411yescharacteristic
101C2×C8×C53C2411yescharacteristic
102C42×C53—21no—
103C42×C53C2×D411yescharacteristic
104C42×C53C2×D411yescharacteristic
105C22⋊Dic53—42no—
106C4⋊Dic53—22no—
107C4⋊Dic53—41no—
108C4⋊Dic53—41no—
109C4⋊Dic53—41no—
110#11719919—42no—
111#11719919—42no—
112C4×Dic53—42no—
113Dic212—24no—
114Dic212—42no—
115#11719908—24no—
116C23×C53—41no—
117C23×C53AGL(1,ℤ/8)11yescharacteristic
118Q8×C53—22no—
119Q8×C53—41no—
120D4×C53—22no—
121D4×C53—41no—
122D4×C53—42no—
123D4×C53—42no—
124C2×C4×C53—21no—
125C2×C4×C53—21no—
126C2×C4×C53—21no—
127C2×C4×C53—21no—
128C2×C4×C53—42no—
129C2×C4×C53—42no—
130C2×C4×C53—41no—
131C2×C4×C53—42no—
132C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
133C2×C4×C53C22×D411yescharacteristic
134C2×Dic53—42no—
135C2×Dic53—42no—
136Dic106—42no—
137Dic106—42no—
138Dic106—82no—
139C8×C53—21no—
140C8×C53—21no—
141C8×C53—42no—
142C8×C53AGL(1,ℤ/8)12yes—
143C8×C53C2×#4112yes—
144#33689—531nomaximal, Sylow
145C22×C53—21no—
146C22×C53—41no—
147C22×C53—42no—
148C22×C53—161no—
149C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
150C4×C53—22no—
151C4×C53—21no—
152C4×C53—41no—
153C4×C53—42no—
154C4×C53#52511yescharacteristic
155C4×C53C2×AGL(1,ℤ/8)11yescharacteristic
156Dic53—82no—
157Dic53—82no—
158#3185—531no—
159#2956D5311yescharacteristic
160#2239—532no—
161#2230—532no—
162#2214—532no—
163#2191—532no—
164#2180—534no—
165#2165—531no—
166C2×C53—21no—
167C2×C53—41no—
168C2×C53—82no—
169C2×C53#310911yescharacteristic
170#575C2×D5311yescharacteristic
171#568—21no—
172#533—1061no—
173#510—534no—
174#498—531no—
175#494—531no—
176#487—1061no—
177#470—532no—
178#470—1061no—
179#444—21no—
180#444C2×D5312yes—
181#443—21no—
182#443C2×D5312yes—
183#441—532no—
184#441—1061no—
185#430C2×D5311yescharacteristic
186C2×#104—21no—
187C4×M16C2×D5311yescharacteristic
188#365—534no—
189#364—532no—
190#362—534no—
191#360—534no—
192#349—532no—
193#347—532no—
194C53#3368911yescharacteristic, complemented, Sylow
195C2×#41—21no—
196#137—22no—
197AGL(1,ℤ/8)—22no—
198#135—21no—
199#135C22×D5312yes—
200C2×Q16—532no—
201C2×Q16—1061no—
202#131—22no—
203#131—22no—
204#131—42no—
205C2×M16—21no—
206C2×M16—42no—
207C2×M16C22⋊D5311yescharacteristic
208C2×M16C22×D5311yescharacteristic
209C4⋊Q8—1061no—
210C4⋊Q8—1061no—
211#125—2121no—
212C22⋊Q8—1062no—
213C22⋊Q8—2121no—
214C4×Q8—1062no—
215C4×Q8—2121no—
216#117—2121no—
217C2×C42C22⋊D5311yescharacteristic
218Q32—1064no—
219SD32—1064no—
220M32—534no—
221#108—21no—
222#108C22×D5312yes—
223#106—532no—
224#106—1061no—
225#104—22no—
226#104—22no—
227#103—1062no—
228#103—1062no—
229#97C22⋊D5311yescharacteristic
230#97C22×D5311yescharacteristic
231C4×C8C22⋊D5311yescharacteristic
232C4×C8C22×D5311yescharacteristic
233#41—22no—
234#41—22no—
235#41—41no—
236#41—41no—
237C2×Q8—21no—
238C2×Q8—1062no—
239C2×Q8—2121no—
240C2×D4—21no—
241C2×D4—41no—
242C22×C4—21no—
243C22×C4—41no—
244C22×C4C2×C22⋊D5311yescharacteristic
245Q16—21no—
246Q16—1064no—
247Q16—2122no—
248Q16#1171994512yes—
249SD16—22no—
250SD16—22no—
251D8—21no—
252D8#1171994512yes—
253M16—22no—
254M16—22no—
255M16—22no—
256M16—42no—
257M16—41no—
258M16—42no—
259C2×C8—42no—
260C2×C8—41no—
261C2×C8—42no—
262C2×C8C2×C22⋊D5311yescharacteristic
263C2×C8C2×C22⋊D5311yescharacteristic
264C2×C8C2×C22⋊D5311yescharacteristic
265C2×C8C23×D5311yescharacteristic
266C4⋊C4—1062no—
267C4⋊C4—2121no—
268C4⋊C4—2121no—
269C4⋊C4—2122no—
270C4⋊C4—2121no—
271C4⋊C4—2122no—
272C22⋊C4—2122no—
273C42—21no—
274C42—2122no—
275C42C2×C22⋊D5311yescharacteristic
276C42C2×C22⋊D5311yescharacteristic
277C16—1064no—
278C23—41no—
279C23#1313324211yescharacteristic
280Q8—22no—
281Q8—41no—
282Q8—4242no—
283Q8—2122no—
284Q8—2122no—
285D4—22no—
286D4—41no—
287D4—42no—
288D4—42no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—42no—
294C2×C4—42no—
295C2×C4—41no—
296C2×C4—42no—
297C2×C4—2122no—
298C2×C4—2122no—
299C2×C4#1313324211yescharacteristic
300C2×C4C22×C22⋊D5311yescharacteristic
301C8—21no—
302C8—21no—
303C8—42no—
304C8#1313324312yes—
305C8C2×#1171994512yes—
306C22—21no—
307C22—41no—
308C22—42no—
309C22—161no—
310C22C2×#1313324211yescharacteristic
311C4—22no—
312C4—21no—
313C4—41no—
314C4—42no—
315C4—4242no—
316C4—4242no—
317C4#1681113411yescharacteristic
318C4C2×#1313324311yescharacteristic
319C2—21no—
320C2—41no—
321C2—82no—
322C2#2028480611yescharacteristic
subgroupG/H
centerC2#20284806
commutator, cosocleC2×C8×C53C24
FrattiniC2×C8C23×D53
FittingC53×#2956C2
socleC2×C53#3109
prime radicalC53×#575C22
hypercenter#2956D53
hypocenterC53#33689
LCSG ▷ C2×C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C2×C4 ◁ C4×M16 ◁ #2956

Complements

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

NH ≅ G/Ntype
1. #20285460C2semidirect
2. #20276934C2semidirect
3. #20276886C2semidirect
4. #20276796C2semidirect
6. #20276606C2semidirect
7. #20276527C2semidirect
11. #16811021C22semidirect
12. #16811003C22semidirect
17. #16810731C22semidirect
18. #16810447C22semidirect
19. #16810444C22semidirect
20. #16810438C22semidirect
23. #16810401C22semidirect
194. C53#33689semidirect

Maximal subgroups

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

subgroupindex
#202854602
#202769342
#202768862
#202767962
#202766662
#202766062
#202765272
C53×#29562
#3368953