Finite Group Databasebeta

← #28310599

Subgroups — #28310599

Order 13568.

470 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 470 conjugacy classes of subgroups — too many to draw. Only its 98 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#20290968C212yesmaximal
2#20290470C212yesmaximal, complemented
3#20289807C211yesmaximal, characteristic, complemented
4#20289804C211yesmaximal, characteristic, complemented
5#20283411C211yesmaximal, characteristic
6#16811609C2212yes—
7#16811602C2212yes—
8#16811586—22no—
9#16811526C414yescomplemented
10#16811524C414yescomplemented
11#16811505—22no—
12C2×#13133223C2211yescharacteristic
13C22×C8×D53C411yescharacteristic, complemented
14#16811135C2211yescharacteristic
15#16810696—22no—
16C8×D212—22no—
17#16810650C411yescharacteristic, complemented
18#16810591C2211yescharacteristic
19#16810586C411yescharacteristic, complemented
20C53×#405C411yescharacteristic, complemented
21D53⋊M16—22no—
22#13133296—22no—
23#13133278—22no—
24#13133277C2×C414yes—
25#13133276—22no—
26#13133275C2×C414yes—
27#13133271—42no—
28C8×#6157—42no—
29C2×#11719945C2311yescharacteristic
30C22×C4×D53C2×C411yescharacteristic
31#13133223—41no—
32#13133223—41no—
33C2×#11719917C2×C411yescharacteristic
34C2×#11719912D412yes—
35C22×#88322C2×C411yescharacteristic
36C2×C8×D53—21no—
37C2×C8×D53—21no—
38C2×C8×D53—41no—
39C2×C8×D53—41no—
40C2×C8×D53D412yes—
41#13133201C2×C412yes—
42#13133197—21no—
43#13133192—21no—
44#13133187C2×C412yes—
45C4×D212—22no—
46C4⋊#88322—22no—
47#13133129—22no—
48#13133129—22no—
49C2×M16×C53C2×C411yescharacteristic
50C22×C8×C53C2×C411yescharacteristic
51#117×C53C2×C411yescharacteristic
52#97×C53—21no—
53C4×C8×C53—21no—
54C23×D53C4211yescharacteristic
55C22×C4×C53C22×C411yescharacteristic
56C2×C22⋊D53C22×C412yes—
57C22×Dic53C4211yescharacteristic
58#11719945—41no—
59#11719945—41no—
60C2×D212—22no—
61C2×D212#4111yescharacteristic
62C2×C4×D53—21no—
63C2×C4×D53—21no—
64C2×C4×D53—41no—
65C2×C4×D53—41no—
66C2×C4×D53C2×D412yes—
67C2×Dic106#4111yescharacteristic
68C22⋊#6157—24no—
69#11719940—24no—
70C4⋊#6157—42no—
71C4×#6157—42no—
72#11719936—42no—
73D53⋊C8—42no—
74M16×C53—41no—
75C2×C8×C53—21no—
76C2×C8×C53—21no—
77C2×C8×C53—21no—
78C2×C8×C53C22⋊C411yescharacteristic
79C2×C8×C53C22⋊C411yescharacteristic
80C4⋊C4×C53—21no—
81C22⋊C4×C53C4212yes—
82C42×C53—21no—
83#11719921—22no—
84#11719921—22no—
85C4⋊Dic53—22no—
86#11719917—41no—
87C2×#88322—21no—
88C2×#88322—41no—
89C2×#88322C22⋊C412yes—
90#11719912—42no—
91C8×D53—22no—
92C8×D53—22no—
93C8×D53—42no—
94C8×D53—41no—
95C8×D53—41no—
96C23×C53C2×C4211yescharacteristic
97C22×D53—21no—
98C22×D53—42no—
99C22×D53—41no—
100C22×D53C4×D412yes—
101C2×#6157—44no—
102C2×C4×C53—21no—
103C2×C4×C53—21no—
104C2×C4×C53—22no—
105C2×C4×C53C2×C22⋊C411yescharacteristic
106C2×C4×C53#11711yescharacteristic
107C22⋊D53—42no—
108C2×Dic53—21no—
109C2×Dic53—41no—
110C2×Dic53C4×D412yes—
111D212—41no—
112D212—44no—
113C4×D53—22no—
114C4×D53—22no—
115C4×D53—42no—
116C4×D53—41no—
117C4×D53—41no—
118Dic106—41no—
119#88324—44no—
120C8×C53—21no—
121C8×C53—41no—
122C8×C53#9911yescharacteristic
123C8×C53#9911yescharacteristic
124#88322—22no—
125#88322—41no—
126#12623—531nomaximal, Sylow
127C22×C53—21no—
128C22×C53—41no—
129C22×C53C4×C22⋊C411yescharacteristic
130C2×D53—22no—
131C2×D53—22no—
132C2×D53—41no—
133C2×D53—42no—
134C2×D53—41no—
135C2×D53—42no—
136C2×D53—42no—
137#6157—84no—
138C4×C53—21no—
139C4×C53—42no—
140C4×C53C2×#9911yescharacteristic
141C4×C53#41011yescharacteristic
142Dic53—22no—
143Dic53—41no—
144#2929—531no—
145#2120—532no—
146#1757—532no—
147#1392—531no—
148#1391—531no—
149C2×C53—21no—
150C2×C53—41no—
151C2×C53C4×#9911yescharacteristic
152D53—42no—
153D53—42no—
154C2×#131—531no—
155C23×C8—531no—
156#525—531no—
157#435—1062no—
158C8×D4—1062no—
159#424—531no—
160#413—532no—
161#410—532no—
162#408—531no—
163#407—531no—
164#405—1062no—
165#405#615711yescharacteristic, complemented
166#352—534no—
167AΣL(1,𝔽16)—534no—
168#344—1062no—
169C53#1262311yescharacteristic, complemented, Sylow
170C2×#41—531no—
171C23×C4—531no—
172#131—2121no—
173#131—2121no—
174C2×M16—531no—
175C2×M16—532no—
176C2×M16—1062no—
177C2×M16C2×#615711yescharacteristic
178C22×C8—532no—
179C22×C8—531no—
180C22×C8—1061no—
181C22×C8—1061no—
182C22×C8—2121no—
183C22×C8—2121no—
184C22×C8C2×#615711yescharacteristic
185#123—532no—
186C22⋊Q8—1061no—
187#121—1061no—
188#120—532no—
189C4×D4—1062no—
190#117—1062no—
191#117C2×#615711yescharacteristic
192C4⋊C8—1062no—
193#101—1062no—
194#100—534no—
195#99—534no—
196#99—1062no—
197C22⋊C8—1062no—
198C22⋊C8—1062no—
199#97—21no—
200#97—2122no—
201C4×C8—21no—
202C4×C8—2122no—
203C24—531no—
204#41—2121no—
205#41—2121no—
206C2×Q8—531no—
207C2×D4—532no—
208C2×D4—531no—
209C2×D4—1062no—
210C22×C4—532no—
211C22×C4—531no—
212C22×C4—1061no—
213C22×C4—1061no—
214C22×C4—2121no—
215C22×C4—2121no—
216C22×C4C22×#615711yescharacteristic
217M16—41no—
218M16—1064no—
219M16—2122no—
220M16—2121no—
221M16—2122no—
222C2×C8—21no—
223C2×C8—21no—
224C2×C8—21no—
225C2×C8—532no—
226C2×C8—1062no—
227C2×C8—1062no—
228C2×C8—1061no—
229C2×C8—2122no—
230C2×C8—2122no—
231C2×C8—2121no—
232C2×C8—2121no—
233C2×C8—2121no—
234C2×C8C22⋊#615711yescharacteristic
235C2×C8C22⋊#615711yescharacteristic
236C4⋊C4—21no—
237C4⋊C4—1062no—
238C4⋊C4—2122no—
239C22⋊C4—1062no—
240C22⋊C4—1062no—
241C22⋊C4—1064no—
242C22⋊C4C4×#615712yes—
243C42—21no—
244C42—2122no—
245C23—532no—
246C23—1061no—
247C23—2121no—
248C23—2122no—
249C23C2×C4×#615711yescharacteristic
250Q8—2121no—
251D4—2121no—
252D4—2124no—
253D4—2122no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—22no—
257C2×C4—532no—
258C2×C4—1062no—
259C2×C4—1062no—
260C2×C4—1061no—
261C2×C4—2122no—
262C2×C4—2124no—
263C2×C4—2121no—
264C2×C4—2121no—
265C2×C4—2121no—
266C2×C4C2×C22⋊#615711yescharacteristic
267C2×C4#1313329611yescharacteristic
268C8—21no—
269C8—41no—
270C8—1062no—
271C8—2124no—
272C8—2121no—
273C8#1313327611yescharacteristic
274C8#1313327611yescharacteristic
275C22—21no—
276C22—41no—
277C22—1062no—
278C22—1062no—
279C22—2122no—
280C22—2121no—
281C22—2122no—
282C22—2122no—
283C22—2121no—
284C22C4×C22⋊#615711yescharacteristic
285C4—21no—
286C4—42no—
287C4—1062no—
288C4—4244no—
289C4—2121no—
290C4C2×#1313327611yescharacteristic
291C4#1681160211yescharacteristic
292C2—21no—
293C2—41no—
294C2—2122no—
295C2—2122no—
296C2C4×#1313327611yescharacteristic
subgroupG/H
centerC8#13133276
commutatorC23×C53C2×C42
FrattiniC22×C4C22×#6157
FittingC53×#405C4
socleC2×C53C4×#99
prime radical#16811135C22
cosocleC2×#11719945C23
hypercenter#405#6157
hypocenterC53#12623
LCSG ▷ C23×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C8 ◁ C2×C8 ◁ C22×C8 ◁ #405

Complements

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

NH ≅ G/Ntype
2. #20290470C2semidirect
3. #20289807C2semidirect
4. #20289804C2semidirect
9. #16811526C4semidirect
10. #16811524C4semidirect
13. C22×C8×D53C4semidirect
17. #16810650C4semidirect
19. #16810586C4semidirect
20. C53×#405C4semidirect
165. #405#6157semidirect
169. C53#12623semidirect

Maximal subgroups

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

subgroupindex
#202909682
#202904702
#202898072
#202898042
#202834112
#1262353