Finite Group Databasebeta

← #28310605

Subgroups — #28310605

Order 13568.

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

normal subgroups only

This group has 456 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#20283420C211yesmaximal, characteristic
6#16811609C2212yes—
7#16811602C2212yes—
8#16811586—22no—
9#16811526C414yescomplemented
10#16811524C414yescomplemented
11#16811505—22no—
12C2×#13133223C2211yescharacteristic
13C22×#11719912C411yescharacteristic, complemented
14#16811135C2211yescharacteristic
15#16810702—22no—
16#16810696—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—42no—
32C2×#11719917C2×C411yescharacteristic
33C2×#11719912—21no—
34C2×#11719912—21no—
35C2×#11719912—42no—
36C2×#11719912D412yes—
37C22×#88322C2×C411yescharacteristic
38C2×C8×D53D412yes—
39#13133201C2×C412yes—
40#13133197—21no—
41#13133192—21no—
42#13133187C2×C412yes—
43C4×D212—22no—
44C4⋊#88322—22no—
45#13133129—22no—
46#13133129—22no—
47C2×M16×C53C2×C411yescharacteristic
48C22×C8×C53C2×C411yescharacteristic
49#117×C53C2×C411yescharacteristic
50#97×C53—21no—
51C4×C8×C53—21no—
52C23×D53C4211yescharacteristic
53C22×C4×C53C22×C411yescharacteristic
54C2×C22⋊D53C22×C412yes—
55C22×Dic53C4211yescharacteristic
56#11719945—41no—
57#11719945—41no—
58C2×D212—22no—
59C2×D212#4111yescharacteristic
60C2×C4×D53—21no—
61C2×C4×D53—21no—
62C2×C4×D53—41no—
63C2×C4×D53—41no—
64C2×C4×D53C2×D412yes—
65C2×Dic106#4111yescharacteristic
66C22⋊#6157—24no—
67#11719940—24no—
68C4⋊#6157—42no—
69C4×#6157—42no—
70#11719936—42no—
71D53⋊C8—42no—
72M16×C53—41no—
73C2×C8×C53—22no—
74C2×C8×C53—21no—
75C2×C8×C53C22⋊C411yescharacteristic
76C2×C8×C53C22⋊C411yescharacteristic
77C4⋊C4×C53—21no—
78C22⋊C4×C53C4212yes—
79C42×C53—21no—
80#11719921—22no—
81#11719921—22no—
82C4⋊Dic53—22no—
83#11719917—41no—
84C2×#88322—21no—
85C2×#88322—41no—
86C2×#88322C22⋊C412yes—
87#11719912—42no—
88#11719912—42no—
89#11719912—42no—
90C8×D53—22no—
91C8×D53—22no—
92C23×C53C2×C4211yescharacteristic
93C22×D53—21no—
94C22×D53—41no—
95C22×D53—81no—
96C22×D53C4×D412yes—
97C2×#6157—44no—
98C2×C4×C53—21no—
99C2×C4×C53—21no—
100C2×C4×C53—22no—
101C2×C4×C53C2×C22⋊C411yescharacteristic
102C2×C4×C53#11711yescharacteristic
103C22⋊D53—42no—
104C2×Dic53—21no—
105C2×Dic53—41no—
106C2×Dic53C4×D412yes—
107D212—41no—
108D212—82no—
109C4×D53—22no—
110C4×D53—22no—
111C4×D53—42no—
112C4×D53—41no—
113C4×D53—41no—
114Dic106—41no—
115#88324—44no—
116C8×C53—21no—
117C8×C53—41no—
118C8×C53#9912yes—
119#88322—22no—
120#88322—41no—
121#12624—531nomaximal, Sylow
122C22×C53—21no—
123C22×C53—41no—
124C22×C53C4×C22⋊C411yescharacteristic
125C2×D53—22no—
126C2×D53—42no—
127C2×D53—41no—
128C2×D53—41no—
129C2×D53—41no—
130C2×D53—82no—
131#6157—84no—
132C4×C53—21no—
133C4×C53—42no—
134C4×C53C2×#9911yescharacteristic
135C4×C53#41011yescharacteristic
136Dic53—22no—
137Dic53—41no—
138#2930—531no—
139#2120—532no—
140#1757—532no—
141#1392—531no—
142#1391—531no—
143C2×C53—21no—
144C2×C53—41no—
145C2×C53C4×#9911yescharacteristic
146D53—42no—
147D53—81no—
148C2×#131—531no—
149C22×M16—531no—
150#525—531no—
151#436—1062no—
152#435—1062no—
153#424—531no—
154#413—532no—
155#410—532no—
156#408—531no—
157#407—531no—
158#405—1062no—
159#405#615711yescharacteristic, complemented
160#352—534no—
161AΣL(1,𝔽16)—534no—
162#344—1062no—
163C53#1262411yescharacteristic, complemented, Sylow
164C2×#41—531no—
165C23×C4—531no—
166#131—2122no—
167C2×M16—531no—
168C2×M16—532no—
169C2×M16—1061no—
170C2×M16—1061no—
171C2×M16—1062no—
172C2×M16—2122no—
173C2×M16C2×#615711yescharacteristic
174C22×C8—532no—
175C22×C8—531no—
176C22×C8C2×#615711yescharacteristic
177#123—532no—
178C22⋊Q8—1061no—
179#121—1061no—
180#120—532no—
181C4×D4—1062no—
182#117—1062no—
183#117C2×#615711yescharacteristic
184C4⋊C8—1062no—
185#101—1062no—
186#100—534no—
187#99—534no—
188#99—1062no—
189C22⋊C8—1062no—
190C22⋊C8—1062no—
191#97—21no—
192#97—2122no—
193C4×C8—21no—
194C4×C8—2122no—
195C24—531no—
196#41—2121no—
197#41—2121no—
198C2×Q8—531no—
199C2×D4—532no—
200C2×D4—531no—
201C2×D4—1062no—
202C22×C4—532no—
203C22×C4—531no—
204C22×C4—1061no—
205C22×C4—1061no—
206C22×C4—2121no—
207C22×C4—2121no—
208C22×C4C22×#615711yescharacteristic
209M16—41no—
210M16—1064no—
211M16—2122no—
212M16—2122no—
213M16—2121no—
214M16—2122no—
215M16—2122no—
216C2×C8—22no—
217C2×C8—21no—
218C2×C8—532no—
219C2×C8—1062no—
220C2×C8—1062no—
221C2×C8—1061no—
222C2×C8—2122no—
223C2×C8—2121no—
224C2×C8C22⋊#615711yescharacteristic
225C2×C8C22⋊#615711yescharacteristic
226C4⋊C4—21no—
227C4⋊C4—1062no—
228C4⋊C4—2122no—
229C22⋊C4—1062no—
230C22⋊C4—1062no—
231C22⋊C4—1064no—
232C22⋊C4C4×#615712yes—
233C42—21no—
234C42—2122no—
235C23—532no—
236C23—1061no—
237C23—4241no—
238C23—2121no—
239C23C2×C4×#615711yescharacteristic
240Q8—2121no—
241D4—4242no—
242D4—2121no—
243D4—2122no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—22no—
247C2×C4—532no—
248C2×C4—1062no—
249C2×C4—1062no—
250C2×C4—1061no—
251C2×C4—2122no—
252C2×C4—2124no—
253C2×C4—2121no—
254C2×C4—2121no—
255C2×C4—2121no—
256C2×C4C2×C22⋊#615711yescharacteristic
257C2×C4#1313329611yescharacteristic
258C8—21no—
259C8—41no—
260C8—1062no—
261C8—2124no—
262C8—2121no—
263C8#1313327612yes—
264C22—21no—
265C22—41no—
266C22—1062no—
267C22—4242no—
268C22—2121no—
269C22—2121no—
270C22—2122no—
271C22—2121no—
272C22C4×C22⋊#615711yescharacteristic
273C4—21no—
274C4—42no—
275C4—1062no—
276C4—4244no—
277C4—2121no—
278C4C2×#1313327611yescharacteristic
279C4#1681160211yescharacteristic
280C2—21no—
281C2—41no—
282C2—4241no—
283C2—2122no—
284C2C4×#1313327611yescharacteristic
subgroupG/H
centerC4C2×#13133276
commutatorC23×C53C2×C42
FrattiniC22×C4C22×#6157
FittingC53×#405C4
socleC2×C53C4×#99
prime radical#16811135C22
cosocleC2×#11719945C23
hypercenter#405#6157
hypocenterC53#12624
LCSG ▷ C23×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C4 ◁ 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×#11719912C4semidirect
17. #16810650C4semidirect
19. #16810586C4semidirect
20. C53×#405C4semidirect
159. #405#6157semidirect
163. C53#12624semidirect

Maximal subgroups

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

subgroupindex
#202909682
#202904702
#202898072
#202898042
#202834202
#1262453