Finite Group Databasebeta

← #27293981

Subgroups — #27293981

Order 13568.

312 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 312 conjugacy classes of subgroups — too many to draw. Only its 91 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#20283693C211yesmaximal, characteristic
2#20276516C211yesmaximal, characteristic, complemented
3C4⋊C16×D53C211yesmaximal, characteristic, complemented
4#20272698C211yesmaximal, characteristic, complemented
5#20272696C211yesmaximal, characteristic, complemented
6#20272673C211yesmaximal, characteristic, complemented
7C53×#1330C211yesmaximal, characteristic, complemented
8#16811155C411yescharacteristic
9C2×C16×D53—21no—
10#16810753C2211yescharacteristic
11#16810747C411yescharacteristic
12#16810692—21no—
13D4×#88322C411yescharacteristic
14#16810686—21no—
15#16810681—21no—
16#16810679C411yescharacteristic
17#16810581—21no—
18C4×C8×D53C2211yescharacteristic
19C4⋊#11719908C2211yescharacteristic
20#16810441—21no—
21#16810441—21no—
22#16810402—21no—
23#16810399C2211yescharacteristic
24C53×#436C2211yescharacteristic
25C53×C4⋊C16C2211yescharacteristic
26C53×#346C2211yescharacteristic
27C22×#88322—21no—
28C2×C8×D53D411yescharacteristic
29C2×C8×D53D411yescharacteristic
30#13133210—21no—
31#13133208C811yescharacteristic
32#13133204C811yescharacteristic
33#13133199—21no—
34#13133197C811yescharacteristic
35#13133188C811yescharacteristic
36C4×Dic106C2×C411yescharacteristic
37C4×C22⋊D53—21no—
38#13133176—21no—
39C42×D53C2×C411yescharacteristic
40C2×#11719908—21no—
41C2×#11719908—21no—
42C16×D53—42no—
43C4⋊#88322C2×C411yescharacteristic
44#13133146—21no—
45#13133146—21no—
46C22⋊#88322—21no—
47#13133129—21no—
48#13133127—21no—
49#13133127—21no—
50C4×#88322C2×C411yescharacteristic
51C8×Dic53—21no—
52C8×Dic53D411yescharacteristic
53C8×Dic53D411yescharacteristic
54C2×M16×C53—21no—
55C4×D4×C53C2×C411yescharacteristic
56C2×C16×C53—21no—
57C2×C16×C53—21no—
58C4⋊C8×C53C2×C411yescharacteristic
59C22⋊C8×C53—21no—
60C4×C8×C53C2311yescharacteristic
61C2×D4×C53C2×C811yescharacteristic
62C22×C4×C53—21no—
63C2×C22⋊D53—21no—
64C2×C4×D53C22⋊C411yescharacteristic
65C2×C4×D53C2×C811yescharacteristic
66C2×C4×D53M1611yescharacteristic
67C2×Dic106C2×C811yescharacteristic
68M16×C53—41no—
69M16×C53—41no—
70C2×C8×C53—21no—
71C2×C8×C53C2×D411yescharacteristic
72C2×C8×C53C2×D411yescharacteristic
73C4⋊C4×C53C2×C811yescharacteristic
74C22⋊C4×C53—21no—
75C42×C53C22×C411yescharacteristic
76C22⋊Dic53—21no—
77C22⋊Dic53—21no—
78C22⋊Dic53—21no—
79#11719921—21no—
80C4⋊Dic53—21no—
81C4⋊Dic53—21no—
82C4⋊Dic53C2×C811yescharacteristic
83#11719919—21no—
84#11719919—21no—
85C4×Dic53—21no—
86C4×Dic53C22⋊C411yescharacteristic
87C4×Dic53C2×C811yescharacteristic
88C4×Dic53M1611yescharacteristic
89C2×#88322—21no—
90C2×#88322—42no—
91C2×#88322C22⋊C411yescharacteristic
92C2×#88322C22⋊C411yescharacteristic
93C8×D53—21no—
94C8×D53—21no—
95C8×D53—41no—
96C16×C53—41no—
97C16×C53—41no—
98#11719908—41no—
99#11719908—41no—
100C23×C53—21no—
101C22×D53C22⋊C811yescharacteristic
102D4×C53—41no—
103C2×C4×C53—21no—
104C2×C4×C53—41no—
105C2×C4×C53—41no—
106C2×C4×C53C2×C22⋊C411yescharacteristic
107C2×C4×C53C22×C811yescharacteristic
108C2×C4×C53C2×M1611yescharacteristic
109C22⋊D53—42no—
110C2×Dic53—21no—
111C2×Dic53—21no—
112C2×Dic53C22⋊C811yescharacteristic
113C2×Dic53C22⋊C811yescharacteristic
114C2×Dic53C22⋊C811yescharacteristic
115C4×D53—22no—
116C4×D53—21no—
117C4×D53—21no—
118C4×D53—41no—
119Dic106—41no—
120C8×C53—21no—
121C8×C53—41no—
122C8×C53AGL(1,ℤ/8)11yescharacteristic
123C8×C53#13711yescharacteristic
124#88322—21no—
125#88322—21no—
126#88322—41no—
127#11625—531nomaximal, Sylow
128C22×C53—41no—
129C22×C53—41no—
130C22×C53—41no—
131C22×C53C2×C22⋊C811yescharacteristic
132C2×D53—21no—
133C2×D53—21no—
134C2×D53#35012yes—
135C4×C53—21no—
136C4×C53—21no—
137C4×C53—41no—
138C4×C53#41711yescharacteristic
139C4×C53#42111yescharacteristic
140Dic53—21no—
141Dic53—21no—
142Dic53—21no—
143Dic53—41no—
144Dic53—41no—
145#2966—531no—
146#2163—531no—
147C2×C4⋊C16—531no—
148#1338—531no—
149#1338—531no—
150#1330—531no—
151#1330D5311yescharacteristic, complemented
152C2×C53—81no—
153C2×C53#147711yescharacteristic
154C2×C53#211511yescharacteristic
155C2×C53C2×#35011yescharacteristic
156D53—22no—
157#529—531no—
158C22×C16—1061no—
159#446—531no—
160C8×Q8—531no—
161#436C2×D5311yescharacteristic
162#435—1061no—
163C8×D4—531no—
164C8×D4—1061no—
165#433—1061no—
166#432—531no—
167#405—1061no—
168C2×C4×C8—531no—
169C4⋊C16—531no—
170C4⋊C16—1061no—
171C4⋊C16—1061no—
172C4⋊C16C2×D5311yescharacteristic
173C22⋊C16—1061no—
174#346—531no—
175#346C2×D5311yescharacteristic
176C53#1162511yescharacteristic, complemented, Sylow
177C2×M16—21no—
178C22×C8—531no—
179C22×C8—531no—
180C22×C8—1061no—
181#126—1061no—
182#125—531no—
183#124—531no—
184#123—1061no—
185C22⋊Q8—531no—
186#121—531no—
187C4×Q8—531no—
188C4×D4—1061no—
189C4×D4C4×D5311yescharacteristic
190#117—1061no—
191C2×C42—531no—
192C2×C16—21no—
193C2×C16—21no—
194C2×C16—1061no—
195C2×C16—1061no—
196C2×C16—2122no—
197C4⋊C8—531no—
198C4⋊C8—1061no—
199C4⋊C8—1061no—
200C4⋊C8C4×D5311yescharacteristic
201C22⋊C8—21no—
202C22⋊C8—1061no—
203C22⋊C8—1061no—
204#97—1061no—
205#97—1061no—
206C4×C8—531no—
207C4×C8—531no—
208C4×C8—531no—
209C4×C8—1061no—
210C4×C8C22×D5311yescharacteristic
211C2×Q8—531no—
212C2×D4—1061no—
213C2×D4C8×D5311yescharacteristic
214C22×C4—21no—
215C22×C4—531no—
216C22×C4—531no—
217C22×C4—531no—
218M16—41no—
219M16—41no—
220C2×C8—21no—
221C2×C8—531no—
222C2×C8—531no—
223C2×C8—1061no—
224C2×C8—1061no—
225C2×C8—1061no—
226C2×C8—2121no—
227C2×C8—2122no—
228C2×C8D4×D5311yescharacteristic
229C2×C8D4×D5311yescharacteristic
230C4⋊C4—531no—
231C4⋊C4—1061no—
232C4⋊C4—1061no—
233C4⋊C4—1061no—
234C4⋊C4—1061no—
235C4⋊C4C8×D5311yescharacteristic
236C22⋊C4—21no—
237C22⋊C4—1061no—
238C22⋊C4—1061no—
239C22⋊C4—1061no—
240C22⋊C4—1061no—
241C42—531no—
242C42—531no—
243C42—531no—
244C42—1061no—
245C42C2×C4×D5311yescharacteristic
246C16—41no—
247C16—41no—
248C16—2121no—
249C16—2121no—
250C23—21no—
251C23—531no—
252Q8—2121no—
253D4—41no—
254D4—2122no—
255C2×C4—21no—
256C2×C4—41no—
257C2×C4—41no—
258C2×C4—531no—
259C2×C4—531no—
260C2×C4—531no—
261C2×C4—1062no—
262C2×C4—1061no—
263C2×C4—1061no—
264C2×C4—1061no—
265C2×C4—1061no—
266C2×C4—2121no—
267C2×C4C22⋊C4×D5311yescharacteristic
268C2×C4C2×C8×D5311yescharacteristic
269C2×C4M16×D5311yescharacteristic
270C8—21no—
271C8—41no—
272C8—1061no—
273C8—1061no—
274C8—2121no—
275C8#1313324111yescharacteristic
276C8#1313324611yescharacteristic
277C22—41no—
278C22—41no—
279C22—41no—
280C22—532no—
281C22—1061no—
282C22—1061no—
283C22C22⋊C8×D5311yescharacteristic
284C4—21no—
285C4—21no—
286C4—41no—
287C4—1061no—
288C4—1061no—
289C4—1061no—
290C4—2121no—
291C4—2121no—
292C4#1681062111yescharacteristic
293C4#1681063811yescharacteristic
294C2—81no—
295C2—1062no—
296C2#2027309911yescharacteristic
297C2#2027633411yescharacteristic
298C2#350×D5311yescharacteristic
subgroupG/H
centerC2×C4C22⋊C4×D53
commutatorC2×C4×C53C22×C8
FrattiniC4×C8C22×D53
FittingC53×#1330C2
socleC22×C53C2×C22⋊C8
prime radical#13133188C8
cosocleC4×C8×C53C23
hypercenter#1330D53
hypocenterC53#11625
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C4×C8 ◁ #1330

Complements

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

NH ≅ G/Ntype
2. #20276516C2semidirect
3. C4⋊C16×D53C2semidirect
4. #20272698C2semidirect
5. #20272696C2semidirect
6. #20272673C2semidirect
7. C53×#1330C2semidirect
151. #1330D53semidirect
176. C53#11625semidirect

Maximal subgroups

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

subgroupindex
#202836932
#202765162
C4⋊C16×D532
#202726982
#202726962
#202726732
C53×#13302
#1162553