Finite Group Databasebeta

← #27293986

Subgroups — #27293986

Order 13568.

328 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 328 conjugacy classes of subgroups — too many to draw. Only its 93 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#20283678C211yesmaximal, 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
8C4×#11719947C411yescharacteristic
9C2×C16×D53—21no—
10#16810753C2211yescharacteristic
11#16810750C411yescharacteristic
12#16810701C411yescharacteristic
13#16810697—21no—
14#16810692—21no—
15#16810675C411yescharacteristic
16#16810673—21no—
17M16×Dic53—21no—
18C4×C8×D53C2211yescharacteristic
19C4⋊#11719908C2211yescharacteristic
20#16810441—21no—
21#16810441—21no—
22#16810402—21no—
23#16810399C2211yescharacteristic
24C53×#436C2211yescharacteristic
25C53×C4⋊C16C2211yescharacteristic
26C53×#346C2211yescharacteristic
27C2×#11719947C811yescharacteristic
28C2×#11719917—21no—
29C2×C8×D53D411yescharacteristic
30C2×C8×D53D411yescharacteristic
31C4×Dic106C2×C411yescharacteristic
32#13133182C811yescharacteristic
33D4×Dic53C811yescharacteristic
34C4×C22⋊D53—21no—
35#13133177—21no—
36#13133175C811yescharacteristic
37#13133173—21no—
38C2×C4×Dic53—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—
64C22×Dic53—21no—
65C22×Dic53—21no—
66#11719947—42no—
67C2×C4×D53C22⋊C411yescharacteristic
68C2×C4×D53C2×C811yescharacteristic
69C2×C4×D53M1611yescharacteristic
70C2×Dic106C2×C811yescharacteristic
71M16×C53—41no—
72M16×C53—41no—
73C2×C8×C53—21no—
74C2×C8×C53C2×D411yescharacteristic
75C2×C8×C53C2×D411yescharacteristic
76C4⋊C4×C53C2×C811yescharacteristic
77C22⋊C4×C53—21no—
78C42×C53C22×C411yescharacteristic
79C22⋊Dic53—21no—
80#11719921—21no—
81C4⋊Dic53C2×C811yescharacteristic
82#11719919—21no—
83#11719919—21no—
84C4×Dic53—21no—
85C4×Dic53—21no—
86C4×Dic53—21no—
87C4×Dic53C22⋊C411yescharacteristic
88C4×Dic53C2×C811yescharacteristic
89C4×Dic53M1611yescharacteristic
90#11719917—42no—
91C2×#88322—21no—
92C2×#88322C22⋊C411yescharacteristic
93C2×#88322C22⋊C411yescharacteristic
94C8×D53—21no—
95C8×D53—21no—
96C8×D53—41no—
97C16×C53—41no—
98C16×C53—41no—
99#11719908—41no—
100#11719908—41no—
101C23×C53—21no—
102C22×D53C22⋊C811yescharacteristic
103D4×C53—41no—
104C2×C4×C53—21no—
105C2×C4×C53—41no—
106C2×C4×C53—41no—
107C2×C4×C53C2×C22⋊C411yescharacteristic
108C2×C4×C53C22×C811yescharacteristic
109C2×C4×C53C2×M1611yescharacteristic
110C22⋊D53—42no—
111C2×Dic53—21no—
112C2×Dic53—21no—
113C2×Dic53—42no—
114C2×Dic53—42no—
115C2×Dic53C22⋊C811yescharacteristic
116C2×Dic53C22⋊C811yescharacteristic
117C2×Dic53C22⋊C811yescharacteristic
118C4×D53—22no—
119C4×D53—21no—
120C4×D53—21no—
121C4×D53—22no—
122Dic106—41no—
123C8×C53—21no—
124C8×C53—41no—
125C8×C53AGL(1,ℤ/8)11yescharacteristic
126C8×C53#13711yescharacteristic
127#88322—21no—
128#88322—21no—
129#88322—41no—
130#11626—531nomaximal, Sylow
131C22×C53—41no—
132C22×C53—41no—
133C22×C53—41no—
134C22×C53C2×C22⋊C811yescharacteristic
135C2×D53—21no—
136C2×D53—21no—
137C2×D53#34912yes—
138C4×C53—21no—
139C4×C53—21no—
140C4×C53—41no—
141C4×C53#41711yescharacteristic
142C4×C53#42111yescharacteristic
143Dic53—21no—
144Dic53—21no—
145Dic53—41no—
146Dic53—41no—
147Dic53#35012yes—
148#2965—531no—
149#2163—531no—
150C2×C4⋊C16—531no—
151#1338—531no—
152#1338—531no—
153#1330—531no—
154#1330D5311yescharacteristic, complemented
155C2×C53—81no—
156C2×C53#147711yescharacteristic
157C2×C53C2×#34911yescharacteristic
158C2×C53C2×#35011yescharacteristic
159D53—22no—
160C4×#41—531no—
161C22×C16—1061no—
162#446—531no—
163#446—531no—
164#436—531no—
165#436C2×D5311yescharacteristic
166#435—1061no—
167#435—1061no—
168#431—531no—
169#431—1061no—
170C4×M16—1061no—
171C2×C4×C8—531no—
172C4⋊C16—531no—
173C4⋊C16—1061no—
174C4⋊C16—1061no—
175C4⋊C16C2×D5311yescharacteristic
176C22⋊C16—1061no—
177#346—531no—
178#346C2×D5311yescharacteristic
179C53#1162611yescharacteristic, complemented, Sylow
180C2×#41—531no—
181C2×M16—21no—
182C2×M16—1061no—
183C22×C8—531no—
184C22×C8—531no—
185C4×Q8—531no—
186C4×Q8—531no—
187C4×D4—531no—
188C4×D4—1061no—
189C4×D4—1061no—
190C4×D4C4×D5311yescharacteristic
191#117—531no—
192#117—1061no—
193C2×C42—531no—
194C2×C42—1061no—
195C2×C16—21no—
196C2×C16—21no—
197C2×C16—1061no—
198C2×C16—1061no—
199C2×C16—2122no—
200C4⋊C8—531no—
201C4⋊C8—1061no—
202C4⋊C8—1061no—
203C4⋊C8C4×D5311yescharacteristic
204C22⋊C8—21no—
205C22⋊C8—1061no—
206C22⋊C8—1061no—
207#97—1061no—
208#97—1061no—
209C4×C8—531no—
210C4×C8—531no—
211C4×C8—531no—
212C4×C8—1061no—
213C4×C8C22×D5311yescharacteristic
214#41—2122no—
215C2×Q8—531no—
216C2×D4—1061no—
217C2×D4C8×D5311yescharacteristic
218C22×C4—21no—
219C22×C4—531no—
220C22×C4—531no—
221C22×C4—531no—
222C22×C4—1061no—
223C22×C4—1061no—
224M16—41no—
225M16—41no—
226M16—2122no—
227C2×C8—21no—
228C2×C8—531no—
229C2×C8—531no—
230C2×C8—1061no—
231C2×C8—1061no—
232C2×C8—1061no—
233C2×C8—2121no—
234C2×C8D4×D5311yescharacteristic
235C2×C8D4×D5311yescharacteristic
236C4⋊C4—531no—
237C4⋊C4—1061no—
238C4⋊C4—1061no—
239C4⋊C4C8×D5311yescharacteristic
240C22⋊C4—21no—
241C22⋊C4—1061no—
242C22⋊C4—1061no—
243C42—531no—
244C42—531no—
245C42—531no—
246C42—1061no—
247C42—1061no—
248C42—1061no—
249C42C2×C4×D5311yescharacteristic
250C16—41no—
251C16—41no—
252C16—2121no—
253C16—2121no—
254C23—21no—
255C23—531no—
256Q8—2121no—
257D4—41no—
258D4—2122no—
259C2×C4—21no—
260C2×C4—41no—
261C2×C4—41no—
262C2×C4—531no—
263C2×C4—531no—
264C2×C4—531no—
265C2×C4—1062no—
266C2×C4—1062no—
267C2×C4—1061no—
268C2×C4—1061no—
269C2×C4—1061no—
270C2×C4—1061no—
271C2×C4—2122no—
272C2×C4—2122no—
273C2×C4C22⋊C4×D5311yescharacteristic
274C2×C4C2×C8×D5311yescharacteristic
275C2×C4M16×D5311yescharacteristic
276C8—21no—
277C8—41no—
278C8—1061no—
279C8—1061no—
280C8—2121no—
281C8#1313324111yescharacteristic
282C8#1313324611yescharacteristic
283C22—41no—
284C22—41no—
285C22—41no—
286C22—532no—
287C22—1061no—
288C22—1061no—
289C22C22⋊C8×D5311yescharacteristic
290C4—21no—
291C4—21no—
292C4—41no—
293C4—532no—
294C4—1061no—
295C4—1061no—
296C4—2121no—
297C4—2121no—
298C4#1681062111yescharacteristic
299C4#1681063811yescharacteristic
300C2—81no—
301C2—1062no—
302C2#2027309911yescharacteristic
303C2#349×D5311yescharacteristic
304C2#350×D5311yescharacteristic
subgroupG/H
centerC2×C4C22⋊C4×D53
commutatorC2×C4×C53C22×C8
FrattiniC4×C8C22×D53
FittingC53×#1330C2
socleC22×C53C2×C22⋊C8
prime radicalC2×#11719947C8
cosocleC4×C8×C53C23
hypercenter#1330D53
hypocenterC53#11626
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
154. #1330D53semidirect
179. C53#11626semidirect

Maximal subgroups

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

subgroupindex
#202836782
#202765162
C4⋊C16×D532
#202726982
#202726962
#202726732
C53×#13302
#1162653