Finite Group Databasebeta

← #27278752

Subgroups — #27278752

Order 13568.

566 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 566 conjugacy classes of subgroups — too many to draw. Only its 143 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
1C4×#13133206C211yesmaximal, characteristic
2#20273105C212yesmaximal
3#20273039C211yesmaximal, characteristic, complemented
4#20272547C212yesmaximal, complemented
5C53×#2982C211yesmaximal, characteristic, complemented
6C2×#13133206C411yescharacteristic
7C2×C4×D212C2211yescharacteristic
8#16810676C2212yes—
9#16810530C411yescharacteristic
10#16810524C411yescharacteristic
11#16810513—22no—
12#16810488C411yescharacteristic
13C4×#11719921—22no—
14C42×Dic53C2211yescharacteristic
15#16810350—24no—
16#16810349—22no—
17C2×C4×Q8×C53C2211yescharacteristic
18C8×Q8×C53—24no—
19C53×#431—22no—
20C53×#431C2212yes—
21C22×D212C2×C411yescharacteristic
22C22×C4×D53—21no—
23#13133206—22no—
24#13133206C814yescomplemented
25C4×D212—22no—
26C4×D212—22no—
27C2×C4⋊Dic53C2×C411yescharacteristic
28C2×#11719921—22no—
29C2×#11719921—22no—
30C2×#11719921—21no—
31C2×C4×Dic53—21no—
32C2×C4×Dic53—21no—
33C2×C4×Dic53C2×C411yescharacteristic
34C2×C4×Dic53C2×C411yescharacteristic
35C4⋊#88322—22no—
36C22⋊#88322—22no—
37C4×#88322—22no—
38#13133124—22no—
39#13133124—22no—
40C22×Q8×C53C2×C411yescharacteristic
41C4×Q8×C53—22no—
42C4×Q8×C53—22no—
43C2×C4⋊C4×C53C2×C411yescharacteristic
44C2×C4⋊C4×C53D412yes—
45C2×C42×C53D412yes—
46C2×C42×C53C2311yescharacteristic
47C4⋊C8×C53—24no—
48C4⋊C8×C53D414yes—
49C22⋊C8×C53—22no—
50C4×C8×C53—24no—
51C4×C8×C53—22no—
52C23×D53—21no—
53C2×Q8×C53—22no—
54C2×Q8×C53—22no—
55C2×Q8×C53C2×C812yes—
56C2×Q8×C53M1612yes—
57C22×C4×C53C22⋊C412yes—
58C22×C4×C53C22⋊C412yes—
59C22×C4×C53C22×C411yescharacteristic
60C22×C4×C53C2×D411yescharacteristic
61C22×C4×C53C2×D411yescharacteristic
62C22×Dic53—21no—
63C22×Dic53—21no—
64C22×Dic53—21no—
65C2×D212—22no—
66C2×D212—22no—
67C2×D212—21no—
68C2×D212C2×C812yes—
69C2×C4×D53—44no—
70C2×C4×D53—41no—
71C2×C4×D53—41no—
72C2×C8×C53—24no—
73C4⋊C4×C53—22no—
74C4⋊C4×C53—24no—
75C42×C53—22no—
76C42×C53—22no—
77C42×C53C2×D412yes—
78C42×C53#4112yes—
79#11719921—28no—
80#11719921—44no—
81#11719921—42no—
82C4⋊Dic53—22no—
83C4×Dic53—21no—
84C4×Dic53—22no—
85C4×Dic53—42no—
86C4×Dic53—42no—
87C4×Dic53C2×C812yes—
88C2×#88322—42no—
89C23×C53C2×C22⋊C411yescharacteristic
90C22×D53—22no—
91C22×D53—42no—
92C22×D53—42no—
93C22×D53—41no—
94C22×D53—41no—
95Q8×C53—24no—
96Q8×C53—24no—
97C2×C4×C53—22no—
98C2×C4×C53—22no—
99C2×C4×C53—22no—
100C2×C4×C53—22no—
101C2×C4×C53—41no—
102C2×C4×C53—42no—
103C2×C4×C53C22⋊C814yes—
104C2×C4×C53C4×D412yes—
105C2×C4×C53#12011yescharacteristic
106C2×C4×C53#12112yes—
107C2×C4×C53#12311yescharacteristic
108C2×C4×C53C22×C811yescharacteristic
109C2×C4×C53C2×M1611yescharacteristic
110C2×C4×C53#13112yes—
111C2×Dic53—22no—
112C2×Dic53—22no—
113C2×Dic53—41no—
114C2×Dic53—41no—
115C2×Dic53—41no—
116C2×Dic53—41no—
117C2×Dic53—42no—
118C2×Dic53—41no—
119D212—24no—
120D212—42no—
121C4×D53—82no—
122C4×D53—82no—
123C8×C53—44no—
124#88322—82no—
125#8960—531nomaximal, Sylow
126C22×C53—21no—
127C22×C53—21no—
128C22×C53#38611yescharacteristic
129C22×C53C2×C22⋊C811yescharacteristic
130C22×C53#40811yescharacteristic
131C2×D53—44no—
132C2×D53—42no—
133C2×D53—82no—
134C2×D53—81no—
135C2×D53—81no—
136C4×C53—24no—
137C4×C53—22no—
138C4×C53—21no—
139C4×C53—42no—
140C4×C53C8×D412yes—
141C4×C53#43512yes—
142C4×C53#45412yes—
143Dic53—42no—
144Dic53—42no—
145Dic53—81no—
146#2982D5311yescharacteristic, complemented
147C4×#124—531no—
148#1478—532no—
149#1461—531no—
150#1276—532no—
151C2×C53—21no—
152C2×C53—21no—
153C2×C53#186911yescharacteristic
154C2×C53#188411yescharacteristic
155C2×C53#188811yescharacteristic
156D53—82no—
157C2×#124—531no—
158C2×C4×Q8C2×D5311yescharacteristic
159C2×C4×D4—531no—
160C8×Q8—24no—
161#431—22no—
162#431—532no—
163#431C2×D5312yes—
164#391—531no—
165#390—531no—
166#388—1062no—
167#382—531no—
168C4×C22⋊C4—1062no—
169C43—531no—
170D4⋊C8—1064no—
171#324—1062no—
172C53#896011yescharacteristic, complemented, Sylow
173C22×Q8C4×D5311yescharacteristic
174C22×D4—531no—
175C23×C4—1061no—
176#124—534no—
177#124—1062no—
178C4×Q8—22no—
179C4×Q8—22no—
180C4×D4—1062no—
181C4×D4—1062no—
182C2×C4⋊C4—531no—
183C2×C4⋊C4C4×D5311yescharacteristic
184C2×C4⋊C4D21212yes—
185C2×C22⋊C4—1062no—
186C2×C22⋊C4—1061no—
187C2×C22⋊C4—1062no—
188C2×C42—531no—
189C2×C42—531no—
190C2×C42—1061no—
191C2×C42—1061no—
192C2×C42C22⋊D5312yes—
193C2×C42C22×D5311yescharacteristic
194C4⋊C8—24no—
195C4⋊C8—1062no—
196C4⋊C8C22⋊D5314yes—
197C22⋊C8—22no—
198C22⋊C8—1062no—
199C4×C8—24no—
200C4×C8—22no—
201C4×C8—1062no—
202#95—1062no—
203#95—1062no—
204C24—1061no—
205C2×Q8—22no—
206C2×Q8—22no—
207C2×Q8C8×D5312yes—
208C2×Q8#1171991212yes—
209C2×D4—532no—
210C2×D4—1062no—
211C2×D4—1061no—
212C2×D4—1062no—
213C22×C4—1061no—
214C22×C4—1061no—
215C22×C4—1061no—
216C22×C4—2124no—
217C22×C4—2121no—
218C22×C4—2121no—
219C22×C4#1171992112yes—
220C22×C4#1171992112yes—
221C22×C4C2×C4×D5311yescharacteristic
222C22×C4C2×D21211yescharacteristic
223C22×C4C2×C22⋊D5311yescharacteristic
224C2×C8—24no—
225C2×C8—2122no—
226C4⋊C4—22no—
227C4⋊C4—24no—
228C4⋊C4—1062no—
229C22⋊C4—1068no—
230C22⋊C4—2124no—
231C22⋊C4—2122no—
232C42—22no—
233C42—22no—
234C42—532no—
235C42—1061no—
236C42—1062no—
237C42—2122no—
238C42—2122no—
239C42#1171994512yes—
240C42C2×C22⋊D5312yes—
241C23—1062no—
242C23—2122no—
243C23—2122no—
244C23—2121no—
245C23—2121no—
246C23C2×#1171992111yescharacteristic
247Q8—24no—
248Q8—24no—
249D4—1064no—
250D4—2122no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—41no—
256C2×C4—42no—
257C2×C4—1062no—
258C2×C4—1062no—
259C2×C4—4242no—
260C2×C4—4242no—
261C2×C4—2122no—
262C2×C4—2121no—
263C2×C4—2121no—
264C2×C4—2121no—
265C2×C4—2121no—
266C2×C4—2121no—
267C2×C4#1313312914yes—
268C2×C4C4×C22⋊D5312yes—
269C2×C4#1313318611yescharacteristic
270C2×C4#1313319112yes—
271C2×C4#1313320011yescharacteristic
272C2×C4C2×C8×D5311yescharacteristic
273C2×C4C2×#1171991211yescharacteristic
274C2×C4#1313322312yes—
275C8—44no—
276C8—4242no—
277C22—21no—
278C22—21no—
279C22—4242no—
280C22—4241no—
281C22—4241no—
282C22—2124no—
283C22—2122no—
284C22#1681050211yescharacteristic
285C22C2×#1313312911yescharacteristic
286C22#1681058911yescharacteristic
287C4—24no—
288C4—22no—
289C4—21no—
290C4—42no—
291C4—4241no—
292C4—2122no—
293C4—2122no—
294C4C8×C22⋊D5312yes—
295C4#1681069412yes—
296C4#1681079612yes—
297C2—21no—
298C2—21no—
299C2—4242no—
300C2#2027500711yescharacteristic
301C2#2027509811yescharacteristic
302C2#2027511811yescharacteristic
subgroupG/H
centerC2×C4#13133186
commutatorC2×C4×C53C22×C8
FrattiniC2×C42C22×D53
FittingC53×#2982C2
socleC22×C53#386
prime radicalC22×D212C2×C4
cosocleC2×C42×C53C23
hypercenter#2982D53
hypocenterC53#8960
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #2982

Complements

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

NH ≅ G/Ntype
3. #20273039C2semidirect
4. #20272547C2semidirect
5. C53×#2982C2semidirect
24. #13133206C8semidirect
146. #2982D53semidirect
172. C53#8960semidirect

Maximal subgroups

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

subgroupindex
C4×#131332062
#202731052
#202730392
#202725472
C53×#29822
#896053