Finite Group Databasebeta

← #27278735

Subgroups — #27278735

Order 13568.

586 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 586 conjugacy classes of subgroups — too many to draw. Only its 139 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#20283739C211yesmaximal, characteristic
2C4×#13133204C211yesmaximal, characteristic
3#20273111C212yesmaximal, complemented
4#20272547C212yesmaximal, complemented
5C53×#1460C211yesmaximal, characteristic
6C2×#13133204C411yescharacteristic
7C2×C4×Dic106C2211yescharacteristic
8D4×#88322—24no—
9#16810676C2212yes—
10C2×C22⋊#88322—22no—
11#16810528C411yescharacteristic
12#16810526C411yescharacteristic
13#16810517—22no—
14#16810488C411yescharacteristic
15C4×C22⋊Dic53—22no—
16C42×Dic53C2211yescharacteristic
17#16810355—24no—
18C2×C4×D4×C53C2211yescharacteristic
19C53×#431C2212yes—
20C53×#323—22no—
21C22×Dic106C2×C411yescharacteristic
22C22×#88322—24no—
23#13133204—22no—
24#13133204C814yescomplemented
25C4×Dic106—22no—
26C4×Dic106—22no—
27C2×C4⋊Dic53C2×C411yescharacteristic
28C2×#11719919—21no—
29C2×C22⋊Dic53—22no—
30C2×C22⋊Dic53—22no—
31C2×C4×Dic53—21no—
32C2×C4×Dic53—21no—
33C2×C4×Dic53—21no—
34C2×C4×Dic53C2×C411yescharacteristic
35C2×C4×Dic53C2×C411yescharacteristic
36C4⋊#88322D414yes—
37C22⋊#88322—24no—
38C22⋊#88322—22no—
39C4×#88322—22no—
40#13133124—22no—
41#13133124—22no—
42C22×D4×C53C2×C411yescharacteristic
43C23×C4×C53D412yes—
44C4×D4×C53—22no—
45C4×D4×C53—22no—
46C2×C4⋊C4×C53C2×C411yescharacteristic
47C2×C22⋊C4×C53D412yes—
48C2×C42×C53C2311yescharacteristic
49C4⋊C8×C53—22no—
50C22⋊C8×C53—22no—
51C4×C8×C53—22no—
52C24×C53C22⋊C412yes—
53C2×D4×C53—22no—
54C2×D4×C53—22no—
55C2×D4×C53C2×C812yes—
56C2×D4×C53M1612yes—
57C22×C4×C53—22no—
58C22×C4×C53—22no—
59C22×C4×C53—44no—
60C22×C4×C53C22⋊C412yes—
61C22×C4×C53C22×C411yescharacteristic
62C22×C4×C53C2×D411yescharacteristic
63C22×C4×C53C2×D411yescharacteristic
64C22×Dic53—21no—
65C22×Dic53—21no—
66C22×Dic53—21no—
67C22×Dic53—21no—
68C2×Dic106—22no—
69C2×Dic106—22no—
70C2×Dic106—21no—
71C2×Dic106C2×C812yescomplemented
72C2×C8×C53—42no—
73C4⋊C4×C53—22no—
74C22⋊C4×C53—24no—
75C42×C53C2×D412yes—
76C42×C53#4112yes—
77C22⋊Dic53—28no—
78C22⋊Dic53—44no—
79C4⋊Dic53—22no—
80#11719919—42no—
81C4×Dic53—21no—
82C4×Dic53—22no—
83C4×Dic53—42no—
84C4×Dic53—42no—
85C4×Dic53—41no—
86C4×Dic53—41no—
87C4×Dic53C2×C812yescomplemented
88C2×#88322—24no—
89C2×#88322—84no—
90C23×C53—22no—
91C23×C53—22no—
92C23×C53—42no—
93C23×C53—42no—
94C23×C53C22⋊C814yes—
95C23×C53C2×C22⋊C411yescharacteristic
96D4×C53—24no—
97D4×C53—24no—
98C2×C4×C53—22no—
99C2×C4×C53—22no—
100C2×C4×C53—41no—
101C2×C4×C53—44no—
102C2×C4×C53—42no—
103C2×C4×C53—82no—
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—22no—
114C2×Dic53—41no—
115C2×Dic53—41no—
116C2×Dic53—41no—
117C2×Dic53—41no—
118C2×Dic53—41no—
119C2×Dic53—41no—
120C2×Dic53—42no—
121C2×Dic53—41no—
122Dic106—24no—
123Dic106—42no—
124C8×C53—82no—
125#88322—44no—
126#8957—531nomaximal, Sylow
127C22×C53—21no—
128C22×C53—21no—
129C22×C53—24no—
130C22×C53—44no—
131C22×C53—42no—
132C22×C53—42no—
133C22×C53—82no—
134C22×C53#38611yescharacteristic
135C22×C53C2×C22⋊C811yescharacteristic
136C22×C53#40811yescharacteristic
137C4×C53—22no—
138C4×C53—21no—
139C4×C53—42no—
140C4×C53C8×D412yes—
141C4×C53#43512yes—
142C4×C53#45512yes—
143Dic53—42no—
144Dic53—42no—
145Dic53—42no—
146Dic53—81no—
147#2972—531no—
148C4×#124—531no—
149#1479—532no—
150#1460D5311yescharacteristic
151#1276—532no—
152C2×C53—21no—
153C2×C53—21no—
154C2×C53—44no—
155C2×C53#186911yescharacteristic
156C2×C53#188411yescharacteristic
157C2×C53#188511yescharacteristic
158C2×#124—531no—
159C2×C4×Q8—531no—
160C2×C4×D4C2×D5311yescharacteristic
161C8×D4—1064no—
162#431—532no—
163#431C2×D5312yes—
164C2×C22⋊C8—1062no—
165#391—531no—
166#390—531no—
167#388—1062no—
168#382—531no—
169C4×C22⋊C4—1062no—
170C43—531no—
171Q8⋊C8—1064no—
172#323—22no—
173C53#895711yescharacteristic, complemented, Sylow
174C22×Q8—531no—
175C22×D4C4×D5311yescharacteristic
176C23×C4D21212yes—
177C22×C8—1064no—
178#124—534no—
179#124—1062no—
180C4×Q8—1062no—
181C4×Q8—1062no—
182C4×D4—22no—
183C4×D4—22no—
184C2×C4⋊C4—531no—
185C2×C4⋊C4—1061no—
186C2×C4⋊C4C4×D5311yescharacteristic
187C2×C22⋊C4—1062no—
188C2×C22⋊C4—1062no—
189C2×C22⋊C4C22⋊D5312yes—
190C2×C42—531no—
191C2×C42—531no—
192C2×C42—1061no—
193C2×C42—1061no—
194C2×C42—1061no—
195C2×C42C22×D5311yescharacteristic
196C4⋊C8—22no—
197C4⋊C8—534no—
198C22⋊C8—22no—
199C22⋊C8—1064no—
200C22⋊C8—1062no—
201C4×C8—22no—
202C4×C8—1062no—
203#95—1062no—
204#95—1062no—
205C24#1171992112yes—
206C2×Q8—532no—
207C2×Q8—1061no—
208C2×Q8—1062no—
209C2×Q8—1062no—
210C2×D4—22no—
211C2×D4—22no—
212C2×D4C8×D5312yes—
213C2×D4#1171991212yes—
214C22×C4—22no—
215C22×C4—22no—
216C22×C4—44no—
217C22×C4—1061no—
218C22×C4—1061no—
219C22×C4—1061no—
220C22×C4—1061no—
221C22×C4#1171992112yes—
222C22×C4C2×C4×D5311yescharacteristic
223C22×C4C2×D21211yescharacteristic
224C22×C4C2×C22⋊D5311yescharacteristic
225C2×C8—42no—
226C2×C8—1064no—
227C2×C8—4244no—
228C4⋊C4—22no—
229C4⋊C4—1062no—
230C4⋊C4—2122no—
231C22⋊C4—24no—
232C22⋊C4—1068no—
233C22⋊C4—2124no—
234C42—532no—
235C42—1061no—
236C42—1062no—
237C42—2122no—
238C42—2122no—
239C42—2121no—
240C42—2121no—
241C42D4×D5312yes—
242C42#1171994712yes—
243C23—22no—
244C23—22no—
245C23—42no—
246C23—42no—
247C23#1313312914yes—
248C23C2×#1171992111yescharacteristic
249Q8—1064no—
250Q8—2122no—
251D4—24no—
252D4—24no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—41no—
256C2×C4—44no—
257C2×C4—42no—
258C2×C4—82no—
259C2×C4—1062no—
260C2×C4—1062no—
261C2×C4—1062no—
262C2×C4—2122no—
263C2×C4—2121no—
264C2×C4—2121no—
265C2×C4—2121no—
266C2×C4—2121no—
267C2×C4—2121no—
268C2×C4—2121no—
269C2×C4—2121no—
270C2×C4#1313317712yes—
271C2×C4#1313318711yescharacteristic
272C2×C4#1313318912yes—
273C2×C4#1313320111yescharacteristic
274C2×C4C2×C8×D5311yescharacteristic
275C2×C4C2×#1171991211yescharacteristic
276C2×C4#1313322512yes—
277C8—82no—
278C8—2124no—
279C22—21no—
280C22—21no—
281C22—24no—
282C22—44no—
283C22—42no—
284C22—42no—
285C22—82no—
286C22#1681050011yescharacteristic
287C22C2×#1313312911yescharacteristic
288C22#1681059111yescharacteristic
289C4—22no—
290C4—21no—
291C4—42no—
292C4—4241no—
293C4—2122no—
294C4—2122no—
295C4—2122no—
296C4#1681068612yes—
297C4#1681069112yes—
298C4#1681080312yes—
299C2—21no—
300C2—21no—
301C2—44no—
302C2#2027500511yescharacteristic
303C2#2027509611yescharacteristic
304C2#2027510111yescharacteristic
subgroupG/H
centerC2×C4#13133187
commutatorC2×C4×C53C22×C8
FrattiniC2×C42C22×D53
FittingC53×#1460C2
socleC22×C53#386
prime radicalC22×D4×C53C2×C4
cosocleC2×C42×C53C23
hypercenter#1460D53
hypocenterC53#8957
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1460

Complements

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

NH ≅ G/Ntype
3. #20273111C2semidirect
4. #20272547C2semidirect
24. #13133204C8semidirect
71. C2×Dic106C2×C8semidirect
87. C4×Dic53C2×C8semidirect
173. C53#8957semidirect

Maximal subgroups

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

subgroupindex
#202837392
C4×#131332042
#202731112
#202725472
C53×#14602
#895753