Finite Group Databasebeta

← #27299266

Subgroups — #27299266

Order 13568.

628 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 628 conjugacy classes of subgroups — too many to draw. Only its 135 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
1C2×#16810416C212yesmaximal
2C4×#13133132C212yesmaximal, complemented
3#20272840C212yesmaximal, complemented
4C53×#2328C211yesmaximal, characteristic
5C2×#13133132C2214yes—
6C4×C22⋊Dic53—22no—
7#16810416C418yescomplemented
8#16810390—22no—
9C2×#127×C53C2211yescharacteristic
10C2×C4×D4×C53C411yescharacteristic, complemented
11C2×C4×D4×C53C2211yescharacteristic
12C53×#390C411yescharacteristic, complemented
13C53×#390C2211yescharacteristic
14C53×#386—22no—
15C53×#386—22no—
16C53×#382C411yescharacteristic, complemented
17C53×#376C411yescharacteristic, complemented
18C2×C22⋊Dic53—24no—
19C2×C4×Dic53—42no—
20#13133132—24no—
21#13133132C2×C418yescomplemented
22#13133124—42no—
23C22×D4×C53—22no—
24C22×D4×C53C2×C411yescharacteristic
25C22×D4×C53C2311yescharacteristic
26C23×C4×C53—21no—
27C23×C4×C53D412yes—
28#127×C53—22no—
29#127×C53C2×C414yes—
30C4×D4×C53—42no—
31C4×D4×C53—42no—
32C2×C4⋊C4×C53C2×C411yescharacteristic
33C2×C4⋊C4×C53C2×C411yescharacteristic
34C2×C22⋊C4×C53—22no—
35C2×C22⋊C4×C53—21no—
36C2×C22⋊C4×C53—22no—
37C2×C22⋊C4×C53D412yes—
38C2×C42×C53C2×C411yescharacteristic
39C2×C42×C53C2×C411yescharacteristic
40C2×C42×C53C2×C411yescharacteristic
41#95×C53—21no—
42#95×C53—21no—
43#95×C53—22no—
44C24×C53—21no—
45C24×C53C2×D412yes—
46C2×D4×C53—28no—
47C2×D4×C53—21no—
48C2×D4×C53—41no—
49C2×D4×C53—41no—
50C2×D4×C53—41no—
51C2×D4×C53—41no—
52C2×D4×C53—82no—
53C2×D4×C53—82no—
54C2×D4×C53C4212yescomplemented
55C2×D4×C53C22×C412yes—
56C2×D4×C53#4112yes—
57C22×C4×C53—21no—
58C22×C4×C53—22no—
59C22×C4×C53—22no—
60C22×C4×C53—21no—
61C22×C4×C53—21no—
62C22×C4×C53—42no—
63C22×C4×C53—82no—
64C22×C4×C53—82no—
65C22×C4×C53C22⋊C411yescharacteristic
66C22×C4×C53C22⋊C412yes—
67C22×C4×C53C22⋊C411yescharacteristic
68C22×C4×C53C22×C411yescharacteristic
69C22×Dic53—44no—
70C4⋊C4×C53—41no—
71C4⋊C4×C53—41no—
72C22⋊C4×C53—42no—
73C22⋊C4×C53—42no—
74C22⋊C4×C53—82no—
75C22⋊C4×C53—82no—
76C42×C53—22no—
77C42×C53—21no—
78C42×C53—41no—
79C42×C53C4212yescomplemented
80C22⋊Dic53—28no—
81C22⋊Dic53—44no—
82C4×Dic53—84no—
83C23×C53—22no—
84C23×C53—22no—
85C23×C53—22no—
86C23×C53—42no—
87C23×C53—42no—
88C23×C53—42no—
89C23×C53—42no—
90C23×C53—41no—
91C23×C53—41no—
92C23×C53C2×C22⋊C411yescharacteristic
93C23×C53C4×D414yes—
94D4×C53—42no—
95D4×C53—42no—
96D4×C53—42no—
97D4×C53—42no—
98D4×C53—88no—
99C2×C4×C53—21no—
100C2×C4×C53—21no—
101C2×C4×C53—22no—
102C2×C4×C53—22no—
103C2×C4×C53—21no—
104C2×C4×C53—41no—
105C2×C4×C53—44no—
106C2×C4×C53—41no—
107C2×C4×C53—41no—
108C2×C4×C53—41no—
109C2×C4×C53—42no—
110C2×C4×C53—42no—
111C2×C4×C53—41no—
112C2×C4×C53—41no—
113C2×C4×C53—41no—
114C2×C4×C53—41no—
115C2×C4×C53—82no—
116C2×C4×C53—84no—
117C2×C4×C53#9912yes—
118C2×C4×C53C2×C4211yescharacteristic
119C2×C4×C53#11711yescharacteristic
120C2×Dic53—48no—
121C2×Dic53—84no—
122C2×Dic53—84no—
123#12628—531nomaximal, Sylow
124C22×C53—21no—
125C22×C53—21no—
126C22×C53—24no—
127C22×C53—44no—
128C22×C53—44no—
129C22×C53—42no—
130C22×C53—42no—
131C22×C53—42no—
132C22×C53—82no—
133C22×C53—82no—
134C22×C53—81no—
135C22×C53—81no—
136C22×C53C4×C22⋊C411yescharacteristic
137C22×C53C2×#9911yescharacteristic
138C22×C53#41011yescharacteristic
139C4×C53—22no—
140C4×C53—21no—
141C4×C53—21no—
142C4×C53—42no—
143C4×C53—42no—
144C4×C53—41no—
145C4×C53—81no—
146C4×C53—81no—
147Dic53—88no—
148#2328D5311yescharacteristic
149C2×#353—532no—
150C4×#99—532no—
151#1393—532no—
152C2×C53—21no—
153C2×C53—21no—
154C2×C53—44no—
155C2×C53—82no—
156C2×C53C4×#9911yescharacteristic
157C2×C53#212711yescharacteristic
158C2×C53#212811yescharacteristic
159C2×#127C2×D5311yescharacteristic
160C2×C4×D4Dic5311yescharacteristic, complemented
161C2×C4×D4C2×D5311yescharacteristic
162C2×#99—534no—
163#390Dic5311yescharacteristic, complemented
164#390C2×D5311yescharacteristic
165#386—22no—
166#386—22no—
167#382Dic5311yescharacteristic, complemented
168C4×C22⋊C4—1062no—
169#376Dic5311yescharacteristic, complemented
170#353—538no—
171#342—1062no—
172C53#1262811yescharacteristic, complemented, Sylow
173C22×D4—22no—
174C22×D4C2×Dic5311yescharacteristic
175C22×D4C22×D5311yescharacteristic
176C23×C4—21no—
177C23×C4C22⋊D5312yes—
178#127—22no—
179#127C4×D5314yes—
180C4×D4—42no—
181C4×D4—42no—
182C2×C4⋊C4C2×Dic5311yescharacteristic
183C2×C4⋊C4C2×Dic5311yescharacteristic
184C2×C22⋊C4—21no—
185C2×C22⋊C4—22no—
186C2×C22⋊C4—22no—
187C2×C22⋊C4—1064no—
188C2×C22⋊C4C22⋊D5312yes—
189C2×C42—2122no—
190C2×C42C2×Dic5311yescharacteristic
191C2×C42C2×Dic5311yescharacteristic
192C2×C42C2×Dic5311yescharacteristic
193#99—538no—
194#99—1064no—
195#95—21no—
196#95—22no—
197#95—21no—
198#95—2122no—
199C24—21no—
200C24C2×C22⋊D5312yes—
201C2×D4—28no—
202C2×D4—21no—
203C2×D4—41no—
204C2×D4—41no—
205C2×D4—41no—
206C2×D4—41no—
207C2×D4—82no—
208C2×D4—82no—
209C2×D4C4×Dic5312yescomplemented
210C2×D4C2×C4×D5312yes—
211C2×D4#1171994512yes—
212C22×C4—21no—
213C22×C4—22no—
214C22×C4—22no—
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—42no—
218C22×C4—82no—
219C22×C4—82no—
220C22×C4—2124no—
221C22×C4C22⋊Dic5311yescharacteristic
222C22×C4C22⋊Dic5312yes—
223C22×C4C22⋊Dic5311yescharacteristic
224C22×C4C22×Dic5311yescharacteristic
225C4⋊C4—41no—
226C4⋊C4—41no—
227C22⋊C4—42no—
228C22⋊C4—42no—
229C22⋊C4—82no—
230C22⋊C4—82no—
231C22⋊C4—1068no—
232C22⋊C4—2124no—
233C42—22no—
234C42—21no—
235C42—41no—
236C42—4244no—
237C42C4×Dic5312yescomplemented
238C23—22no—
239C23—22no—
240C23—22no—
241C23—42no—
242C23—42no—
243C23—42no—
244C23—42no—
245C23—41no—
246C23—41no—
247C23C2×C22⋊Dic5311yescharacteristic
248C23C4×C22⋊D5314yes—
249D4—42no—
250D4—42no—
251D4—42no—
252D4—42no—
253D4—88no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—22no—
258C2×C4—22no—
259C2×C4—41no—
260C2×C4—44no—
261C2×C4—41no—
262C2×C4—41no—
263C2×C4—42no—
264C2×C4—41no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—42no—
268C2×C4—41no—
269C2×C4—41no—
270C2×C4—82no—
271C2×C4—84no—
272C2×C4—4244no—
273C2×C4—4244no—
274C2×C4—2128no—
275C2×C4#1313313212yes—
276C2×C4C2×C4×Dic5311yescharacteristic
277C2×C4#1313317611yescharacteristic
278C22—21no—
279C22—21no—
280C22—24no—
281C22—44no—
282C22—44no—
283C22—42no—
284C22—42no—
285C22—42no—
286C22—82no—
287C22—82no—
288C22—81no—
289C22—81no—
290C22C4×C22⋊Dic5311yescharacteristic
291C22C2×#1313313211yescharacteristic
292C22#1681059711yescharacteristic
293C4—21no—
294C4—21no—
295C4—22no—
296C4—42no—
297C4—42no—
298C4—41no—
299C4—81no—
300C4—81no—
301C4—4248no—
302C2—21no—
303C2—21no—
304C2—44no—
305C2—82no—
306C2C4×#1313313211yescharacteristic
307C2#2027638411yescharacteristic
308C2#2027638711yescharacteristic
subgroupG/H
centerC22C2×#13133132
commutatorC2×C4×C53C2×C42
FrattiniC22×D4C22×D53
FittingC53×#2328C2
socleC22×C53C2×#99
prime radicalC2×#127×C53C22
cosocleC22×D4×C53C23
hypercenter#2328D53
hypocenterC53#12628
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2328

Complements

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

NH ≅ G/Ntype
2. C4×#13133132C2semidirect
3. #20272840C2semidirect
7. #16810416C4semidirect
10. C2×C4×D4×C53C4semidirect
12. C53×#390C4semidirect
16. C53×#382C4semidirect
17. C53×#376C4semidirect
21. #13133132C2×C4semidirect
54. C2×D4×C53C42semidirect
79. C42×C53C42semidirect
160. C2×C4×D4Dic53semidirect
163. #390Dic53semidirect
167. #382Dic53semidirect
169. #376Dic53semidirect
172. C53#12628semidirect
209. C2×D4C4×Dic53semidirect
237. C42C4×Dic53semidirect

Maximal subgroups

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

subgroupindex
C2×#168104162
C4×#131331322
#202728402
C53×#23282
#1262853