Finite Group Databasebeta

← #28308256

Subgroups — #28308256

Order 13568.

596 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 596 conjugacy classes of subgroups — too many to draw. Only its 76 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#20289971C212yesmaximal, complemented
2#20289657C212yesmaximal, complemented
3#20289643C212yesmaximal, complemented
4#127×Dic53C211yesmaximal, characteristic
5#16811593—22no—
6#16811468C2214yes—
7#16811466—22no—
8C2×D4×Dic53—22no—
9C2×D4×Dic53C411yescharacteristic
10C2×D4×Dic53C2211yescharacteristic
11#16810526—22no—
12#16810526C411yescharacteristic
13#16810526C2211yescharacteristic
14#16810493C2211yescharacteristic
15C42×Dic53C411yescharacteristic
16C2×#127×C53C411yescharacteristic
17C2×#11719940—42no—
18C22⋊#88324—24no—
19C22⋊#88324—42no—
20C23×Dic53—21no—
21C23×Dic53D412yes—
22D4×Dic53—41no—
23D4×Dic53—41no—
24D4×Dic53—48no—
25D4×Dic53—41no—
26D4×Dic53—41no—
27C2×C4⋊Dic53—22no—
28C2×C4⋊Dic53C2×C411yescharacteristic
29C2×C4⋊Dic53C2311yescharacteristic
30C2×C22⋊Dic53—22no—
31C2×C22⋊Dic53—21no—
32C2×C22⋊Dic53—22no—
33C2×C22⋊Dic53D412yes—
34C2×C4×Dic53—21no—
35C2×C4×Dic53—21no—
36C2×C4×Dic53C2×C411yescharacteristic
37C2×C4×Dic53C2×C411yescharacteristic
38C22×D4×C53—22no—
39C22×D4×C53C2×C411yescharacteristic
40C22×D4×C53C2×C411yescharacteristic
41#127×C53—22no—
42#127×C53—22no—
43C2×C42×C53C2×C411yescharacteristic
44C24×C53—21no—
45C24×C53C22⋊C412yes—
46C2×D4×C53—28no—
47C2×D4×C53—21no—
48C2×D4×C53—21no—
49C2×D4×C53—21no—
50C2×D4×C53—41no—
51C2×D4×C53—41no—
52C2×D4×C53—44no—
53C2×D4×C53—44no—
54C2×D4×C53—41no—
55C2×D4×C53—41no—
56C2×D4×C53M1612yescomplemented
57C22×C4×C53—21no—
58C22×C4×C53C22×C411yescharacteristic
59C22×Dic53—21no—
60C22×Dic53—22no—
61C22×Dic53—22no—
62C22×Dic53—21no—
63C22×Dic53—44no—
64C22×Dic53—44no—
65C22×Dic53—41no—
66C22×Dic53—41no—
67C22×Dic53C22⋊C411yescharacteristic
68C22×Dic53C22⋊C411yescharacteristic
69C22×Dic53C2×D412yes—
70#11719940—84no—
71C2×#88324—44no—
72C42×C53—21no—
73C42×C53M1612yescomplemented
74C22⋊Dic53—42no—
75C22⋊Dic53—44no—
76C22⋊Dic53—44no—
77C22⋊Dic53—42no—
78C4⋊Dic53—41no—
79C4⋊Dic53—44no—
80C4⋊Dic53—41no—
81C4×Dic53—24no—
82C4×Dic53—21no—
83C4×Dic53—21no—
84C4×Dic53—42no—
85C4×Dic53—41no—
86C23×C53—22no—
87C23×C53—22no—
88C23×C53—22no—
89C23×C53—22no—
90C23×C53—42no—
91C23×C53—42no—
92C23×C53—42no—
93C23×C53—42no—
94C23×C53—41no—
95C23×C53—41no—
96C23×C53C2×C22⋊C411yescharacteristic
97D4×C53—42no—
98D4×C53—416no—
99D4×C53—42no—
100D4×C53—42no—
101D4×C53—42no—
102C2×C4×C53—22no—
103C2×C4×C53—22no—
104C2×C4×C53—22no—
105C2×C4×C53—21no—
106C2×C4×C53—21no—
107C2×C4×C53C2×M1611yescharacteristic
108C2×C4×C53C2×M1611yescharacteristic
109C2×Dic53—22no—
110C2×Dic53—22no—
111C2×Dic53—22no—
112C2×Dic53—21no—
113C2×Dic53—41no—
114C2×Dic53—41no—
115C2×Dic53—44no—
116C2×Dic53—42no—
117C2×Dic53—41no—
118C2×Dic53—42no—
119C2×Dic53—41no—
120C2×Dic53—82no—
121C2×Dic53—82no—
122C2×Dic53—82no—
123C2×Dic53#9911yescharacteristic
124C2×Dic53#9911yescharacteristic
125#88324—84no—
126#11529—531nomaximal, Sylow
127C22×C53—21no—
128C22×C53—21no—
129C22×C53—44no—
130C22×C53—44no—
131C22×C53—42no—
132C22×C53—42no—
133C22×C53—42no—
134C22×C53—42no—
135C22×C53—82no—
136C22×C53—82no—
137C22×C53—81no—
138C22×C53—81no—
139C22×C53#40711yescharacteristic
140C22×C53C2×#9911yescharacteristic
141C22×C53#41311yescharacteristic
142C4×C53—24no—
143C4×C53—21no—
144C4×C53—21no—
145Dic53—21no—
146Dic53—42no—
147Dic53—42no—
148Dic53—41no—
149Dic53—81no—
150Dic53#35312yes—
151C4×#127—531no—
152#1465—532no—
153#1325—532no—
154#1318—532no—
155C2×C53—21no—
156C2×C53—21no—
157C2×C53—82no—
158C2×C53—82no—
159C2×C53#146511yescharacteristic
160C2×C53C2×#35311yescharacteristic
161C2×C53#213211yescharacteristic
162C2×#127#615711yescharacteristic
163C2×C4×D4—531no—
164C2×C4×D4—531no—
165C2×C4×D4—1062no—
166#407—1062no—
167#390—531no—
168#390—531no—
169#390—1062no—
170#384—531no—
171C43—531no—
172#324—534no—
173#323—1062no—
174C53#1152911yescharacteristic, complemented, Sylow
175C22×D4—22no—
176C22×D4C2×#615711yescharacteristic
177C22×D4C2×#615711yescharacteristic
178C23×C4—532no—
179C23×C4—1061no—
180C2×M16—2122no—
181#127—22no—
182#127—22no—
183C4×D4—2121no—
184C4×D4—2121no—
185C4×D4—2128no—
186C4×D4—2121no—
187C4×D4—2121no—
188C2×C4⋊C4—531no—
189C2×C4⋊C4—531no—
190C2×C4⋊C4—1062no—
191C2×C22⋊C4—532no—
192C2×C22⋊C4—1061no—
193C2×C22⋊C4—1062no—
194C2×C22⋊C4—1062no—
195C2×C42—531no—
196C2×C42—531no—
197C2×C42—1061no—
198C2×C42—1061no—
199C2×C42C2×#615711yescharacteristic
200C22⋊C8—1064no—
201C22⋊C8—2122no—
202C24—21no—
203C24C22⋊#615712yes—
204C2×D4—28no—
205C2×D4—21no—
206C2×D4—21no—
207C2×D4—21no—
208C2×D4—41no—
209C2×D4—41no—
210C2×D4—41no—
211C2×D4—41no—
212C2×D4—44no—
213C2×D4—44no—
214C2×D4#1171994012yescomplemented
215C22×C4—21no—
216C22×C4—531no—
217C22×C4—531no—
218C22×C4—532no—
219C22×C4—1062no—
220C22×C4—1062no—
221C22×C4—1061no—
222C22×C4—1061no—
223C22×C4—2121no—
224C22×C4—2121no—
225C22×C4—2124no—
226C22×C4—2124no—
227C22×C4C22×#615711yescharacteristic
228M16—4244no—
229C2×C8—2124no—
230C4⋊C4—2121no—
231C4⋊C4—2121no—
232C4⋊C4—2124no—
233C22⋊C4—2122no—
234C22⋊C4—2124no—
235C22⋊C4—2124no—
236C22⋊C4—2122no—
237C42—21no—
238C42—1061no—
239C42—1061no—
240C42—1064no—
241C42—2121no—
242C42—2122no—
243C42#1171994012yescomplemented
244C23—22no—
245C23—22no—
246C23—22no—
247C23—22no—
248C23—42no—
249C23—42no—
250C23—42no—
251C23—42no—
252C23—41no—
253C23—41no—
254C23C2×C22⋊#615711yescharacteristic
255D4—42no—
256D4—416no—
257D4—42no—
258D4—42no—
259D4—42no—
260C2×C4—22no—
261C2×C4—22no—
262C2×C4—22no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—531no—
266C2×C4—531no—
267C2×C4—1062no—
268C2×C4—1062no—
269C2×C4—1061no—
270C2×C4—1062no—
271C2×C4—4242no—
272C2×C4—4242no—
273C2×C4—4242no—
274C2×C4—2124no—
275C2×C4—2122no—
276C2×C4—2121no—
277C2×C4—2121no—
278C2×C4—2121no—
279C2×C4—2121no—
280C2×C4—2122no—
281C2×C4C2×#1171994011yescharacteristic
282C2×C4C2×#1171994011yescharacteristic
283C8—4244no—
284C22—21no—
285C22—21no—
286C22—44no—
287C22—44no—
288C22—42no—
289C22—42no—
290C22—42no—
291C22—42no—
292C22—82no—
293C22—82no—
294C22—81no—
295C22—81no—
296C22#1681159311yescharacteristic
297C22D53⋊#9911yescharacteristic
298C22#1681161011yescharacteristic
299C4—24no—
300C4—21no—
301C4—21no—
302C4—532no—
303C4—1061no—
304C4—4241no—
305C4—2122no—
306C4—2122no—
307C4—2121no—
308C2—21no—
309C2—21no—
310C2—82no—
311C2—82no—
312C2#2028997111yescharacteristic
313C2D53⋊#35311yescharacteristic
314C2#2029100711yescharacteristic
subgroupG/H
centerC22D53⋊#99
commutatorC22×C4×C53C22×C4
FrattiniC22×C4C22×#6157
Fitting, prime radicalC2×#127×C53C4
socleC22×C53C2×#99
cosocleC2×C4⋊Dic53C23
hypercenterC2×#127#6157
hypocenterC53#11529
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23 ◁ C22×D4 ◁ C2×#127

Complements

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

NH ≅ G/Ntype
1. #20289971C2semidirect
2. #20289657C2semidirect
3. #20289643C2semidirect
56. C2×D4×C53M16semidirect
73. C42×C53M16semidirect
174. C53#11529semidirect
214. C2×D4#11719940semidirect
243. C42#11719940semidirect

Maximal subgroups

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

subgroupindex
#202899712
#202896572
#202896432
#127×Dic532
#1152953