Finite Group Databasebeta

← #28299703

Subgroups — #28299703

Order 13568.

646 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 646 conjugacy classes of subgroups — too many to draw. Only its 118 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#20290570C212yesmaximal
2#20289536C212yesmaximal, complemented
3#20289535C212yesmaximal, complemented
4#391×D53C211yesmaximal, characteristic
5C2×C4⋊#88324—22no—
6C2×#13133274—22no—
7#16811489C2214yes—
8C22×Q8×D53C411yescharacteristic
9C2×C4⋊C4×D53C411yescharacteristic
10C2×C42×D53C2211yescharacteristic
11#16810530C2211yescharacteristic
12#16810529C411yescharacteristic
13#16810528C2211yescharacteristic
14#16810527—22no—
15#95×D53—22no—
16C53×#391C411yescharacteristic
17C22×#88324—24no—
18C4⋊#88324—44no—
19#13133274—44no—
20C22×Dic106—21no—
21C22×Dic106C2×C411yescharacteristic
22C2×Q8×D53—22no—
23C2×Q8×D53—22no—
24C2×Q8×D53—42no—
25C2×Q8×D53—42no—
26C22×C4×D53—21no—
27C22×C4×D53—21no—
28C22×C4×D53C2×C411yescharacteristic
29C22×C4×D53D412yes—
30C2×C4⋊Dic53C2×C411yescharacteristic
31C2×#11719919—21no—
32C4⋊C4×D53—42no—
33C2×C4×Dic53D412yes—
34C2×C4×Dic53C2311yescharacteristic
35C42×D53—42no—
36#13133124—22no—
37#13133124—22no—
38#13133124—22no—
39C22×Q8×C53C2×C411yescharacteristic
40C2×C4⋊C4×C53C2×C411yescharacteristic
41C2×C42×C53C2×C411yescharacteristic
42#95×C53—22no—
43C23×D53C22⋊C411yescharacteristic
44C2×Q8×C53—21no—
45C2×Q8×C53—41no—
46C2×Q8×C53—41no—
47C2×Q8×C53M1612yes—
48C22×C4×C53—21no—
49C22×C4×C53—21no—
50C22×C4×C53C22⋊C412yes—
51C22×C4×C53C22×C411yescharacteristic
52C22×Dic53—21no—
53C22×Dic53—21no—
54C22×Dic53C22⋊C411yescharacteristic
55C22×Dic53C2×D412yes—
56Q8×D53—48no—
57Q8×D53—48no—
58C2×C4×D53—22no—
59C2×C4×D53—21no—
60C2×C4×D53—21no—
61C2×C4×D53—22no—
62C2×C4×D53—24no—
63C2×C4×D53—42no—
64C2×C4×D53—42no—
65C2×C4×D53—42no—
66C2×C4×D53—42no—
67C2×C4×D53—44no—
68C2×C4×D53—42no—
69C2×C4×D53SD1614yes—
70C2×C4×D53Q1614yes—
71C2×Dic106—24no—
72C2×Dic106—21no—
73C2×Dic106—42no—
74C2×Dic106—42no—
75C2×Dic106—41no—
76C2×Dic106—41no—
77C2×Dic106M1612yes—
78C2×#88324—44no—
79C2×#88324—44no—
80C2×#88324—44no—
81C4⋊C4×C53—41no—
82C42×C53—41no—
83C4⋊Dic53—41no—
84#11719919—42no—
85C4×Dic53—24no—
86C4×Dic53—41no—
87C23×C53C2×C22⋊C411yescharacteristic
88C22×D53—21no—
89C22×D53—21no—
90C22×D53—22no—
91C22×D53#10012yes—
92C22×D53#10314yes—
93Q8×C53—42no—
94Q8×C53—42no—
95C2×C4×C53—21no—
96C2×C4×C53—22no—
97C2×C4×C53—21no—
98C2×C4×C53—22no—
99C2×C4×C53—22no—
100C2×C4×C53—41no—
101C2×C4×C53—41no—
102C2×C4×C53—41no—
103C2×C4×C53—42no—
104C2×C4×C53#10314yes—
105C2×C4×C53C2×M1611yescharacteristic
106C2×C4×C53C2×M1611yescharacteristic
107C2×Dic53—21no—
108C2×Dic53—22no—
109C2×Dic53—21no—
110C2×Dic53—22no—
111C2×Dic53—22no—
112C2×Dic53—41no—
113C2×Dic53—41no—
114C2×Dic53—41no—
115C2×Dic53—42no—
116C2×Dic53#9912yes—
117C2×Dic53C2×SD1612yes—
118C2×Dic53C2×Q1612yes—
119C4×D53—24no—
120C4×D53—24no—
121C4×D53—44no—
122C4×D53—44no—
123C4×D53—48no—
124C4×D53—84no—
125Dic106—48no—
126Dic106—42no—
127Dic106—42no—
128#88324—84no—
129#7643—531nomaximal, Sylow
130C22×C53#40711yescharacteristic
131C22×C53C2×#9911yescharacteristic
132C22×C53C2×#10011yescharacteristic
133C22×C53C2×#10312yes—
134C22×C53#42112yes—
135C2×D53—22no—
136C2×D53—24no—
137C2×D53—24no—
138C2×D53—22no—
139C2×D53#33314yes—
140C4×C53—21no—
141C4×C53—21no—
142C4×C53—24no—
143C4×C53—42no—
144C4×C53—81no—
145Dic53—22no—
146Dic53—22no—
147Dic53—22no—
148Dic53—42no—
149Dic53—81no—
150C2×#391—531no—
151#1797—532no—
152#1272—532no—
153#1272—532no—
154C2×C53#146211yescharacteristic
155C2×C53#148712yes—
156C2×C53#152012yes—
157C2×C53#153311yescharacteristic
158C2×C53C2×#33311yescharacteristic
159D53—24no—
160C23×Q8—531no—
161C22×C4⋊C4—531no—
162C22×C42—531no—
163C2×C4⋊C8—1062no—
164C2×C22⋊C8—1062no—
165#391—531no—
166#391—531no—
167#391—531no—
168#391—1062no—
169#391#615711yescharacteristic
170C2×#95—1062no—
171#336—534no—
172C53#764311yescharacteristic, complemented, Sylow
173C22×Q8—531no—
174C22×Q8—1061no—
175C22×Q8—1062no—
176C22×Q8—1062no—
177C22×Q8—2122no—
178C22×Q8—2122no—
179C22×Q8C2×#615711yescharacteristic
180C23×C4—532no—
181C23×C4—531no—
182C23×C4—1061no—
183C23×C4—1061no—
184C22×C8—1064no—
185C2×C4⋊C4—531no—
186C2×C4⋊C4—1061no—
187C2×C4⋊C4—2122no—
188C2×C4⋊C4C2×#615711yescharacteristic
189C2×C42—532no—
190C2×C42—531no—
191C2×C42—2122no—
192C2×C42C2×#615711yescharacteristic
193C4⋊C8—2124no—
194C22⋊C8—2124no—
195#95—22no—
196#95—1062no—
197#95—1062no—
198#95—1062no—
199C24—531no—
200C2×Q8—21no—
201C2×Q8—41no—
202C2×Q8—41no—
203C2×Q8—532no—
204C2×Q8—1064no—
205C2×Q8—1061no—
206C2×Q8—2121no—
207C2×Q8—2121no—
208C2×Q8—2122no—
209C2×Q8—2122no—
210C2×Q8—2128no—
211C2×Q8—2128no—
212C2×Q8#1171993612yes—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—531no—
216C22×C4—532no—
217C22×C4—534no—
218C22×C4—534no—
219C22×C4—1062no—
220C22×C4—1062no—
221C22×C4—1064no—
222C22×C4—1061no—
223C22×C4—1061no—
224C22×C4—1061no—
225C22×C4—1061no—
226C22×C4—2122no—
227C22×C4—2122no—
228C22×C4—2122no—
229C22×C4—2124no—
230C22×C4—2122no—
231C22×C4—2122no—
232C22×C4C22⋊#615712yes—
233C22×C4C22×#615711yescharacteristic
234C2×C8—2124no—
235C2×C8—2124no—
236C2×C8—2124no—
237C4⋊C4—41no—
238C4⋊C4—2121no—
239C4⋊C4—2122no—
240C42—41no—
241C42—1064no—
242C42—2121no—
243C23—532no—
244C23—534no—
245C23—1061no—
246C23—1061no—
247C23—1062no—
248C23C2×C22⋊#615711yescharacteristic
249Q8—42no—
250Q8—42no—
251Q8—2122no—
252Q8—2122no—
253Q8—2128no—
254C2×C4—22no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—22no—
258C2×C4—22no—
259C2×C4—41no—
260C2×C4—41no—
261C2×C4—41no—
262C2×C4—42no—
263C2×C4—532no—
264C2×C4—532no—
265C2×C4—532no—
266C2×C4—1064no—
267C2×C4—1064no—
268C2×C4—1061no—
269C2×C4—1062no—
270C2×C4—1061no—
271C2×C4—1062no—
272C2×C4—1062no—
273C2×C4—4244no—
274C2×C4—2124no—
275C2×C4—2124no—
276C2×C4—2128no—
277C2×C4—2121no—
278C2×C4—2122no—
279C2×C4—2121no—
280C2×C4—2121no—
281C2×C4#1313328014yes—
282C2×C4D53⋊M1611yescharacteristic
283C2×C4C2×#1171993611yescharacteristic
284C8—4244no—
285C22—534no—
286C22—1064no—
287C22—1062no—
288C22—1062no—
289C22—1064no—
290C22#1681159411yescharacteristic
291C22D53⋊#9911yescharacteristic
292C22D53⋊#10011yescharacteristic
293C22C2×#1313328012yes—
294C22#1681162012yes—
295C4—24no—
296C4—21no—
297C4—21no—
298C4—42no—
299C4—81no—
300C4—1062no—
301C4—1062no—
302C4—1062no—
303C4—4241no—
304C4—2122no—
305C2—1064no—
306C2#2028996111yescharacteristic
307C2#2029004412yes—
308C2#2029015312yes—
309C2#2029019511yescharacteristic
310C2D53⋊#33311yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6157
commutatorC22×C4×C53C22×C4
FrattiniC22×C4C22×#6157
FittingC53×#391C4
socleC23×C53C2×C22⋊C4
prime radicalC23×D53C22⋊C4
cosocleC2×C4×Dic53C23
hypercenter#391#6157
hypocenterC53#7643
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #391

Complements

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

NH ≅ G/Ntype
2. #20289536C2semidirect
3. #20289535C2semidirect
172. C53#7643semidirect

Maximal subgroups

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

subgroupindex
#202905702
#202895362
#202895352
#391×D532
#764353