Finite Group Databasebeta

← #28299424

Subgroups — #28299424

Order 13568.

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

normal subgroups only

This group has 690 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
1C4×#13133274C212yesmaximal, complemented
2#20289535C212yesmaximal, complemented
3#20289532C212yesmaximal, complemented
4#20277204C211yesmaximal, characteristic
5C2×#13133274—22no—
6C2×C4×#88324—22no—
7#16811489C2214yes—
8C22×#11719949C411yescharacteristic
9C2×#13133175C411yescharacteristic
10C2×C42×D53C2211yescharacteristic
11#16810530C2211yescharacteristic
12#16810525C411yescharacteristic
13#16810524C2211yescharacteristic
14#16810501—22no—
15#16810470—22no—
16C53×#391C411yescharacteristic
17C22×#88324—24no—
18#13133274—28no—
19C4×#88324—44no—
20C2×#11719949—22no—
21C2×#11719949—42no—
22C2×#11719949—42no—
23C2×#11719949C814yescomplemented
24C22×D212—21no—
25C22×D212C2×C411yescharacteristic
26C22×C4×D53—21no—
27C22×C4×D53C2×C411yescharacteristic
28C22×C4×D53D412yes—
29#13133175—42no—
30C2×C4⋊Dic53C2×C411yescharacteristic
31C2×#11719921—21no—
32C2×#11719921—22no—
33C2×#11719921—22no—
34C2×C4×Dic53—21no—
35C2×C4×Dic53D412yes—
36C2×C4×Dic53C2311yescharacteristic
37C42×D53—42no—
38#13133124—22no—
39C22×Q8×C53C2×C411yescharacteristic
40C2×C4⋊C4×C53C2×C411yescharacteristic
41C2×C42×C53C2×C411yescharacteristic
42#95×C53—22no—
43C23×D53—21no—
44C23×D53C22⋊C411yescharacteristic
45C2×Q8×C53—21no—
46C2×Q8×C53—41no—
47C2×Q8×C53—41no—
48C2×Q8×C53C2×C812yes—
49C22×C4×C53—21no—
50C22×C4×C53—21no—
51C22×C4×C53C22⋊C412yes—
52C22×C4×C53C22×C411yescharacteristic
53C22×Dic53—21no—
54C22×Dic53C22⋊C411yescharacteristic
55C22×Dic53C2×D412yes—
56#11719949—48no—
57#11719949—48no—
58C2×D212—24no—
59C2×D212—21no—
60C2×D212—41no—
61C2×D212—41no—
62C2×D212—42no—
63C2×D212—42no—
64C2×D212C2×C812yes—
65C2×C4×D53—22no—
66C2×C4×D53—22no—
67C2×C4×D53—24no—
68C2×C4×D53—24no—
69C2×C4×D53—24no—
70C2×C4×D53—42no—
71C2×C4×D53—42no—
72C2×C4×D53—42no—
73C2×C4×D53C2×C812yes—
74C2×C4×D53M1612yes—
75C2×#88324—28no—
76C2×#88324—44no—
77C2×#88324—44no—
78C4⋊C4×C53—41no—
79C42×C53—41no—
80#11719921—42no—
81#11719921—44no—
82#11719921—44no—
83C4⋊Dic53—41no—
84C4×Dic53—24no—
85C4×Dic53—41no—
86C4×Dic53—42no—
87C23×C53C2×C22⋊C411yescharacteristic
88C22×D53—21no—
89C22×D53—21no—
90C22×D53—22no—
91C22×D53—22no—
92C22×D53—22no—
93C22×D53—41no—
94C22×D53—41no—
95C22×D53—42no—
96C22×D53—42no—
97C22×D53C22⋊C812yes—
98Q8×C53—42no—
99Q8×C53—42no—
100C2×C4×C53—21no—
101C2×C4×C53—22no—
102C2×C4×C53—21no—
103C2×C4×C53—22no—
104C2×C4×C53—22no—
105C2×C4×C53—41no—
106C2×C4×C53—41no—
107C2×C4×C53—41no—
108C2×C4×C53—42no—
109C2×C4×C53#10414yes—
110C2×C4×C53C22×C811yescharacteristic
111C2×C4×C53C2×M1611yescharacteristic
112C2×Dic53—22no—
113C2×Dic53—22no—
114C2×Dic53—41no—
115C2×Dic53—42no—
116C2×Dic53C22⋊C812yes—
117C2×Dic53#9912yes—
118C2×Dic53#10012yes—
119C2×Dic53#13514yes—
120D212—42no—
121D212—42no—
122D212—48no—
123C4×D53—24no—
124C4×D53—24no—
125C4×D53—48no—
126C4×D53—48no—
127#88324—48no—
128#7563—531nomaximal, Sylow
129C22×C53C2×C22⋊C811yescharacteristic
130C22×C53C2×#9911yescharacteristic
131C22×C53C2×#10011yescharacteristic
132C22×C53#41612yes—
133C22×C53C2×#10412yes—
134C2×D53—22no—
135C2×D53—24no—
136C2×D53—41no—
137C2×D53—42no—
138C2×D53—41no—
139C2×D53—42no—
140C2×D53—44no—
141C2×D53—81no—
142C2×D53—82no—
143C2×D53—81no—
144C4×C53—21no—
145C4×C53—21no—
146C4×C53—24no—
147C4×C53—42no—
148C4×C53—81no—
149Dic53—24no—
150Dic53—81no—
151Dic53#32314yes—
152#2297—531no—
153C4×C22⋊C8—532no—
154#1272—532no—
155#1270—532no—
156C2×C53C2×#32311yescharacteristic
157C2×C53#147612yes—
158C2×C53#150112yes—
159C2×C53#152511yescharacteristic
160C2×C53#154311yescharacteristic
161D53—42no—
162D53—82no—
163C22×#41—531no—
164C2×#117—531no—
165C22×C42—531no—
166C2×C22⋊C8—1062no—
167C2×C4×C8—1062no—
168#391—531no—
169#391#615711yescharacteristic
170#390—531no—
171#390—531no—
172#386—1062no—
173C4×C22⋊C4—1062no—
174#336—534no—
175C53#756311yescharacteristic, complemented, Sylow
176C2×#41—534no—
177C2×#41—1062no—
178C2×#41—2122no—
179C2×#41—2122no—
180C22×Q8C2×#615711yescharacteristic
181C22×D4—531no—
182C22×D4—1061no—
183C23×C4—531no—
184C23×C4—532no—
185C23×C4—1061no—
186C22×C8—1064no—
187#117—2122no—
188C2×C4⋊C4—531no—
189C2×C4⋊C4C2×#615711yescharacteristic
190C2×C22⋊C4—1062no—
191C2×C22⋊C4—1061no—
192C2×C22⋊C4—1062no—
193C2×C42—532no—
194C2×C42—531no—
195C2×C42—1061no—
196C2×C42—2122no—
197C2×C42C2×#615711yescharacteristic
198C22⋊C8—1068no—
199C4×C8—2124no—
200#95—22no—
201#95—1062no—
202C24—531no—
203C24—1061no—
204#41—2128no—
205#41—2128no—
206C2×Q8—21no—
207C2×Q8—41no—
208C2×Q8—41no—
209C2×Q8D53⋊C812yes—
210C2×D4—532no—
211C2×D4—1064no—
212C2×D4—1061no—
213C2×D4—2121no—
214C2×D4—2121no—
215C2×D4—2122no—
216C2×D4—2122no—
217C22×C4—21no—
218C22×C4—21no—
219C22×C4—531no—
220C22×C4—532no—
221C22×C4—532no—
222C22×C4—532no—
223C22×C4—1062no—
224C22×C4—1062no—
225C22×C4—1064no—
226C22×C4—1064no—
227C22×C4—1064no—
228C22×C4—1061no—
229C22×C4—2122no—
230C22×C4—2122no—
231C22×C4—2122no—
232C22×C4C22⋊#615712yes—
233C22×C4C22×#615711yescharacteristic
234C2×C8—1068no—
235C2×C8—2124no—
236C2×C8—2124no—
237C4⋊C4—41no—
238C4⋊C4—2121no—
239C22⋊C4—2124no—
240C22⋊C4—2122no—
241C22⋊C4—2124no—
242C42—41no—
243C42—1064no—
244C42—2121no—
245C42—2122no—
246C23—532no—
247C23—1061no—
248C23—1061no—
249C23—1062no—
250C23—1062no—
251C23—1062no—
252C23—2121no—
253C23—2121no—
254C23—2122no—
255C23—2122no—
256C23C2×C22⋊#615711yescharacteristic
257Q8—42no—
258Q8—42no—
259D4—2122no—
260D4—2122no—
261D4—2128no—
262C2×C4—22no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—22no—
266C2×C4—22no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—41no—
270C2×C4—42no—
271C2×C4—532no—
272C2×C4—532no—
273C2×C4—532no—
274C2×C4—534no—
275C2×C4—1064no—
276C2×C4—1064no—
277C2×C4—1062no—
278C2×C4—1062no—
279C2×C4—2128no—
280C2×C4—2128no—
281C2×C4—2121no—
282C2×C4—2122no—
283C2×C4#1313328214yes—
284C2×C4C2×D53⋊C811yescharacteristic
285C2×C4D53⋊M1611yescharacteristic
286C8—2128no—
287C22—1062no—
288C22—1064no—
289C22—4242no—
290C22—4241no—
291C22—4241no—
292C22—2122no—
293C22—2124no—
294C22—2122no—
295C22—2121no—
296C22—2121no—
297C22D53⋊(C22⋊C8)11yescharacteristic
298C22D53⋊#9911yescharacteristic
299C22D53⋊#10011yescharacteristic
300C22#1681161312yes—
301C22C2×#1313328212yes—
302C4—24no—
303C4—21no—
304C4—21no—
305C4—42no—
306C4—81no—
307C4—534no—
308C4—1064no—
309C4—4241no—
310C2—4242no—
311C2—2122no—
312C2D53⋊#32311yescharacteristic
313C2#2029000912yes—
314C2#2029009312yes—
315C2#2029017111yescharacteristic
316C2#2029021911yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6157
commutatorC2×C4×C53C22×C8
FrattiniC22×C4C22×#6157
FittingC53×#391C4
socleC23×C53C2×C22⋊C4
prime radicalC22×#11719949C4
cosocleC2×C4×Dic53C23
hypercenter#391#6157
hypocenterC53#7563
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #391

Complements

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

NH ≅ G/Ntype
1. C4×#13133274C2semidirect
2. #20289535C2semidirect
3. #20289532C2semidirect
23. C2×#11719949C8semidirect
175. C53#7563semidirect

Maximal subgroups

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

subgroupindex
C4×#131332742
#202895352
#202895322
#202772042
#756353