Finite Group Databasebeta

← #28307919

Subgroups — #28307919

Order 13568.

594 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 594 conjugacy classes of subgroups — too many to draw. Only its 112 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×#16811468C212yesmaximal, complemented
2#20289645C212yesmaximal, complemented
3#20289643C212yesmaximal, complemented
4#20277307C211yesmaximal, characteristic
5C2×C22⋊#88324—22no—
6#16811468—22no—
7#16811468C2214yes—
8C2×#13133176C411yescharacteristic
9C22×C4⋊Dic53C2211yescharacteristic
10#16810530C411yescharacteristic
11#16810530C2211yescharacteristic
12#16810526C411yescharacteristic
13#16810526C2211yescharacteristic
14#16810503—22no—
15#16810483—22no—
16C22×#41×C53C411yescharacteristic
17C22×#88324—42no—
18C22⋊#88324—24no—
19C22⋊#88324—42no—
20C23×Dic53D412yes—
21#13133176—42no—
22C2×C4⋊Dic53—21no—
23C2×C4⋊Dic53—42no—
24C2×C4⋊Dic53C2×C411yescharacteristic
25C2×C4⋊Dic53D412yes—
26C2×C4⋊Dic53C2311yescharacteristic
27C2×C22⋊Dic53—21no—
28C2×C22⋊Dic53—22no—
29C2×C22⋊Dic53—22no—
30C2×C4×Dic53C2×C411yescharacteristic
31C2×C4×Dic53C2×C411yescharacteristic
32#13133124—22no—
33#13133124—22no—
34C2×#41×C53—22no—
35C2×#41×C53—42no—
36C2×#41×C53—42no—
37C2×#41×C53C814yescomplemented
38C22×Q8×C53C2×C411yescharacteristic
39C22×D4×C53—21no—
40C22×D4×C53C2×C411yescharacteristic
41C23×C4×C53—21no—
42C23×C4×C53C2×C411yescharacteristic
43C24×C53—21no—
44C24×C53C22⋊C411yescharacteristic
45#41×C53—48no—
46#41×C53—48no—
47C2×Q8×C53—21no—
48C2×Q8×C53—41no—
49C2×Q8×C53—41no—
50C2×Q8×C53C2×C812yescomplemented
51C2×D4×C53—24no—
52C2×D4×C53—21no—
53C2×D4×C53—41no—
54C2×D4×C53—41no—
55C2×D4×C53—42no—
56C2×D4×C53—42no—
57C2×D4×C53C2×C812yescomplemented
58C22×C4×C53—22no—
59C22×C4×C53—22no—
60C22×C4×C53—21no—
61C22×C4×C53—24no—
62C22×C4×C53—42no—
63C22×C4×C53—42no—
64C22×C4×C53C22⋊C411yescharacteristic
65C22×C4×C53C2×C812yes—
66C22×C4×C53M1612yes—
67C22×C4×C53C22×C411yescharacteristic
68C22×Dic53—22no—
69C22×Dic53—22no—
70C22×Dic53—21no—
71C22×Dic53—21no—
72C22×Dic53—82no—
73C22×Dic53C22⋊C412yes—
74C22×Dic53C2×D412yes—
75C2×#88324—44no—
76C2×#88324—84no—
77C22⋊Dic53—42no—
78C22⋊Dic53—82no—
79C22⋊Dic53—82no—
80C4⋊Dic53—41no—
81C4⋊Dic53—42no—
82C4⋊Dic53—41no—
83C4⋊Dic53—42no—
84C4×Dic53—41no—
85C4×Dic53—41no—
86C23×C53—22no—
87C23×C53—41no—
88C23×C53—41no—
89C23×C53—42no—
90C23×C53—42no—
91C23×C53—41no—
92C23×C53—41no—
93C23×C53C22⋊C812yes—
94C23×C53#9912yes—
95C23×C53#10012yes—
96C23×C53C2×C22⋊C411yescharacteristic
97Q8×C53—42no—
98Q8×C53—42no—
99D4×C53—42no—
100D4×C53—42no—
101D4×C53—48no—
102C2×C4×C53—24no—
103C2×C4×C53—24no—
104C2×C4×C53—21no—
105C2×C4×C53—21no—
106C2×C4×C53—21no—
107C2×C4×C53—21no—
108C2×C4×C53—22no—
109C2×C4×C53—48no—
110C2×C4×C53—41no—
111C2×C4×C53—41no—
112C2×C4×C53C22⋊C812yes—
113C2×C4×C53C22×C811yescharacteristic
114C2×C4×C53C2×M1611yescharacteristic
115C2×Dic53—22no—
116C2×Dic53—22no—
117C2×Dic53—44no—
118C2×Dic53—42no—
119C2×Dic53—42no—
120C2×Dic53—41no—
121C2×Dic53—41no—
122C2×Dic53—41no—
123C2×Dic53—41no—
124C2×Dic53—41no—
125C2×Dic53—41no—
126C2×Dic53—82no—
127#88324—84no—
128#11432—531nomaximal, Sylow
129C22×C53—21no—
130C22×C53—22no—
131C22×C53—21no—
132C22×C53—22no—
133C22×C53—42no—
134C22×C53—42no—
135C22×C53—44no—
136C22×C53—42no—
137C22×C53—82no—
138C22×C53—81no—
139C22×C53—81no—
140C22×C53#32314yes—
141C22×C53C2×C22⋊C811yescharacteristic
142C22×C53C2×#9911yescharacteristic
143C22×C53C2×#10011yescharacteristic
144C4×C53—22no—
145C4×C53—21no—
146C4×C53—21no—
147C4×C53—42no—
148Dic53—42no—
149Dic53—42no—
150Dic53—81no—
151Dic53—81no—
152#2311—531no—
153C2×#324—532no—
154#1319—532no—
155#1318—532no—
156C2×C53—21no—
157C2×C53—21no—
158C2×C53—24no—
159C2×C53—82no—
160C2×C53C2×#32311yescharacteristic
161C2×C53#212711yescharacteristic
162C2×C53#213311yescharacteristic
163C22×#41#615711yescharacteristic
164C2×#117—531no—
165C22×C4⋊C4—531no—
166C2×C22⋊C8—1062no—
167#391—531no—
168#391—531no—
169#390—531no—
170#390—531no—
171#386—1062no—
172#380—1062no—
173#324—534no—
174#324—1062no—
175C53#1143211yescharacteristic, complemented, Sylow
176C2×#41—22no—
177C2×#41—42no—
178C2×#41—42no—
179C2×#41#8832414yescomplemented
180C22×Q8C2×#615711yescharacteristic
181C22×D4—21no—
182C22×D4C2×#615711yescharacteristic
183C23×C4—21no—
184C23×C4—532no—
185C23×C4C2×#615711yescharacteristic
186C22×C8—2122no—
187#117—2122no—
188C2×C4⋊C4—532no—
189C2×C4⋊C4—531no—
190C2×C4⋊C4—531no—
191C2×C4⋊C4—1061no—
192C2×C4⋊C4—2122no—
193C2×C22⋊C4—1062no—
194C2×C22⋊C4—1062no—
195C2×C22⋊C4—1061no—
196C2×C42—531no—
197C2×C42—531no—
198C22⋊C8—1064no—
199C22⋊C8—2122no—
200#95—1062no—
201#95—1062no—
202C24—21no—
203C24C22⋊#615711yescharacteristic
204#41—48no—
205#41—48no—
206C2×Q8—21no—
207C2×Q8—41no—
208C2×Q8—41no—
209C2×Q8C2×#8832412yescomplemented
210C2×D4—24no—
211C2×D4—21no—
212C2×D4—41no—
213C2×D4—41no—
214C2×D4—42no—
215C2×D4—42no—
216C2×D4C2×#8832412yescomplemented
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—21no—
220C22×C4—24no—
221C22×C4—42no—
222C22×C4—42no—
223C22×C4—532no—
224C22×C4—532no—
225C22×C4—1062no—
226C22×C4—1062no—
227C22×C4—1061no—
228C22×C4—1061no—
229C22×C4—4242no—
230C22×C4C2×#8832412yes—
231C22×C4#1171994012yes—
232C22×C4C22⋊#615711yescharacteristic
233C22×C4C22×#615711yescharacteristic
234C2×C8—4244no—
235C2×C8—2124no—
236C4⋊C4—2122no—
237C4⋊C4—2121no—
238C4⋊C4—2121no—
239C4⋊C4—2122no—
240C22⋊C4—4242no—
241C22⋊C4—4242no—
242C22⋊C4—2122no—
243C42—2121no—
244C42—2121no—
245C23—22no—
246C23—41no—
247C23—41no—
248C23—42no—
249C23—42no—
250C23—41no—
251C23—41no—
252C23C22⋊#8832412yes—
253C23#1313327512yes—
254C23#1313327712yes—
255C23C2×C22⋊#615711yescharacteristic
256Q8—42no—
257Q8—42no—
258D4—42no—
259D4—42no—
260D4—48no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—24no—
266C2×C4—24no—
267C2×C4—22no—
268C2×C4—48no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—1062no—
272C2×C4—1062no—
273C2×C4—4242no—
274C2×C4—2121no—
275C2×C4—2121no—
276C2×C4—2121no—
277C2×C4—2121no—
278C2×C4—2121no—
279C2×C4—2121no—
280C2×C4—2124no—
281C2×C4—2122no—
282C2×C4—2122no—
283C2×C4C22⋊#8832412yes—
284C2×C4C22×#8832411yescharacteristic
285C2×C4C2×#1171994011yescharacteristic
286C8—4244no—
287C22—22no—
288C22—21no—
289C22—21no—
290C22—22no—
291C22—42no—
292C22—42no—
293C22—44no—
294C22—42no—
295C22—82no—
296C22—81no—
297C22—81no—
298C22#1681146614yes—
299C22C2×C22⋊#8832411yescharacteristic
300C22C2×#1313327511yescharacteristic
301C22C2×#1313327711yescharacteristic
302C4—22no—
303C4—21no—
304C4—21no—
305C4—42no—
306C4—4241no—
307C4—4241no—
308C4—2122no—
309C4—2122no—
310C2—21no—
311C2—21no—
312C2—24no—
313C2—82no—
314C2C2×#1681146611yescharacteristic
315C2#2029099011yescharacteristic
316C2#2029101011yescharacteristic
subgroupG/H
centerC22C2×#13133275
commutatorC2×C4×C53C22×C8
FrattiniC22×C4C22×#6157
Fitting, prime radicalC22×#41×C53C4
socleC22×C53C2×#99
cosocleC2×C4⋊Dic53C23
hypercenterC22×#41#6157
hypocenterC53#11432
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C24 ◁ C23×C4 ◁ C22×#41

Complements

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

NH ≅ G/Ntype
1. C2×#16811468C2semidirect
2. #20289645C2semidirect
3. #20289643C2semidirect
37. C2×#41×C53C8semidirect
50. C2×Q8×C53C2×C8semidirect
57. C2×D4×C53C2×C8semidirect
175. C53#11432semidirect
179. C2×#41#88324semidirect
209. C2×Q8C2×#88324semidirect
216. C2×D4C2×#88324semidirect

Maximal subgroups

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

subgroupindex
C2×#168114682
#202896452
#202896432
#202773072
#1143253