Finite Group Databasebeta

← #28307895

Subgroups — #28307895

Order 13568.

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

normal subgroups only

This group has 678 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
1#20289946C212yesmaximal, complemented
2#20289641C212yesmaximal, complemented
3#20289639C212yesmaximal
4#20277098C211yesmaximal, characteristic
5#16811639—22no—
6#16811468—22no—
7#16811466C2214yes—
8C2×#13133176C411yescharacteristic
9C2×#13133176C2211yescharacteristic
10#16810486C411yescharacteristic
11#16810486C2211yescharacteristic
12#16810483—22no—
13#16810479C411yescharacteristic
14#16810479C2211yescharacteristic
15C4×C22⋊Dic53—22no—
16C24×C4×C53C411yescharacteristic
17C4⋊#88324—42no—
18C22⋊#88324—24no—
19C4×#88324—42no—
20#13133176—41no—
21#13133176—41no—
22#13133176—42no—
23C2×C4⋊Dic53—21no—
24C2×C4⋊Dic53D412yes—
25C2×C22⋊Dic53—22no—
26C2×C22⋊Dic53—22no—
27C2×C22⋊Dic53C2×C411yescharacteristic
28C2×C22⋊Dic53C2×C411yescharacteristic
29C2×C22⋊Dic53C2×C411yescharacteristic
30C2×C22⋊Dic53C2311yescharacteristic
31C2×C4×Dic53—21no—
32C2×C4×Dic53D412yes—
33#13133124—22no—
34#13133124—22no—
35C25×C53C2×C411yescharacteristic
36C23×C4×C53—21no—
37C23×C4×C53—21no—
38C23×C4×C53—22no—
39C23×C4×C53—41no—
40C23×C4×C53—41no—
41C23×C4×C53—42no—
42C23×C4×C53C814yescomplemented
43C23×C4×C53C2×C411yescharacteristic
44C23×C4×C53C2×C411yescharacteristic
45C24×C53—21no—
46C24×C53—21no—
47C24×C53—41no—
48C24×C53—41no—
49C24×C53—44no—
50C24×C53C2×C812yes—
51C24×C53C22×C411yescharacteristic
52C22×C4×C53—21no—
53C22×C4×C53—21no—
54C22×C4×C53—22no—
55C22×C4×C53—21no—
56C22×C4×C53—21no—
57C22×C4×C53—24no—
58C22×C4×C53—24no—
59C22×C4×C53—42no—
60C22×C4×C53—42no—
61C22×C4×C53—44no—
62C22×C4×C53—44no—
63C22×C4×C53—44no—
64C22×C4×C53—44no—
65C22×C4×C53—41no—
66C22×C4×C53—41no—
67C22×C4×C53—42no—
68C22×C4×C53—41no—
69C22×C4×C53—41no—
70C22×C4×C53C22⋊C411yescharacteristic
71C22×C4×C53C22⋊C411yescharacteristic
72C22×C4×C53C2×C812yes—
73C22×C4×C53C2×C812yes—
74C22×C4×C53M1612yes—
75C22×Dic53—21no—
76C22×Dic53—21no—
77C22×Dic53C22⋊C412yes—
78C22×Dic53C2×D412yes—
79C2×#88324—44no—
80C22⋊Dic53—41no—
81C22⋊Dic53—41no—
82C22⋊Dic53—44no—
83C22⋊Dic53—44no—
84C22⋊Dic53—41no—
85C22⋊Dic53—41no—
86C4⋊Dic53—42no—
87C4⋊Dic53—42no—
88C4×Dic53—22no—
89C4×Dic53—22no—
90C4×Dic53—42no—
91C23×C53—24no—
92C23×C53—22no—
93C23×C53—21no—
94C23×C53—21no—
95C23×C53—42no—
96C23×C53—44no—
97C23×C53—42no—
98C23×C53—42no—
99C23×C53—42no—
100C23×C53—44no—
101C23×C53—44no—
102C23×C53—48no—
103C23×C53—44no—
104C23×C53—41no—
105C23×C53—41no—
106C23×C53C2×C22⋊C411yescharacteristic
107C23×C53C22×C811yescharacteristic
108C23×C53C2×M1611yescharacteristic
109C2×C4×C53—21no—
110C2×C4×C53—21no—
111C2×C4×C53—22no—
112C2×C4×C53—22no—
113C2×C4×C53—22no—
114C2×C4×C53—22no—
115C2×C4×C53—22no—
116C2×C4×C53—48no—
117C2×C4×C53—42no—
118C2×C4×C53—41no—
119C2×C4×C53—41no—
120C2×C4×C53—44no—
121C2×C4×C53—44no—
122C2×C4×C53—42no—
123C2×C4×C53C22⋊C812yes—
124C2×C4×C53C22⋊C812yes—
125C2×C4×C53#9911yescharacteristic
126C2×C4×C53#9911yescharacteristic
127C2×C4×C53#10012yes—
128C2×Dic53—22no—
129C2×Dic53—22no—
130C2×Dic53—42no—
131C2×Dic53—42no—
132C2×Dic53—42no—
133C2×Dic53—42no—
134C2×Dic53—41no—
135C2×Dic53—41no—
136#88324—84no—
137#11425—531nomaximal, Sylow
138C22×C53—22no—
139C22×C53—24no—
140C22×C53—21no—
141C22×C53—21no—
142C22×C53—41no—
143C22×C53—41no—
144C22×C53—44no—
145C22×C53—42no—
146C22×C53—42no—
147C22×C53—44no—
148C22×C53—44no—
149C22×C53—48no—
150C22×C53—44no—
151C22×C53—42no—
152C22×C53—42no—
153C22×C53C2×C22⋊C811yescharacteristic
154C22×C53C2×#9911yescharacteristic
155C22×C53C2×#10011yescharacteristic
156C4×C53—22no—
157C4×C53—42no—
158C4×C53#32312yes—
159C4×C53#32312yes—
160Dic53—42no—
161Dic53—42no—
162Dic53—81no—
163Dic53—81no—
164#2280—531no—
165#1458—532no—
166#1317—532no—
167#1316—532no—
168C2×C53—21no—
169C2×C53—21no—
170C2×C53—44no—
171C2×C53—41no—
172C2×C53—41no—
173C2×C53C2×#32311yescharacteristic
174C2×C53#212012yes—
175C24×C4#615711yescharacteristic
176C2×#117—531no—
177C2×#117—531no—
178#431—1062no—
179#381—531no—
180#381—531no—
181#380—1062no—
182#379—531no—
183#379—531no—
184C4×C22⋊C4—1062no—
185#324—1062no—
186#323—534no—
187C53#1142511yescharacteristic, complemented, Sylow
188C25C2×#615711yescharacteristic
189C23×C4—22no—
190C23×C4—21no—
191C23×C4—21no—
192C23×C4—41no—
193C23×C4—41no—
194C23×C4—42no—
195C23×C4#8832414yescomplemented
196C23×C4C2×#615711yescharacteristic
197C23×C4C2×#615711yescharacteristic
198#117—2121no—
199#117—2121no—
200#117—2122no—
201C2×C4⋊C4—532no—
202C2×C4⋊C4—1061no—
203C2×C22⋊C4—531no—
204C2×C22⋊C4—531no—
205C2×C22⋊C4—531no—
206C2×C22⋊C4—531no—
207C2×C22⋊C4—1062no—
208C2×C22⋊C4—1062no—
209C2×C42—532no—
210C2×C42—1061no—
211C4⋊C8—2122no—
212C22⋊C8—1064no—
213C4×C8—2122no—
214#95—1062no—
215#95—1062no—
216C24—21no—
217C24—21no—
218C24—41no—
219C24—41no—
220C24—44no—
221C24C2×#8832412yes—
222C24C22×#615711yescharacteristic
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—22no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—24no—
229C22×C4—24no—
230C22×C4—42no—
231C22×C4—42no—
232C22×C4—44no—
233C22×C4—44no—
234C22×C4—44no—
235C22×C4—44no—
236C22×C4—41no—
237C22×C4—41no—
238C22×C4—42no—
239C22×C4—41no—
240C22×C4—41no—
241C22×C4—532no—
242C22×C4—532no—
243C22×C4—1061no—
244C22×C4—1061no—
245C22×C4C2×#8832412yes—
246C22×C4C2×#8832412yes—
247C22×C4#1171994012yes—
248C22×C4C22⋊#615711yescharacteristic
249C22×C4C22⋊#615711yescharacteristic
250C2×C8—2124no—
251C4⋊C4—2122no—
252C4⋊C4—2122no—
253C22⋊C4—2121no—
254C22⋊C4—2121no—
255C22⋊C4—2124no—
256C22⋊C4—2124no—
257C22⋊C4—2121no—
258C22⋊C4—2121no—
259C42—1062no—
260C42—1062no—
261C42—2122no—
262C23—24no—
263C23—22no—
264C23—21no—
265C23—21no—
266C23—42no—
267C23—44no—
268C23—42no—
269C23—42no—
270C23—42no—
271C23—44no—
272C23—44no—
273C23—48no—
274C23—44no—
275C23—41no—
276C23—41no—
277C23C2×C22⋊#615711yescharacteristic
278C23C22×#8832411yescharacteristic
279C23C2×#1171994011yescharacteristic
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—22no—
283C2×C4—22no—
284C2×C4—22no—
285C2×C4—22no—
286C2×C4—22no—
287C2×C4—48no—
288C2×C4—42no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—44no—
292C2×C4—44no—
293C2×C4—42no—
294C2×C4—1062no—
295C2×C4—1062no—
296C2×C4—2122no—
297C2×C4—2122no—
298C2×C4—2121no—
299C2×C4—2121no—
300C2×C4—2122no—
301C2×C4—2122no—
302C2×C4C22⋊#8832412yes—
303C2×C4C22⋊#8832412yes—
304C2×C4#1313327511yescharacteristic
305C2×C4#1313327511yescharacteristic
306C2×C4#1313327712yes—
307C8—4244no—
308C22—22no—
309C22—24no—
310C22—21no—
311C22—21no—
312C22—41no—
313C22—41no—
314C22—44no—
315C22—42no—
316C22—42no—
317C22—44no—
318C22—44no—
319C22—48no—
320C22—42no—
321C22—44no—
322C22—42no—
323C22C2×C22⋊#8832411yescharacteristic
324C22C2×#1313327511yescharacteristic
325C22C2×#1313327711yescharacteristic
326C4—22no—
327C4—42no—
328C4—4241no—
329C4—4241no—
330C4—2122no—
331C4—2122no—
332C4#1681146612yes—
333C4#1681146612yes—
334C2—21no—
335C2—21no—
336C2—44no—
337C2—41no—
338C2—41no—
339C2C2×#1681146611yescharacteristic
340C2#2029096612yes—
subgroupG/H
centerC2×C4#13133275
commutatorC23×C53C22×C8
FrattiniC24C22×#6157
FittingC24×C4×C53C4
socleC22×C53C2×#99
prime radicalC25×C53C2×C4
cosocleC2×C22⋊Dic53C23
hypercenterC24×C4#6157
hypocenterC53#11425
LCSG ▷ C23×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C23×C4 ◁ C24×C4

Complements

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

NH ≅ G/Ntype
1. #20289946C2semidirect
2. #20289641C2semidirect
42. C23×C4×C53C8semidirect
187. C53#11425semidirect
195. C23×C4#88324semidirect

Maximal subgroups

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

subgroupindex
#202899462
#202896412
#202896392
#202770982
#1142553