Finite Group Databasebeta

← #28303041

Subgroups — #28303041

Order 13568.

478 conjugacy classes of subgroups; only the 128 normal subgroups are shown.

normal subgroups only

This group has 478 conjugacy classes of subgroups — too many to draw. Only its 128 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
2C2×#16811486C211yesmaximal, characteristic, complemented
3C2×#16811485C211yesmaximal, characteristic, complemented
4#20289545C212yesmaximal, complemented
5C2×#16810675C211yesmaximal, characteristic
6C2×C4⋊#88324C2212yes—
7C2×C4⋊#88324C2212yes—
8C2×#13133274—22no—
9#16811489—22no—
10#16811486—22no—
11#16811485—22no—
12C2×C42×D53C2211yescharacteristic
13#16810675—22no—
14#16810675—22no—
15C2×C4⋊#88322C411yescharacteristic
16C2×#13133129C411yescharacteristic
17C2×#13133129C411yescharacteristic
18C2×C8×Dic53C2211yescharacteristic
19C2×C8×Dic53C2211yescharacteristic
20C2×C4⋊C8×C53C411yescharacteristic
21C22×#88324—22no—
22C22×#88324—22no—
23C4⋊#88324—22no—
24C4⋊#88324—22no—
25C4⋊#88324—22no—
26C4⋊#88324—22no—
27#13133274—44no—
28C22×C4×D53C2×C411yescharacteristic
29C22×C4×D53D411yescharacteristic
30C22×C4×D53Q811yescharacteristic
31C22×#88322C2×C411yescharacteristic
32C22×#88322C2×C411yescharacteristic
33C2×C4×Dic53D411yescharacteristic
34C2×C4×Dic53Q811yescharacteristic
35C2×C4×Dic53C2311yescharacteristic
36C42×D53—21no—
37C42×D53—21no—
38C42×D53—21no—
39C42×D53—21no—
40C4⋊#88322—22no—
41#13133129—22no—
42#13133129—22no—
43C8×Dic53—21no—
44C8×Dic53—21no—
45C8×Dic53D412yes—
46C8×Dic53D412yes—
47C22×C8×C53C2×C411yescharacteristic
48C22×C8×C53C2×C411yescharacteristic
49C2×C42×C53C2×C411yescharacteristic
50C4⋊C8×C53—22no—
51C23×D53C4⋊C411yescharacteristic
52C22×C4×C53C4⋊C411yescharacteristic
53C22×C4×C53C4⋊C411yescharacteristic
54C22×C4×C53C22×C411yescharacteristic
55C22×Dic53C4⋊C411yescharacteristic
56C22×Dic53C2×D411yescharacteristic
57C22×Dic53C2×Q811yescharacteristic
58C2×C4×D53—22no—
59C2×C4×D53—22no—
60C2×C4×D53—21no—
61C2×C4×D53—21no—
62C2×C4×D53—21no—
63C2×C4×D53—21no—
64C2×C4×D53—22no—
65C2×C4×D53—22no—
66C2×C4×D53—21no—
67C2×C4×D53—21no—
68C2×C4×D53—22no—
69C2×C4×D53—22no—
70C2×#88324—42no—
71C2×#88324—42no—
72C2×#88324—42no—
73C2×#88324—42no—
74C2×#88324—42no—
75C2×#88324—42no—
76C2×C8×C53—22no—
77C2×C8×C53—21no—
78C2×C8×C53—22no—
79C2×C8×C53C22⋊C412yes—
80C42×C53—21no—
81C42×C53—21no—
82C4×Dic53—21no—
83C4×Dic53—21no—
84C4×Dic53—21no—
85C4×Dic53—21no—
86C4×Dic53C2×D411yescharacteristic
87C4×Dic53C2×D411yescharacteristic
88C4×Dic53#4111yescharacteristic
89C4×Dic53#4111yescharacteristic
90C2×#88322—22no—
91C2×#88322—21no—
92C2×#88322—22no—
93C2×#88322C22⋊C412yes—
94C23×C53C2×C4⋊C411yescharacteristic
95C22×D53—21no—
96C22×D53—22no—
97C22×D53—22no—
98C22×D53—21no—
99C22×D53#10812yes—
100C2×C4×C53—21no—
101C2×C4×C53—21no—
102C2×C4×C53—21no—
103C2×C4×C53—22no—
104C2×C4×C53#10812yes—
105C2×C4×C53C2×C22⋊C411yescharacteristic
106C2×C4×C53#11711yescharacteristic
107C2×C4×C53#13112yes—
108C2×C4×C53#13112yes—
109C2×Dic53—21no—
110C2×Dic53—21no—
111C2×Dic53—21no—
112C2×Dic53—22no—
113C2×Dic53#12111yescharacteristic
114C2×Dic53#12111yescharacteristic
115C2×Dic53C22⋊Q811yescharacteristic
116C2×Dic53C22⋊Q811yescharacteristic
117C2×Dic53AGL(1,ℤ/8)12yes—
118C2×Dic53#13712yes—
119C4×D53—24no—
120C4×D53—24no—
121C4×D53—41no—
122C4×D53—41no—
123C4×D53—42no—
124C4×D53—42no—
125C4×D53—41no—
126C4×D53—41no—
127#88324—82no—
128#88324—82no—
129C8×C53—22no—
130C8×C53—41no—
131#88322—22no—
132#88322—41no—
133#8600—531nomaximal, Sylow
134C22×C53#38011yescharacteristic
135C22×C53#42412yes—
136C22×C53#42812yes—
137C22×C53C2×#10811yescharacteristic
138C22×C53C2×#10811yescharacteristic
139C2×D53—22no—
140C2×D53—24no—
141C2×D53—41no—
142C2×D53—41no—
143C2×D53—42no—
144C4×C53—22no—
145C4×C53—41no—
146C4×C53#40811yescharacteristic
147C4×C53#40811yescharacteristic
148C4×C53#43312yes—
149Dic53—21no—
150Dic53—21no—
151Dic53—41no—
152Dic53#46812yes—
153Dic53#46911yescharacteristic
154Dic53#47011yescharacteristic
155C2×#431—531no—
156#1797—532no—
157C2×#335—531no—
158C2×#334—531no—
159#1277—532no—
160C2×C53#157312yes—
161C2×C53#157412yes—
162C2×C53#179811yescharacteristic
163C2×C53#180911yescharacteristic
164C2×C53#181011yescharacteristic
165D53—42no—
166C22×C42—531no—
167#431—1062no—
168#431—1062no—
169C2×C4⋊C8—532no—
170C2×C4⋊C8—532no—
171C2×C4⋊C8—531no—
172C2×C4⋊C8#615711yescharacteristic
173C2×C22⋊C8—531no—
174C2×C22⋊C8—531no—
175C2×C22⋊C8—1062no—
176C2×C4×C8—531no—
177C2×C4×C8—531no—
178#336—1062no—
179#335—1062no—
180#334—1062no—
181C53#860011yescharacteristic, complemented, Sylow
182C23×C4—531no—
183C23×C4—531no—
184C23×C4—531no—
185C22×C8—531no—
186C22×C8—531no—
187C22×C8—1062no—
188C22×C8—1062no—
189C22×C8C2×#615711yescharacteristic
190C22×C8C2×#615711yescharacteristic
191C2×C42—531no—
192C2×C42—531no—
193C2×C42—531no—
194C2×C42—1061no—
195C2×C42—1061no—
196C2×C42—1061no—
197C2×C42—1061no—
198C2×C42C2×#615711yescharacteristic
199C4⋊C8—22no—
200C4⋊C8—1062no—
201C4⋊C8—1062no—
202C4⋊C8—1062no—
203C4⋊C8—1062no—
204C4⋊C8—1062no—
205C22⋊C8—1062no—
206C22⋊C8—1062no—
207C22⋊C8—2124no—
208C4×C8—532no—
209C4×C8—532no—
210C4×C8—1061no—
211C4×C8—1061no—
212C24—531no—
213C22×C4—531no—
214C22×C4—531no—
215C22×C4—531no—
216C22×C4—1061no—
217C22×C4—1061no—
218C22×C4—1062no—
219C22×C4—1061no—
220C22×C4—1061no—
221C22×C4—1062no—
222C22×C4—1062no—
223C22×C4—1062no—
224C22×C4—1061no—
225C22×C4—1061no—
226C22×C4—1062no—
227C22×C4—1062no—
228C22×C4C4⋊#615711yescharacteristic
229C22×C4C4⋊#615711yescharacteristic
230C22×C4C22×#615711yescharacteristic
231C2×C8—22no—
232C2×C8—21no—
233C2×C8—22no—
234C2×C8—532no—
235C2×C8—1062no—
236C2×C8—1061no—
237C2×C8—1062no—
238C2×C8—2122no—
239C2×C8—2122no—
240C2×C8—2122no—
241C2×C8—2122no—
242C2×C8—2122no—
243C2×C8—2122no—
244C2×C8C22⋊#615712yes—
245C42—21no—
246C42—21no—
247C42—531no—
248C42—531no—
249C42—531no—
250C42—531no—
251C42—1061no—
252C42—1061no—
253C42—1061no—
254C42—1061no—
255C23—532no—
256C23—1061no—
257C23—1062no—
258C23—1062no—
259C23—1061no—
260C23C2×C4⋊#615711yescharacteristic
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—22no—
265C2×C4—531no—
266C2×C4—531no—
267C2×C4—532no—
268C2×C4—532no—
269C2×C4—531no—
270C2×C4—531no—
271C2×C4—1064no—
272C2×C4—1064no—
273C2×C4—1061no—
274C2×C4—1061no—
275C2×C4—1061no—
276C2×C4—1062no—
277C2×C4—2121no—
278C2×C4—2122no—
279C2×C4—2121no—
280C2×C4—2121no—
281C2×C4—2121no—
282C2×C4—2122no—
283C2×C4#1313328812yes—
284C2×C4C2×C22⋊#615711yescharacteristic
285C2×C4#1313329611yescharacteristic
286C2×C4#1313330412yes—
287C2×C4#1313330512yes—
288C8—22no—
289C8—41no—
290C8—1062no—
291C8—4242no—
292C8—4242no—
293C8—2121no—
294C22—1062no—
295C22—1064no—
296C22—2121no—
297C22—2121no—
298C22—2122no—
299C22#1681156511yescharacteristic
300C22#1681163012yes—
301C22#1681163412yes—
302C22D53⋊#10811yescharacteristic
303C22C2×#1313328811yescharacteristic
304C4—22no—
305C4—41no—
306C4—531no—
307C4—532no—
308C4—531no—
309C4—1061no—
310C4—1061no—
311C4—2121no—
312C4#1681159611yescharacteristic
313C4#1681159811yescharacteristic
314C4#1681164812yes—
315C2—2122no—
316C2#2029028912yes—
317C2#2029029312yes—
318C2#2029057411yescharacteristic
319C2#2029059311yescharacteristic
320C2#2029059511yescharacteristic
subgroupG/H
centerC23C2×C4⋊#6157
commutatorC22×C4×C53C22×C4
FrattiniC22×C4C22×#6157
FittingC2×C4⋊C8×C53C4
socleC23×C53C2×C4⋊C4
prime radicalC23×D53C4⋊C4
cosocleC2×C4×Dic53C23
hypercenterC2×C4⋊C8#6157
hypocenterC53#8600
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×C4⋊C8

Complements

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

NH ≅ G/Ntype
2. C2×#16811486C2semidirect
3. C2×#16811485C2semidirect
4. #20289545C2semidirect
181. C53#8600semidirect

Maximal subgroups

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

subgroupindex
#202905702
C2×#168114862
C2×#168114852
#202895452
C2×#168106752
#860053