Finite Group Databasebeta

← #28299560

Subgroups — #28299560

Order 13568.

408 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 408 conjugacy classes of subgroups — too many to draw. Only its 106 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#20290631C211yesmaximal, characteristic
2#20290631C211yesmaximal, characteristic
3#20289534C212yesmaximal
4#20289533C212yesmaximal
5C4×#13133208C211yesmaximal, characteristic
6C2×#13133284—21no—
7C2×#13133284—21no—
8#16811489—21no—
9#16811489—21no—
10#16811489C2212yes—
11#16811489C2212yes—
12C2×#13133208C411yescharacteristic
13#16810522C2211yescharacteristic
14#16810519C2211yescharacteristic
15#16810510—21no—
16#16810506—21no—
17#16810488C411yescharacteristic
18C4×#11719919—21no—
19C4×C4⋊Dic53—21no—
20C4×C4⋊Dic53C411yescharacteristic
21C42×Dic53C2211yescharacteristic
22C4×C4⋊C4×C53C411yescharacteristic
23C22×#88324—22no—
24C22×#88324—22no—
25#13133284—42no—
26#13133284—42no—
27#13133208—22no—
28#13133208C814yescomplemented
29C2×C4⋊Dic53—21no—
30C2×C4⋊Dic53—21no—
31C2×C4⋊Dic53C2×C411yescharacteristic
32C2×C4⋊Dic53C2×C411yescharacteristic
33C2×#11719919—21no—
34C2×#11719919—21no—
35C2×C4×Dic53—21no—
36C2×C4×Dic53C2×C411yescharacteristic
37C2×C4×Dic53D411yescharacteristic
38C2×C4×Dic53D411yescharacteristic
39C2×C4×Dic53D411yescharacteristic
40C2×C4×Dic53D411yescharacteristic
41C2×C4×Dic53C2311yescharacteristic
42#13133124—21no—
43#13133124—21no—
44#13133124—21no—
45#13133124—21no—
46#13133124—21no—
47C2×C4⋊C4×C53C2×C411yescharacteristic
48C2×C4⋊C4×C53C2×C411yescharacteristic
49C2×C42×C53—21no—
50C2×C42×C53C2×C411yescharacteristic
51#95×C53—21no—
52C22×C4×C53—21no—
53C22×C4×C53—21no—
54C22×C4×C53C22⋊C411yescharacteristic
55C22×C4×C53C22⋊C411yescharacteristic
56C22×C4×C53C22×C411yescharacteristic
57C22×Dic53—21no—
58C22×Dic53—21no—
59C22×Dic53C22⋊C411yescharacteristic
60C22×Dic53C22⋊C411yescharacteristic
61C22×Dic53C2×D411yescharacteristic
62C22×Dic53C2×D411yescharacteristic
63C2×#88324—42no—
64C2×#88324—42no—
65C2×#88324—42no—
66C2×#88324—42no—
67C2×#88324—42no—
68C2×#88324—42no—
69C4⋊C4×C53—21no—
70C4⋊C4×C53—41no—
71C4⋊C4×C53C2×C812yes—
72C42×C53—22no—
73C42×C53—42no—
74C4⋊Dic53—21no—
75C4⋊Dic53—24no—
76C4⋊Dic53—41no—
77C4⋊Dic53—41no—
78C4⋊Dic53—41no—
79C4⋊Dic53C2×C812yes—
80#11719919—24no—
81#11719919—41no—
82#11719919—41no—
83C4×Dic53—21no—
84C4×Dic53—21no—
85C4×Dic53—21no—
86C4×Dic53—21no—
87C4×Dic53—41no—
88C4×Dic53—41no—
89C4×Dic53—41no—
90C4×Dic53—42no—
91C4×Dic53C2×C812yes—
92C4×Dic53M1612yes—
93C23×C53C2×C22⋊C411yescharacteristic
94C2×C4×C53—21no—
95C2×C4×C53—21no—
96C2×C4×C53—21no—
97C2×C4×C53—21no—
98C2×C4×C53—22no—
99C2×C4×C53—21no—
100C2×C4×C53—21no—
101C2×C4×C53—21no—
102C2×C4×C53—41no—
103C2×C4×C53—41no—
104C2×C4×C53—41no—
105C2×C4×C53—41no—
106C2×C4×C53—41no—
107C2×C4×C53#10112yes—
108C2×C4×C53C22×C811yescharacteristic
109C2×C4×C53C2×M1611yescharacteristic
110C2×Dic53—21no—
111C2×Dic53—21no—
112C2×Dic53—21no—
113C2×Dic53—21no—
114C2×Dic53—21no—
115C2×Dic53—21no—
116C2×Dic53—21no—
117C2×Dic53—21no—
118C2×Dic53—22no—
119C2×Dic53—41no—
120C2×Dic53—41no—
121C2×Dic53—41no—
122C2×Dic53—41no—
123C2×Dic53—41no—
124C2×Dic53C22⋊C812yes—
125C2×Dic53C22⋊C812yes—
126C2×Dic53AGL(1,ℤ/8)11yescharacteristic
127C2×Dic53AGL(1,ℤ/8)11yescharacteristic
128C2×Dic53#13711yescharacteristic
129C2×Dic53#13711yescharacteristic
130#88324—82no—
131#88324—82no—
132#7602—531nomaximal, Sylow
133C22×C53C2×C22⋊C811yescharacteristic
134C22×C53#41011yescharacteristic
135C22×C53C2×#10111yescharacteristic
136C22×C53#41711yescharacteristic
137C22×C53#41711yescharacteristic
138C22×C53#42111yescharacteristic
139C22×C53#42111yescharacteristic
140C4×C53—21no—
141C4×C53—21no—
142C4×C53—22no—
143C4×C53—41no—
144C4×C53—42no—
145C4×C53—81no—
146Dic53—22no—
147Dic53—22no—
148Dic53—21no—
149Dic53—21no—
150Dic53—21no—
151Dic53—21no—
152Dic53—42no—
153Dic53—81no—
154C4×#125—531no—
155#1831—531no—
156#1831—531no—
157#1271—532no—
158#1271—532no—
159C2×C53#146111yescharacteristic
160C2×C53#147711yescharacteristic
161C2×C53#147711yescharacteristic
162C2×C53#151211yescharacteristic
163C2×C53#151211yescharacteristic
164C2×C53#153411yescharacteristic
165C2×C53#156111yescharacteristic
166C2×#125—531no—
167C2×C4⋊C8—1061no—
168C2×C4⋊C8—1061no—
169#389—531no—
170#389—531no—
171#387—1061no—
172#387—1061no—
173#382—531no—
174C4×C4⋊C4—531no—
175C4×C4⋊C4—1061no—
176C4×C4⋊C4—1061no—
177C4×C4⋊C4#615711yescharacteristic
178C43—531no—
179#336—532no—
180#336—532no—
181#336—1061no—
182#336—1061no—
183C53#760211yescharacteristic, complemented, Sylow
184C22×C8—1062no—
185C22×C8—1062no—
186#125—534no—
187#125—1062no—
188C2×C4⋊C4—531no—
189C2×C4⋊C4—531no—
190C2×C4⋊C4—1061no—
191C2×C4⋊C4—1061no—
192C2×C4⋊C4—1061no—
193C2×C4⋊C4—1061no—
194C2×C4⋊C4C2×#615711yescharacteristic
195C2×C4⋊C4C2×#615711yescharacteristic
196C2×C42—21no—
197C2×C42—531no—
198C2×C42—531no—
199C2×C42—531no—
200C2×C42—531no—
201C2×C42—531no—
202C2×C42—531no—
203C2×C42—1061no—
204C2×C42C2×#615711yescharacteristic
205C4⋊C8—2122no—
206C4⋊C8—2122no—
207#95—21no—
208#95—1061no—
209#95—1061no—
210#95—1061no—
211#95—1061no—
212#95—1061no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—531no—
216C22×C4—531no—
217C22×C4—531no—
218C22×C4—531no—
219C22×C4—1061no—
220C22×C4—1061no—
221C22×C4C22⋊#615711yescharacteristic
222C22×C4C22⋊#615711yescharacteristic
223C22×C4C22×#615711yescharacteristic
224C2×C8—2122no—
225C2×C8—2122no—
226C2×C8—2122no—
227C2×C8—2122no—
228C2×C8—2122no—
229C2×C8—2122no—
230C4⋊C4—21no—
231C4⋊C4—41no—
232C4⋊C4—532no—
233C4⋊C4—1064no—
234C4⋊C4—1064no—
235C4⋊C4—1061no—
236C4⋊C4—2121no—
237C4⋊C4—2121no—
238C4⋊C4—2121no—
239C4⋊C4—2121no—
240C4⋊C4—2121no—
241C4⋊C4D53⋊C812yes—
242C42—22no—
243C42—42no—
244C42—532no—
245C42—532no—
246C42—1061no—
247C42—1061no—
248C42—1061no—
249C42—1061no—
250C42—2121no—
251C42—2121no—
252C42—2121no—
253C42—2122no—
254C23C2×C22⋊#615711yescharacteristic
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—22no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—41no—
264C2×C4—41no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—41no—
268C2×C4—532no—
269C2×C4—532no—
270C2×C4—531no—
271C2×C4—531no—
272C2×C4—531no—
273C2×C4—531no—
274C2×C4—1061no—
275C2×C4—1061no—
276C2×C4—1061no—
277C2×C4—1061no—
278C2×C4—1061no—
279C2×C4—1061no—
280C2×C4—1061no—
281C2×C4—1061no—
282C2×C4—1062no—
283C2×C4—2121no—
284C2×C4—2121no—
285C2×C4—2121no—
286C2×C4—2121no—
287C2×C4—2121no—
288C2×C4#1313327812yes—
289C2×C4C2×D53⋊C811yescharacteristic
290C2×C4D53⋊M1611yescharacteristic
291C8—4242no—
292C8—4242no—
293C22D53⋊(C22⋊C8)11yescharacteristic
294C22#1681160211yescharacteristic
295C22C2×#1313327811yescharacteristic
296C22#1681161411yescharacteristic
297C22#1681161411yescharacteristic
298C22#1681162111yescharacteristic
299C22#1681162211yescharacteristic
300C4—22no—
301C4—21no—
302C4—21no—
303C4—41no—
304C4—42no—
305C4—81no—
306C4—1061no—
307C4—1061no—
308C4—1061no—
309C4—1061no—
310C4—1062no—
311C4—1062no—
312C4—4241no—
313C4—2122no—
314C2#2028995711yescharacteristic
315C2#2029001111yescharacteristic
316C2#2029001311yescharacteristic
317C2#2029012611yescharacteristic
318C2#2029012811yescharacteristic
319C2#2029019711yescharacteristic
320C2#2029025411yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6157
commutatorC2×C4×C53C22×C8
FrattiniC22×C4C22×#6157
FittingC4×C4⋊C4×C53C4
socle, prime radicalC23×C53C2×C22⋊C4
cosocleC2×C4×Dic53C23
hypercenterC4×C4⋊C4#6157
hypocenterC53#7602
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ C4×C4⋊C4

Complements

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

NH ≅ G/Ntype
28. #13133208C8semidirect
183. C53#7602semidirect

Maximal subgroups

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

subgroupindex
#202906312
#202895342
#202895332
C4×#131332082
#760253