Finite Group Databasebeta

← #28299527

Subgroups — #28299527

Order 13568.

566 conjugacy classes of subgroups; only the 132 normal subgroups are shown.

normal subgroups only

This group has 566 conjugacy classes of subgroups — too many to draw. Only its 132 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#20290605C211yesmaximal, characteristic
2#20290605C211yesmaximal, characteristic
3#20289533C214yesmaximal, complemented
4C2×#16810522C211yesmaximal, characteristic
5C2×C22⋊#88324—21no—
6C2×C22⋊#88324—21no—
7#16811489—21no—
8#16811489—21no—
9#16811489C2212yes—
10#16811489C2212yes—
11C22×C4⋊Dic53—21no—
12C22×C4⋊Dic53C411yescharacteristic
13C22×C4×Dic53C2211yescharacteristic
14#16810522—21no—
15#16810522—21no—
16#16810522C412yes—
17#16810522C2212yes—
18C2×#13133124—21no—
19C22×C4⋊C4×C53C411yescharacteristic
20C22×#88324—22no—
21C22×#88324—22no—
22C22⋊#88324—42no—
23C22⋊#88324—42no—
24C23×Dic53—21no—
25C23×Dic53D411yescharacteristic
26C23×Dic53D411yescharacteristic
27C2×C4⋊Dic53—21no—
28C2×C4⋊Dic53—22no—
29C2×C4⋊Dic53—22no—
30C2×C4⋊Dic53—44no—
31C2×C4⋊Dic53—42no—
32C2×C4⋊Dic53C2×C412yes—
33C2×C4×Dic53—42no—
34C2×C4×Dic53C2×C411yescharacteristic
35C2×C4×Dic53D411yescharacteristic
36C2×C4×Dic53D411yescharacteristic
37C2×C4×Dic53C2311yescharacteristic
38#13133124—22no—
39#13133124—22no—
40C23×C4×C53—21no—
41C23×C4×C53C2×C411yescharacteristic
42C2×C4⋊C4×C53—21no—
43C2×C4⋊C4×C53—22no—
44C2×C4⋊C4×C53C814yescomplemented
45C2×C4⋊C4×C53C2×C412yes—
46C24×C53C22⋊C411yescharacteristic
47C22×C4×C53—22no—
48C22×C4×C53—22no—
49C22×C4×C53—24no—
50C22×C4×C53—22no—
51C22×C4×C53—42no—
52C22×C4×C53—42no—
53C22×C4×C53C22⋊C411yescharacteristic
54C22×C4×C53C2×C812yes—
55C22×C4×C53M1612yes—
56C22×C4×C53C22×C411yescharacteristic
57C22×Dic53—22no—
58C22×Dic53—22no—
59C22×Dic53—22no—
60C22×Dic53—22no—
61C22×Dic53—22no—
62C22×Dic53—41no—
63C22×Dic53—41no—
64C22×Dic53—42no—
65C22×Dic53—42no—
66C22×Dic53—42no—
67C22×Dic53C22⋊C411yescharacteristic
68C22×Dic53C22⋊C411yescharacteristic
69C22×Dic53C2×D411yescharacteristic
70C22×Dic53C2×D411yescharacteristic
71C2×#88324—42no—
72C2×#88324—42no—
73C2×#88324—42no—
74C2×#88324—42no—
75C2×#88324—42no—
76C2×#88324—42no—
77C4⋊C4×C53—24no—
78C4⋊C4×C53—22no—
79C4⋊C4×C53C2×C814yes—
80C4⋊Dic53—44no—
81C4⋊Dic53—42no—
82C4⋊Dic53—44no—
83C4⋊Dic53—42no—
84C4×Dic53—41no—
85C4×Dic53—41no—
86C4×Dic53—41no—
87C4×Dic53—41no—
88C23×C53—21no—
89C23×C53—21no—
90C23×C53—21no—
91C23×C53—21no—
92C23×C53C22⋊C812yes—
93C23×C53#9912yes—
94C23×C53#10112yes—
95C23×C53C2×C22⋊C411yescharacteristic
96C2×C4×C53—24no—
97C2×C4×C53—24no—
98C2×C4×C53—21no—
99C2×C4×C53—21no—
100C2×C4×C53—21no—
101C2×C4×C53—21no—
102C2×C4×C53—24no—
103C2×C4×C53—48no—
104C2×C4×C53—42no—
105C2×C4×C53—42no—
106C2×C4×C53C22⋊C812yes—
107C2×C4×C53C22×C811yescharacteristic
108C2×C4×C53C2×M1611yescharacteristic
109C2×Dic53—21no—
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—21no—
119C2×Dic53—44no—
120C2×Dic53—44no—
121C2×Dic53—42no—
122C2×Dic53—42no—
123C2×Dic53—42no—
124C2×Dic53—84no—
125C2×Dic53AGL(1,ℤ/8)12yes—
126C2×Dic53#13712yes—
127#88324—82no—
128#88324—82no—
129#7593—531nomaximal, Sylow
130C22×C53—22no—
131C22×C53—22no—
132C22×C53—22no—
133C22×C53—22no—
134C22×C53—22no—
135C22×C53—22no—
136C22×C53#32414yes—
137C22×C53C2×C22⋊C811yescharacteristic
138C22×C53C2×#9911yescharacteristic
139C22×C53C2×#10111yescharacteristic
140C22×C53#41811yescharacteristic
141C22×C53#41911yescharacteristic
142C22×C53#42111yescharacteristic
143C22×C53#42111yescharacteristic
144C4×C53—22no—
145C4×C53—21no—
146C4×C53—21no—
147C4×C53—44no—
148Dic53—22no—
149Dic53—22no—
150Dic53—41no—
151Dic53—41no—
152Dic53—82no—
153C2×#389—531no—
154#1817—531no—
155#1817—531no—
156#1271—534no—
157C2×C53—24no—
158C2×C53C2×#32411yescharacteristic
159C2×C53#147811yescharacteristic
160C2×C53#147911yescharacteristic
161C2×C53#151011yescharacteristic
162C2×C53#151111yescharacteristic
163C2×C53#153411yescharacteristic
164C2×C53#155611yescharacteristic
165C22×C4⋊C4—531no—
166C22×C4⋊C4—1061no—
167C22×C4⋊C4#615711yescharacteristic
168C22×C42—531no—
169C2×C22⋊C8—1061no—
170C2×C22⋊C8—1061no—
171#389—532no—
172#389—532no—
173#389—1061no—
174#389—1061no—
175C2×#95—1061no—
176#336—532no—
177#336—532no—
178#336—1061no—
179#336—1061no—
180C53#759311yescharacteristic, complemented, Sylow
181C23×C4—21no—
182C23×C4—531no—
183C23×C4—531no—
184C23×C4—1061no—
185C23×C4C2×#615711yescharacteristic
186C22×C8—1062no—
187C22×C8—1062no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—532no—
191C2×C4⋊C4—1062no—
192C2×C4⋊C4—1062no—
193C2×C4⋊C4—1061no—
194C2×C4⋊C4—2122no—
195C2×C4⋊C4—2124no—
196C2×C4⋊C4#8832414yescomplemented
197C2×C4⋊C4C2×#615712yes—
198C2×C42—531no—
199C2×C42—531no—
200C2×C42—531no—
201C2×C42—531no—
202C2×C42—2122no—
203C22⋊C8—2122no—
204C22⋊C8—2122no—
205#95—1062no—
206#95—1062no—
207C24C22⋊#615711yescharacteristic
208C22×C4—22no—
209C22×C4—22no—
210C22×C4—24no—
211C22×C4—22no—
212C22×C4—42no—
213C22×C4—42no—
214C22×C4—531no—
215C22×C4—531no—
216C22×C4—531no—
217C22×C4—531no—
218C22×C4—1062no—
219C22×C4—1062no—
220C22×C4—1062no—
221C22×C4—1062no—
222C22×C4—1062no—
223C22×C4—2121no—
224C22×C4—2121no—
225C22×C4—2122no—
226C22×C4—2122no—
227C22×C4—2122no—
228C22×C4C2×#8832412yes—
229C22×C4#1171994012yes—
230C22×C4C22⋊#615711yescharacteristic
231C22×C4C22×#615711yescharacteristic
232C2×C8—2122no—
233C2×C8—2122no—
234C2×C8—2122no—
235C2×C8—2122no—
236C2×C8—2122no—
237C2×C8—2122no—
238C4⋊C4—24no—
239C4⋊C4—22no—
240C4⋊C4—2122no—
241C4⋊C4—2122no—
242C4⋊C4—2124no—
243C4⋊C4—2124no—
244C4⋊C4C2×#8832414yes—
245C42—2121no—
246C42—2121no—
247C42—2121no—
248C42—2121no—
249C23—21no—
250C23—21no—
251C23—21no—
252C23—21no—
253C23C22⋊#8832412yes—
254C23#1313327612yes—
255C23#1313327812yes—
256C23C2×C22⋊#615711yescharacteristic
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—24no—
262C2×C4—24no—
263C2×C4—24no—
264C2×C4—48no—
265C2×C4—42no—
266C2×C4—42no—
267C2×C4—532no—
268C2×C4—532no—
269C2×C4—1061no—
270C2×C4—1061no—
271C2×C4—1061no—
272C2×C4—1061no—
273C2×C4—1061no—
274C2×C4—1061no—
275C2×C4—1061no—
276C2×C4—1061no—
277C2×C4—1061no—
278C2×C4—1061no—
279C2×C4—4244no—
280C2×C4—2124no—
281C2×C4—2124no—
282C2×C4—2122no—
283C2×C4—2122no—
284C2×C4—2122no—
285C2×C4C22⋊#8832412yes—
286C2×C4C22×#8832411yescharacteristic
287C2×C4C2×#1171994011yescharacteristic
288C8—4242no—
289C8—4242no—
290C22—22no—
291C22—22no—
292C22—22no—
293C22—22no—
294C22—22no—
295C22—22no—
296C22#1681146814yes—
297C22C2×C22⋊#8832411yescharacteristic
298C22C2×#1313327611yescharacteristic
299C22C2×#1313327811yescharacteristic
300C22#1681161511yescharacteristic
301C22#1681161611yescharacteristic
302C22#1681162111yescharacteristic
303C22#1681162211yescharacteristic
304C4—22no—
305C4—21no—
306C4—21no—
307C4—44no—
308C4—1062no—
309C4—1062no—
310C4—4242no—
311C4—2121no—
312C4—2121no—
313C2—24no—
314C2C2×#1681146811yescharacteristic
315C2#2029001411yescharacteristic
316C2#2029001711yescharacteristic
317C2#2029012211yescharacteristic
318C2#2029012311yescharacteristic
319C2#2029019711yescharacteristic
320C2#2029024411yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6157
commutatorC2×C4×C53C22×C8
FrattiniC22×C4C22×#6157
FittingC22×C4⋊C4×C53C4
socleC23×C53C2×C22⋊C4
prime radicalC24×C53C22⋊C4
cosocleC2×C4×Dic53C23
hypercenterC22×C4⋊C4#6157
hypocenterC53#7593
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×C4⋊C4

Complements

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

NH ≅ G/Ntype
3. #20289533C2semidirect
44. C2×C4⋊C4×C53C8semidirect
180. C53#7593semidirect
196. C2×C4⋊C4#88324semidirect

Maximal subgroups

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

subgroupindex
#202906052
#202895332
C2×#168105222
#759353