Finite Group Databasebeta

← #20274997

Subgroups — #20274997

Order 6784.

374 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 374 conjugacy classes of subgroups — too many to draw. Only its 83 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#16811096C211yesmaximal, characteristic
2C2×#13133131C211yesmaximal, characteristic
3#16810500C211yesmaximal, characteristic
4#16810498C211yesmaximal, characteristic, complemented
5#16810495C211yesmaximal, characteristic, complemented
6#16810389C211yesmaximal, characteristic, complemented
7C53×#342C211yesmaximal, characteristic, complemented
8C22×C22⋊D53C2211yescharacteristic
9C23×Dic53C2211yescharacteristic
10C4×C22⋊D53—21no—
11C4×C22⋊D53—21no—
12#13133177—21no—
13#13133177—21no—
14#13133173—21no—
15C2×C4⋊Dic53—21no—
16C2×#11719919—21no—
17C2×C22⋊Dic53C2211yescharacteristic
18C2×C22⋊Dic53C2211yescharacteristic
19C2×#11719921—21no—
20C22⋊C4×D53—21no—
21#13133131C414yescomplemented
22#13133124—21no—
23#13133124—21no—
24#13133124—21no—
25C2×C22⋊C4×C53C2211yescharacteristic
26C2×C22⋊C4×C53C2211yescharacteristic
27C2×C22⋊C4×C53C2211yescharacteristic
28C24×C53C2311yescharacteristic
29C23×D53D411yescharacteristic
30C22×C4×C53—21no—
31C22×C4×C53—21no—
32C22×C4×C53D411yescharacteristic
33C22×C4×C53D411yescharacteristic
34C2×C22⋊D53—21no—
35C2×C22⋊D53—21no—
36C2×C22⋊D53—21no—
37C2×C22⋊D53—21no—
38C2×C22⋊D53C2×C412yes—
39C2×C22⋊D53Q812yes—
40C22×Dic53—21no—
41C22×Dic53—21no—
42C22×Dic53—21no—
43C22×Dic53—21no—
44C22×Dic53—42no—
45C22×Dic53D411yescharacteristic
46C22×Dic53D411yescharacteristic
47C22×Dic53D411yescharacteristic
48C2×C4×D53—21no—
49C2×C4×D53—21no—
50C22⋊C4×C53—21no—
51C22⋊C4×C53—21no—
52C22⋊C4×C53—21no—
53C22⋊C4×C53—21no—
54C22⋊C4×C53—21no—
55C22⋊C4×C53C2×C412yescomplemented
56C22⋊Dic53—21no—
57C22⋊Dic53—21no—
58C22⋊Dic53—21no—
59C22⋊Dic53C2×C412yescomplemented
60#11719921—21no—
61#11719921—21no—
62#11719921—42no—
63C4⋊Dic53—42no—
64#11719919—21no—
65#11719919—21no—
66#11719919—42no—
67C4×Dic53—21no—
68C4×Dic53—21no—
69C23×C53—41no—
70C23×C53—41no—
71C23×C53C22×C411yescharacteristic
72C23×C53C2×D411yescharacteristic
73C23×C53C2×D411yescharacteristic
74C23×C53C2×D411yescharacteristic
75C23×C53C2×Q811yescharacteristic
76C23×C53#4111yescharacteristic
77C23×C53#4111yescharacteristic
78C22×D53—21no—
79C22×D53—21no—
80C22×D53—41no—
81C22×D53—41no—
82C22×D53C4⋊C412yescomplemented
83C2×C4×C53—22no—
84C2×C4×C53—21no—
85C2×C4×C53—21no—
86C2×C4×C53—41no—
87C2×C4×C53—41no—
88C2×C4×C53—41no—
89C2×C4×C53—41no—
90C2×C4×C53—41no—
91C2×C4×C53—41no—
92C2×C4×C53—41no—
93C22⋊D53—22no—
94C22⋊D53—22no—
95C22⋊D53—22no—
96C22⋊D53—22no—
97C2×Dic53—21no—
98C2×Dic53—21no—
99C2×Dic53—22no—
100C2×Dic53—22no—
101C2×Dic53—22no—
102C2×Dic53—22no—
103C2×Dic53—21no—
104C2×Dic53—21no—
105C2×Dic53—41no—
106C2×Dic53—41no—
107C2×Dic53—41no—
108C2×Dic53—41no—
109C2×Dic53—41no—
110C2×Dic53—41no—
111C2×Dic53—41no—
112C2×Dic53—41no—
113C2×Dic53C4⋊C412yescomplemented
114C4×D53—81no—
115C4×D53—81no—
116C22×C53—21no—
117C22×C53—21no—
118C22×C53—21no—
119C22×C53—21no—
120C22×C53—21no—
121C22×C53—21no—
122C22×C53—41no—
123C22×C53—41no—
124C22×C53—41no—
125C22×C53—41no—
126C22×C53C2×C4⋊C411yescharacteristic
127C22×C53C4×D411yescharacteristic
128C22×C53C4×D411yescharacteristic
129C22×C53#12011yescharacteristic
130C22×C53C22⋊Q811yescharacteristic
131C22×C53C22⋊Q811yescharacteristic
132C22×C53#12311yescharacteristic
133C2×D53—22no—
134C2×D53—41no—
135C2×D53—41no—
136C2×D53—42no—
137C2×D53—81no—
138C4×C53—42no—
139C4×C53—41no—
140C4×C53—41no—
141C4×C53—81no—
142Dic53—22no—
143Dic53—42no—
144Dic53—41no—
145Dic53—42no—
146Dic53—41no—
147#1867—531nomaximal, Sylow
148C2×C53—21no—
149C2×C53—21no—
150C2×C53—21no—
151C2×C53—21no—
152C2×C53—21no—
153C2×C53—21no—
154C2×C53#38511yescharacteristic
155C2×C53B(4,𝔽2)11yescharacteristic
156C2×C53#45811yescharacteristic
157D53—42no—
158#518—531no—
159C2×#99—531no—
160#386—531no—
161#385—531no—
162#385—531no—
163#342—531no—
164#342D5311yescharacteristic, complemented
165C53#186711yescharacteristic, complemented, Sylow
166C22×D4—531no—
167C23×C4—531no—
168C4×D4—1061no—
169C4×D4—1061no—
170C4×D4—1061no—
171C4×D4—1061no—
172#117—1061no—
173C2×C4⋊C4—1061no—
174C2×C4⋊C4—1061no—
175C2×C22⋊C4—531no—
176C2×C22⋊C4—531no—
177C2×C22⋊C4—1061no—
178C2×C22⋊C4—1061no—
179C2×C22⋊C4C2×D5311yescharacteristic
180C2×C22⋊C4C2×D5311yescharacteristic
181C2×C22⋊C4C2×D5311yescharacteristic
182#99—534no—
183#95—1061no—
184#95—1061no—
185#95—1061no—
186C24—531no—
187C24C22×D5311yescharacteristic
188C2×D4—532no—
189C2×D4—532no—
190C2×D4—1061no—
191C2×D4—1061no—
192C2×D4—1061no—
193C2×D4—1061no—
194C22×C4—21no—
195C22×C4—21no—
196C22×C4—531no—
197C22×C4—531no—
198C22×C4—531no—
199C22×C4—1061no—
200C22×C4—1061no—
201C22×C4—1061no—
202C22×C4—1061no—
203C22×C4—1061no—
204C22×C4—1061no—
205C22×C4—2122no—
206C22×C4D21211yescharacteristic
207C22×C4D21211yescharacteristic
208C4⋊C4—1061no—
209C4⋊C4—1061no—
210C4⋊C4—2122no—
211C4⋊C4—2122no—
212C22⋊C4—21no—
213C22⋊C4—21no—
214C22⋊C4—21no—
215C22⋊C4—21no—
216C22⋊C4—21no—
217C22⋊C4—532no—
218C22⋊C4—1061no—
219C22⋊C4—1061no—
220C22⋊C4—1061no—
221C22⋊C4—1061no—
222C22⋊C4—1061no—
223C22⋊C4—2122no—
224C22⋊C4C4×D5312yescomplemented
225C42—1061no—
226C42—1061no—
227C23—41no—
228C23—41no—
229C23—532no—
230C23—1061no—
231C23—1061no—
232C23—2121no—
233C23—2121no—
234C23C2×C4×D5311yescharacteristic
235C23C2×D21211yescharacteristic
236C23#1171994511yescharacteristic
237C23D4×D5311yescharacteristic
238C23D4×D5311yescharacteristic
239C23#1171994711yescharacteristic
240C23Q8×D5311yescharacteristic
241D4—1062no—
242D4—1062no—
243D4—1062no—
244D4—1062no—
245C2×C4—22no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—41no—
249C2×C4—41no—
250C2×C4—41no—
251C2×C4—41no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—41no—
255C2×C4—532no—
256C2×C4—1061no—
257C2×C4—1061no—
258C2×C4—1062no—
259C2×C4—1062no—
260C2×C4—1062no—
261C2×C4—1062no—
262C2×C4—1061no—
263C2×C4—1061no—
264C2×C4—4241no—
265C2×C4—4241no—
266C2×C4—2121no—
267C2×C4—2121no—
268C2×C4—2121no—
269C2×C4—2121no—
270C2×C4—2121no—
271C2×C4—2121no—
272C2×C4—2121no—
273C2×C4—2121no—
274C22—21no—
275C22—21no—
276C22—21no—
277C22—21no—
278C22—21no—
279C22—21no—
280C22—41no—
281C22—41no—
282C22—41no—
283C22—41no—
284C22—1062no—
285C22—4241no—
286C22—2122no—
287C22—2121no—
288C22—2121no—
289C22C4⋊C4×D5311yescharacteristic
290C22#1313317711yescharacteristic
291C22C4×D21211yescharacteristic
292C22#1313318711yescharacteristic
293C22#1313319511yescharacteristic
294C22#1313319711yescharacteristic
295C22#1313319911yescharacteristic
296C4—42no—
297C4—41no—
298C4—41no—
299C4—81no—
300C4—1062no—
301C4—2122no—
302C4—2122no—
303C4—2121no—
304C4—2121no—
305C2—21no—
306C2—21no—
307C2—21no—
308C2—21no—
309C2—21no—
310C2—21no—
311C2—2122no—
312C2#1681049711yescharacteristic
313C2#1681081111yescharacteristic
314C2#1681081411yescharacteristic
subgroupG/H
centerC22#13133187
commutatorC23×C53C22×C4
FrattiniC24C22×D53
FittingC53×#342C2
socleC22×C53#120
hypercenter#342D53
hypocenterC53#1867
LCSG ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C24 ◁ #342

Complements

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

NH ≅ G/Ntype
4. #16810498C2semidirect
5. #16810495C2semidirect
6. #16810389C2semidirect
7. C53×#342C2semidirect
21. #13133131C4semidirect
55. C22⋊C4×C53C2×C4semidirect
59. C22⋊Dic53C2×C4semidirect
82. C22×D53C4⋊C4semidirect
113. C2×Dic53C4⋊C4semidirect
164. #342D53semidirect
165. C53#1867semidirect
224. C22⋊C4C4×D53semidirect