Finite Group Databasebeta

← #20278351

Subgroups — #20278351

Order 6784.

1348 conjugacy classes of subgroups; only the 193 normal subgroups are shown.

normal subgroups only

This group has 1348 conjugacy classes of subgroups — too many to draw. Only its 193 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
1C22×D4×D53C211yesmaximal, characteristic, complemented
2C2×#13133214C211yesmaximal, characteristic, complemented
3C2×#13133213C211yesmaximal, characteristic, complemented
4C2×#13133192C211yesmaximal, characteristic, complemented
5C2×#13133187C214yesmaximal, complemented
6#16810532C214yesmaximal, complemented
7#16810524C211yesmaximal, characteristic, complemented
8C4×#11719921C211yesmaximal, characteristic, complemented
9C53×#390C211yesmaximal, characteristic, complemented
10C24×D53C2212yes—
11C22×D212C2212yes—
12C22×D212C2211yescharacteristic
13C22×D212C2211yescharacteristic, complemented
14C22×D212C2214yescomplemented
15C22×C22⋊D53C2212yes—
16C22×C22⋊D53C2214yescomplemented
17C2×D4×D53—24no—
18C2×D4×D53—24no—
19C2×D4×D53—24no—
20C22×C4×D53C2211yescharacteristic
21#13133214—24no—
22#13133213—24no—
23#13133192—24no—
24#13133187—216no—
25C2×#11719921C2212yes—
26C2×#11719921C2214yescomplemented
27C2×#11719921C2212yescomplemented
28C2×C4×Dic53C2211yescharacteristic, complemented
29#13133124C2212yescomplemented
30C22×D4×C53C2211yescharacteristic
31C2×C4⋊C4×C53C2211yescharacteristic, complemented
32C2×C22⋊C4×C53C2214yescomplemented
33C2×C42×C53C2211yescharacteristic, complemented
34C24×C53C2312yes—
35C23×D53—24no—
36C23×D53—24no—
37C23×D53—24no—
38C23×D53—28no—
39C23×D53—44no—
40C23×D53C2312yes—
41C23×D53C2312yes—
42C23×D53C2311yescharacteristic
43C2×D4×C53—22no—
44C2×D4×C53—22no—
45C2×D4×C53D414yescomplemented
46C22×C4×C53C2312yes—
47C22×C4×C53C2312yes—
48C22×C4×C53C2311yescharacteristic
49C2×C22⋊D53—24no—
50C2×C22⋊D53—24no—
51C2×C22⋊D53—28no—
52C2×C22⋊D53—24no—
53C2×C22⋊D53—28no—
54C2×C22⋊D53—44no—
55C22×Dic53C2311yescharacteristic
56C22×Dic53C2312yes—
57D4×D53—232no—
58C2×D212—22no—
59C2×D212—22no—
60C2×D212—28no—
61C2×D212—24no—
62C2×D212—28no—
63C2×D212—22no—
64C2×D212—22no—
65C2×D212—22no—
66C2×D212—28no—
67C2×D212—44no—
68C2×D212D414yescomplemented
69C2×C4×D53—22no—
70C2×C4×D53—22no—
71C2×C4×D53D414yescomplemented
72C4⋊C4×C53—22no—
73C22⋊C4×C53—28no—
74C42×C53—22no—
75#11719921—24no—
76#11719921—28no—
77#11719921D418yescomplemented
78C4×Dic53—22no—
79C23×C53—22no—
80C23×C53—22no—
81C23×C53—24no—
82C23×C53—24no—
83C23×C53C2×D414yes—
84C23×C53C2411yescharacteristic
85C22×D53—21no—
86C22×D53—21no—
87C22×D53—22no—
88C22×D53—216no—
89C22×D53—22no—
90C22×D53—22no—
91C22×D53—24no—
92C22×D53—24no—
93C22×D53—22no—
94C22×D53—24no—
95C22×D53—22no—
96C22×D53—22no—
97C22×D53—24no—
98C22×D53—416no—
99C22×D53—416no—
100C22×D53—48no—
101C22×D53—48no—
102C22×D53C2×D414yes—
103C22×D53C2×D412yes—
104C22×D53C2×D414yes—
105D4×C53—28no—
106C2×C4×C53—22no—
107C2×C4×C53—22no—
108C2×C4×C53—24no—
109C2×C4×C53—22no—
110C2×C4×C53C2×D412yes—
111C2×C4×C53C2×D412yes—
112C2×C4×C53C2×D412yes—
113C2×C4×C53C2×D414yescomplemented
114C22⋊D53—216no—
115C22⋊D53—416no—
116C2×Dic53—21no—
117C2×Dic53—21no—
118C2×Dic53—22no—
119C2×Dic53—22no—
120C2×Dic53C2×D412yes—
121C2×Dic53C2×D414yescomplemented
122D212—28no—
123D212—416no—
124D212—48no—
125D212—44no—
126C4×D53—28no—
127C22×C53—24no—
128C22×C53—28no—
129C22×C53—42no—
130C22×C53—42no—
131C22×C53—44no—
132C22×C53C22×D411yescharacteristic
133C22×C53C22×D411yescharacteristic
134C22×C53C22×D412yes—
135C22×C53C22×D411yescharacteristic
136C22×C53H(5,𝔽2)12yes—
137C2×D53—22no—
138C2×D53—22no—
139C2×D53—24no—
140C2×D53—24no—
141C2×D53—24no—
142C2×D53—28no—
143C2×D53—432no—
144C2×D53—42no—
145C2×D53—42no—
146C2×D53—44no—
147C2×D53—42no—
148C2×D53—42no—
149C2×D53—42no—
150C2×D53—44no—
151C2×D53—44no—
152C2×D53#12714yescomplemented
153C4×C53—24no—
154C4×C53—42no—
155C4×C53#12014yescomplemented
156Dic53—22no—
157Dic53—24no—
158#2433—531nomaximal, Sylow
159C2×C53—44no—
160C2×C53C2×#12011yescharacteristic
161C2×C53C2×#12711yescharacteristic
162C2×C53#53511yescharacteristic
163C2×C53D4212yes—
164C2×C53D4212yes—
165D53—24no—
166D53—44no—
167D53—82no—
168C23×D4—531no—
169C2×#127—531no—
170C2×#127—531no—
171C2×#121—531no—
172C2×#120—534no—
173#392—534no—
174#390—531no—
175#390D5311yescharacteristic, complemented
176C4×C22⋊C4—531no—
177C53#243311yescharacteristic, complemented, Sylow
178C25—532no—
179C22×D4—532no—
180C22×D4—534no—
181C22×D4—531no—
182C22×D4—532no—
183C22×D4—531no—
184C22×D4—534no—
185C22×D4—1064no—
186C22×D4—1064no—
187C22×D4—1064no—
188C22×D4C2×D5311yescharacteristic
189C23×C4—531no—
190#127—1064no—
191#127—1064no—
192#121—1064no—
193#120—10616no—
194C2×C4⋊C4C2×D5311yescharacteristic, complemented
195C2×C22⋊C4—532no—
196C2×C22⋊C4—534no—
197C2×C22⋊C4—532no—
198C2×C22⋊C4C2×D5314yescomplemented
199C2×C42—531no—
200C2×C42C2×D5311yescharacteristic, complemented
201#95—532no—
202C24—531no—
203C24—532no—
204C24—532no—
205C24—1064no—
206C24—1064no—
207C24—1068no—
208C24—1064no—
209C24—2124no—
210C24C22×D5312yes—
211C2×D4—22no—
212C2×D4—22no—
213C2×D4—534no—
214C2×D4—1068no—
215C2×D4—1064no—
216C2×D4—1068no—
217C2×D4—1064no—
218C2×D4—1068no—
219C2×D4—10632no—
220C2×D4—1064no—
221C2×D4—1064no—
222C2×D4—1068no—
223C2×D4—1062no—
224C2×D4—1062no—
225C2×D4—1062no—
226C2×D4—1062no—
227C2×D4—1062no—
228C2×D4—1068no—
229C2×D4—2124no—
230C2×D4—2124no—
231C2×D4D21214yescomplemented
232C22×C4—531no—
233C22×C4—532no—
234C22×C4—534no—
235C22×C4—1062no—
236C22×C4—1062no—
237C22×C4C22×D5312yes—
238C22×C4C22×D5312yes—
239C22×C4C22×D5311yescharacteristic
240C4⋊C4—22no—
241C22⋊C4—28no—
242C22⋊C4—538no—
243C22⋊C4—1068no—
244C22⋊C4—1064no—
245C42—22no—
246C42—1062no—
247C23—24no—
248C23—22no—
249C23—22no—
250C23—24no—
251C23—534no—
252C23—532no—
253C23—534no—
254C23—10616no—
255C23—1061no—
256C23—1061no—
257C23—1062no—
258C23—1064no—
259C23—1062no—
260C23—1062no—
261C23—1064no—
262C23—1062no—
263C23—1062no—
264C23—1064no—
265C23—1062no—
266C23—1064no—
267C23—21216no—
268C23—2128no—
269C23—21216no—
270C23—2128no—
271C23C2×D21214yes—
272C23C23×D5311yescharacteristic
273D4—28no—
274D4—10616no—
275D4—1068no—
276D4—21216no—
277D4—2128no—
278D4—21216no—
279D4—2124no—
280C2×C4—22no—
281C2×C4—22no—
282C2×C4—24no—
283C2×C4—22no—
284C2×C4—532no—
285C2×C4—534no—
286C2×C4—1061no—
287C2×C4—1062no—
288C2×C4—1061no—
289C2×C4—1062no—
290C2×C4—1068no—
291C2×C4C2×D21212yes—
292C2×C4D4×D5312yes—
293C2×C4D4×D5312yes—
294C2×C4D4×D5314yescomplemented
295C22—24no—
296C22—28no—
297C22—42no—
298C22—42no—
299C22—44no—
300C22—534no—
301C22—1062no—
302C22—1062no—
303C22—1064no—
304C22—1064no—
305C22—1064no—
306C22—1068no—
307C22—21232no—
308C22—2122no—
309C22—2122no—
310C22—2122no—
311C22—2124no—
312C22—2124no—
313C22—2122no—
314C22—2122no—
315C22—2124no—
316C22C2×D4×D5311yescharacteristic
317C22C2×D4×D5311yescharacteristic
318C22C2×D4×D5312yes—
319C22C22×D21211yescharacteristic
320C22#1313326312yes—
321C4—24no—
322C4—42no—
323C4—1062no—
324C4—1064no—
325C4#1313318714yescomplemented
326C2—44no—
327C2—1064no—
328C2—4242no—
329C2—2124no—
330C2C2×#1313318711yescharacteristic
331C2#127×D5311yescharacteristic
332C2#1681119111yescharacteristic
333C2#1681125712yes—
334C2D4×D21212yes—
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#390C2
hypercenter#390D53
hypocenterC53#2433
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #390

Complements

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

NH ≅ G/Ntype
1. C22×D4×D53C2semidirect
2. C2×#13133214C2semidirect
3. C2×#13133213C2semidirect
4. C2×#13133192C2semidirect
5. C2×#13133187C2semidirect
6. #16810532C2semidirect
7. #16810524C2semidirect
8. C4×#11719921C2semidirect
9. C53×#390C2semidirect
13. C22×D212C22semidirect
14. C22×D212C22semidirect
16. C22×C22⋊D53C22semidirect
26. C2×#11719921C22semidirect
27. C2×#11719921C22semidirect
28. C2×C4×Dic53C22semidirect
29. #13133124C22semidirect
31. C2×C4⋊C4×C53C22semidirect
32. C2×C22⋊C4×C53C22semidirect
33. C2×C42×C53C22semidirect
45. C2×D4×C53D4semidirect
68. C2×D212D4semidirect
71. C2×C4×D53D4semidirect
77. #11719921D4semidirect
113. C2×C4×C53C2×D4semidirect
121. C2×Dic53C2×D4semidirect
152. C2×D53#127semidirect
155. C4×C53#120semidirect
175. #390D53semidirect
177. C53#2433semidirect
194. C2×C4⋊C4C2×D53semidirect
198. C2×C22⋊C4C2×D53semidirect
200. C2×C42C2×D53semidirect
231. C2×D4D212semidirect
294. C2×C4D4×D53semidirect
325. C4#13133187semidirect