Finite Group Databasebeta

← #20279648

Subgroups — #20279648

Order 6784.

760 conjugacy classes of subgroups; only the 161 normal subgroups are shown.

normal subgroups only

This group has 760 conjugacy classes of subgroups — too many to draw. Only its 161 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
1C2×#13133203C211yesmaximal, characteristic, complemented
2C2×#13133194C212yesmaximal, complemented
3C2×#13133189C212yesmaximal, complemented
4C2×D4×Dic53C211yesmaximal, characteristic, complemented
5#16810560C211yesmaximal, characteristic, complemented
6#16810559C211yesmaximal, characteristic, complemented
7#16810531C211yesmaximal, characteristic, complemented
8#16810526C212yesmaximal, complemented
9#16810523C212yesmaximal, complemented
10#16810474C211yesmaximal, characteristic, complemented
11C53×#392C211yesmaximal, characteristic, complemented
12C22×Dic106C2211yescharacteristic
13C22×C22⋊D53C2211yescharacteristic
14C22×C22⋊D53C2212yes—
15C23×Dic53C2212yes—
16#13133203—24no—
17#13133194—28no—
18#13133189—28no—
19D4×Dic53—24no—
20C2×C4⋊Dic53C2212yes—
21C2×C4⋊Dic53C2211yescharacteristic
22C2×#11719919C2212yes—
23C2×#11719919C2212yes—
24C2×C22⋊Dic53C2212yes—
25C2×C22⋊Dic53C2211yescharacteristic
26C2×C22⋊Dic53C2212yes—
27C2×C22⋊Dic53C2212yes—
28C2×#11719921C2211yescharacteristic
29C2×#11719921C2212yes—
30C2×#11719921C2211yescharacteristic
31C2×C4×Dic53C2212yes—
32C2×C4×Dic53C2211yescharacteristic
33#13133124C2211yescharacteristic
34C22×D4×C53C2211yescharacteristic
35C22×D4×C53C2212yes—
36C2×C22⋊C4×C53C2212yes—
37C2×C22⋊C4×C53C2211yescharacteristic
38#95×C53C2211yescharacteristic
39C24×C53C2311yescharacteristic
40C24×C53C2312yes—
41C23×D53C2311yescharacteristic
42C2×D4×C53—24no—
43C2×D4×C53—24no—
44C2×D4×C53—22no—
45C2×D4×C53—24no—
46C2×D4×C53—42no—
47C22×C4×C53C2311yescharacteristic
48C22×C4×C53C2311yescharacteristic
49C22×C4×C53C2312yes—
50C2×C22⋊D53—24no—
51C2×C22⋊D53—24no—
52C2×C22⋊D53—24no—
53C2×C22⋊D53—22no—
54C2×C22⋊D53—42no—
55C22×Dic53—24no—
56C22×Dic53—24no—
57C22×Dic53—24no—
58C22×Dic53C2311yescharacteristic
59C22×Dic53C2312yes—
60C22×Dic53C2312yes—
61C22×Dic53C2311yescharacteristic
62C22×Dic53C2311yescharacteristic
63C2×Dic106—24no—
64C2×Dic106—22no—
65C22⋊C4×C53—24no—
66C22⋊C4×C53—22no—
67C22⋊Dic53—24no—
68C22⋊Dic53—24no—
69C22⋊Dic53—24no—
70C22⋊Dic53—22no—
71#11719921—24no—
72#11719921—22no—
73#11719921—22no—
74C4⋊Dic53—22no—
75C4⋊Dic53—24no—
76#11719919—24no—
77#11719919D418yescomplemented
78C4×Dic53—42no—
79C4×Dic53D414yescomplemented
80C23×C53—22no—
81C23×C53—22no—
82C23×C53—22no—
83C23×C53—22no—
84C23×C53—22no—
85C23×C53—22no—
86C23×C53—22no—
87C23×C53—21no—
88C23×C53—22no—
89C23×C53—21no—
90C23×C53—21no—
91C23×C53#4114yes—
92C23×C53C2411yescharacteristic
93C22×D53—22no—
94C22×D53—21no—
95C22×D53—22no—
96C22×D53—21no—
97C22×D53—21no—
98D4×C53—28no—
99D4×C53—48no—
100C2×C4×C53—22no—
101C2×C4×C53—22no—
102C2×C4×C53—22no—
103C2×C4×C53—22no—
104C2×C4×C53—21no—
105C2×C4×C53C2×D412yes—
106C2×C4×C53C2×D414yes—
107C22⋊D53—48no—
108C22⋊D53—44no—
109C2×Dic53—22no—
110C2×Dic53—22no—
111C2×Dic53—21no—
112C2×Dic53—21no—
113C2×Dic53—22no—
114C2×Dic53—21no—
115C2×Dic53—22no—
116C2×Dic53—22no—
117C2×Dic53—22no—
118C2×Dic53—48no—
119C2×Dic53C2×D414yes—
120C2×Dic53C2×D412yes—
121C2×Dic53C2×D412yes—
122C2×Dic53C2×D414yes—
123Dic106—44no—
124C22×C53—24no—
125C22×C53—24no—
126C22×C53—24no—
127C22×C53—42no—
128C22×C53—42no—
129C22×C53—42no—
130C22×C53—42no—
131C22×C53—41no—
132C22×C53—41no—
133C22×C53—41no—
134C22×C53—42no—
135C22×C53—42no—
136C22×C53C22×D411yescharacteristic
137C22×C53C22×D412yes—
138C22×C53C2×#4112yes—
139C22×C53H(5,𝔽2)11yescharacteristic
140C22×C532−1+411yescharacteristic
141C2×D53—41no—
142C2×D53—41no—
143C2×D53—41no—
144C2×D53—42no—
145C2×D53—42no—
146C4×C53—22no—
147C4×C53—24no—
148C4×C53—41no—
149Dic53—22no—
150Dic53—24no—
151Dic53—41no—
152Dic53—42no—
153Dic53#12714yes—
154#2540—531nomaximal, Sylow
155C2×C53—44no—
156C2×C53—81no—
157C2×C53#52511yescharacteristic
158C2×C53C2×#12711yescharacteristic
159C2×C53#53711yescharacteristic
160C2×C53#54612yes—
161C2×C53#54812yes—
162D53—81no—
163C2×#124—531no—
164C2×C22⋊Q8—532no—
165C2×#121—532no—
166C2×C4×D4—531no—
167#399—531no—
168#399—531no—
169#392—531no—
170#392D5311yescharacteristic, complemented
171#390—532no—
172#390—532no—
173C4×C4⋊C4—531no—
174C53#254011yescharacteristic, complemented, Sylow
175C22×Q8—531no—
176C22×D4—531no—
177C22×D4—532no—
178C22×D4C2×D5311yescharacteristic
179C22×D4C2×D5312yes—
180C23×C4—532no—
181#124—1064no—
182C22⋊Q8—1068no—
183#121—1068no—
184C4×D4—1064no—
185C2×C4⋊C4—532no—
186C2×C4⋊C4—531no—
187C2×C4⋊C4—532no—
188C2×C4⋊C4—532no—
189C2×C22⋊C4—532no—
190C2×C22⋊C4—532no—
191C2×C22⋊C4—532no—
192C2×C22⋊C4—531no—
193C2×C22⋊C4—532no—
194C2×C22⋊C4—531no—
195C2×C22⋊C4—531no—
196C2×C22⋊C4C2×D5312yes—
197C2×C22⋊C4C2×D5311yescharacteristic
198C2×C42—531no—
199C2×C42—532no—
200#95—531no—
201#95C2×D5311yescharacteristic
202C24—531no—
203C24C22×D5311yescharacteristic
204C24C22×D5312yes—
205C2×Q8—1064no—
206C2×Q8—1062no—
207C2×D4—24no—
208C2×D4—24no—
209C2×D4—22no—
210C2×D4—24no—
211C2×D4—42no—
212C2×D4—1064no—
213C2×D4—1064no—
214C2×D4—1062no—
215C2×D4—1064no—
216C2×D4—2122no—
217C22×C4—531no—
218C22×C4—532no—
219C22×C4—531no—
220C22×C4—531no—
221C22×C4—532no—
222C22×C4—1064no—
223C22×C4—1064no—
224C22×C4—1064no—
225C22×C4C22×D5311yescharacteristic
226C22×C4C22×D5311yescharacteristic
227C22×C4C22×D5312yes—
228C4⋊C4—538no—
229C4⋊C4—1064no—
230C4⋊C4—1062no—
231C4⋊C4—1064no—
232C22⋊C4—24no—
233C22⋊C4—22no—
234C22⋊C4—1062no—
235C22⋊C4—1064no—
236C22⋊C4—1064no—
237C22⋊C4—1064no—
238C22⋊C4—1062no—
239C22⋊C4—1064no—
240C22⋊C4—1062no—
241C42—534no—
242C42—2122no—
243C23—22no—
244C23—22no—
245C23—22no—
246C23—22no—
247C23—22no—
248C23—22no—
249C23—21no—
250C23—22no—
251C23—22no—
252C23—21no—
253C23—21no—
254C23—1061no—
255C23—1062no—
256C23—1062no—
257C23—1061no—
258C23—1061no—
259C23#1171994714yes—
260C23C23×D5311yescharacteristic
261Q8—2124no—
262D4—28no—
263D4—48no—
264D4—2124no—
265D4—2128no—
266C2×C4—22no—
267C2×C4—22no—
268C2×C4—22no—
269C2×C4—21no—
270C2×C4—22no—
271C2×C4—532no—
272C2×C4—534no—
273C2×C4—532no—
274C2×C4—534no—
275C2×C4—1062no—
276C2×C4—1061no—
277C2×C4—1062no—
278C2×C4—1061no—
279C2×C4—1062no—
280C2×C4—1061no—
281C2×C4—1062no—
282C2×C4—1062no—
283C2×C4—1062no—
284C2×C4—2128no—
285C2×C4D4×D5312yes—
286C2×C4D4×D5314yes—
287C22—24no—
288C22—24no—
289C22—24no—
290C22—42no—
291C22—42no—
292C22—42no—
293C22—42no—
294C22—42no—
295C22—41no—
296C22—41no—
297C22—41no—
298C22—42no—
299C22—2122no—
300C22—2121no—
301C22—2121no—
302C22—2121no—
303C22—2122no—
304C22C2×D4×D5312yes—
305C22C2×D4×D5311yescharacteristic
306C22C2×#1171994712yes—
307C22#1313326211yescharacteristic
308C22#1313326411yescharacteristic
309C4—22no—
310C4—24no—
311C4—41no—
312C4—534no—
313C4—1062no—
314C4—1064no—
315C4—2122no—
316C4—2121no—
317C2—44no—
318C2—81no—
319C2—4241no—
320C2#1681112911yescharacteristic
321C2#127×D5311yescharacteristic
322C2#1681120211yescharacteristic
323C2#1681126012yes—
324C2#1681127912yes—
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#392C2
hypercenter#392D53
hypocenterC53#2540
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #392

Complements

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

NH ≅ G/Ntype
1. C2×#13133203C2semidirect
2. C2×#13133194C2semidirect
3. C2×#13133189C2semidirect
4. C2×D4×Dic53C2semidirect
5. #16810560C2semidirect
6. #16810559C2semidirect
7. #16810531C2semidirect
8. #16810526C2semidirect
9. #16810523C2semidirect
10. #16810474C2semidirect
11. C53×#392C2semidirect
77. #11719919D4semidirect
79. C4×Dic53D4semidirect
170. #392D53semidirect
174. C53#2540semidirect