Finite Group Databasebeta

← #20279677

Subgroups — #20279677

Order 6784.

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

normal subgroups only

This group has 724 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×#13133189C212yesmaximal, complemented
2C2×C4×C22⋊D53C211yesmaximal, characteristic, complemented
3#16810554C211yesmaximal, characteristic, complemented
4#16810549C211yesmaximal, characteristic, complemented
5#16810537C211yesmaximal, characteristic, complemented
6#16810531C211yesmaximal, characteristic, complemented
7#16810511C212yesmaximal, complemented
8#16810503C212yesmaximal, complemented
9#16810499C212yesmaximal, complemented
10C2×#13133124C211yesmaximal, characteristic, complemented
11C2×#121×C53C211yesmaximal, characteristic, complemented
12C22×C22⋊D53C2212yes—
13C22×C22⋊D53C2211yescharacteristic
14C23×Dic53C2212yes—
15C22×C4×D53C2211yescharacteristic
16#13133189—28no—
17C4×C22⋊D53—24no—
18C2×#11719919C2212yes—
19C2×#11719919C2211yescharacteristic
20C2×C22⋊Dic53C2211yescharacteristic
21C2×C22⋊Dic53C2212yes—
22C2×C22⋊Dic53C2212yes—
23C2×C22⋊Dic53C2212yes—
24C2×#11719921C2212yes—
25C2×#11719921C2211yescharacteristic
26C2×C4×Dic53C2211yescharacteristic
27#13133124C2212yes—
28#13133124C2212yes—
29#13133124C2212yes—
30#13133124C2211yescharacteristic
31#13133124C2211yescharacteristic
32C22×D4×C53C2212yes—
33C22×D4×C53C2211yescharacteristic
34C23×C4×C53C2211yescharacteristic
35#121×C53—24no—
36C2×C4⋊C4×C53C2211yescharacteristic
37C2×C22⋊C4×C53C2212yes—
38C24×C53C2312yes—
39C24×C53C2311yescharacteristic
40C23×D53C2311yescharacteristic
41C2×D4×C53—24no—
42C2×D4×C53—24no—
43C2×D4×C53—24no—
44C2×D4×C53—22no—
45C2×D4×C53—42no—
46C22×C4×C53—42no—
47C22×C4×C53D414yescomplemented
48C22×C4×C53C2312yes—
49C22×C4×C53C2311yescharacteristic
50C22×C4×C53C2311yescharacteristic
51C2×C22⋊D53—22no—
52C2×C22⋊D53—22no—
53C2×C22⋊D53—24no—
54C2×C22⋊D53—24no—
55C2×C22⋊D53—42no—
56C2×C22⋊D53D414yescomplemented
57C22×Dic53—24no—
58C22×Dic53—24no—
59C22×Dic53—42no—
60C22×Dic53C2312yes—
61C22×Dic53C2311yescharacteristic
62C22×Dic53C2311yescharacteristic
63C22×Dic53C2311yescharacteristic
64C22×Dic53C2312yes—
65C2×C4×D53—22no—
66C2×C4×D53—42no—
67C4⋊C4×C53—22no—
68C22⋊C4×C53—24no—
69C22⋊Dic53—22no—
70C22⋊Dic53—24no—
71C22⋊Dic53—24no—
72C22⋊Dic53—42no—
73#11719921—24no—
74#11719921—22no—
75#11719919—22no—
76#11719919—24no—
77C4×Dic53—22no—
78C23×C53—22no—
79C23×C53—22no—
80C23×C53—22no—
81C23×C53—22no—
82C23×C53—22no—
83C23×C53—22no—
84C23×C53—22no—
85C23×C53—22no—
86C23×C53—22no—
87C23×C53C2×D412yes—
88C23×C53C2×D412yes—
89C23×C53#4112yes—
90C23×C53C2411yescharacteristic
91C22×D53—21no—
92C22×D53—22no—
93C22×D53—21no—
94C22×D53—22no—
95C22×D53C2×D412yes—
96D4×C53—44no—
97D4×C53—48no—
98C2×C4×C53—22no—
99C2×C4×C53—22no—
100C2×C4×C53—22no—
101C2×C4×C53—22no—
102C2×C4×C53—22no—
103C2×C4×C53—44no—
104C2×C4×C53C2×D412yes—
105C2×C4×C53C2×D412yes—
106C22⋊D53—28no—
107C22⋊D53—48no—
108C2×Dic53—22no—
109C2×Dic53—21no—
110C2×Dic53—21no—
111C2×Dic53—22no—
112C2×Dic53—21no—
113C2×Dic53—22no—
114C2×Dic53—21no—
115C2×Dic53—22no—
116C2×Dic53—22no—
117C2×Dic53—22no—
118C2×Dic53—22no—
119C2×Dic53—48no—
120C2×Dic53C2×D412yes—
121C2×Dic53#4114yes—
122C4×D53—44no—
123C22×C53—22no—
124C22×C53—22no—
125C22×C53—24no—
126C22×C53—24no—
127C22×C53—42no—
128C22×C53—42no—
129C22×C53—42no—
130C22×C53—42no—
131C22×C53—42no—
132C22×C53—42no—
133C22×C53—42no—
134C22×C53#12114yes—
135C22×C53C22×D411yescharacteristic
136C22×C53C22×D411yescharacteristic
137C22×C53C2×#4112yes—
138C22×C53C2×#4111yescharacteristic
139C22×C53H(5,𝔽2)12yes—
140C2×D53—22no—
141C2×D53—24no—
142C2×D53—41no—
143C2×D53—41no—
144C2×D53—42no—
145C4×C53—22no—
146C4×C53—22no—
147C4×C53—42no—
148Dic53—22no—
149Dic53—24no—
150Dic53—41no—
151Dic53—41no—
152Dic53—42no—
153#2543—531nomaximal, Sylow
154C2×C53—24no—
155C2×C53—82no—
156C2×C53C2×#12111yescharacteristic
157C2×C53#53211yescharacteristic
158C2×C53#53411yescharacteristic
159C2×C53#54612yes—
160C2×C53#55312yes—
161D53—42no—
162C2×#121—532no—
163C2×#121D5311yescharacteristic, complemented
164C2×C4×D4—531no—
165#398—531no—
166#397—531no—
167#394—531no—
168#392—531no—
169#388—532no—
170#386—532no—
171#386—532no—
172C2×#95—531no—
173C53#254311yescharacteristic, complemented, Sylow
174C22×D4—531no—
175C22×D4—532no—
176C22×D4C2×D5312yes—
177C22×D4C2×D5311yescharacteristic
178C23×C4—532no—
179C23×C4—531no—
180C23×C4C2×D5311yescharacteristic
181#121—24no—
182#121—1068no—
183C4×D4—1064no—
184C2×C4⋊C4—531no—
185C2×C4⋊C4—532no—
186C2×C4⋊C4C2×D5311yescharacteristic
187C2×C22⋊C4—531no—
188C2×C22⋊C4—532no—
189C2×C22⋊C4—532no—
190C2×C22⋊C4—531no—
191C2×C22⋊C4—532no—
192C2×C22⋊C4—532no—
193C2×C22⋊C4C2×D5312yes—
194C2×C42—531no—
195#95—532no—
196#95—531no—
197#95—531no—
198#95—532no—
199#95—532no—
200C24—531no—
201C24C22×D5312yes—
202C24C22×D5311yescharacteristic
203C2×D4—24no—
204C2×D4—24no—
205C2×D4—24no—
206C2×D4—22no—
207C2×D4—42no—
208C2×D4—534no—
209C2×D4—1064no—
210C2×D4—1062no—
211C2×D4—1062no—
212C2×D4—1064no—
213C2×D4—2122no—
214C22×C4—42no—
215C22×C4—532no—
216C22×C4—531no—
217C22×C4—532no—
218C22×C4—531no—
219C22×C4—531no—
220C22×C4—1064no—
221C22×C4—1064no—
222C22×C4—1062no—
223C22×C4—2122no—
224C22×C4—2122no—
225C22×C4C22⋊D5314yescomplemented
226C22×C4C22×D5312yes—
227C22×C4C22×D5311yescharacteristic
228C22×C4C22×D5311yescharacteristic
229C4⋊C4—22no—
230C4⋊C4—1064no—
231C4⋊C4—1062no—
232C22⋊C4—24no—
233C22⋊C4—1064no—
234C22⋊C4—1064no—
235C22⋊C4—1064no—
236C22⋊C4—1062no—
237C22⋊C4—1062no—
238C22⋊C4—2122no—
239C42—1062no—
240C23—22no—
241C23—22no—
242C23—22no—
243C23—22no—
244C23—22no—
245C23—22no—
246C23—22no—
247C23—22no—
248C23—22no—
249C23—532no—
250C23—1061no—
251C23—1062no—
252C23—1061no—
253C23—1062no—
254C23D4×D5312yes—
255C23#1171994712yes—
256C23C2×C22⋊D5312yes—
257C23C23×D5311yescharacteristic
258D4—44no—
259D4—48no—
260D4—1068no—
261D4—2128no—
262C2×C4—22no—
263C2×C4—22no—
264C2×C4—22no—
265C2×C4—22no—
266C2×C4—22no—
267C2×C4—44no—
268C2×C4—532no—
269C2×C4—534no—
270C2×C4—1062no—
271C2×C4—1061no—
272C2×C4—1061no—
273C2×C4—1062no—
274C2×C4—1062no—
275C2×C4—1061no—
276C2×C4—1061no—
277C2×C4—1062no—
278C2×C4—1062no—
279C2×C4—1062no—
280C2×C4—1062no—
281C2×C4—2124no—
282C2×C4—2128no—
283C2×C4C2×C22⋊D5312yes—
284C2×C4C2×C22⋊D5312yes—
285C22—24no—
286C22—22no—
287C22—22no—
288C22—24no—
289C22—42no—
290C22—42no—
291C22—42no—
292C22—42no—
293C22—42no—
294C22—42no—
295C22—42no—
296C22—1062no—
297C22—1064no—
298C22—2122no—
299C22—2121no—
300C22—2121no—
301C22#1313318814yes—
302C22C2×D4×D5311yescharacteristic
303C22C22×C22⋊D5311yescharacteristic
304C22#41×D5312yes—
305C22C2×#1171994711yescharacteristic
306C22#1313326212yes—
307C4—22no—
308C4—22no—
309C4—42no—
310C4—1062no—
311C4—1064no—
312C4—2121no—
313C4—2121no—
314C4—2122no—
315C2—24no—
316C2—82no—
317C2—2122no—
318C2C2×#1313318811yescharacteristic
319C2#1681116911yescharacteristic
320C2#1681118211yescharacteristic
321C2#1681126212yes—
322C2#1681131612yes—
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC2×#121×C53C2
hypercenterC2×#121D53
hypocenterC53#2543
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×#121

Complements

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

NH ≅ G/Ntype
1. C2×#13133189C2semidirect
2. C2×C4×C22⋊D53C2semidirect
3. #16810554C2semidirect
4. #16810549C2semidirect
5. #16810537C2semidirect
6. #16810531C2semidirect
7. #16810511C2semidirect
8. #16810503C2semidirect
9. #16810499C2semidirect
10. C2×#13133124C2semidirect
11. C2×#121×C53C2semidirect
47. C22×C4×C53D4semidirect
56. C2×C22⋊D53D4semidirect
163. C2×#121D53semidirect
173. C53#2543semidirect
225. C22×C4C22⋊D53semidirect