Finite Group Databasebeta

← #20279672

Subgroups — #20279672

Order 6784.

724 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 724 conjugacy classes of subgroups — too many to draw. Only its 147 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×#13133192C211yesmaximal, characteristic, complemented
2C2×#13133190C212yesmaximal, complemented
3C2×#13133177C211yesmaximal, characteristic, complemented
4#16810554C211yesmaximal, characteristic, complemented
5#16810541C211yesmaximal, characteristic, complemented
6#16810532C211yesmaximal, characteristic, complemented
7#16810512C212yesmaximal, complemented
8#16810501C212yesmaximal, complemented
9#16810500C212yesmaximal, complemented
10#95×D53C211yesmaximal, characteristic, complemented
11C53×#397C211yesmaximal, characteristic, complemented
12C22×D212C2212yes—
13C22×D212C2211yescharacteristic
14C22×C22⋊D53C2212yes—
15C22×C22⋊D53C2211yescharacteristic
16C23×Dic53C2211yescharacteristic
17C22×C4×D53C2212yes—
18C22×C4×D53C2211yescharacteristic
19#13133192—24no—
20#13133190—28no—
21#13133177—24no—
22C2×#11719919C2212yes—
23C2×#11719919C2211yescharacteristic
24C2×#11719921C2212yes—
25C2×#11719921C2211yescharacteristic
26C2×#11719921C2212yes—
27C2×#11719921C2212yes—
28C2×#11719921C2212yes—
29C2×C4×Dic53C2211yescharacteristic
30#13133124C2212yes—
31#13133124C2212yes—
32#13133124C2211yescharacteristic
33C2×C4⋊C4×C53C2211yescharacteristic
34C2×C22⋊C4×C53C2212yes—
35C2×C22⋊C4×C53C2211yescharacteristic
36#95×C53C2212yes—
37#95×C53C2211yescharacteristic
38C24×C53C2311yescharacteristic
39C23×D53C2312yes—
40C23×D53C2311yescharacteristic
41C22×C4×C53C2312yes—
42C22×C4×C53C2311yescharacteristic
43C22×C4×C53C2312yes—
44C22×C4×C53C2311yescharacteristic
45C2×C22⋊D53—22no—
46C2×C22⋊D53—22no—
47C2×C22⋊D53—24no—
48C2×C22⋊D53—24no—
49C2×C22⋊D53—42no—
50C2×C22⋊D53D414yescomplemented
51C22×Dic53—22no—
52C22×Dic53—42no—
53C22×Dic53C2311yescharacteristic
54C22×Dic53C2311yescharacteristic
55C22×Dic53C2311yescharacteristic
56C22×Dic53C2312yes—
57C2×D212—24no—
58C2×D212—24no—
59C2×D212—24no—
60C2×D212—22no—
61C2×D212—42no—
62C2×C4×D53—24no—
63C2×C4×D53—24no—
64C2×C4×D53—42no—
65C2×C4×D53—42no—
66C2×C4×D53D414yescomplemented
67C4⋊C4×C53—22no—
68C22⋊C4×C53—24no—
69C22⋊C4×C53—22no—
70#11719921—22no—
71#11719921—24no—
72#11719921—24no—
73#11719921—24no—
74#11719921—42no—
75#11719919—22no—
76#11719919—24no—
77C4×Dic53—22no—
78C23×C53—21no—
79C23×C53—21no—
80C23×C53—22no—
81C23×C53—22no—
82C23×C53C2×D412yes—
83C23×C53C2411yescharacteristic
84C22×D53—22no—
85C22×D53—22no—
86C22×D53—22no—
87C22×D53—22no—
88C22×D53—22no—
89C22×D53—22no—
90C22×D53—22no—
91C22×D53—22no—
92C22×D53—22no—
93C22×D53C2×D412yes—
94C22×D53C2×D412yes—
95C22×D53#4112yes—
96C2×C4×C53—22no—
97C2×C4×C53—22no—
98C2×C4×C53—22no—
99C2×C4×C53—22no—
100C2×C4×C53—21no—
101C2×C4×C53—22no—
102C2×C4×C53—22no—
103C2×C4×C53—22no—
104C2×C4×C53—22no—
105C2×C4×C53C2×D412yes—
106C22⋊D53—28no—
107C22⋊D53—48no—
108C2×Dic53—22no—
109C2×Dic53—21no—
110C2×Dic53—21no—
111C2×Dic53—22no—
112C2×Dic53—22no—
113C2×Dic53—21no—
114C2×Dic53—22no—
115C2×Dic53—44no—
116C2×Dic53C2×D412yes—
117C2×Dic53C2×D412yes—
118C2×Dic53#4114yes—
119D212—44no—
120D212—48no—
121C4×D53—44no—
122C4×D53—48no—
123C22×C53—22no—
124C22×C53—24no—
125C22×C53—41no—
126C22×C53—41no—
127C22×C53—42no—
128C22×C53C22×D411yescharacteristic
129C22×C53C22×D411yescharacteristic
130C22×C53C2×#4111yescharacteristic
131C22×C53C2×#4112yes—
132C22×C53H(5,𝔽2)12yes—
133C2×D53—22no—
134C2×D53—22no—
135C2×D53—24no—
136C2×D53—24no—
137C2×D53—42no—
138C2×D53—42no—
139C2×D53—42no—
140C2×D53—42no—
141C2×D53—42no—
142C2×D53—42no—
143C2×D53—42no—
144C2×D53#12114yes—
145C4×C53—22no—
146C4×C53—42no—
147C4×C53—41no—
148C4×C53—42no—
149Dic53—22no—
150Dic53—22no—
151Dic53—24no—
152Dic53—41no—
153#2543—531nomaximal, Sylow
154C2×C53—42no—
155C2×C53C2×#12111yescharacteristic
156C2×C53#53211yescharacteristic
157C2×C53#53411yescharacteristic
158C2×C53#54612yes—
159C2×C53#55312yes—
160D53—24no—
161D53—82no—
162C2×#121—531no—
163C2×#121—532no—
164C2×C4×D4—531no—
165#398—531no—
166#397D5311yescharacteristic, complemented
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×D4—532no—
177C22×D4—531no—
178C23×C4—531no—
179C23×C4—531no—
180C23×C4—532no—
181#121—1068no—
182#121—1064no—
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—532no—
191C2×C22⋊C4—532no—
192C2×C22⋊C4C2×D5311yescharacteristic
193C2×C22⋊C4C2×D5312yes—
194C2×C42—531no—
195#95—532no—
196#95—531no—
197#95—532no—
198#95C2×D5311yescharacteristic
199#95C2×D5312yes—
200C24—531no—
201C24—532no—
202C24C22×D5311yescharacteristic
203C2×D4—534no—
204C2×D4—1064no—
205C2×D4—1064no—
206C2×D4—1064no—
207C2×D4—1064no—
208C2×D4—1062no—
209C2×D4—1062no—
210C2×D4—1064no—
211C2×D4—1062no—
212C2×D4—2122no—
213C2×D4—2122no—
214C22×C4—532no—
215C22×C4—531no—
216C22×C4—531no—
217C22×C4—534no—
218C22×C4—531no—
219C22×C4—1064no—
220C22×C4—1064no—
221C22×C4—1062no—
222C22×C4—2122no—
223C22×C4—2122no—
224C22×C4—2122no—
225C22×C4C22×D5312yes—
226C22×C4C22×D5312yes—
227C22×C4C22×D5311yescharacteristic
228C22×C4C22×D5311yescharacteristic
229C4⋊C4—22no—
230C4⋊C4—1064no—
231C4⋊C4—1062no—
232C22⋊C4—24no—
233C22⋊C4—22no—
234C22⋊C4—1064no—
235C22⋊C4—1064no—
236C22⋊C4—1064no—
237C22⋊C4—1062no—
238C22⋊C4—2122no—
239C42—1062no—
240C23—21no—
241C23—22no—
242C23—21no—
243C23—22no—
244C23—532no—
245C23—532no—
246C23—532no—
247C23—1062no—
248C23—1062no—
249C23—1062no—
250C23—1062no—
251C23—1062no—
252C23—1062no—
253C23—1062no—
254C23—1062no—
255C23—1062no—
256C23D4×D5312yes—
257C23C23×D5311yescharacteristic
258D4—1068no—
259D4—2124no—
260D4—2128no—
261D4—2128no—
262C2×C4—22no—
263C2×C4—21no—
264C2×C4—22no—
265C2×C4—22no—
266C2×C4—22no—
267C2×C4—22no—
268C2×C4—22no—
269C2×C4—22no—
270C2×C4—22no—
271C2×C4—532no—
272C2×C4—532no—
273C2×C4—534no—
274C2×C4—1061no—
275C2×C4—1062no—
276C2×C4—1061no—
277C2×C4—1062no—
278C2×C4—1062no—
279C2×C4—1061no—
280C2×C4—1062no—
281C2×C4—2124no—
282C2×C4—2128no—
283C2×C4—2124no—
284C2×C4D4×D5312yes—
285C22—24no—
286C22—22no—
287C22—42no—
288C22—41no—
289C22—41no—
290C22—534no—
291C22—1064no—
292C22—1064no—
293C22—1062no—
294C22—1062no—
295C22—2122no—
296C22—2122no—
297C22—2122no—
298C22—2122no—
299C22—2122no—
300C22—2122no—
301C22—2122no—
302C22C2×D4×D5311yescharacteristic
303C22C2×D4×D5311yescharacteristic
304C22#41×D5312yes—
305C22#41×D5311yescharacteristic
306C22#1313326312yes—
307C4—22no—
308C4—42no—
309C4—42no—
310C4—41no—
311C4—1064no—
312C4—1062no—
313C4—1062no—
314C4—2121no—
315C2—42no—
316C2—1064no—
317C2—4242no—
318C2#121×D5311yescharacteristic
319C2#1681117211yescharacteristic
320C2#1681118411yescharacteristic
321C2#1681126512yes—
322C2#1681132412yes—
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#397C2
hypercenter#397D53
hypocenterC53#2543
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #397

Complements

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

NH ≅ G/Ntype
1. C2×#13133192C2semidirect
2. C2×#13133190C2semidirect
3. C2×#13133177C2semidirect
4. #16810554C2semidirect
5. #16810541C2semidirect
6. #16810532C2semidirect
7. #16810512C2semidirect
8. #16810501C2semidirect
9. #16810500C2semidirect
10. #95×D53C2semidirect
11. C53×#397C2semidirect
50. C2×C22⋊D53D4semidirect
66. C2×C4×D53D4semidirect
166. #397D53semidirect
173. C53#2543semidirect