Finite Group Databasebeta

← #20279650

Subgroups — #20279650

Order 6784.

760 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 760 conjugacy classes of subgroups — too many to draw. Only its 157 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×#13133195C212yesmaximal, complemented
3C2×#13133190C212yesmaximal, complemented
4C2×#13133179C211yesmaximal, characteristic, complemented
5#16810558C211yesmaximal, characteristic, complemented
6#16810533C211yesmaximal, characteristic, complemented
7#16810532C211yesmaximal, characteristic, complemented
8#16810524C212yesmaximal, complemented
9#16810523C212yesmaximal, complemented
10#16810474C211yesmaximal, characteristic, complemented
11C53×#399C211yesmaximal, characteristic, complemented
12C22×Dic106C2211yescharacteristic
13C22×D212C2211yescharacteristic
14C22×D212C2212yes—
15C22×C22⋊D53C2211yescharacteristic
16C22×C22⋊D53C2212yes—
17C22×C4×D53C2212yes—
18#13133203—24no—
19#13133195—28no—
20#13133190—28no—
21#13133179—24no—
22C2×#11719919C2212yes—
23C2×#11719919C2212yes—
24C2×C22⋊Dic53C2211yescharacteristic
25C2×#11719921C2212yes—
26C2×#11719921C2212yes—
27C2×#11719921C2211yescharacteristic
28C2×#11719921C2212yes—
29C2×#11719921C2212yes—
30C2×#11719921C2211yescharacteristic
31C2×C4×Dic53C2212yes—
32C2×C4×Dic53C2211yescharacteristic
33#13133124C2211yescharacteristic
34C2×C4⋊C4×C53C2212yes—
35C2×C4⋊C4×C53C2211yescharacteristic
36C2×C22⋊C4×C53C2212yes—
37C2×C22⋊C4×C53C2211yescharacteristic
38#95×C53C2211yescharacteristic
39C24×C53C2311yescharacteristic
40C23×D53C2311yescharacteristic
41C23×D53C2312yes—
42C22×C4×C53C2311yescharacteristic
43C22×C4×C53C2312yes—
44C22×C4×C53C2311yescharacteristic
45C22×C4×C53C2312yes—
46C2×C22⋊D53—24no—
47C2×C22⋊D53—24no—
48C2×C22⋊D53—24no—
49C2×C22⋊D53—22no—
50C2×C22⋊D53—42no—
51C22×Dic53C2311yescharacteristic
52C22×Dic53C2312yes—
53C22×Dic53C2311yescharacteristic
54C22×Dic53C2311yescharacteristic
55C2×D212—24no—
56C2×D212—22no—
57C2×D212—24no—
58C2×D212—24no—
59C2×D212—42no—
60C2×C4×D53—24no—
61C2×C4×D53—24no—
62C2×C4×D53—24no—
63C2×Dic106—24no—
64C2×Dic106—22no—
65C4⋊C4×C53—22no—
66C4⋊C4×C53—24no—
67C22⋊C4×C53—24no—
68C22⋊C4×C53—22no—
69C22⋊Dic53—22no—
70#11719921—24no—
71#11719921—22no—
72#11719921—24no—
73#11719921—24no—
74#11719921—24no—
75#11719921—22no—
76#11719919—24no—
77#11719919D418yescomplemented
78C4×Dic53—42no—
79C4×Dic53D414yescomplemented
80C23×C53—22no—
81C23×C53—21no—
82C23×C53—22no—
83C23×C53—21no—
84C23×C53—21no—
85C23×C53C2411yescharacteristic
86C22×D53—22no—
87C22×D53—22no—
88C22×D53—22no—
89C22×D53—22no—
90C22×D53—22no—
91C22×D53—22no—
92C22×D53—22no—
93C22×D53—21no—
94C22×D53—22no—
95C22×D53—21no—
96C22×D53—21no—
97C22×D53#4114yes—
98C2×C4×C53—22no—
99C2×C4×C53—22no—
100C2×C4×C53—22no—
101C2×C4×C53—21no—
102C2×C4×C53—22no—
103C2×C4×C53—22no—
104C2×C4×C53—22no—
105C2×C4×C53—22no—
106C2×C4×C53C2×D412yes—
107C2×C4×C53C2×D414yes—
108C22⋊D53—48no—
109C22⋊D53—44no—
110C2×Dic53—22no—
111C2×Dic53—22no—
112C2×Dic53—21no—
113C2×Dic53—21no—
114C2×Dic53—22no—
115C2×Dic53—21no—
116C2×Dic53C2×D414yes—
117C2×Dic53C2×D412yes—
118C2×Dic53C2×D412yes—
119C2×Dic53C2×D414yes—
120D212—28no—
121D212—48no—
122C4×D53—48no—
123Dic106—44no—
124C22×C53—41no—
125C22×C53—41no—
126C22×C53—41no—
127C22×C53—42no—
128C22×C53—42no—
129C22×C53C22×D411yescharacteristic
130C22×C53C22×D412yes—
131C22×C53C2×#4112yes—
132C22×C53H(5,𝔽2)11yescharacteristic
133C22×C532−1+411yescharacteristic
134C2×D53—24no—
135C2×D53—24no—
136C2×D53—24no—
137C2×D53—42no—
138C2×D53—42no—
139C2×D53—42no—
140C2×D53—42no—
141C2×D53—41no—
142C2×D53—41no—
143C2×D53—41no—
144C2×D53—42no—
145C2×D53—42no—
146C4×C53—22no—
147C4×C53—24no—
148C4×C53—41no—
149C4×C53—42no—
150Dic53—22no—
151Dic53—24no—
152Dic53—41no—
153Dic53#12714yes—
154#2540—531nomaximal, Sylow
155C2×C53—81no—
156C2×C53#52511yescharacteristic
157C2×C53C2×#12711yescharacteristic
158C2×C53#53711yescharacteristic
159C2×C53#54612yes—
160C2×C53#54812yes—
161D53—44no—
162D53—81no—
163C2×#124—531no—
164C2×C22⋊Q8—532no—
165C2×#121—532no—
166C2×C4×D4—531no—
167#399—531no—
168#399D5311yescharacteristic, complemented
169#392—531no—
170#392—531no—
171#390—532no—
172#390—532no—
173C4×C4⋊C4—531no—
174C53#254011yescharacteristic, complemented, Sylow
175C22×Q8—531no—
176C22×D4—531no—
177C22×D4—532no—
178C22×D4—531no—
179C22×D4—532no—
180C23×C4—532no—
181#124—1064no—
182C22⋊Q8—1068no—
183#121—1068no—
184C4×D4—1064no—
185C2×C4⋊C4—532no—
186C2×C4⋊C4—532no—
187C2×C4⋊C4C2×D5311yescharacteristic
188C2×C4⋊C4C2×D5312yes—
189C2×C22⋊C4—531no—
190C2×C22⋊C4—532no—
191C2×C22⋊C4—532no—
192C2×C22⋊C4—532no—
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—532no—
203C24—531no—
204C24C22×D5311yescharacteristic
205C2×Q8—1064no—
206C2×Q8—1062no—
207C2×D4—1064no—
208C2×D4—1064no—
209C2×D4—1064no—
210C2×D4—1064no—
211C2×D4—1062no—
212C2×D4—1064no—
213C2×D4—1062no—
214C2×D4—1064no—
215C2×D4—2122no—
216C2×D4—2122no—
217C22×C4—531no—
218C22×C4—532no—
219C22×C4—531no—
220C22×C4—531no—
221C22×C4—1064no—
222C22×C4—1064no—
223C22×C4—1064no—
224C22×C4C22×D5311yescharacteristic
225C22×C4C22×D5312yes—
226C22×C4C22×D5311yescharacteristic
227C22×C4C22×D5312yes—
228C4⋊C4—22no—
229C4⋊C4—24no—
230C4⋊C4—538no—
231C4⋊C4—1064no—
232C22⋊C4—24no—
233C22⋊C4—22no—
234C22⋊C4—1064no—
235C22⋊C4—1064no—
236C22⋊C4—1064no—
237C22⋊C4—1062no—
238C22⋊C4—1062no—
239C22⋊C4—1064no—
240C22⋊C4—1062no—
241C42—534no—
242C42—2122no—
243C23—21no—
244C23—22no—
245C23—22no—
246C23—21no—
247C23—21no—
248C23—534no—
249C23—1062no—
250C23—1062no—
251C23—1062no—
252C23—1062no—
253C23—1062no—
254C23—1062no—
255C23—1061no—
256C23—1062no—
257C23—1062no—
258C23—1061no—
259C23—1061no—
260C23C23×D5311yescharacteristic
261Q8—2124no—
262D4—1068no—
263D4—2128no—
264D4—2124no—
265D4—2128no—
266C2×C4—22no—
267C2×C4—22no—
268C2×C4—22no—
269C2×C4—22no—
270C2×C4—21no—
271C2×C4—22no—
272C2×C4—22no—
273C2×C4—22no—
274C2×C4—532no—
275C2×C4—534no—
276C2×C4—532no—
277C2×C4—534no—
278C2×C4—1062no—
279C2×C4—1061no—
280C2×C4—1062no—
281C2×C4—1061no—
282C2×C4—1061no—
283C2×C4—1062no—
284C2×C4—2128no—
285C2×C4D4×D5312yes—
286C2×C4D4×D5314yes—
287C22—42no—
288C22—42no—
289C22—41no—
290C22—41no—
291C22—41no—
292C22—1064no—
293C22—1064no—
294C22—1064no—
295C22—2122no—
296C22—2122no—
297C22—2121no—
298C22—2121no—
299C22—2121no—
300C22—2122no—
301C22—2122no—
302C22—2122no—
303C22—2122no—
304C22C2×D4×D5312yes—
305C22C2×D4×D5311yescharacteristic
306C22#41×D5312yes—
307C22#1313326311yescharacteristic
308C22#1313326511yescharacteristic
309C4—24no—
310C4—22no—
311C4—42no—
312C4—41no—
313C4—534no—
314C4—1064no—
315C4—1062no—
316C4—2121no—
317C2—81no—
318C2—4241no—
319C2—2124no—
320C2#1681113111yescharacteristic
321C2#127×D5311yescharacteristic
322C2#1681120311yescharacteristic
323C2#1681126512yes—
324C2#1681128312yes—
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#399C2
hypercenter#399D53
hypocenterC53#2540
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #399

Complements

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

NH ≅ G/Ntype
1. C2×#13133203C2semidirect
2. C2×#13133195C2semidirect
3. C2×#13133190C2semidirect
4. C2×#13133179C2semidirect
5. #16810558C2semidirect
6. #16810533C2semidirect
7. #16810532C2semidirect
8. #16810524C2semidirect
9. #16810523C2semidirect
10. #16810474C2semidirect
11. C53×#399C2semidirect
77. #11719919D4semidirect
79. C4×Dic53D4semidirect
168. #399D53semidirect
174. C53#2540semidirect