Finite Group Databasebeta

← #20282757

Subgroups — #20282757

Order 6784.

630 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 630 conjugacy classes of subgroups — too many to draw. Only its 125 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×#13133213C211yesmaximal, characteristic, complemented
2C2×#13133212C211yesmaximal, characteristic, complemented
3#16810540C211yesmaximal, characteristic, complemented
4#16810539C212yesmaximal, complemented
5#16810538C212yesmaximal, complemented
6#16810533C211yesmaximal, characteristic, complemented
7#16810532C212yesmaximal, complemented
8#16810532C212yesmaximal, complemented
9#16810531C211yesmaximal, characteristic, complemented
10#16810490C211yesmaximal, characteristic, complemented
11C53×#394C211yesmaximal, characteristic, complemented
12C22×D212C2211yescharacteristic
13C22×D212C2212yes—
14C22×C22⋊D53C2212yes—
15C22×C22⋊D53C2212yes—
16C22×C22⋊D53C2212yes—
17#13133213—24no—
18#13133212—24no—
19C2×C4⋊Dic53C2211yescharacteristic
20C2×#11719919C2211yescharacteristic
21C2×C22⋊Dic53C2212yes—
22C2×#11719921C2211yescharacteristic
23C2×#11719921C2211yescharacteristic
24C2×#11719921C2211yescharacteristic
25C2×#11719921C2212yes—
26C2×#11719921C2212yes—
27C2×#11719921C2212yes—
28C2×C4×Dic53C2211yescharacteristic
29#13133124C2211yescharacteristic
30#13133124C2212yes—
31#13133124C2212yes—
32C22×D4×C53C2211yescharacteristic
33C2×C4⋊C4×C53C2211yescharacteristic
34C2×C22⋊C4×C53C2212yes—
35C2×C22⋊C4×C53C2212yes—
36#95×C53C2211yescharacteristic
37C24×C53C2312yes—
38C23×D53C2312yes—
39C23×D53C2311yescharacteristic
40C2×D4×C53—22no—
41C2×D4×C53—42no—
42C22×C4×C53C2312yes—
43C22×C4×C53C2311yescharacteristic
44C22×C4×C53C2311yescharacteristic
45C22×C4×C53C2311yescharacteristic
46C2×C22⋊D53—24no—
47C2×C22⋊D53—24no—
48C2×C22⋊D53—42no—
49C2×C22⋊D53—42no—
50C2×C22⋊D53—42no—
51C2×C22⋊D53—42no—
52C2×C22⋊D53—42no—
53C2×C22⋊D53—42no—
54C2×C22⋊D53—42no—
55C22×Dic53C2312yes—
56C22×Dic53C2311yescharacteristic
57C22×Dic53C2311yescharacteristic
58C22×Dic53C2311yescharacteristic
59C2×D212—22no—
60C2×D212—42no—
61C2×D212—42no—
62C2×D212—42no—
63C2×D212—42no—
64C4⋊C4×C53—22no—
65C22⋊C4×C53—42no—
66C22⋊C4×C53—42no—
67C22⋊Dic53—42no—
68#11719921—22no—
69#11719921—22no—
70#11719921—22no—
71#11719921—42no—
72#11719921—42no—
73#11719921—42no—
74C4⋊Dic53—22no—
75#11719919—22no—
76C4×Dic53D414yescomplemented
77C23×C53—22no—
78C23×C53—22no—
79C23×C53—22no—
80C23×C53—22no—
81C23×C53—22no—
82C23×C53—22no—
83C23×C53—22no—
84C23×C53C2411yescharacteristic
85C22×D53—22no—
86C22×D53—21no—
87C22×D53—22no—
88C22×D53—21no—
89C22×D53—22no—
90C22×D53—22no—
91C22×D53—22no—
92C22×D53—22no—
93C22×D53—22no—
94C22×D53—21no—
95C22×D53—22no—
96C22×D53—22no—
97D4×C53—44no—
98C2×C4×C53—22no—
99C2×C4×C53—21no—
100C2×C4×C53—21no—
101C2×C4×C53—22no—
102C2×C4×C53—22no—
103C2×C4×C53—22no—
104C2×C4×C53—21no—
105C2×C4×C53—22no—
106C2×C4×C53—21no—
107C2×C4×C53C2×D412yes—
108C22⋊D53—48no—
109C22⋊D53—48no—
110C22⋊D53—48no—
111C2×Dic53—21no—
112C2×Dic53—21no—
113C2×Dic53—22no—
114C2×Dic53—21no—
115C2×Dic53—22no—
116C2×Dic53—22no—
117C2×Dic53—22no—
118C2×Dic53—22no—
119C2×Dic53C2×D412yes—
120C2×Dic53C2×D412yes—
121D212—44no—
122D212—48no—
123C22×C53—42no—
124C22×C53—42no—
125C22×C53—42no—
126C22×C53—42no—
127C22×C53—42no—
128C22×C53—42no—
129C22×C53—42no—
130C22×C53C22×D411yescharacteristic
131C22×C53H(5,𝔽2)11yescharacteristic
132C22×C53H(5,𝔽2)12yes—
133C22×C53H(5,𝔽2)12yes—
134C22×C53H(5,𝔽2)11yescharacteristic
135C2×D53—42no—
136C2×D53—42no—
137C2×D53—42no—
138C2×D53—41no—
139C2×D53—41no—
140C2×D53—41no—
141C2×D53—42no—
142C2×D53—42no—
143C2×D53—42no—
144C2×D53—42no—
145C2×D53—42no—
146C2×D53—42no—
147C4×C53—22no—
148C4×C53—41no—
149C4×C53—41no—
150C4×C53—42no—
151Dic53—22no—
152Dic53—22no—
153Dic53—41no—
154Dic53—42no—
155#2818—531nomaximal, Sylow
156C2×C53—82no—
157C2×C53#53511yescharacteristic
158C2×C53#53511yescharacteristic
159C2×C53#53511yescharacteristic
160C2×C53#56012yes—
161C2×C53#56012yes—
162D53—82no—
163D53—81no—
164C2×#127—531no—
165C2×#126—531no—
166#394—531no—
167#394—532no—
168#394—532no—
169#394D5311yescharacteristic, complemented
170#392—531no—
171#392—532no—
172#392—532no—
173#392—531no—
174#383—531no—
175C53#281811yescharacteristic, complemented, Sylow
176C22×D4—532no—
177C22×D4—532no—
178C22×D4—532no—
179C22×D4—531no—
180C22×D4—532no—
181C22×D4C2×D5311yescharacteristic
182#127—1064no—
183#126—1064no—
184C2×C4⋊C4—531no—
185C2×C4⋊C4—531no—
186C2×C4⋊C4C2×D5311yescharacteristic
187C2×C22⋊C4—531no—
188C2×C22⋊C4—531no—
189C2×C22⋊C4—531no—
190C2×C22⋊C4—532no—
191C2×C22⋊C4—532no—
192C2×C22⋊C4—532no—
193C2×C22⋊C4—532no—
194C2×C22⋊C4C2×D5312yes—
195C2×C22⋊C4C2×D5312yes—
196C2×C42—531no—
197#95—532no—
198#95—532no—
199#95—531no—
200#95C2×D5311yescharacteristic
201C24—531no—
202C24—532no—
203C24C22×D5312yes—
204C2×D4—22no—
205C2×D4—42no—
206C2×D4—1064no—
207C2×D4—1064no—
208C2×D4—1062no—
209C2×D4—2122no—
210C2×D4—2122no—
211C2×D4—2122no—
212C2×D4—2122no—
213C2×D4—2122no—
214C2×D4—2122no—
215C2×D4—2122no—
216C2×D4—2122no—
217C2×D4—2122no—
218C2×D4—2122no—
219C2×D4—2122no—
220C22×C4—531no—
221C22×C4—531no—
222C22×C4—532no—
223C22×C4—531no—
224C22×C4C22×D5312yes—
225C22×C4C22×D5311yescharacteristic
226C22×C4C22×D5311yescharacteristic
227C22×C4C22×D5311yescharacteristic
228C4⋊C4—22no—
229C4⋊C4—1062no—
230C4⋊C4—1062no—
231C22⋊C4—42no—
232C22⋊C4—42no—
233C22⋊C4—1062no—
234C22⋊C4—1062no—
235C22⋊C4—1062no—
236C22⋊C4—2122no—
237C22⋊C4—2122no—
238C22⋊C4—2122no—
239C22⋊C4—2122no—
240C42—534no—
241C23—22no—
242C23—22no—
243C23—22no—
244C23—22no—
245C23—22no—
246C23—22no—
247C23—22no—
248C23—1061no—
249C23—1062no—
250C23—1062no—
251C23—1061no—
252C23—1062no—
253C23—1062no—
254C23—1062no—
255C23—1062no—
256C23—1062no—
257C23—1062no—
258C23—1062no—
259C23—1061no—
260C23C23×D5311yescharacteristic
261D4—44no—
262D4—2124no—
263D4—2128no—
264D4—2128no—
265D4—2128no—
266D4—2128no—
267C2×C4—22no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—22no—
272C2×C4—22no—
273C2×C4—22no—
274C2×C4—21no—
275C2×C4—22no—
276C2×C4—532no—
277C2×C4—532no—
278C2×C4—1061no—
279C2×C4—1061no—
280C2×C4—1062no—
281C2×C4—1061no—
282C2×C4—1062no—
283C2×C4—1062no—
284C2×C4—1062no—
285C2×C4—1062no—
286C2×C4D4×D5312yes—
287C22—42no—
288C22—42no—
289C22—42no—
290C22—42no—
291C22—42no—
292C22—42no—
293C22—42no—
294C22—2122no—
295C22—2122no—
296C22—2122no—
297C22—2122no—
298C22—2122no—
299C22—2122no—
300C22—2122no—
301C22—2121no—
302C22—2122no—
303C22—2121no—
304C22—2122no—
305C22—2121no—
306C22C2×D4×D5311yescharacteristic
307C22#1313326211yescharacteristic
308C22#1313326212yes—
309C22#1313326312yes—
310C22#1313326311yescharacteristic
311C4—22no—
312C4—42no—
313C4—41no—
314C4—41no—
315C4—1062no—
316C4—1062no—
317C4—2122no—
318C4—2121no—
319C2—82no—
320C2—4241no—
321C2—4242no—
322C2#1681118811yescharacteristic
323C2#1681118911yescharacteristic
324C2#1681119111yescharacteristic
325C2#1681136112yes—
326C2#1681136312yes—
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#394C2
hypercenter#394D53
hypocenterC53#2818
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #394

Complements

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

NH ≅ G/Ntype
1. C2×#13133213C2semidirect
2. C2×#13133212C2semidirect
3. #16810540C2semidirect
4. #16810539C2semidirect
5. #16810538C2semidirect
6. #16810533C2semidirect
7. #16810532C2semidirect
8. #16810532C2semidirect
9. #16810531C2semidirect
10. #16810490C2semidirect
11. C53×#394C2semidirect
76. C4×Dic53D4semidirect
169. #394D53semidirect
175. C53#2818semidirect