Finite Group Databasebeta

← #20278051

Subgroups — #20278051

Order 6784.

864 conjugacy classes of subgroups; only the 159 normal subgroups are shown.

normal subgroups only

This group has 864 conjugacy classes of subgroups — too many to draw. Only its 159 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
1C22×#11719949C211yesmaximal, characteristic
2C2×#13133202C212yesmaximal, complemented
3C2×C4⋊C4×D53C211yesmaximal, characteristic, complemented
4#16810542C211yesmaximal, characteristic, complemented
5#16810541C211yesmaximal, characteristic, complemented
6#16810540C211yesmaximal, characteristic, complemented
7#16810501C212yesmaximal, complemented
8#16810497C212yesmaximal, complemented
9#16810494C212yesmaximal, complemented
10#16810485C211yesmaximal, characteristic, complemented
11C53×#395C211yesmaximal, characteristic, complemented
12C2×#11719949—28no—
13C2×#11719949—24no—
14C22×D212C2211yescharacteristic
15C22×D212C2212yes—
16C22×C4×D53C2212yes—
17C22×C4×D53C2211yescharacteristic
18C22×C4×D53C2211yescharacteristic
19C22×C4×D53C2211yescharacteristic
20#13133202—28no—
21C2×C4⋊Dic53C2211yescharacteristic
22C2×#11719919C2212yes—
23C2×#11719919C2211yescharacteristic
24C2×#11719919C2211yescharacteristic
25C4⋊C4×D53—24no—
26C2×#11719921C2212yes—
27C2×#11719921C2212yes—
28C2×#11719921C2212yes—
29C2×#11719921C2212yes—
30C2×#11719921C2212yes—
31#13133124C2212yes—
32#13133124C2212yes—
33#13133124C2211yescharacteristic
34C22×Q8×C53C2211yescharacteristic
35C2×C4⋊C4×C53C2211yescharacteristic
36C2×C4⋊C4×C53C2212yes—
37#95×C53C2212yes—
38#95×C53C2211yescharacteristic
39C23×D53C2312yes—
40C23×D53C2311yescharacteristic
41C2×Q8×C53—24no—
42C2×Q8×C53—22no—
43C22×C4×C53C2312yes—
44C22×C4×C53C2311yescharacteristic
45C22×C4×C53C2312yes—
46C22×C4×C53C2311yescharacteristic
47C22×C4×C53C2311yescharacteristic
48C22×Dic53C2312yes—
49C22×Dic53C2311yescharacteristic
50C22×Dic53C2311yescharacteristic
51C22×Dic53C2311yescharacteristic
52#11719949—232no—
53C2×D212—24no—
54C2×D212—24no—
55C2×D212—24no—
56C2×D212—22no—
57C2×D212D418yes—
58C2×C4×D53—22no—
59C2×C4×D53—24no—
60C2×C4×D53—24no—
61C2×C4×D53—24no—
62C2×C4×D53—22no—
63C2×C4×D53—24no—
64C2×C4×D53—22no—
65C2×C4×D53—22no—
66C2×C4×D53—24no—
67C2×C4×D53D414yes—
68C4⋊C4×C53—22no—
69C4⋊C4×C53—24no—
70#11719921—24no—
71#11719921—24no—
72#11719921—24no—
73#11719921—24no—
74#11719921—24no—
75C4⋊Dic53—22no—
76#11719919—22no—
77#11719919—24no—
78#11719919—22no—
79C23×C53C2411yescharacteristic
80C22×D53—21no—
81C22×D53—21no—
82C22×D53—22no—
83C22×D53—22no—
84C22×D53—22no—
85C22×D53—22no—
86C22×D53—22no—
87C22×D53—22no—
88C22×D53—22no—
89C22×D53C2×D414yes—
90C22×D53C2×D412yes—
91C22×D53C2×D414yes—
92Q8×C53—28no—
93C2×C4×C53—22no—
94C2×C4×C53—22no—
95C2×C4×C53—22no—
96C2×C4×C53—21no—
97C2×C4×C53—21no—
98C2×C4×C53—21no—
99C2×C4×C53—22no—
100C2×C4×C53—22no—
101C2×C4×C53—22no—
102C2×C4×C53—22no—
103C2×C4×C53C2×D414yes—
104C2×C4×C53C2×D412yes—
105C2×C4×C53#4112yescomplemented
106C2×Dic53—22no—
107C2×Dic53—21no—
108C2×Dic53—21no—
109C2×Dic53—21no—
110C2×Dic53—22no—
111C2×Dic53—22no—
112C2×Dic53—22no—
113C2×Dic53—22no—
114C2×Dic53C2×D412yes—
115C2×Dic53#4112yescomplemented
116D212—28no—
117D212—216no—
118C4×D53—28no—
119C4×D53—216no—
120C4×D53—28no—
121C4×D53—44no—
122C22×C53C22×D411yescharacteristic
123C22×C53C22×D412yes—
124C22×C53C2×#4111yescharacteristic
125C22×C53C2×#4111yescharacteristic
126C22×C532−1+412yes—
127C2×D53—22no—
128C2×D53—22no—
129C2×D53—24no—
130C2×D53—24no—
131C2×D53—24no—
132C2×D53—24no—
133C2×D53—24no—
134C2×D53—42no—
135C2×D53—42no—
136C2×D53—42no—
137C2×D53—42no—
138C2×D53#12014yes—
139C4×C53—24no—
140C4×C53—22no—
141C4×C53—22no—
142C4×C53—41no—
143C4×C53—42no—
144Dic53—22no—
145Dic53—22no—
146Dic53—41no—
147Dic53—42no—
148#2406—531nomaximal, Sylow
149C2×C53C2×#12011yescharacteristic
150C2×C53#52511yescharacteristic
151C2×C53#53611yescharacteristic
152C2×C53#54712yes—
153C2×C53#54712yes—
154D53—24no—
155D53—44no—
156C22×#41—531no—
157C2×#123—532no—
158C22×C4⋊C4—531no—
159#395—531no—
160#395D5311yescharacteristic, complemented
161#394—531no—
162#394—531no—
163#386—532no—
164#385—532no—
165#385—532no—
166#381—531no—
167C53#240611yescharacteristic, complemented, Sylow
168C2×#41—1064no—
169C2×#41—1068no—
170C22×Q8C2×D5311yescharacteristic
171C22×D4—532no—
172C22×D4—531no—
173C23×C4—531no—
174C23×C4—531no—
175C23×C4—531no—
176C23×C4—532no—
177#123—1068no—
178C2×C4⋊C4—531no—
179C2×C4⋊C4—531no—
180C2×C4⋊C4—531no—
181C2×C4⋊C4—532no—
182C2×C4⋊C4—1064no—
183C2×C4⋊C4C2×D5311yescharacteristic
184C2×C4⋊C4C2×D5312yes—
185C2×C22⋊C4—532no—
186C2×C22⋊C4—532no—
187C2×C22⋊C4—532no—
188C2×C22⋊C4—532no—
189C2×C22⋊C4—532no—
190#95—532no—
191#95—531no—
192#95—532no—
193#95C2×D5311yescharacteristic
194#95C2×D5312yes—
195C24—531no—
196C24—532no—
197#41—10632no—
198C2×Q8—22no—
199C2×Q8—24no—
200C2×D4—538no—
201C2×D4—1064no—
202C2×D4—1064no—
203C2×D4—1064no—
204C2×D4—1062no—
205C22×C4—531no—
206C22×C4—531no—
207C22×C4—534no—
208C22×C4—532no—
209C22×C4—531no—
210C22×C4—1062no—
211C22×C4—1064no—
212C22×C4—1062no—
213C22×C4—1064no—
214C22×C4—1064no—
215C22×C4—1062no—
216C22×C4—1064no—
217C22×C4—1064no—
218C22×C4—1062no—
219C22×C4C22×D5312yes—
220C22×C4C22×D5311yescharacteristic
221C22×C4C22×D5311yescharacteristic
222C22×C4C22×D5312yes—
223C22×C4C22×D5311yescharacteristic
224C4⋊C4—22no—
225C4⋊C4—24no—
226C4⋊C4—1062no—
227C4⋊C4—1062no—
228C4⋊C4—1062no—
229C4⋊C4—1064no—
230C22⋊C4—1064no—
231C22⋊C4—1064no—
232C22⋊C4—1064no—
233C22⋊C4—1064no—
234C22⋊C4—1064no—
235C23—534no—
236C23—532no—
237C23—534no—
238C23—1061no—
239C23—1061no—
240C23—1062no—
241C23—1062no—
242C23—1062no—
243C23—1062no—
244C23—1062no—
245C23—1062no—
246C23—1062no—
247C23C23×D5311yescharacteristic
248Q8—28no—
249D4—10616no—
250D4—1068no—
251C2×C4—21no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—21no—
256C2×C4—22no—
257C2×C4—22no—
258C2×C4—21no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—532no—
262C2×C4—532no—
263C2×C4—1068no—
264C2×C4—1068no—
265C2×C4—10616no—
266C2×C4—1061no—
267C2×C4—1062no—
268C2×C4—1061no—
269C2×C4—1062no—
270C2×C4—1062no—
271C2×C4—1061no—
272C2×C4—1062no—
273C2×C4—1062no—
274C2×C4—2124no—
275C2×C4D4×D5312yes—
276C2×C4D4×D5314yes—
277C2×C4#1171994912yescomplemented
278C22—534no—
279C22—1064no—
280C22—1064no—
281C22—1062no—
282C22—1062no—
283C22—1064no—
284C22—1064no—
285C22—1064no—
286C22—2122no—
287C22—2122no—
288C22—2122no—
289C22—2122no—
290C22C2×D4×D5312yes—
291C22C2×D4×D5311yescharacteristic
292C22#41×D5311yescharacteristic
293C22C2×#1171994911yescharacteristic
294C22#1313326512yes—
295C4—22no—
296C4—24no—
297C4—22no—
298C4—42no—
299C4—41no—
300C4—1062no—
301C4—1062no—
302C4—2122no—
303C4—2121no—
304C2—1064no—
305C2—2124no—
306C2#120×D5311yescharacteristic
307C2#1681113611yescharacteristic
308C2#1681119811yescharacteristic
309C2#1681127512yes—
310C2#1681127612yes—
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#395C2
hypercenter#395D53
hypocenterC53#2406
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #395

Complements

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

NH ≅ G/Ntype
2. C2×#13133202C2semidirect
3. C2×C4⋊C4×D53C2semidirect
4. #16810542C2semidirect
5. #16810541C2semidirect
6. #16810540C2semidirect
7. #16810501C2semidirect
8. #16810497C2semidirect
9. #16810494C2semidirect
10. #16810485C2semidirect
11. C53×#395C2semidirect
105. C2×C4×C53#41semidirect
115. C2×Dic53#41semidirect
160. #395D53semidirect
167. C53#2406semidirect
277. C2×C4#11719949semidirect