Finite Group Databasebeta

← #20280605

Subgroups — #20280605

Order 6784.

620 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 620 conjugacy classes of subgroups — too many to draw. Only its 131 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×#13133177C211yesmaximal, characteristic, complemented
3#16810561C211yesmaximal, characteristic, complemented
4#16810541C211yesmaximal, characteristic, complemented
5#16810540C212yesmaximal, complemented
6#16810532C211yesmaximal, characteristic, complemented
7#16810532C212yesmaximal, complemented
8#16810513C212yesmaximal, complemented
9#16810500C212yesmaximal, complemented
10#16810485C211yesmaximal, characteristic, complemented
11C53×#399C211yesmaximal, characteristic, complemented
12C22×D212C2212yes—
13C22×D212C2211yescharacteristic
14C22×C22⋊D53C2211yescharacteristic
15C22×C22⋊D53C2212yes—
16C23×Dic53C2211yescharacteristic
17C22×C4×D53C2211yescharacteristic
18#13133192—24no—
19#13133177—24no—
20C2×#11719919C2211yescharacteristic
21C2×#11719921C2212yes—
22C2×#11719921C2212yes—
23C2×#11719921C2211yescharacteristic
24C2×#11719921C2212yes—
25C2×#11719921C2212yes—
26C2×#11719921C2212yes—
27C2×C4×Dic53C2211yescharacteristic
28#13133124C2212yes—
29#13133124C2212yes—
30#13133124C2212yes—
31#13133124C2211yescharacteristic
32C2×C4⋊C4×C53C2211yescharacteristic
33C2×C4⋊C4×C53C2212yes—
34C2×C22⋊C4×C53C2211yescharacteristic
35C2×C22⋊C4×C53C2212yes—
36#95×C53C2211yescharacteristic
37C24×C53C2311yescharacteristic
38C23×D53C2312yes—
39C23×D53C2311yescharacteristic
40C22×C4×C53C2312yes—
41C22×C4×C53C2312yes—
42C22×C4×C53C2311yescharacteristic
43C22×C4×C53C2311yescharacteristic
44C2×C22⋊D53—22no—
45C2×C22⋊D53—22no—
46C2×C22⋊D53—24no—
47C2×C22⋊D53—22no—
48C2×C22⋊D53—42no—
49C2×C22⋊D53—42no—
50C22×Dic53—22no—
51C22×Dic53—42no—
52C22×Dic53C2312yes—
53C22×Dic53C2311yescharacteristic
54C22×Dic53C2311yescharacteristic
55C22×Dic53C2311yescharacteristic
56C2×D212—24no—
57C2×D212—22no—
58C2×D212—42no—
59C2×D212—42no—
60C2×D212—42no—
61C2×C4×D53—42no—
62C2×C4×D53D414yescomplemented
63C4⋊C4×C53—22no—
64C4⋊C4×C53—42no—
65C22⋊C4×C53—22no—
66C22⋊C4×C53—42no—
67#11719921—22no—
68#11719921—24no—
69#11719921—24no—
70#11719921—42no—
71#11719921—42no—
72#11719921—42no—
73#11719919—22no—
74C4×Dic53—22no—
75C23×C53—21no—
76C23×C53—21no—
77C23×C53—22no—
78C23×C53—22no—
79C23×C53#4112yes—
80C23×C53C2411yescharacteristic
81C22×D53—21no—
82C22×D53—21no—
83C22×D53—22no—
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×D53C2×D412yes—
93C2×C4×C53—22no—
94C2×C4×C53—22no—
95C2×C4×C53—21no—
96C2×C4×C53—22no—
97C2×C4×C53—22no—
98C2×C4×C53—22no—
99C2×C4×C53—22no—
100C2×C4×C53—22no—
101C2×C4×C53—22no—
102C2×C4×C53C2×D412yes—
103C22⋊D53—28no—
104C22⋊D53—48no—
105C2×Dic53—21no—
106C2×Dic53—21no—
107C2×Dic53—22no—
108C2×Dic53—21no—
109C2×Dic53—22no—
110C2×Dic53—22no—
111C2×Dic53—22no—
112C2×Dic53—22no—
113C2×Dic53—44no—
114C2×Dic53C2×D412yes—
115C2×Dic53#4112yes—
116D212—44no—
117D212—48no—
118C4×D53—44no—
119C22×C53—22no—
120C22×C53—24no—
121C22×C53—41no—
122C22×C53—41no—
123C22×C53—42no—
124C22×C53C22×D411yescharacteristic
125C22×C53C2×#4111yescharacteristic
126C22×C53C2×#4111yescharacteristic
127C22×C53H(5,𝔽2)12yes—
128C22×C53H(5,𝔽2)12yes—
129C2×D53—22no—
130C2×D53—22no—
131C2×D53—22no—
132C2×D53—42no—
133C2×D53—42no—
134C2×D53—42no—
135C2×D53—42no—
136C2×D53—42no—
137C2×D53—42no—
138C2×D53—42no—
139C2×D53—42no—
140C2×D53—42no—
141C4×C53—22no—
142C4×C53—42no—
143C4×C53—41no—
144C4×C53—42no—
145Dic53—22no—
146Dic53—22no—
147Dic53—41no—
148Dic53—42no—
149#2635—531nomaximal, Sylow
150C2×C53—42no—
151C2×C53#52511yescharacteristic
152C2×C53#53411yescharacteristic
153C2×C53#53511yescharacteristic
154C2×C53#53812yes—
155C2×C53#54012yes—
156D53—42no—
157D53—82no—
158C2×#121—531no—
159C2×C4×D4—531no—
160#400—531no—
161#399D5311yescharacteristic, complemented
162#394—532no—
163#394—531no—
164#392—531no—
165#392—532no—
166#388—532no—
167#386—532no—
168#381—531no—
169C53#263511yescharacteristic, complemented, Sylow
170C22×D4—531no—
171C22×D4—532no—
172C22×D4—532no—
173C22×D4—531no—
174C23×C4—531no—
175C23×C4—531no—
176#121—1064no—
177C4×D4—1064no—
178C2×C4⋊C4—531no—
179C2×C4⋊C4C2×D5311yescharacteristic
180C2×C4⋊C4C2×D5312yes—
181C2×C22⋊C4—531no—
182C2×C22⋊C4—532no—
183C2×C22⋊C4—532no—
184C2×C22⋊C4—532no—
185C2×C22⋊C4—532no—
186C2×C22⋊C4—532no—
187C2×C22⋊C4C2×D5311yescharacteristic
188C2×C22⋊C4C2×D5312yes—
189C2×C42—531no—
190#95—532no—
191#95—531no—
192#95—532no—
193#95—532no—
194#95C2×D5311yescharacteristic
195C24—531no—
196C24—532no—
197C24C22×D5311yescharacteristic
198C2×D4—1064no—
199C2×D4—1062no—
200C2×D4—1062no—
201C2×D4—1062no—
202C2×D4—1064no—
203C2×D4—1062no—
204C2×D4—2122no—
205C2×D4—2122no—
206C2×D4—2122no—
207C2×D4—2122no—
208C2×D4—2122no—
209C22×C4—531no—
210C22×C4—531no—
211C22×C4—534no—
212C22×C4—532no—
213C22×C4—531no—
214C22×C4—1062no—
215C22×C4—2122no—
216C22×C4—2122no—
217C22×C4C22×D5312yes—
218C22×C4C22×D5312yes—
219C22×C4C22×D5311yescharacteristic
220C22×C4C22×D5311yescharacteristic
221C4⋊C4—22no—
222C4⋊C4—42no—
223C4⋊C4—1062no—
224C22⋊C4—22no—
225C22⋊C4—42no—
226C22⋊C4—1064no—
227C22⋊C4—1064no—
228C22⋊C4—1062no—
229C22⋊C4—2122no—
230C22⋊C4—2122no—
231C22⋊C4—2122no—
232C42—1062no—
233C23—21no—
234C23—21no—
235C23—22no—
236C23—22no—
237C23—532no—
238C23—1061no—
239C23—1061no—
240C23—1062no—
241C23—1062no—
242C23—1062no—
243C23—1062no—
244C23—1062no—
245C23—1062no—
246C23—1062no—
247C23—1062no—
248C23—1062no—
249C23#1171994912yes—
250C23C23×D5311yescharacteristic
251D4—1068no—
252D4—2124no—
253D4—2128no—
254D4—2128no—
255C2×C4—22no—
256C2×C4—21no—
257C2×C4—22no—
258C2×C4—22no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—22no—
262C2×C4—22no—
263C2×C4—22no—
264C2×C4—532no—
265C2×C4—532no—
266C2×C4—1061no—
267C2×C4—1061no—
268C2×C4—1062no—
269C2×C4—1061no—
270C2×C4—1062no—
271C2×C4—1062no—
272C2×C4—1062no—
273C2×C4—1062no—
274C2×C4—2124no—
275C2×C4—2124no—
276C2×C4D4×D5312yes—
277C22—22no—
278C22—24no—
279C22—41no—
280C22—41no—
281C22—42no—
282C22—1062no—
283C22—1062no—
284C22—1062no—
285C22—2122no—
286C22—2122no—
287C22—2122no—
288C22—2122no—
289C22—2122no—
290C22—2122no—
291C22—2122no—
292C22—2122no—
293C22—2122no—
294C22C2×D4×D5311yescharacteristic
295C22#41×D5311yescharacteristic
296C22C2×#1171994911yescharacteristic
297C22#1313326212yes—
298C22#1313326312yes—
299C4—22no—
300C4—42no—
301C4—41no—
302C4—42no—
303C4—1062no—
304C4—1062no—
305C4—2121no—
306C4—2122no—
307C2—42no—
308C2—4242no—
309C2—2122no—
310C2#1681113611yescharacteristic
311C2#1681118411yescharacteristic
312C2#1681118611yescharacteristic
313C2#1681121312yes—
314C2#1681122512yes—
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#399C2
hypercenter#399D53
hypocenterC53#2635
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #399

Complements

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

NH ≅ G/Ntype
1. C2×#13133192C2semidirect
2. C2×#13133177C2semidirect
3. #16810561C2semidirect
4. #16810541C2semidirect
5. #16810540C2semidirect
6. #16810532C2semidirect
7. #16810532C2semidirect
8. #16810513C2semidirect
9. #16810500C2semidirect
10. #16810485C2semidirect
11. C53×#399C2semidirect
62. C2×C4×D53D4semidirect
161. #399D53semidirect
169. C53#2635semidirect