Finite Group Databasebeta

← #20279804

Subgroups — #20279804

Order 6784.

492 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#13133207C211yesmaximal, characteristic, complemented
2C2×#13133193C211yesmaximal, characteristic
3#16810558C211yesmaximal, characteristic
4#16810557C211yesmaximal, characteristic, complemented
5#16810547C211yesmaximal, characteristic
6#16810542C211yesmaximal, characteristic, complemented
7#16810520C211yesmaximal, characteristic, complemented
8#16810520C211yesmaximal, characteristic, complemented
9#16810514C211yesmaximal, characteristic
10#16810512C211yesmaximal, characteristic
11#16810505C211yesmaximal, characteristic, complemented
12#16810497C211yesmaximal, characteristic
13C4×#11719919C211yesmaximal, characteristic, complemented
14#16810470C211yesmaximal, characteristic
15C53×#391C211yesmaximal, characteristic, complemented
16C22×C4×D53C2211yescharacteristic
17#13133207—24no—
18#13133193—24no—
19C2×C4⋊Dic53C2211yescharacteristic
20C2×C4⋊Dic53C2211yescharacteristic
21C2×#11719919C2211yescharacteristic
22C2×#11719919C2211yescharacteristic
23C2×#11719919C2211yescharacteristic
24C2×#11719919C2211yescharacteristic
25C2×#11719919C2211yescharacteristic
26C2×#11719919C2211yescharacteristic
27C2×#11719919C2211yescharacteristic
28C2×#11719919C2211yescharacteristic
29C2×#11719919C2211yescharacteristic
30C2×#11719919C2211yescharacteristic
31C2×#11719921C2211yescharacteristic
32C2×#11719921C2211yescharacteristic
33C2×#11719921C2211yescharacteristic
34C2×#11719921C2211yescharacteristic
35C2×#11719921C2211yescharacteristic
36C2×#11719921C2211yescharacteristic
37C2×C4×Dic53C2211yescharacteristic
38C2×C4×Dic53C2211yescharacteristic
39C2×C4×Dic53C2211yescharacteristic
40#13133124C2211yescharacteristic
41#13133124C2211yescharacteristic
42#13133124C2211yescharacteristic
43#13133124C2211yescharacteristic
44#13133124C2211yescharacteristic
45#13133124C2211yescharacteristic
46C22×Q8×C53C2211yescharacteristic
47C2×C4⋊C4×C53C2211yescharacteristic
48C2×C42×C53C2211yescharacteristic
49#95×C53C2211yescharacteristic
50#95×C53C2211yescharacteristic
51#95×C53C2211yescharacteristic
52#95×C53C2211yescharacteristic
53C23×D53C2311yescharacteristic
54C2×Q8×C53—22no—
55C2×Q8×C53—22no—
56C2×Q8×C53—41no—
57C22×C4×C53C2311yescharacteristic
58C22×C4×C53C2311yescharacteristic
59C22×C4×C53C2311yescharacteristic
60C22×C4×C53C2311yescharacteristic
61C22×C4×C53C2311yescharacteristic
62C22×C4×C53C2311yescharacteristic
63C22×C4×C53C2311yescharacteristic
64C22×Dic53C2311yescharacteristic
65C22×Dic53C2311yescharacteristic
66C22×Dic53C2311yescharacteristic
67C22×Dic53C2311yescharacteristic
68C22×Dic53C2311yescharacteristic
69C22×Dic53C2311yescharacteristic
70C22×Dic53C2311yescharacteristic
71C2×C4×D53—22no—
72C2×C4×D53—22no—
73C2×C4×D53—41no—
74C4⋊C4×C53—22no—
75C42×C53—41no—
76#11719921—22no—
77#11719921—22no—
78#11719921—22no—
79#11719921—22no—
80#11719921—41no—
81#11719921Q814yescomplemented
82C4⋊Dic53—22no—
83C4⋊Dic53—22no—
84#11719919—22no—
85#11719919—22no—
86#11719919—22no—
87#11719919—22no—
88#11719919—22no—
89#11719919—22no—
90#11719919—22no—
91#11719919—22no—
92#11719919—41no—
93#11719919D414yes—
94C4×Dic53—22no—
95C4×Dic53—22no—
96C4×Dic53—41no—
97C23×C53C2411yescharacteristic
98C22×D53—21no—
99C22×D53—21no—
100C22×D53—21no—
101C22×D53—21no—
102C22×D53—21no—
103C22×D53—21no—
104C22×D53C2×Q812yes—
105Q8×C53—44no—
106C2×C4×C53—21no—
107C2×C4×C53—21no—
108C2×C4×C53—21no—
109C2×C4×C53—21no—
110C2×C4×C53—21no—
111C2×C4×C53—21no—
112C2×C4×C53—21no—
113C2×C4×C53—21no—
114C2×C4×C53—21no—
115C2×C4×C53—21no—
116C2×C4×C53—21no—
117C2×C4×C53—21no—
118C2×C4×C53—21no—
119C2×C4×C53—21no—
120C2×C4×C53—21no—
121C2×C4×C53—21no—
122C2×C4×C53—21no—
123C2×C4×C53—21no—
124C2×C4×C53C2×D412yes—
125C2×C4×C53C2×Q812yes—
126C2×C4×C53#4112yes—
127C2×Dic53—21no—
128C2×Dic53—21no—
129C2×Dic53—21no—
130C2×Dic53—21no—
131C2×Dic53—21no—
132C2×Dic53—21no—
133C2×Dic53—21no—
134C2×Dic53—21no—
135C2×Dic53—21no—
136C2×Dic53—21no—
137C2×Dic53—21no—
138C2×Dic53—21no—
139C2×Dic53—21no—
140C2×Dic53—21no—
141C2×Dic53—21no—
142C2×Dic53—21no—
143C2×Dic53C2×D412yes—
144C2×Dic53C2×D412yes—
145C2×Dic53C2×Q812yes—
146C2×Dic53#4112yes—
147C2×Dic53#4112yes—
148C4×D53—44no—
149C22×C53C22×D411yescharacteristic
150C22×C53C22×Q811yescharacteristic
151C22×C53C2×#4111yescharacteristic
152C22×C53C2×#4111yescharacteristic
153C22×C53C2×#4111yescharacteristic
154C22×C53H(5,𝔽2)11yescharacteristic
155C22×C532−1+411yescharacteristic
156C2×D53—22no—
157C2×D53—22no—
158C2×D53—22no—
159C2×D53—41no—
160C2×D53—41no—
161C2×D53—41no—
162C2×D53—41no—
163C4×C53—22no—
164C4×C53—22no—
165C4×C53—22no—
166C4×C53—41no—
167C4×C53—41no—
168C4×C53—41no—
169C4×C53—41no—
170Dic53—22no—
171Dic53—22no—
172Dic53—22no—
173Dic53—22no—
174Dic53—22no—
175Dic53—41no—
176Dic53—41no—
177#2556—531nomaximal, Sylow
178C2×C53#53211yescharacteristic
179C2×C53#53311yescharacteristic
180C2×C53#53911yescharacteristic
181C2×C53#54611yescharacteristic
182C2×C53D4×Q811yescharacteristic
183C2×C53#55211yescharacteristic
184C2×C53#55411yescharacteristic
185D53—42no—
186C2×#125—531no—
187C2×C22⋊Q8—531no—
188#399—531no—
189#398—531no—
190#396—531no—
191#395—531no—
192#391D5311yescharacteristic, complemented
193#389—531no—
194#389—531no—
195#388—531no—
196#388—531no—
197#387—531no—
198#385—531no—
199C4×C4⋊C4—531no—
200C4×C22⋊C4—531no—
201C53#255611yescharacteristic, complemented, Sylow
202C22×Q8C2×D5311yescharacteristic
203C23×C4—531no—
204#125—1064no—
205C22⋊Q8—1064no—
206C2×C4⋊C4—531no—
207C2×C4⋊C4—531no—
208C2×C4⋊C4—531no—
209C2×C4⋊C4—531no—
210C2×C4⋊C4—531no—
211C2×C4⋊C4—531no—
212C2×C4⋊C4—531no—
213C2×C4⋊C4—531no—
214C2×C4⋊C4—531no—
215C2×C4⋊C4—531no—
216C2×C4⋊C4—531no—
217C2×C4⋊C4—531no—
218C2×C4⋊C4C2×D5311yescharacteristic
219C2×C22⋊C4—531no—
220C2×C22⋊C4—531no—
221C2×C22⋊C4—531no—
222C2×C22⋊C4—531no—
223C2×C22⋊C4—531no—
224C2×C22⋊C4—531no—
225C2×C42—531no—
226C2×C42—531no—
227C2×C42—531no—
228C2×C42C2×D5311yescharacteristic
229#95—531no—
230#95—531no—
231#95—531no—
232#95—531no—
233#95—531no—
234#95—531no—
235#95C2×D5311yescharacteristic
236#95C2×D5311yescharacteristic
237#95C2×D5311yescharacteristic
238#95C2×D5311yescharacteristic
239C24—531no—
240C2×Q8—22no—
241C2×Q8—22no—
242C2×Q8—41no—
243C22×C4—531no—
244C22×C4—531no—
245C22×C4—531no—
246C22×C4—531no—
247C22×C4—531no—
248C22×C4—531no—
249C22×C4—531no—
250C22×C4—1062no—
251C22×C4—1062no—
252C22×C4—2121no—
253C22×C4C22×D5311yescharacteristic
254C22×C4C22×D5311yescharacteristic
255C22×C4C22×D5311yescharacteristic
256C22×C4C22×D5311yescharacteristic
257C22×C4C22×D5311yescharacteristic
258C22×C4C22×D5311yescharacteristic
259C22×C4C22×D5311yescharacteristic
260C4⋊C4—22no—
261C4⋊C4—534no—
262C4⋊C4—1062no—
263C4⋊C4—1062no—
264C4⋊C4—1062no—
265C4⋊C4—1062no—
266C4⋊C4—1062no—
267C4⋊C4—1062no—
268C4⋊C4—1062no—
269C4⋊C4—1062no—
270C4⋊C4—1062no—
271C4⋊C4—1062no—
272C4⋊C4—2121no—
273C22⋊C4—534no—
274C22⋊C4—1062no—
275C22⋊C4—1062no—
276C22⋊C4—1062no—
277C22⋊C4—1062no—
278C22⋊C4—2121no—
279C42—41no—
280C42—1062no—
281C42—1062no—
282C42—2121no—
283C23—532no—
284C23—1061no—
285C23—1061no—
286C23—1061no—
287C23—1061no—
288C23—1061no—
289C23—1061no—
290C23C23×D5311yescharacteristic
291Q8—44no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—532no—
311C2×C4—532no—
312C2×C4—532no—
313C2×C4—532no—
314C2×C4—532no—
315C2×C4—1061no—
316C2×C4—1061no—
317C2×C4—1061no—
318C2×C4—1061no—
319C2×C4—1061no—
320C2×C4—1061no—
321C2×C4—1061no—
322C2×C4—1061no—
323C2×C4—1061no—
324C2×C4—1061no—
325C2×C4—1061no—
326C2×C4—1061no—
327C2×C4—1061no—
328C2×C4—1061no—
329C2×C4—1061no—
330C2×C4—1061no—
331C2×C4—2124no—
332C2×C4#1171994512yes—
333C2×C4D4×D5312yes—
334C2×C4Q8×D5312yes—
335C22—1062no—
336C22—1062no—
337C22—1062no—
338C22—2121no—
339C22—2121no—
340C22—2121no—
341C22—2121no—
342C22C2×D4×D5311yescharacteristic
343C22C2×Q8×D5311yescharacteristic
344C22#41×D5311yescharacteristic
345C22#41×D5311yescharacteristic
346C22C2×#1171994511yescharacteristic
347C22#1313326211yescharacteristic
348C22#1313326511yescharacteristic
349C4—22no—
350C4—22no—
351C4—22no—
352C4—41no—
353C4—41no—
354C4—41no—
355C4—41no—
356C4—1062no—
357C4—1062no—
358C4—1062no—
359C4—1062no—
360C4—1062no—
361C4—2121no—
362C4—2121no—
363C2—2122no—
364C2#1681117211yescharacteristic
365C2#1681117711yescharacteristic
366C2#1681121911yescharacteristic
367C2#1681126311yescharacteristic
368C2#1681128811yescharacteristic
369C2#1681131411yescharacteristic
370C2#1681132711yescharacteristic
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#391C2
hypercenter#391D53
hypocenterC53#2556
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #391

Complements

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

NH ≅ G/Ntype
1. C2×#13133207C2semidirect
4. #16810557C2semidirect
6. #16810542C2semidirect
7. #16810520C2semidirect
8. #16810520C2semidirect
11. #16810505C2semidirect
13. C4×#11719919C2semidirect
15. C53×#391C2semidirect
81. #11719921Q8semidirect
192. #391D53semidirect
201. C53#2556semidirect