Finite Group Databasebeta

← #20278397

Subgroups — #20278397

Order 6784.

528 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#13133194C211yesmaximal, characteristic
2C2×#13133173C211yesmaximal, characteristic
3#16810564C211yesmaximal, characteristic, complemented
4#16810557C211yesmaximal, characteristic, complemented
5#16810554C211yesmaximal, characteristic, complemented
6#16810543C211yesmaximal, characteristic, complemented
7#16810527C211yesmaximal, characteristic, complemented
8#16810518C211yesmaximal, characteristic, complemented
9#16810509C211yesmaximal, characteristic, complemented
10#16810505C211yesmaximal, characteristic, complemented
11#16810496C211yesmaximal, characteristic
12#16810495C211yesmaximal, characteristic
13#16810484C211yesmaximal, characteristic
14C2×#13133124C211yesmaximal, characteristic
15C53×#380C211yesmaximal, characteristic
16C22×Dic106C2211yescharacteristic
17C23×Dic53C2211yescharacteristic
18C23×Dic53C2211yescharacteristic
19#13133194—24no—
20#13133173—24no—
21C2×C4⋊Dic53C2211yescharacteristic
22C2×#11719919C2211yescharacteristic
23C2×#11719919C2211yescharacteristic
24C2×#11719919C2211yescharacteristic
25C2×#11719919C2211yescharacteristic
26C2×#11719919C2211yescharacteristic
27C2×#11719919C2211yescharacteristic
28C2×#11719919C2211yescharacteristic
29C2×#11719919C2211yescharacteristic
30C2×C22⋊Dic53C2211yescharacteristic
31C2×C22⋊Dic53C2211yescharacteristic
32C2×C4×Dic53C2211yescharacteristic
33C2×C4×Dic53C2211yescharacteristic
34#13133124C2211yescharacteristic
35#13133124C2211yescharacteristic
36#13133124C2211yescharacteristic
37#13133124C2211yescharacteristic
38#13133124C2211yescharacteristic
39#13133124C2211yescharacteristic
40#13133124C2211yescharacteristic
41#13133124C2211yescharacteristic
42#13133124C2211yescharacteristic
43#13133124C2211yescharacteristic
44#13133124C2211yescharacteristic
45#13133124C2211yescharacteristic
46C23×C4×C53C2211yescharacteristic
47C2×C4⋊C4×C53C2211yescharacteristic
48C2×C4⋊C4×C53C2211yescharacteristic
49C2×C22⋊C4×C53C2211yescharacteristic
50C2×C22⋊C4×C53C2211yescharacteristic
51#95×C53C2211yescharacteristic
52#95×C53C2211yescharacteristic
53C24×C53C2311yescharacteristic
54C22×C4×C53—22no—
55C22×C4×C53—22no—
56C22×C4×C53—41no—
57C22×C4×C53C2311yescharacteristic
58C22×C4×C53C2311yescharacteristic
59C22×C4×C53C2311yescharacteristic
60C22×C4×C53C2311yescharacteristic
61C22×C4×C53C2311yescharacteristic
62C22×C4×C53C2311yescharacteristic
63C22×Dic53—22no—
64C22×Dic53—22no—
65C22×Dic53—22no—
66C22×Dic53—22no—
67C22×Dic53—41no—
68C22×Dic53Q814yescomplemented
69C22×Dic53C2311yescharacteristic
70C22×Dic53C2311yescharacteristic
71C22×Dic53C2311yescharacteristic
72C22×Dic53C2311yescharacteristic
73C22×Dic53C2311yescharacteristic
74C22×Dic53C2311yescharacteristic
75C22×Dic53C2311yescharacteristic
76C22×Dic53C2311yescharacteristic
77C2×Dic106—22no—
78C2×Dic106—22no—
79C2×Dic106—41no—
80C4⋊C4×C53—22no—
81C4⋊C4×C53—41no—
82C22⋊C4×C53—22no—
83C22⋊C4×C53—22no—
84C22⋊Dic53—22no—
85C22⋊Dic53D414yes—
86C4⋊Dic53—22no—
87#11719919—22no—
88#11719919—22no—
89#11719919—22no—
90#11719919—22no—
91#11719919—22no—
92#11719919—22no—
93#11719919—41no—
94#11719919—41no—
95C4×Dic53—22no—
96C4×Dic53—22no—
97C23×C53—21no—
98C23×C53—21no—
99C23×C53—21no—
100C23×C53—21no—
101C23×C53C2×D412yes—
102C23×C53C2×Q812yes—
103C23×C53#4112yes—
104C23×C53C2411yescharacteristic
105C2×C4×C53—21no—
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—44no—
123C2×C4×C53#4112yes—
124C2×Dic53—28no—
125C2×Dic53—21no—
126C2×Dic53—21no—
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×Dic53—21no—
144C2×Dic53—44no—
145C2×Dic53C2×D412yes—
146C2×Dic53C2×D412yes—
147C2×Dic53C2×Q812yes—
148C2×Dic53C2×Q812yes—
149C2×Dic53#4112yes—
150Dic106—44no—
151C22×C53—22no—
152C22×C53—22no—
153C22×C53—22no—
154C22×C53—22no—
155C22×C53—22no—
156C22×C53—22no—
157C22×C53C22⋊Q814yes—
158C22×C53C22×D411yescharacteristic
159C22×C53C22×Q811yescharacteristic
160C22×C53C2×#4111yescharacteristic
161C22×C53C2×#4111yescharacteristic
162C22×C53C2×#4111yescharacteristic
163C22×C53H(5,𝔽2)11yescharacteristic
164C22×C532−1+411yescharacteristic
165C4×C53—22no—
166C4×C53—41no—
167C4×C53—41no—
168C4×C53—41no—
169C4×C53—41no—
170C4×C53—41no—
171Dic53—22no—
172Dic53—22no—
173Dic53—22no—
174Dic53—22no—
175Dic53—22no—
176Dic53—41no—
177Dic53—41no—
178Dic53—41no—
179#2438—531nomaximal, Sylow
180C2×C53—24no—
181C2×C53C2×C22⋊Q811yescharacteristic
182C2×C53#53311yescharacteristic
183C2×C53#54411yescharacteristic
184C2×C53#54611yescharacteristic
185C2×C53#54711yescharacteristic
186C2×C53#55111yescharacteristic
187C2×C53#55211yescharacteristic
188C2×C22⋊Q8—531no—
189C2×#117—531no—
190#400—531no—
191#398—531no—
192#398—531no—
193#395—531no—
194#391—531no—
195#389—531no—
196#387—531no—
197#387—531no—
198#385—531no—
199#385—531no—
200#381—531no—
201#380D5311yescharacteristic
202C2×#95—531no—
203C53#243811yescharacteristic, complemented, Sylow
204C22×Q8—531no—
205C23×C4—531no—
206C23×C4—531no—
207C23×C4C2×D5311yescharacteristic
208C22⋊Q8—1064no—
209#117—1064no—
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⋊C4—531no—
219C2×C4⋊C4C2×D5311yescharacteristic
220C2×C4⋊C4C2×D5311yescharacteristic
221C2×C22⋊C4—531no—
222C2×C22⋊C4—531no—
223C2×C22⋊C4C2×D5311yescharacteristic
224C2×C22⋊C4C2×D5311yescharacteristic
225C2×C42—531no—
226C2×C42—531no—
227#95—531no—
228#95—531no—
229#95—531no—
230#95—531no—
231#95—531no—
232#95—531no—
233#95—531no—
234#95—531no—
235#95—531no—
236#95—531no—
237#95—531no—
238#95—531no—
239#95C2×D5311yescharacteristic
240#95C2×D5311yescharacteristic
241C24C22×D5311yescharacteristic
242C2×Q8—1062no—
243C2×Q8—1062no—
244C2×Q8—2121no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—41no—
248C22×C4—534no—
249C22×C4—531no—
250C22×C4—531no—
251C22×C4—531no—
252C22×C4—531no—
253C22×C4—531no—
254C22×C4—531no—
255C22×C4—531no—
256C22×C4—531no—
257C22×C4—1062no—
258C22×C4—1062no—
259C22×C4—1062no—
260C22×C4—1062no—
261C22×C4—2121no—
262C22×C4C22×D5311yescharacteristic
263C22×C4C22×D5311yescharacteristic
264C22×C4C22×D5311yescharacteristic
265C22×C4C22×D5311yescharacteristic
266C22×C4C22×D5311yescharacteristic
267C22×C4C22×D5311yescharacteristic
268C4⋊C4—22no—
269C4⋊C4—41no—
270C4⋊C4—1062no—
271C4⋊C4—1062no—
272C4⋊C4—1062no—
273C4⋊C4—1062no—
274C4⋊C4—1062no—
275C4⋊C4—1062no—
276C4⋊C4—1062no—
277C4⋊C4—2121no—
278C4⋊C4—2121no—
279C22⋊C4—22no—
280C22⋊C4—22no—
281C22⋊C4—534no—
282C22⋊C4—1062no—
283C42—1062no—
284C42—1062no—
285C23—21no—
286C23—21no—
287C23—21no—
288C23—21no—
289C23#1171994512yes—
290C23D4×D5312yes—
291C23Q8×D5312yes—
292C23C23×D5311yescharacteristic
293Q8—2124no—
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—21no—
311C2×C4—44no—
312C2×C4—532no—
313C2×C4—532no—
314C2×C4—532no—
315C2×C4—532no—
316C2×C4—532no—
317C2×C4—1068no—
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—1061no—
332C2×C4—1061no—
333C2×C4—1061no—
334C2×C4—1061no—
335C2×C4—1061no—
336C2×C4—1061no—
337C2×C4—2124no—
338C2×C4#1171994712yes—
339C22—22no—
340C22—22no—
341C22—22no—
342C22—22no—
343C22—22no—
344C22—22no—
345C22#1313319514yes—
346C22C2×D4×D5311yescharacteristic
347C22C2×Q8×D5311yescharacteristic
348C22#41×D5311yescharacteristic
349C22C2×#1171994711yescharacteristic
350C22C2×#1171994511yescharacteristic
351C22#1313326211yescharacteristic
352C22#1313326411yescharacteristic
353C4—22no—
354C4—41no—
355C4—41no—
356C4—41no—
357C4—41no—
358C4—41no—
359C4—1062no—
360C4—1062no—
361C4—1062no—
362C4—1062no—
363C4—1062no—
364C4—2121no—
365C4—2121no—
366C4—2121no—
367C2—24no—
368C2C2×#1313319511yescharacteristic
369C2#1681117511yescharacteristic
370C2#1681125211yescharacteristic
371C2#1681126211yescharacteristic
372C2#1681127311yescharacteristic
373C2#1681129911yescharacteristic
374C2#1681130611yescharacteristic
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#380C2
hypercenter#380D53
hypocenterC53#2438
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #380

Complements

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

NH ≅ G/Ntype
3. #16810564C2semidirect
4. #16810557C2semidirect
5. #16810554C2semidirect
6. #16810543C2semidirect
7. #16810527C2semidirect
8. #16810518C2semidirect
9. #16810509C2semidirect
10. #16810505C2semidirect
68. C22×Dic53Q8semidirect
203. C53#2438semidirect