Finite Group Databasebeta

← #20280381

Subgroups — #20280381

Order 6784.

428 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#16810546C211yesmaximal, characteristic
2#16810546C211yesmaximal, characteristic
3#16810546C211yesmaximal, characteristic
4#16810522C211yesmaximal, characteristic, complemented
5#16810519C211yesmaximal, characteristic, complemented
6#16810518C211yesmaximal, characteristic, complemented
7#16810518C211yesmaximal, characteristic, complemented
8#16810509C211yesmaximal, characteristic, complemented
9#16810508C211yesmaximal, characteristic, complemented
10#16810505C211yesmaximal, characteristic, complemented
11#16810504C211yesmaximal, characteristic, complemented
12#16810484C211yesmaximal, characteristic
13C22⋊C4×Dic53C211yesmaximal, characteristic
14C22⋊C4×Dic53C211yesmaximal, characteristic
15C53×#396C211yesmaximal, characteristic
16C23×Dic53C2211yescharacteristic
17C2×C4⋊Dic53C2211yescharacteristic
18C2×C4⋊Dic53C2211yescharacteristic
19C2×C4⋊Dic53C2211yescharacteristic
20C2×#11719919C2211yescharacteristic
21C2×#11719919C2211yescharacteristic
22C2×#11719919C2211yescharacteristic
23C2×#11719919C2211yescharacteristic
24C2×#11719919C2211yescharacteristic
25C2×#11719919C2211yescharacteristic
26C2×C22⋊Dic53C2211yescharacteristic
27C2×C22⋊Dic53C2211yescharacteristic
28C2×C22⋊Dic53C2211yescharacteristic
29C2×C4×Dic53C2211yescharacteristic
30C2×C4×Dic53C2211yescharacteristic
31C2×C4×Dic53C2211yescharacteristic
32C2×C4×Dic53C2211yescharacteristic
33#13133124C2211yescharacteristic
34#13133124C2211yescharacteristic
35#13133124C2211yescharacteristic
36#13133124C2211yescharacteristic
37#13133124C2211yescharacteristic
38#13133124C2211yescharacteristic
39#13133124C2211yescharacteristic
40#13133124C2211yescharacteristic
41#13133124C2211yescharacteristic
42#13133124C2211yescharacteristic
43#13133124C2211yescharacteristic
44C2×C4⋊C4×C53C2211yescharacteristic
45C2×C4⋊C4×C53C2211yescharacteristic
46C2×C4⋊C4×C53C2211yescharacteristic
47C2×C22⋊C4×C53C2211yescharacteristic
48C2×C22⋊C4×C53C2211yescharacteristic
49C2×C22⋊C4×C53C2211yescharacteristic
50#95×C53C2211yescharacteristic
51C24×C53C2311yescharacteristic
52C22×C4×C53C2311yescharacteristic
53C22×C4×C53C2311yescharacteristic
54C22×C4×C53C2311yescharacteristic
55C22×C4×C53C2311yescharacteristic
56C22×C4×C53C2311yescharacteristic
57C22×C4×C53C2311yescharacteristic
58C22×Dic53—41no—
59C22×Dic53—41no—
60C22×Dic53Q814yes—
61C22×Dic53C2311yescharacteristic
62C22×Dic53C2311yescharacteristic
63C22×Dic53C2311yescharacteristic
64C22×Dic53C2311yescharacteristic
65C22×Dic53C2311yescharacteristic
66C22×Dic53C2311yescharacteristic
67C22×Dic53C2311yescharacteristic
68C22×Dic53C2311yescharacteristic
69C4⋊C4×C53—22no—
70C4⋊C4×C53—22no—
71C4⋊C4×C53—41no—
72C22⋊C4×C53—22no—
73C22⋊C4×C53—22no—
74C22⋊C4×C53—41no—
75C22⋊Dic53—22no—
76C22⋊Dic53—22no—
77C22⋊Dic53—41no—
78C4⋊Dic53—22no—
79C4⋊Dic53—22no—
80C4⋊Dic53—41no—
81#11719919—22no—
82#11719919—22no—
83#11719919—22no—
84#11719919—22no—
85#11719919—41no—
86#11719919—41no—
87C4×Dic53—41no—
88C4×Dic53—41no—
89C4×Dic53—41no—
90C4×Dic53—41no—
91C23×C53—21no—
92C23×C53—21no—
93C23×C53—21no—
94C23×C53—21no—
95C23×C53—21no—
96C23×C53—21no—
97C23×C53C2×Q812yes—
98C23×C53C2411yescharacteristic
99C2×C4×C53—21no—
100C2×C4×C53—21no—
101C2×C4×C53—21no—
102C2×C4×C53—21no—
103C2×C4×C53—21no—
104C2×C4×C53—21no—
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#4112yes—
116C2×C4×C53#4112yes—
117C2×Dic53—21no—
118C2×Dic53—21no—
119C2×Dic53—21no—
120C2×Dic53—21no—
121C2×Dic53—21no—
122C2×Dic53—21no—
123C2×Dic53—21no—
124C2×Dic53—21no—
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—44no—
138C2×Dic53C2×Q812yes—
139C2×Dic53C2×Q812yes—
140C2×Dic53#4112yes—
141C2×Dic53#4112yes—
142C22×C53—22no—
143C22×C53—22no—
144C22×C53—22no—
145C22×C53—41no—
146C22×C53—41no—
147C22×C53—41no—
148C22×C53—41no—
149C22×C53C22×Q811yescharacteristic
150C22×C53C2×#4111yescharacteristic
151C22×C53C2×#4111yescharacteristic
152C22×C53C2×#4111yescharacteristic
153C22×C53C2×#4111yescharacteristic
154C22×C53H(5,𝔽2)11yescharacteristic
155C22×C53H(5,𝔽2)11yescharacteristic
156C4×C53—22no—
157C4×C53—22no—
158C4×C53—41no—
159C4×C53—41no—
160C4×C53—41no—
161C4×C53—41no—
162Dic53—22no—
163Dic53—22no—
164Dic53—22no—
165Dic53—22no—
166Dic53—41no—
167Dic53—41no—
168Dic53—41no—
169Dic53—41no—
170#2608—531nomaximal, Sylow
171C2×C53—42no—
172C2×C53#53311yescharacteristic
173C2×C53#53311yescharacteristic
174C2×C53#54311yescharacteristic
175C2×C53#55311yescharacteristic
176C2×C53#55311yescharacteristic
177C2×C53#55311yescharacteristic
178C2×C53#55311yescharacteristic
179#396—531no—
180#396—531no—
181#396—531no—
182#396D5311yescharacteristic
183#389—531no—
184#389—531no—
185#389—531no—
186#389—531no—
187#387—531no—
188#387—531no—
189#387—531no—
190#387—531no—
191#381—531no—
192C4×C22⋊C4—531no—
193C4×C22⋊C4—531no—
194C53#260811yescharacteristic, complemented, Sylow
195C23×C4—531no—
196C2×C4⋊C4—531no—
197C2×C4⋊C4—531no—
198C2×C4⋊C4—531no—
199C2×C4⋊C4—531no—
200C2×C4⋊C4—531no—
201C2×C4⋊C4—531no—
202C2×C4⋊C4—531no—
203C2×C4⋊C4—531no—
204C2×C4⋊C4—531no—
205C2×C4⋊C4C2×D5311yescharacteristic
206C2×C4⋊C4C2×D5311yescharacteristic
207C2×C4⋊C4C2×D5311yescharacteristic
208C2×C22⋊C4—531no—
209C2×C22⋊C4—531no—
210C2×C22⋊C4—531no—
211C2×C22⋊C4C2×D5311yescharacteristic
212C2×C22⋊C4C2×D5311yescharacteristic
213C2×C22⋊C4C2×D5311yescharacteristic
214C2×C42—531no—
215C2×C42—531no—
216C2×C42—531no—
217C2×C42—531no—
218#95—531no—
219#95—531no—
220#95—531no—
221#95—531no—
222#95—531no—
223#95—531no—
224#95—531no—
225#95—531no—
226#95—531no—
227#95—531no—
228#95—531no—
229#95C2×D5311yescharacteristic
230C24C22×D5311yescharacteristic
231C22×C4—531no—
232C22×C4—531no—
233C22×C4—531no—
234C22×C4—531no—
235C22×C4—534no—
236C22×C4—531no—
237C22×C4—531no—
238C22×C4—531no—
239C22×C4—531no—
240C22×C4—2121no—
241C22×C4—2121no—
242C22×C4C22×D5311yescharacteristic
243C22×C4C22×D5311yescharacteristic
244C22×C4C22×D5311yescharacteristic
245C22×C4C22×D5311yescharacteristic
246C22×C4C22×D5311yescharacteristic
247C22×C4C22×D5311yescharacteristic
248C4⋊C4—22no—
249C4⋊C4—22no—
250C4⋊C4—41no—
251C4⋊C4—1062no—
252C4⋊C4—1062no—
253C4⋊C4—1062no—
254C4⋊C4—1062no—
255C4⋊C4—1062no—
256C4⋊C4—1062no—
257C4⋊C4—2121no—
258C4⋊C4—2121no—
259C4⋊C4—2121no—
260C22⋊C4—22no—
261C22⋊C4—22no—
262C22⋊C4—41no—
263C22⋊C4—1062no—
264C22⋊C4—1062no—
265C22⋊C4—2121no—
266C42—2121no—
267C42—2121no—
268C42—2121no—
269C42—2121no—
270C23—21no—
271C23—21no—
272C23—21no—
273C23—21no—
274C23—21no—
275C23—21no—
276C23Q8×D5312yes—
277C23C23×D5311yescharacteristic
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—532no—
295C2×C4—532no—
296C2×C4—532no—
297C2×C4—532no—
298C2×C4—1061no—
299C2×C4—1061no—
300C2×C4—1061no—
301C2×C4—1061no—
302C2×C4—1061no—
303C2×C4—1061no—
304C2×C4—1061no—
305C2×C4—1061no—
306C2×C4—1061no—
307C2×C4—1061no—
308C2×C4—1061no—
309C2×C4—1061no—
310C2×C4—1061no—
311C2×C4—1061no—
312C2×C4—1061no—
313C2×C4—1061no—
314C2×C4—1061no—
315C2×C4—1061no—
316C2×C4—1061no—
317C2×C4—1061no—
318C2×C4—2124no—
319C2×C4#1171994712yes—
320C2×C4#1171994712yes—
321C22—22no—
322C22—22no—
323C22—22no—
324C22—41no—
325C22—41no—
326C22—41no—
327C22—41no—
328C22C2×Q8×D5311yescharacteristic
329C22#41×D5311yescharacteristic
330C22#41×D5311yescharacteristic
331C22C2×#1171994711yescharacteristic
332C22C2×#1171994711yescharacteristic
333C22#1313326211yescharacteristic
334C22#1313326311yescharacteristic
335C4—22no—
336C4—22no—
337C4—41no—
338C4—41no—
339C4—41no—
340C4—41no—
341C4—1062no—
342C4—1062no—
343C4—1062no—
344C4—1062no—
345C4—2121no—
346C4—2121no—
347C4—2121no—
348C4—2121no—
349C2—42no—
350C2#1681117511yescharacteristic
351C2#1681117511yescharacteristic
352C2#1681124811yescharacteristic
353C2#1681131611yescharacteristic
354C2#1681132011yescharacteristic
355C2#1681132411yescharacteristic
356C2#1681132511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#396C2
hypercenter#396D53
hypocenterC53#2608
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #396

Complements

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

NH ≅ G/Ntype
4. #16810522C2semidirect
5. #16810519C2semidirect
6. #16810518C2semidirect
7. #16810518C2semidirect
8. #16810509C2semidirect
9. #16810508C2semidirect
10. #16810505C2semidirect
11. #16810504C2semidirect
194. C53#2608semidirect