Finite Group Databasebeta

← #20285591

Subgroups — #20285591

Order 6784.

390 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#16811194C211yesmaximal, characteristic, complemented
2#16810961C211yesmaximal, characteristic, complemented
3#16810958C211yesmaximal, characteristic, complemented
4#16810953C211yesmaximal, characteristic, complemented
5#16810944C211yesmaximal, characteristic, complemented
6#16810879C211yesmaximal, characteristic, complemented
7#16810867C211yesmaximal, characteristic, complemented
8#16810862C211yesmaximal, characteristic, complemented
9#16810850C211yesmaximal, characteristic, complemented
10#16810785C211yesmaximal, characteristic, complemented
11#16810785C211yesmaximal, characteristic, complemented
12#16810778C211yesmaximal, characteristic, complemented
13#16810761C211yesmaximal, characteristic, complemented
14#16810590C211yesmaximal, characteristic, complemented
15C53×#537C211yesmaximal, characteristic
16C22×Dic106C2211yescharacteristic
17C2×#11719925C2211yescharacteristic
18C2×#11719925C2211yescharacteristic
19C2×#11719923C2211yescharacteristic
20C2×#11719923C2211yescharacteristic
21C2×#11719917C2211yescharacteristic
22C22×#88322C2211yescharacteristic
23#13133215—21no—
24#13133204—21no—
25#13133198—21no—
26#13133198—21no—
27#13133196C2211yescharacteristic
28#13133196C2211yescharacteristic
29#13133196C2211yescharacteristic
30#13133196C2211yescharacteristic
31#13133176C2211yescharacteristic
32#13133150C2211yescharacteristic
33#13133150C2211yescharacteristic
34#13133148C2211yescharacteristic
35#13133148C2211yescharacteristic
36#13133141C2211yescharacteristic
37#13133141C2211yescharacteristic
38#13133140C2211yescharacteristic
39#13133140C2211yescharacteristic
40#13133140C2211yescharacteristic
41#13133140C2211yescharacteristic
42#13133138C2211yescharacteristic
43#13133138C2211yescharacteristic
44C22⋊#88322C2211yescharacteristic
45C22⋊#88322C2211yescharacteristic
46C22⋊#88322C2211yescharacteristic
47C22⋊#88322C2211yescharacteristic
48C2×#41×C53C2211yescharacteristic
49C2×#41×C53C2211yescharacteristic
50C22⋊Q8×C53—21no—
51C22⋊Q8×C53C2211yescharacteristic
52C22⋊Q8×C53C2211yescharacteristic
53#121×C53—21no—
54#121×C53—21no—
55#121×C53—21no—
56#121×C53C2211yescharacteristic
57#121×C53C2211yescharacteristic
58C2×C4⋊C4×C53C2211yescharacteristic
59#41×C53—41no—
60#41×C53—41no—
61#41×C53—41no—
62#41×C53—41no—
63C2×Q8×C53D411yescharacteristic
64C2×Q8×C53C2311yescharacteristic
65C2×D4×C53—21no—
66C2×D4×C53—21no—
67C2×D4×C53—21no—
68C2×D4×C53D411yescharacteristic
69C2×D4×C53D411yescharacteristic
70C2×D4×C53D411yescharacteristic
71C2×D4×C53C2311yescharacteristic
72C22×C4×C53—21no—
73C22×C4×C53—21no—
74C22×C4×C53C2311yescharacteristic
75C22×C4×C53C2311yescharacteristic
76C22×C4×C53C2311yescharacteristic
77C22×Dic53—21no—
78C2×Dic106—21no—
79C2×Dic106—41no—
80C2×Dic106—41no—
81C2×Dic106C2311yescharacteristic
82C2×Dic106C2311yescharacteristic
83C4⋊C4×C53—21no—
84C4⋊C4×C53—21no—
85C4⋊C4×C53—21no—
86C4⋊C4×C53C2311yescharacteristic
87C4⋊C4×C53C2311yescharacteristic
88C22⋊C4×C53—21no—
89C22⋊C4×C53—21no—
90C22⋊C4×C53—21no—
91C22⋊C4×C53—21no—
92C22⋊Dic53—21no—
93C22⋊Dic53—21no—
94C22⋊Dic53—21no—
95C22⋊Dic53—21no—
96C22⋊Dic53—21no—
97#11719925—41no—
98#11719925—41no—
99#11719923—41no—
100#11719923—41no—
101C4⋊Dic53C2311yescharacteristic
102C4⋊Dic53C2311yescharacteristic
103#11719919—21no—
104#11719919—21no—
105#11719919—21no—
106#11719919—21no—
107C4×Dic53—21no—
108#11719917—41no—
109C2×#88322—41no—
110C2×#88322C2311yescharacteristic
111C2×#88322C2311yescharacteristic
112C2×#88322C2311yescharacteristic
113C2×#88322C2311yescharacteristic
114C23×C53—21no—
115C23×C53C2×D411yescharacteristic
116C23×C53C2×D411yescharacteristic
117C23×C53C2×D411yescharacteristic
118Q8×C53—21no—
119Q8×C53—21no—
120Q8×C53—41no—
121D4×C53—21no—
122D4×C53—21no—
123D4×C53—41no—
124D4×C53—41no—
125D4×C53—41no—
126D4×C53—42no—
127D4×C53—81no—
128D4×C53—81no—
129C2×C4×C53—21no—
130C2×C4×C53—21no—
131C2×C4×C53—21no—
132C2×C4×C53—21no—
133C2×C4×C53—21no—
134C2×C4×C53—21no—
135C2×C4×C53—21no—
136C2×C4×C53—21no—
137C2×C4×C53—21no—
138C2×C4×C53—42no—
139C2×C4×C53—81no—
140C2×C4×C53C2×D411yescharacteristic
141C2×C4×C53C2×D411yescharacteristic
142C2×C4×C53C2×D411yescharacteristic
143C2×C4×C53C2411yescharacteristic
144C2×Dic53—21no—
145C2×Dic53—21no—
146C2×Dic53—21no—
147C2×Dic53—21no—
148C2×Dic53—42no—
149Dic106—21no—
150Dic106—21no—
151Dic106—21no—
152Dic106—21no—
153Dic106—42no—
154#88322—21no—
155#88322—21no—
156#88322—21no—
157#88322—21no—
158C22×C53—21no—
159C22×C53—21no—
160C22×C53—21no—
161C22×C53—21no—
162C22×C53—21no—
163C22×C53—21no—
164C22×C53—21no—
165C22×C53—21no—
166C22×C53—21no—
167C22×C53—41no—
168C22×C53—41no—
169C22×C53—41no—
170C22×C53C22×D411yescharacteristic
171C4×C53—21no—
172C4×C53—21no—
173C4×C53—21no—
174C4×C53—41no—
175C4×C53—41no—
176C4×C53—41no—
177C4×C53H(5,𝔽2)11yescharacteristic
178C4×C53H(5,𝔽2)11yescharacteristic
179Dic53—41no—
180Dic53—41no—
181Dic53—41no—
182Dic53—41no—
183#3201—531nomaximal, Sylow
184C2×C53—41no—
185C2×C53—41no—
186C2×C53—41no—
187C2×C53—81no—
188C2×C53#53411yescharacteristic
189C2×C53#57711yescharacteristic
190C2×C53#57811yescharacteristic
191#537D5311yescharacteristic
192#536—531no—
193#485—531no—
194#484—531no—
195#483—531no—
196#482—531no—
197#470—531no—
198#468—531no—
199#467—531no—
200#465—531no—
201#452—531no—
202#452—531no—
203#451—531no—
204#448—531no—
205#408—531no—
206C53#320111yescharacteristic, complemented, Sylow
207C2×#41C2×D5311yescharacteristic
208C2×#41C2×D5311yescharacteristic
209C22×Q8—531no—
210C2×Q16—531no—
211C2×Q16—531no—
212C2×SD16—531no—
213C2×SD16—531no—
214C2×M16—531no—
215C22×C8—531no—
216C4⋊Q8—1061no—
217#124—1061no—
218#123—1061no—
219#123—1061no—
220C22⋊Q8—21no—
221C22⋊Q8—531no—
222C22⋊Q8—531no—
223C22⋊Q8—531no—
224C22⋊Q8—531no—
225C22⋊Q8C2×D5311yescharacteristic
226C22⋊Q8C2×D5311yescharacteristic
227#121—21no—
228#121—21no—
229#121—21no—
230#121C2×D5311yescharacteristic
231#121C2×D5311yescharacteristic
232#117—531no—
233C2×C4⋊C4C2×D5311yescharacteristic
234#107—531no—
235#107—531no—
236#106—531no—
237#106—531no—
238#103—531no—
239#103—531no—
240#103—531no—
241#103—531no—
242#103—531no—
243#103—531no—
244#102—531no—
245#102—531no—
246C22⋊C8—531no—
247C22⋊C8—531no—
248C22⋊C8—531no—
249C22⋊C8—531no—
250#41—41no—
251#41—41no—
252#41—41no—
253#41—41no—
254C2×Q8—531no—
255C2×Q8—531no—
256C2×Q8—1061no—
257C2×Q8—2121no—
258C2×Q8—2121no—
259C2×Q8C22⋊D5311yescharacteristic
260C2×Q8C22×D5311yescharacteristic
261C2×D4—21no—
262C2×D4—21no—
263C2×D4—21no—
264C2×D4C22⋊D5311yescharacteristic
265C2×D4C22⋊D5311yescharacteristic
266C2×D4C22⋊D5311yescharacteristic
267C2×D4C22×D5311yescharacteristic
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—1061no—
271C22×C4C22×D5311yescharacteristic
272C22×C4C22×D5311yescharacteristic
273C22×C4C22×D5311yescharacteristic
274Q16—2121no—
275Q16—2121no—
276SD16—2121no—
277SD16—2121no—
278M16—2121no—
279C2×C8—531no—
280C2×C8—531no—
281C2×C8—531no—
282C2×C8—531no—
283C2×C8—2121no—
284C4⋊C4—21no—
285C4⋊C4—21no—
286C4⋊C4—21no—
287C4⋊C4—531no—
288C4⋊C4—531no—
289C4⋊C4—1061no—
290C4⋊C4—1061no—
291C4⋊C4—1061no—
292C4⋊C4—1061no—
293C4⋊C4C22×D5311yescharacteristic
294C4⋊C4C22×D5311yescharacteristic
295C22⋊C4—21no—
296C22⋊C4—21no—
297C22⋊C4—21no—
298C22⋊C4—21no—
299C22⋊C4—1061no—
300C22⋊C4—1061no—
301C22⋊C4—1061no—
302C22⋊C4—1061no—
303C22⋊C4—1061no—
304C42—1061no—
305C23—21no—
306C23C2×C22⋊D5311yescharacteristic
307C23C2×C22⋊D5311yescharacteristic
308C23C2×C22⋊D5311yescharacteristic
309Q8—21no—
310Q8—21no—
311Q8—41no—
312Q8—1061no—
313Q8—1061no—
314Q8—1061no—
315Q8—1061no—
316Q8—2122no—
317D4—21no—
318D4—21no—
319D4—41no—
320D4—41no—
321D4—41no—
322D4—42no—
323D4—81no—
324D4—81no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—42no—
335C2×C4—81no—
336C2×C4—1061no—
337C2×C4—1061no—
338C2×C4—1061no—
339C2×C4—1061no—
340C2×C4—2122no—
341C2×C4C2×C22⋊D5311yescharacteristic
342C2×C4C2×C22⋊D5311yescharacteristic
343C2×C4C2×C22⋊D5311yescharacteristic
344C2×C4C23×D5311yescharacteristic
345C8—1061no—
346C8—1061no—
347C8—1061no—
348C8—1061no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—21no—
353C22—21no—
354C22—21no—
355C22—21no—
356C22—21no—
357C22—21no—
358C22—41no—
359C22—41no—
360C22—41no—
361C22C22×C22⋊D5311yescharacteristic
362C4—21no—
363C4—21no—
364C4—21no—
365C4—41no—
366C4—41no—
367C4—41no—
368C4—2121no—
369C4—2121no—
370C4—2121no—
371C4—2121no—
372C4#1313326211yescharacteristic
373C4#1313326211yescharacteristic
374C2—41no—
375C2—41no—
376C2—41no—
377C2—81no—
378C2#1681118211yescharacteristic
379C2#1681143411yescharacteristic
380C2#1681143811yescharacteristic
subgroupG/H
centerC22C22×C22⋊D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#537C2
socleC22×C53C22×D4
hypercenter#537D53
hypocenterC53#3201
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×#41 ◁ #537

Complements

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

NH ≅ G/Ntype
1. #16811194C2semidirect
2. #16810961C2semidirect
3. #16810958C2semidirect
4. #16810953C2semidirect
5. #16810944C2semidirect
6. #16810879C2semidirect
7. #16810867C2semidirect
8. #16810862C2semidirect
9. #16810850C2semidirect
10. #16810785C2semidirect
11. #16810785C2semidirect
12. #16810778C2semidirect
13. #16810761C2semidirect
14. #16810590C2semidirect
206. C53#3201semidirect