Finite Group Databasebeta

← #35226596

Subgroups — #35226596

Order 20864.

380 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#23782324C211yesmaximal, characteristic
2#23782027C211yesmaximal, characteristic, complemented
3#23782006C211yesmaximal, characteristic, complemented
4#23781989C211yesmaximal, characteristic, complemented
5#23781957C211yesmaximal, characteristic, complemented
6#23781913C211yesmaximal, characteristic, complemented
7#23781904C211yesmaximal, characteristic, complemented
8#23781887C211yesmaximal, characteristic, complemented
9#23781825C211yesmaximal, characteristic, complemented
10#23781820C211yesmaximal, characteristic, complemented
11#23781770C211yesmaximal, characteristic, complemented
12#23781737C211yesmaximal, characteristic, complemented
13#23781673C211yesmaximal, characteristic, complemented
14#23781659C211yesmaximal, characteristic, complemented
15#556×C163C211yesmaximal, characteristic, complemented
16C2×#14579699C2211yescharacteristic
17C22×Dic326C2211yescharacteristic
18C2×#14579689C2211yescharacteristic
19C2×#14579689C2211yescharacteristic
20C2×#14579686C2211yescharacteristic
21C2×#14579683C2211yescharacteristic
22C22×C163⋊C8C2211yescharacteristic
23#19104723C2211yescharacteristic
24#19104721—21no—
25#19104714C2211yescharacteristic
26#19104713—21no—
27#19104709C2211yescharacteristic
28#19104707—21no—
29C4×Dic326C2211yescharacteristic
30(C22⋊D163)⋊C4—21no—
31#19104668C2211yescharacteristic
32#19104668C2211yescharacteristic
33#19104666C2211yescharacteristic
34C4⋊(C163⋊C8)C2211yescharacteristic
35#19104659C2211yescharacteristic
36#19104658C2211yescharacteristic
37#19104658C2211yescharacteristic
38#19104658C2211yescharacteristic
39#19104658C2211yescharacteristic
40#19104658C2211yescharacteristic
41#19104658C2211yescharacteristic
42#19104655C2211yescharacteristic
43#19104655C2211yescharacteristic
44#19104655C2211yescharacteristic
45C22⋊(C163⋊C8)C2211yescharacteristic
46C22⋊(C163⋊C8)C2211yescharacteristic
47#19104646C2211yescharacteristic
48C4⋊Q8×C163C2211yescharacteristic
49#126×C163—21no—
50#126×C163—21no—
51#124×C163—21no—
52C22⋊Q8×C163C2211yescharacteristic
53C22⋊Q8×C163C2211yescharacteristic
54C4×Q8×C163—21no—
55C4×Q8×C163C2211yescharacteristic
56C4×D4×C163C2211yescharacteristic
57#117×C163C2211yescharacteristic
58#117×C163C2211yescharacteristic
59#14579699—22no—
60#14579699—41no—
61C2×Dic326—22no—
62C2×Dic326—21no—
63C2×Dic326—41no—
64C2×Dic326C2311yescharacteristic
65C2×Dic326C2311yescharacteristic
66C2×D652C2311yescharacteristic
67C2×C22⋊D163—21no—
68C2×C22⋊D163—21no—
69C22×Dic163—21no—
70C2×C4×D163—21no—
71#14579689—22no—
72#14579689—41no—
73#14579686—41no—
74#14579683—41no—
75C2×C163⋊C8—41no—
76C2×C163⋊C8C2311yescharacteristic
77C2×C163⋊C8C2311yescharacteristic
78C2×C163⋊C8C2311yescharacteristic
79C2×C163⋊C8C2311yescharacteristic
80C4⋊Dic163C2311yescharacteristic
81Dic163⋊C4—21no—
82Dic163⋊C4—21no—
83C22⋊Dic163—21no—
84(C2×D163)⋊C4—21no—
85(C2×D163)⋊C4—21no—
86(C2×D163)⋊C4—21no—
87C4×Dic163—21no—
88C2×Q8×C163—21no—
89C2×Q8×C163C2311yescharacteristic
90C2×D4×C163D411yescharacteristic
91C22×C4×C163C2311yescharacteristic
92C22×C4×C163C2311yescharacteristic
93C4⋊C4×C163—21no—
94C4⋊C4×C163—21no—
95C4⋊C4×C163—21no—
96C4⋊C4×C163—21no—
97C4⋊C4×C163D411yescharacteristic
98C4⋊C4×C163C2311yescharacteristic
99C4⋊C4×C163C2311yescharacteristic
100C4⋊C4×C163C2311yescharacteristic
101C22⋊C4×C163—21no—
102C22⋊C4×C163—21no—
103C22⋊C4×C163—21no—
104C22⋊C4×C163—21no—
105C22⋊C4×C163D411yescharacteristic
106C22⋊C4×C163D411yescharacteristic
107C42×C163—21no—
108C42×C163—21no—
109C42×C163—21no—
110C42×C163C2311yescharacteristic
111C22×D163—21no—
112Dic326—21no—
113Dic326—21no—
114Dic326—21no—
115Dic326—42no—
116Dic326#4112yes—
117D652—21no—
118D652—21no—
119C22⋊D163—42no—
120C22⋊D163—42no—
121C2×Dic163—21no—
122C2×Dic163—21no—
123C2×Dic163—21no—
124C2×Dic163—42no—
125C4×D163—42no—
126C163⋊C8—21no—
127C163⋊C8—21no—
128C163⋊C8—21no—
129C163⋊C8—21no—
130C23×C163C2×D411yescharacteristic
131C23×C163C2×D411yescharacteristic
132Q8×C163—21no—
133Q8×C163—21no—
134Q8×C163—42no—
135D4×C163—41no—
136C2×C4×C163—21no—
137C2×C4×C163—21no—
138C2×C4×C163—21no—
139C2×C4×C163—21no—
140C2×C4×C163—21no—
141C2×C4×C163—21no—
142C2×C4×C163—21no—
143C2×C4×C163—21no—
144C2×C4×C163C2×D411yescharacteristic
145C2×C4×C163C2×D411yescharacteristic
146C2×C4×C163C2×D411yescharacteristic
147C2×C4×C163C2×D411yescharacteristic
148C2×C4×C163C2411yescharacteristic
149C22×C163—21no—
150C22×C163—21no—
151C22×C163—21no—
152C22×C163—21no—
153C22×C163—21no—
154C22×C163—21no—
155C22×C163C22×D411yescharacteristic
156C2×D163—41no—
157C2×D163—41no—
158C2×D163—41no—
159C4×C163—21no—
160C4×C163—21no—
161C4×C163—21no—
162C4×C163—21no—
163C4×C163—41no—
164C4×C163—41no—
165C4×C163—41no—
166C4×C163—41no—
167C4×C163C2×#4111yescharacteristic
168C4×C163H(5,𝔽2)11yescharacteristic
169Dic163—22no—
170Dic163—41no—
171Dic163—41no—
172C2×C163—41no—
173C2×C163—41no—
174C2×C163#54611yescharacteristic
175C2×C163#57511yescharacteristic
176C2×C163#57811yescharacteristic
177D163—81no—
178C163#331311yescharacteristic, complemented, Sylow
179#3313—1631nomaximal, Sylow
180#556D16311yescharacteristic, complemented
181#548—1631no—
182#489—1631no—
183#485—1631no—
184#482—1631no—
185#477—1631no—
186#468—1631no—
187#467—1631no—
188#464—1631no—
189#452—1631no—
190#451—1631no—
191#441—1631no—
192#435—1631no—
193#419—1631no—
194#416—1631no—
195C2×#41—1631no—
196C22×Q8—1631no—
197C2×Q16—1631no—
198C2×Q16—1631no—
199C2×SD16—1631no—
200C2×M16—1631no—
201C22×C8—1631no—
202C4⋊Q8C2×D16311yescharacteristic
203#126—21no—
204#126—21no—
205#124—21no—
206#124—3261no—
207#124—1631no—
208C22⋊Q8—3261no—
209C22⋊Q8—1631no—
210C22⋊Q8C2×D16311yescharacteristic
211C22⋊Q8C2×D16311yescharacteristic
212#121—3261no—
213#121—1631no—
214C4×Q8—21no—
215C4×Q8—1631no—
216C4×Q8C2×D16311yescharacteristic
217C4×D4—3261no—
218C4×D4C2×D16311yescharacteristic
219#117C2×D16311yescharacteristic
220#117C2×D16311yescharacteristic
221#107—1631no—
222#107—1631no—
223#106—1631no—
224C4⋊C8—1631no—
225#103—1631no—
226#103—1631no—
227#103—1631no—
228#103—1631no—
229#103—1631no—
230#103—1631no—
231#103—1631no—
232#102—1631no—
233#102—1631no—
234#102—1631no—
235C22⋊C8—1631no—
236C22⋊C8—1631no—
237#97—1631no—
238#41—3262no—
239#41—6521no—
240C2×Q8—21no—
241C2×Q8—3262no—
242C2×Q8—3261no—
243C2×Q8—6521no—
244C2×Q8—1631no—
245C2×Q8—1631no—
246C2×Q8C22×D16311yescharacteristic
247C2×D4—3261no—
248C2×D4—3261no—
249C2×D4—1631no—
250C2×D4C22⋊D16311yescharacteristic
251C22×C4—3261no—
252C22×C4—3261no—
253C22×C4C22×D16311yescharacteristic
254C22×C4C22×D16311yescharacteristic
255Q16—3262no—
256Q16—6521no—
257SD16—6521no—
258M16—6521no—
259C2×C8—6521no—
260C2×C8—1631no—
261C2×C8—1631no—
262C2×C8—1631no—
263C2×C8—1631no—
264C4⋊C4—21no—
265C4⋊C4—21no—
266C4⋊C4—21no—
267C4⋊C4—21no—
268C4⋊C4—3261no—
269C4⋊C4—3261no—
270C4⋊C4—1631no—
271C4⋊C4C22⋊D16311yescharacteristic
272C4⋊C4C22×D16311yescharacteristic
273C4⋊C4C22×D16311yescharacteristic
274C4⋊C4C22×D16311yescharacteristic
275C22⋊C4—21no—
276C22⋊C4—21no—
277C22⋊C4—21no—
278C22⋊C4—21no—
279C22⋊C4—3261no—
280C22⋊C4—3261no—
281C22⋊C4—3261no—
282C22⋊C4—3261no—
283C22⋊C4C22⋊D16311yescharacteristic
284C22⋊C4C22⋊D16311yescharacteristic
285C42—21no—
286C42—21no—
287C42—21no—
288C42—3261no—
289C42C22×D16311yescharacteristic
290C23—3261no—
291C23C2×C22⋊D16311yescharacteristic
292C23C2×C22⋊D16311yescharacteristic
293Q8—21no—
294Q8—21no—
295Q8—42no—
296Q8—3261no—
297Q8—3261no—
298Q8—3261no—
299Q8—6522no—
300Q8—1632no—
301D4—41no—
302D4—3261no—
303D4—3261no—
304D4—6522no—
305D4—6522no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—3261no—
315C2×C4—3261no—
316C2×C4—3261no—
317C2×C4—6522no—
318C2×C4—6522no—
319C2×C4C2×C22⋊D16311yescharacteristic
320C2×C4C2×C22⋊D16311yescharacteristic
321C2×C4C2×C22⋊D16311yescharacteristic
322C2×C4C2×C22⋊D16311yescharacteristic
323C2×C4C23×D16311yescharacteristic
324C8—3261no—
325C8—3261no—
326C8—3261no—
327C8—3261no—
328C22—21no—
329C22—21no—
330C22—21no—
331C22—21no—
332C22—21no—
333C22—21no—
334C22—6521no—
335C22—6521no—
336C22—6521no—
337C22C22×C22⋊D16311yescharacteristic
338C4—21no—
339C4—21no—
340C4—21no—
341C4—21no—
342C4—41no—
343C4—41no—
344C4—41no—
345C4—41no—
346C4—3262no—
347C4—6521no—
348C4—6521no—
349C4#41×D16311yescharacteristic
350C4#1910478111yescharacteristic
351C2—41no—
352C2—41no—
353C2—13041no—
354C2#2378230311yescharacteristic
355C2#2378246311yescharacteristic
356C2#2378248011yescharacteristic
subgroupG/H
centerC22C22×C22⋊D163
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
Fitting#556×C163C2
socleC22×C163C22×D4
prime radical#23782324C2
hypercenter#556D163
hypocenterC163#3313
LCSG ▷ C2×C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C22 ◁ C4×D4 ◁ #556

Complements

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

NH ≅ G/Ntype
2. #23782027C2semidirect
3. #23782006C2semidirect
4. #23781989C2semidirect
5. #23781957C2semidirect
6. #23781913C2semidirect
7. #23781904C2semidirect
8. #23781887C2semidirect
9. #23781825C2semidirect
10. #23781820C2semidirect
11. #23781770C2semidirect
12. #23781737C2semidirect
13. #23781673C2semidirect
14. #23781659C2semidirect
15. #556×C163C2semidirect
178. C163#3313semidirect
180. #556D163semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#237823242
#237820272
#237820062
#237819892
#237819572
#237819132
#237819042
#237818872
#237818252
#237818202
#237817702
#237817372
#237816732
#237816592
#556×C1632
#3313163