Finite Group Databasebeta

← #35226429

Subgroups — #35226429

Order 20864.

386 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#23782357C211yesmaximal, characteristic, complemented
2#23782321C211yesmaximal, characteristic
3#23782018C211yesmaximal, characteristic, complemented
4#23782009C211yesmaximal, characteristic, complemented
5#23781976C211yesmaximal, characteristic, complemented
6#23781975C211yesmaximal, characteristic, complemented
7#23781905C211yesmaximal, characteristic, complemented
8#23781885C211yesmaximal, characteristic, complemented
9#23781824C211yesmaximal, characteristic, complemented
10#23781820C211yesmaximal, characteristic, complemented
11#23781757C211yesmaximal, characteristic, complemented
12C8×C22⋊D163C211yesmaximal, characteristic, complemented
13#23781658C211yesmaximal, characteristic, complemented
14C2×#19104660C211yesmaximal, characteristic, complemented
15#465×C163C211yesmaximal, characteristic, complemented
16C2×#14579698C2211yescharacteristic
17C22×Dic326C2211yescharacteristic
18C2×#14579689C2211yescharacteristic
19C2×#14579689C2211yescharacteristic
20C2×C8×D163C2211yescharacteristic
21#19104730—21no—
22#19104726—21no—
23#19104726C2211yescharacteristic
24#19104722C2211yescharacteristic
25#19104721—21no—
26#19104719—21no—
27#19104715C2211yescharacteristic
28#19104712—21no—
29#19104711C2211yescharacteristic
30#19104707—21no—
31Q8×Dic163C2211yescharacteristic
32#19104700C2211yescharacteristic
33D4×Dic163—21no—
34C4×C22⋊D163C2211yescharacteristic
35#19104693C2211yescharacteristic
36#19104691—21no—
37C2×C4⋊Dic163C2211yescharacteristic
38#19104668C2211yescharacteristic
39#19104668C2211yescharacteristic
40Dic163⋊C8C2211yescharacteristic
41#19104660C2211yescharacteristic
42#19104660C2211yescharacteristic
43#19104660C2211yescharacteristic
44#19104660C2211yescharacteristic
45#19104658C2211yescharacteristic
46#19104658C2211yescharacteristic
47#19104656C2211yescharacteristic
48#19104656C2211yescharacteristic
49C22⋊(C163⋊C8)C2211yescharacteristic
50(C2×D163)⋊C8C2211yescharacteristic
51C8×Dic163C2211yescharacteristic
52C2×SD16×C163C2211yescharacteristic
53C22×C8×C163C2211yescharacteristic
54C22⋊Q8×C163C2211yescharacteristic
55#121×C163C2211yescharacteristic
56#106×C163C2211yescharacteristic
57#103×C163C2211yescharacteristic
58#102×C163C2211yescharacteristic
59#14579698—22no—
60#14579698—41no—
61C2×Dic326—22no—
62C2×Dic326—21no—
63C2×Dic326—41no—
64C2×Dic326C2311yescharacteristic
65C2×Dic326C2311yescharacteristic
66C2×C22⋊D163—21no—
67C2×C22⋊D163D411yescharacteristic
68C22×Dic163—21no—
69C22×Dic163—21no—
70C22×Dic163—21no—
71C2×C4×D163C2311yescharacteristic
72#14579689—22no—
73#14579689—41no—
74C2×C163⋊C8C2311yescharacteristic
75C2×C163⋊C8C2311yescharacteristic
76C8×D163—41no—
77C4⋊Dic163—21no—
78C4⋊Dic163—21no—
79C4⋊Dic163—21no—
80C4⋊Dic163—21no—
81C4⋊Dic163C2311yescharacteristic
82C4⋊Dic163C2311yescharacteristic
83Dic163⋊C4—21no—
84Dic163⋊C4—21no—
85Dic163⋊C4—21no—
86Dic163⋊C4D411yescharacteristic
87C22⋊Dic163—21no—
88C22⋊Dic163—21no—
89C22⋊Dic163D411yescharacteristic
90(C2×D163)⋊C4—21no—
91(C2×D163)⋊C4—21no—
92(C2×D163)⋊C4—21no—
93(C2×D163)⋊C4D411yescharacteristic
94C4×Dic163—21no—
95C4×Dic163—21no—
96C4×Dic163—21no—
97C4×Dic163C2311yescharacteristic
98C2×Q8×C163C2311yescharacteristic
99C2×D4×C163—21no—
100C2×D4×C163C2311yescharacteristic
101C22×C4×C163C2311yescharacteristic
102SD16×C163—41no—
103C2×C8×C163—41no—
104C2×C8×C163C2311yescharacteristic
105C2×C8×C163C2311yescharacteristic
106C4⋊C4×C163—21no—
107C4⋊C4×C163C2311yescharacteristic
108C4⋊C4×C163C2311yescharacteristic
109C22⋊C4×C163—21no—
110C22⋊C4×C163—21no—
111C22×D163C2×D411yescharacteristic
112Dic326—21no—
113Dic326—21no—
114Dic326—21no—
115Dic326—42no—
116Dic326#4112yes—
117C22⋊D163—22no—
118C22⋊D163—42no—
119C2×Dic163—21no—
120C2×Dic163—21no—
121C2×Dic163—21no—
122C2×Dic163—21no—
123C2×Dic163—42no—
124C2×Dic163—42no—
125C2×Dic163—42no—
126C2×Dic163C2×D411yescharacteristic
127C2×Dic163C2×D411yescharacteristic
128C2×Dic163C2×D411yescharacteristic
129C4×D163—21no—
130C4×D163—21no—
131C163⋊C8—21no—
132C163⋊C8—21no—
133C23×C163—21no—
134C23×C163C2×D411yescharacteristic
135Q8×C163—21no—
136Q8×C163—21no—
137D4×C163—21no—
138D4×C163—21no—
139D4×C163—42no—
140C2×C4×C163—21no—
141C2×C4×C163—21no—
142C2×C4×C163—21no—
143C2×C4×C163—21no—
144C2×C4×C163—21no—
145C2×C4×C163C2×D411yescharacteristic
146C2×C4×C163C2411yescharacteristic
147C8×C163—21no—
148C8×C163—21no—
149C22×C163—21no—
150C22×C163—21no—
151C22×C163—41no—
152C22×C163—41no—
153C22×C163—41no—
154C22×C163#13512yes—
155C22×C163C22×D411yescharacteristic
156C2×D163—21no—
157C2×D163—21no—
158C2×D163—21no—
159C4×C163—21no—
160C4×C163—41no—
161C4×C163—41no—
162C4×C163—41no—
163C4×C163C2×#4111yescharacteristic
164C4×C163H(5,𝔽2)11yescharacteristic
165Dic163—21no—
166Dic163—22no—
167Dic163—21no—
168Dic163—21no—
169Dic163—41no—
170Dic163—41no—
171Dic163—41no—
172C2×C163—22no—
173C2×C163—81no—
174C2×C163#54611yescharacteristic
175C2×C163C2×#13511yescharacteristic
176C2×C163#57811yescharacteristic
177D163—41no—
178C163#330011yescharacteristic, complemented, Sylow
179#3300—1631nomaximal, Sylow
180#552—1631no—
181#548—1631no—
182#488—1631no—
183#485—1631no—
184#480—1631no—
185#479—1631no—
186#467—1631no—
187#465D16311yescharacteristic, complemented
188#464—1631no—
189#452—1631no—
190#451—1631no—
191C4×Q16—1631no—
192C8×D4—1631no—
193#416—1631no—
194C2×#103—1631no—
195C2×#41—1631no—
196C22×Q8—1631no—
197C2×Q16—1631no—
198C2×Q16—1631no—
199C2×SD16C2×D16311yescharacteristic
200C22×C8—1631no—
201C22×C8C2×D16311yescharacteristic
202#126—3261no—
203#125—3261no—
204#125—1631no—
205#124—3261no—
206#124—1631no—
207#123—3261no—
208C22⋊Q8—3261no—
209C22⋊Q8—1631no—
210C22⋊Q8—1631no—
211C22⋊Q8C2×D16311yescharacteristic
212#121—3261no—
213#121C2×D16311yescharacteristic
214C4×Q8—1631no—
215C4×Q8—1631no—
216C4×D4—3261no—
217C4×D4—1631no—
218#117—3261no—
219#117—1631no—
220C2×C4⋊C4—1631no—
221#107—1631no—
222#107—1631no—
223#106C2×D16311yescharacteristic
224C4⋊C8—1631no—
225#103—1631no—
226#103—1631no—
227#103—1631no—
228#103—1631no—
229#103—1631no—
230#103—1631no—
231#103C2×D16311yescharacteristic
232#102—1631no—
233#102—1631no—
234#102C2×D16311yescharacteristic
235C22⋊C8—1631no—
236C22⋊C8—1631no—
237C4×C8—1631no—
238#41—3262no—
239#41—6521no—
240C2×Q8—3262no—
241C2×Q8—3261no—
242C2×Q8—6521no—
243C2×Q8—1631no—
244C2×Q8—1631no—
245C2×Q8C22×D16311yescharacteristic
246C2×D4—21no—
247C2×D4—3261no—
248C2×D4—1631no—
249C2×D4C22×D16311yescharacteristic
250C22×C4—3261no—
251C22×C4—3261no—
252C22×C4—3261no—
253C22×C4—1631no—
254C22×C4C22×D16311yescharacteristic
255Q16—3262no—
256Q16—6521no—
257SD16—41no—
258C2×C8—41no—
259C2×C8—6521no—
260C2×C8—1631no—
261C2×C8—1631no—
262C2×C8C22×D16311yescharacteristic
263C2×C8C22×D16311yescharacteristic
264C4⋊C4—21no—
265C4⋊C4—3261no—
266C4⋊C4—3261no—
267C4⋊C4—3261no—
268C4⋊C4—3261no—
269C4⋊C4—3261no—
270C4⋊C4—3261no—
271C4⋊C4—3261no—
272C4⋊C4—1631no—
273C4⋊C4—1631no—
274C4⋊C4—1631no—
275C4⋊C4C22×D16311yescharacteristic
276C4⋊C4C22×D16311yescharacteristic
277C22⋊C4—21no—
278C22⋊C4—21no—
279C22⋊C4—3261no—
280C22⋊C4—3261no—
281C22⋊C4—3261no—
282C22⋊C4—3261no—
283C22⋊C4—3261no—
284C22⋊C4—1631no—
285C22⋊C4—1631no—
286C42—3261no—
287C42—3261no—
288C42—3261no—
289C42—1631no—
290C23—21no—
291C23—1631no—
292C23D4×D16311yescharacteristic
293Q8—21no—
294Q8—21no—
295Q8—3261no—
296Q8—3261no—
297Q8—3261no—
298Q8—6522no—
299Q8—1632no—
300D4—21no—
301D4—21no—
302D4—42no—
303D4—3262no—
304D4—6522no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—3261no—
311C2×C4—3261no—
312C2×C4—3261no—
313C2×C4—3261no—
314C2×C4—3261no—
315C2×C4—3261no—
316C2×C4—6522no—
317C2×C4—6522no—
318C2×C4—6522no—
319C2×C4—1631no—
320C2×C4—1631no—
321C2×C4—1631no—
322C2×C4D4×D16311yescharacteristic
323C2×C4C23×D16311yescharacteristic
324C8—21no—
325C8—21no—
326C8—3261no—
327C8—3261no—
328C22—21no—
329C22—21no—
330C22—41no—
331C22—41no—
332C22—41no—
333C22—3261no—
334C22—3261no—
335C22—3261no—
336C22#1910475612yes—
337C22C2×D4×D16311yescharacteristic
338C4—21no—
339C4—41no—
340C4—41no—
341C4—41no—
342C4—3261no—
343C4—3261no—
344C4—3261no—
345C4—3262no—
346C4—6521no—
347C4—6521no—
348C4—6521no—
349C4#41×D16311yescharacteristic
350C4#1910478011yescharacteristic
351C2—22no—
352C2—81no—
353C2—6521no—
354C2#2378230411yescharacteristic
355C2C2×#1910475611yescharacteristic
356C2#2378247911yescharacteristic
subgroupG/H
centerC22C2×D4×D163
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
Fitting#465×C163C2
socleC22×C163C22×D4
prime radical#23782321C2
hypercenter#465D163
hypocenterC163#3300
LCSG ▷ C2×C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C22 ◁ C22×C4 ◁ #465

Complements

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

NH ≅ G/Ntype
1. #23782357C2semidirect
3. #23782018C2semidirect
4. #23782009C2semidirect
5. #23781976C2semidirect
6. #23781975C2semidirect
7. #23781905C2semidirect
8. #23781885C2semidirect
9. #23781824C2semidirect
10. #23781820C2semidirect
11. #23781757C2semidirect
12. C8×C22⋊D163C2semidirect
13. #23781658C2semidirect
14. C2×#19104660C2semidirect
15. #465×C163C2semidirect
178. C163#3300semidirect
187. #465D163semidirect

Maximal subgroups

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

subgroupindex
#237823572
#237823212
#237820182
#237820092
#237819762
#237819752
#237819052
#237818852
#237818242
#237818202
#237817572
C8×C22⋊D1632
#237816582
C2×#191046602
#465×C1632
#3300163