Finite Group Databasebeta

← #35226626

Subgroups — #35226626

Order 20864.

382 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#23782320C211yesmaximal, characteristic
2#23782035C211yesmaximal, characteristic, complemented
3#23782006C211yesmaximal, characteristic, complemented
4#23782000C211yesmaximal, characteristic, complemented
5#23781962C211yesmaximal, characteristic, complemented
6#23781921C211yesmaximal, characteristic, complemented
7#23781904C211yesmaximal, characteristic, complemented
8#23781875C211yesmaximal, characteristic, complemented
9#23781825C211yesmaximal, characteristic, complemented
10#23781825C211yesmaximal, characteristic, complemented
11#23781763C211yesmaximal, characteristic, complemented
12#23781737C211yesmaximal, characteristic, complemented
13#23781664C211yesmaximal, characteristic, complemented
14#23781659C211yesmaximal, characteristic, complemented
15#552×C163C211yesmaximal, characteristic, complemented
16C2×#14579699C2211yescharacteristic
17C2×#14579699C2211yescharacteristic
18C2×#14579689C2211yescharacteristic
19C2×#14579689C2211yescharacteristic
20C2×#14579686C2211yescharacteristic
21C2×#14579683C2211yescharacteristic
22C22×C163⋊C8C2211yescharacteristic
23#19104735C2211yescharacteristic
24#19104717—21no—
25#19104717—21no—
26#19104714C2211yescharacteristic
27#19104714C2211yescharacteristic
28#19104706—21no—
29C4×D652C2211yescharacteristic
30C4⋊C4×D163—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
48#126×C163—21no—
49#125×C163—21no—
50#125×C163C2211yescharacteristic
51#123×C163—21no—
52C22⋊Q8×C163C2211yescharacteristic
53C22⋊Q8×C163C2211yescharacteristic
54C4×Q8×C163C2211yescharacteristic
55C4×D4×C163C2211yescharacteristic
56#117×C163—21no—
57#117×C163C2211yescharacteristic
58C2×C4⋊C4×C163C2211yescharacteristic
59#14579699—22no—
60#14579699—22no—
61#14579699—41no—
62#14579699—41no—
63C2×Dic326C2311yescharacteristic
64C2×Dic326C2311yescharacteristic
65C2×D652C2311yescharacteristic
66C2×C22⋊D163—21no—
67C2×C22⋊D163—21no—
68C2×C4×D163—21no—
69C2×C4×D163—21no—
70C2×C4×D163—21no—
71#14579689—41no—
72#14579689—41no—
73#14579686—22no—
74#14579683—41no—
75C2×C163⋊C8—41no—
76C2×C163⋊C8C2311yescharacteristic
77C2×C163⋊C8C2311yescharacteristic
78C2×C163⋊C8C2311yescharacteristic
79C2×C163⋊C8C2311yescharacteristic
80C4⋊Dic163—21no—
81C4⋊Dic163—21no—
82C4⋊Dic163C2311yescharacteristic
83Dic163⋊C4—21no—
84Dic163⋊C4—21no—
85C22⋊Dic163—21no—
86C22⋊Dic163—21no—
87(C2×D163)⋊C4—21no—
88C2×Q8×C163C2311yescharacteristic
89C2×D4×C163D411yescharacteristic
90C22×C4×C163—21no—
91C22×C4×C163C2311yescharacteristic
92C22×C4×C163C2311yescharacteristic
93C4⋊C4×C163—21no—
94C4⋊C4×C163—21no—
95C4⋊C4×C163—21no—
96C4⋊C4×C163—21no—
97C4⋊C4×C163—21no—
98C4⋊C4×C163—21no—
99C4⋊C4×C163D411yescharacteristic
100C4⋊C4×C163C2311yescharacteristic
101C4⋊C4×C163C2311yescharacteristic
102C4⋊C4×C163C2311yescharacteristic
103C22⋊C4×C163—21no—
104C22⋊C4×C163—21no—
105C22⋊C4×C163—21no—
106C22⋊C4×C163D411yescharacteristic
107C22⋊C4×C163D411yescharacteristic
108C42×C163—21no—
109C42×C163—21no—
110C42×C163C2311yescharacteristic
111C22×D163—21no—
112Dic326—21no—
113Dic326—21no—
114Dic326—21no—
115Dic326—21no—
116D652—21no—
117D652#4112yes—
118C22⋊D163—42no—
119C22⋊D163—42no—
120C2×Dic163—21no—
121C2×Dic163—21no—
122C2×Dic163—21no—
123C4×D163—42no—
124C4×D163—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—
134D4×C163—41no—
135C2×C4×C163—21no—
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—42no—
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—22no—
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—41no—
170Dic163—41no—
171Dic163—41no—
172C2×C163—41no—
173C2×C163—41no—
174C2×C163#54611yescharacteristic
175C2×C163#57511yescharacteristic
176C2×C163#57811yescharacteristic
177D163—42no—
178C163#331511yescharacteristic, complemented, Sylow
179#3315—1631nomaximal, Sylow
180#552D16311yescharacteristic, complemented
181#547—1631no—
182#491—1631no—
183#485—1631no—
184#484—1631no—
185#478—1631no—
186#470—1631no—
187#467—1631no—
188#462—1631no—
189#452—1631no—
190#452—1631no—
191#440—1631no—
192#435—1631no—
193#417—1631no—
194#416—1631no—
195C2×#41—1631no—
196C2×#41—1631no—
197C2×Q16—1631no—
198C2×Q16—1631no—
199C2×SD16—1631no—
200C2×M16—1631no—
201C22×C8—1631no—
202C4⋊Q8—1631no—
203#126—21no—
204#125—21no—
205#125C2×D16311yescharacteristic
206#123—21no—
207#123—3261no—
208#123—3261no—
209C22⋊Q8—1631no—
210C22⋊Q8—1631no—
211C22⋊Q8C2×D16311yescharacteristic
212C22⋊Q8C2×D16311yescharacteristic
213#121—3261no—
214C4×Q8C2×D16311yescharacteristic
215C4×D4—1631no—
216C4×D4C2×D16311yescharacteristic
217#117—21no—
218#117C2×D16311yescharacteristic
219C2×C4⋊C4—3261no—
220C2×C4⋊C4C2×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—3262no—
240#41—6521no—
241#41—6521no—
242C2×Q8—1631no—
243C2×Q8—1631no—
244C2×Q8C22×D16311yescharacteristic
245C2×D4—3261no—
246C2×D4—3261no—
247C2×D4—1631no—
248C2×D4C22⋊D16311yescharacteristic
249C22×C4—21no—
250C22×C4—3261no—
251C22×C4—3261no—
252C22×C4—3261no—
253C22×C4C22×D16311yescharacteristic
254C22×C4C22×D16311yescharacteristic
255Q16—6521no—
256Q16—6521no—
257SD16—3262no—
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—21no—
269C4⋊C4—21no—
270C4⋊C4—3261no—
271C4⋊C4—3261no—
272C4⋊C4—3261no—
273C4⋊C4—3261no—
274C4⋊C4—1631no—
275C4⋊C4C22⋊D16311yescharacteristic
276C4⋊C4C22×D16311yescharacteristic
277C4⋊C4C22×D16311yescharacteristic
278C4⋊C4C22×D16311yescharacteristic
279C22⋊C4—21no—
280C22⋊C4—21no—
281C22⋊C4—21no—
282C22⋊C4—3261no—
283C22⋊C4—3261no—
284C22⋊C4—3261no—
285C22⋊C4C22⋊D16311yescharacteristic
286C22⋊C4C22⋊D16311yescharacteristic
287C42—21no—
288C42—21no—
289C42C22×D16311yescharacteristic
290C23—3261no—
291C23C2×C22⋊D16311yescharacteristic
292C23C2×C22⋊D16311yescharacteristic
293Q8—21no—
294Q8—21no—
295Q8—3261no—
296Q8—3261no—
297Q8—3261no—
298Q8—3261no—
299D4—41no—
300D4—3261no—
301D4—6522no—
302D4—6522no—
303D4—1632no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—42no—
313C2×C4—3261no—
314C2×C4—3261no—
315C2×C4—3261no—
316C2×C4—6522no—
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—3262no—
335C22—6521no—
336C22—6521no—
337C22C22×C22⋊D16311yescharacteristic
338C4—21no—
339C4—21no—
340C4—21no—
341C4—21no—
342C4—41no—
343C4—41no—
344C4—41no—
345C4—41no—
346C4—6521no—
347C4—6521no—
348C4—6521no—
349C4#41×D16311yescharacteristic
350C4#1910478111yescharacteristic
351C2—41no—
352C2—41no—
353C2—6522no—
354C2#2378230311yescharacteristic
355C2#2378246311yescharacteristic
356C2#2378248011yescharacteristic
subgroupG/H
centerC22C22×C22⋊D163
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
Fitting#552×C163C2
socleC22×C163C22×D4
prime radical#23782320C2
hypercenter#552D163
hypocenterC163#3315
LCSG ▷ C2×C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C22 ◁ C4×D4 ◁ #552

Complements

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

NH ≅ G/Ntype
2. #23782035C2semidirect
3. #23782006C2semidirect
4. #23782000C2semidirect
5. #23781962C2semidirect
6. #23781921C2semidirect
7. #23781904C2semidirect
8. #23781875C2semidirect
9. #23781825C2semidirect
10. #23781825C2semidirect
11. #23781763C2semidirect
12. #23781737C2semidirect
13. #23781664C2semidirect
14. #23781659C2semidirect
15. #552×C163C2semidirect
178. C163#3315semidirect
180. #552D163semidirect

Maximal subgroups

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

subgroupindex
#237823202
#237820352
#237820062
#237820002
#237819622
#237819212
#237819042
#237818752
#237818252
#237817632
#237817372
#237816642
#237816592
#552×C1632
#3315163