Finite Group Databasebeta

← #35226603

Subgroups — #35226603

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#23782383C211yesmaximal, characteristic, complemented
2#23782326C211yesmaximal, characteristic
3#23782026C211yesmaximal, characteristic, complemented
4#23782005C211yesmaximal, characteristic, complemented
5#23781989C211yesmaximal, characteristic, complemented
6#23781958C211yesmaximal, characteristic, complemented
7#23781911C211yesmaximal, characteristic, complemented
8#23781905C211yesmaximal, characteristic, complemented
9#23781890C211yesmaximal, characteristic, complemented
10#23781830C211yesmaximal, characteristic, complemented
11#23781821C211yesmaximal, characteristic, complemented
12#23781769C211yesmaximal, characteristic, complemented
13#23781734C211yesmaximal, characteristic, complemented
14#23781660C211yesmaximal, characteristic, complemented
15#419×C163C211yesmaximal, characteristic, complemented
16C2×#14579700C2211yescharacteristic
17C2×#14579689C2211yescharacteristic
18C2×#14579689C2211yescharacteristic
19C2×#14579688C2211yescharacteristic
20C2×C8×D163C2211yescharacteristic
21#19104734C2211yescharacteristic
22#19104729—21no—
23#19104728—21no—
24#19104724—21no—
25#19104724C2211yescharacteristic
26#19104721—21no—
27#19104715C2211yescharacteristic
28#19104714C2211yescharacteristic
29#19104711—21no—
30#19104711C2211yescharacteristic
31#19104709—21no—
32#19104709C2211yescharacteristic
33Q8×Dic163C2211yescharacteristic
34C4×Dic326—21no—
35#19104700C2211yescharacteristic
36C4×C22⋊D163—21no—
37C4×C22⋊D163C2211yescharacteristic
38#19104693C2211yescharacteristic
39#19104669C2211yescharacteristic
40#19104668C2211yescharacteristic
41#19104666C2211yescharacteristic
42Dic163⋊C8C2211yescharacteristic
43#19104660C2211yescharacteristic
44#19104659C2211yescharacteristic
45#19104659C2211yescharacteristic
46#19104657C2211yescharacteristic
47#19104655C2211yescharacteristic
48#19104655C2211yescharacteristic
49C22⋊(C163⋊C8)C2211yescharacteristic
50(C2×D163)⋊C8C2211yescharacteristic
51#19104645C2211yescharacteristic
52C22×Q8×C163C2211yescharacteristic
53C2×M16×C163C2211yescharacteristic
54#117×C163C2211yescharacteristic
55#103×C163C2211yescharacteristic
56#103×C163C2211yescharacteristic
57#103×C163C2211yescharacteristic
58#103×C163C2211yescharacteristic
59#14579700—22no—
60#14579700—41no—
61C2×Dic326—21no—
62C2×Dic326C2311yescharacteristic
63C2×D652—21no—
64C2×D652C2311yescharacteristic
65C2×C22⋊D163—21no—
66C2×C22⋊D163D411yescharacteristic
67C2×C4×D163—21no—
68C2×C4×D163C2311yescharacteristic
69#14579689—22no—
70#14579689—41no—
71#14579688—41no—
72C2×C163⋊C8C2311yescharacteristic
73C2×C163⋊C8C2311yescharacteristic
74C8×D163—41no—
75C4⋊Dic163—21no—
76C4⋊Dic163—21no—
77C4⋊Dic163C2311yescharacteristic
78C4⋊Dic163C2311yescharacteristic
79Dic163⋊C4—21no—
80Dic163⋊C4—21no—
81Dic163⋊C4—21no—
82Dic163⋊C4—21no—
83Dic163⋊C4D411yescharacteristic
84C22⋊Dic163—21no—
85C22⋊Dic163D411yescharacteristic
86(C2×D163)⋊C4—21no—
87(C2×D163)⋊C4—21no—
88(C2×D163)⋊C4—21no—
89(C2×D163)⋊C4—21no—
90(C2×D163)⋊C4—21no—
91(C2×D163)⋊C4—21no—
92(C2×D163)⋊C4D411yescharacteristic
93C4×Dic163—21no—
94C4×Dic163—21no—
95C4×Dic163—21no—
96C4×Dic163C2311yescharacteristic
97C2×Q8×C163—21no—
98C2×Q8×C163—22no—
99C2×Q8×C163—41no—
100C2×Q8×C163C2311yescharacteristic
101C2×Q8×C163C2311yescharacteristic
102C22×C4×C163—21no—
103C22×C4×C163C2311yescharacteristic
104M16×C163—41no—
105C2×C8×C163C2311yescharacteristic
106C2×C8×C163C2311yescharacteristic
107C4⋊C4×C163C2311yescharacteristic
108C4⋊C4×C163C2311yescharacteristic
109C22⋊C4×C163—21no—
110C42×C163—21no—
111C22×D163—21no—
112C22×D163C2×D411yescharacteristic
113Dic326—21no—
114Dic326—21no—
115Dic326—42no—
116D652—21no—
117D652—21no—
118D652—42no—
119C22⋊D163—41no—
120C22⋊D163—42no—
121C2×Dic163—21no—
122C2×Dic163—21no—
123C2×Dic163—21no—
124C2×Dic163C2×D411yescharacteristic
125C2×Dic163C2×D411yescharacteristic
126C2×Dic163C2×D411yescharacteristic
127C4×D163—21no—
128C4×D163—21no—
129C4×D163—42no—
130C163⋊C8—21no—
131C163⋊C8—21no—
132C23×C163C2×D411yescharacteristic
133Q8×C163—21no—
134Q8×C163—21no—
135Q8×C163—21no—
136Q8×C163—42no—
137Q8×C163#4112yes—
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×C163—42no—
145C2×C4×C163C2×D411yescharacteristic
146C2×C4×C163C2411yescharacteristic
147C8×C163—21no—
148C8×C163—21no—
149C22×C163—21no—
150C22×C163—21no—
151C22×C163—21no—
152C22×C163C22×D411yescharacteristic
153C2×D163—21no—
154C2×D163—21no—
155C2×D163—21no—
156C2×D163—41no—
157C2×D163—41no—
158C2×D163—41no—
159C4×C163—21no—
160C4×C163—22no—
161C4×C163—41no—
162C4×C163—41no—
163C4×C163—41no—
164C4×C163C2×#4111yescharacteristic
165C4×C163H(5,𝔽2)11yescharacteristic
166Dic163—21no—
167Dic163—21no—
168Dic163—21no—
169Dic163—41no—
170Dic163—41no—
171Dic163—41no—
172C2×C163—41no—
173C2×C163#54611yescharacteristic
174C2×C163#57511yescharacteristic
175C2×C163#57811yescharacteristic
176D163—41no—
177D163—81no—
178C163#331311yescharacteristic, complemented, Sylow
179#3313—1631nomaximal, Sylow
180#556—1631no—
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#419D16311yescharacteristic, complemented
194#416—1631no—
195C2×#41—1631no—
196C22×Q8C2×D16311yescharacteristic
197C2×Q16—1631no—
198C2×Q16—1631no—
199C2×SD16—1631no—
200C2×M16C2×D16311yescharacteristic
201C22×C8—1631no—
202C4⋊Q8—1631no—
203#126—3261no—
204#126—3261no—
205#124—3261no—
206#124—3261no—
207#124—1631no—
208C22⋊Q8—3261no—
209C22⋊Q8—1631no—
210C22⋊Q8—1631no—
211C22⋊Q8—1631no—
212#121—3261no—
213#121—1631no—
214C4×Q8—3261no—
215C4×Q8—1631no—
216C4×Q8—1631no—
217C4×D4—3261no—
218C4×D4—1631no—
219#117—1631no—
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#103C2×D16311yescharacteristic
229#103C2×D16311yescharacteristic
230#103C2×D16311yescharacteristic
231#103C2×D16311yescharacteristic
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—22no—
242C2×Q8—41no—
243C2×Q8—3261no—
244C2×Q8—1631no—
245C2×Q8C22×D16311yescharacteristic
246C2×Q8C22×D16311yescharacteristic
247C2×D4—3261no—
248C2×D4—3261no—
249C2×D4—1631no—
250C2×D4—1631no—
251C22×C4—21no—
252C22×C4—3261no—
253C22×C4—1631no—
254C22×C4C22×D16311yescharacteristic
255Q16—3262no—
256Q16—6521no—
257SD16—6521no—
258M16—41no—
259C2×C8—6521no—
260C2×C8—1631no—
261C2×C8—1631no—
262C2×C8C22×D16311yescharacteristic
263C2×C8C22×D16311yescharacteristic
264C4⋊C4—3261no—
265C4⋊C4—3261no—
266C4⋊C4—3261no—
267C4⋊C4—3261no—
268C4⋊C4—3261no—
269C4⋊C4—3261no—
270C4⋊C4—1631no—
271C4⋊C4—1631no—
272C4⋊C4—1631no—
273C4⋊C4C22×D16311yescharacteristic
274C4⋊C4C22×D16311yescharacteristic
275C22⋊C4—21no—
276C22⋊C4—3261no—
277C22⋊C4—3261no—
278C22⋊C4—3261no—
279C22⋊C4—3261no—
280C22⋊C4—3261no—
281C22⋊C4—3261no—
282C22⋊C4—3261no—
283C22⋊C4—1631no—
284C22⋊C4—1631no—
285C42—21no—
286C42—3261no—
287C42—3261no—
288C42—3261no—
289C42—1631no—
290C23—3261no—
291C23—1631no—
292C23D4×D16311yescharacteristic
293Q8—21no—
294Q8—21no—
295Q8—21no—
296Q8—42no—
297Q8—3261no—
298Q8—3261no—
299Q8—6522no—
300Q8#1457969912yes—
301D4—3261no—
302D4—3261no—
303D4—6522no—
304D4—6521no—
305D4—6522no—
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—3261no—
317C2×C4—3261no—
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—21no—
331C22—3261no—
332C22—3261no—
333C22—3261no—
334C22—6521no—
335C22—6521no—
336C22—6521no—
337C22C2×D4×D16311yescharacteristic
338C4—21no—
339C4—22no—
340C4—41no—
341C4—41no—
342C4—41no—
343C4—3261no—
344C4—3261no—
345C4—3261no—
346C4—6521no—
347C4—6521no—
348C4—6521no—
349C4C2×#1457969911yescharacteristic
350C4#1910478011yescharacteristic
351C2—41no—
352C2—13041no—
353C2—6521no—
354C2#2378230511yescharacteristic
355C2#2378246411yescharacteristic
356C2#2378248111yescharacteristic
subgroupG/H
centerC22C2×D4×D163
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
Fitting#419×C163C2
socleC22×C163C22×D4
prime radical#23782326C2
hypercenter#419D163
hypocenterC163#3313
LCSG ▷ C2×C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C22 ◁ C22×C4 ◁ #419

Complements

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

NH ≅ G/Ntype
1. #23782383C2semidirect
3. #23782026C2semidirect
4. #23782005C2semidirect
5. #23781989C2semidirect
6. #23781958C2semidirect
7. #23781911C2semidirect
8. #23781905C2semidirect
9. #23781890C2semidirect
10. #23781830C2semidirect
11. #23781821C2semidirect
12. #23781769C2semidirect
13. #23781734C2semidirect
14. #23781660C2semidirect
15. #419×C163C2semidirect
178. C163#3313semidirect
193. #419D163semidirect

Maximal subgroups

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

subgroupindex
#237823832
#237823262
#237820262
#237820052
#237819892
#237819582
#237819112
#237819052
#237818902
#237818302
#237818212
#237817692
#237817342
#237816602
#419×C1632
#3313163