Finite Group Databasebeta

← #35224765

Subgroups — #35224765

Order 20864.

370 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23782286C211yesmaximal, characteristic
2#23782006C211yesmaximal, characteristic, complemented
3#23781989C211yesmaximal, characteristic
4#23781960C211yesmaximal, characteristic, complemented
5#23781950C211yesmaximal, characteristic, complemented
6#23781887C211yesmaximal, characteristic, complemented
7#23781887C211yesmaximal, characteristic, complemented
8#23781821C211yesmaximal, characteristic, complemented
9#23781804C211yesmaximal, characteristic
10#23781770C211yesmaximal, characteristic, complemented
11#23781763C211yesmaximal, characteristic, complemented
12C4×#14579689C211yesmaximal, characteristic, complemented
13C4×#14579686C211yesmaximal, characteristic, complemented
14#23781726C211yesmaximal, characteristic, complemented
15#519×C163C211yesmaximal, characteristic, complemented
16C2×#14579689C2211yescharacteristic
17C2×#14579689C2211yescharacteristic
18C2×#14579686C2211yescharacteristic
19C2×#14579686C2211yescharacteristic
20#19104735C2211yescharacteristic
21#19104730—21no—
22#19104723C2211yescharacteristic
23#19104721—21no—
24#19104717—21no—
25#19104714C2211yescharacteristic
26#19104713—21no—
27#19104709C2211yescharacteristic
28C4×Dic326C2211yescharacteristic
29C4×D652C2211yescharacteristic
30#19104668C2211yescharacteristic
31#19104668C2211yescharacteristic
32#19104666C2211yescharacteristic
33#19104666C2211yescharacteristic
34C4⋊(C163⋊C8)C2211yescharacteristic
35C4⋊(C163⋊C8)C2211yescharacteristic
36#19104659C2211yescharacteristic
37#19104659C2211yescharacteristic
38#19104659C2211yescharacteristic
39#19104659C2211yescharacteristic
40#19104658C2211yescharacteristic
41#19104658C2211yescharacteristic
42#19104655C2211yescharacteristic
43#19104655C2211yescharacteristic
44C22⋊(C163⋊C8)C2211yescharacteristic
45C22⋊(C163⋊C8)C2211yescharacteristic
46#19104646C2211yescharacteristic
47C4×C163⋊C8C2211yescharacteristic
48C22×Q8×C163C2211yescharacteristic
49C4×Q8×C163—21no—
50C4×Q8×C163—21no—
51C4×Q8×C163C2211yescharacteristic
52C4×Q8×C163C2211yescharacteristic
53C4×Q8×C163C2211yescharacteristic
54C4×Q8×C163C2211yescharacteristic
55#117×C163—21no—
56#117×C163—21no—
57#117×C163C2211yescharacteristic
58#117×C163C2211yescharacteristic
59C2×Dic326—21no—
60C2×Dic326C2311yescharacteristic
61C2×D652C2311yescharacteristic
62C2×C22⋊D163—21no—
63C2×C4×D163—21no—
64#14579689—22no—
65#14579689—22no—
66#14579686—22no—
67#14579686—22no—
68C2×C163⋊C8C2311yescharacteristic
69C2×C163⋊C8C2311yescharacteristic
70C2×C163⋊C8C2311yescharacteristic
71C2×C163⋊C8C2311yescharacteristic
72C4⋊Dic163—21no—
73C4⋊Dic163C2311yescharacteristic
74C4⋊Dic163C2311yescharacteristic
75Dic163⋊C4—21no—
76Dic163⋊C4—21no—
77C22⋊Dic163—21no—
78(C2×D163)⋊C4—21no—
79(C2×D163)⋊C4—21no—
80(C2×D163)⋊C4—21no—
81(C2×D163)⋊C4—21no—
82C4×Dic163—21no—
83C2×Q8×C163—22no—
84C2×Q8×C163—22no—
85C2×Q8×C163—21no—
86C2×Q8×C163C2311yescharacteristic
87C2×Q8×C163C2311yescharacteristic
88C22×C4×C163—21no—
89C22×C4×C163C2311yescharacteristic
90C4⋊C4×C163—21no—
91C4⋊C4×C163—21no—
92C4⋊C4×C163—21no—
93C4⋊C4×C163—21no—
94C4⋊C4×C163D411yescharacteristic
95C4⋊C4×C163D411yescharacteristic
96C4⋊C4×C163C2311yescharacteristic
97C4⋊C4×C163C2311yescharacteristic
98C22⋊C4×C163—21no—
99C22⋊C4×C163D411yescharacteristic
100C22⋊C4×C163D411yescharacteristic
101C42×C163—21no—
102C42×C163—21no—
103C42×C163—21no—
104C42×C163—21no—
105C42×C163—21no—
106C42×C163C2311yescharacteristic
107C42×C163C2311yescharacteristic
108C22×D163—21no—
109Dic326—21no—
110Dic326—21no—
111Dic326—81no—
112D652—21no—
113D652—21no—
114C22⋊D163—81no—
115C2×Dic163—21no—
116C2×Dic163—21no—
117C2×Dic163—21no—
118C4×D163—81no—
119C163⋊C8—21no—
120C163⋊C8—21no—
121C163⋊C8—21no—
122C163⋊C8—21no—
123C23×C163C2×D411yescharacteristic
124Q8×C163—24no—
125Q8×C163—21no—
126Q8×C163—21no—
127Q8×C163#4112yes—
128Q8×C163#4112yes—
129C2×C4×C163—24no—
130C2×C4×C163—21no—
131C2×C4×C163—21no—
132C2×C4×C163—21no—
133C2×C4×C163—21no—
134C2×C4×C163—21no—
135C2×C4×C163—21no—
136C2×C4×C163C2×D411yescharacteristic
137C2×C4×C163C2×D411yescharacteristic
138C2×C4×C163C2×D411yescharacteristic
139C2×C4×C163C2×D411yescharacteristic
140C2×C4×C163C2×D411yescharacteristic
141C2×C4×C163C2411yescharacteristic
142C22×C163—21no—
143C22×C163—21no—
144C22×C163—21no—
145C22×C163C22×D411yescharacteristic
146C2×D163—41no—
147C2×D163—41no—
148C2×D163—41no—
149C4×C163—22no—
150C4×C163—22no—
151C4×C163—21no—
152C4×C163—21no—
153C4×C163—21no—
154C4×C163—21no—
155C4×C163—21no—
156C4×C163—41no—
157C4×C163—41no—
158C4×C163C2×#4111yescharacteristic
159C4×C163C2×#4111yescharacteristic
160Dic163—41no—
161Dic163—41no—
162Dic163—41no—
163C2×C163—41no—
164C2×C163#52511yescharacteristic
165C2×C163#57711yescharacteristic
166C2×C163#57811yescharacteristic
167D163—81no—
168C163#312111yescharacteristic, complemented, Sylow
169#3121—1631nomaximal, Sylow
170#542—1631no—
171#519D16311yescharacteristic, complemented
172#485—1631no—
173#482—1631no—
174#477—1631no—
175#475—1631no—
176#464—1631no—
177#464—1631no—
178#451—1631no—
179#448—1631no—
180#441—1631no—
181#440—1631no—
182C4×Q16—1631no—
183C4×SD16—1631no—
184#433—1631no—
185C22×Q8C2×D16311yescharacteristic
186C2×Q16—1631no—
187C2×Q16—1631no—
188C2×SD16—1631no—
189C2×SD16—1631no—
190C4⋊Q8—1631no—
191#126—3261no—
192#124—3261no—
193#124—1631no—
194#123—3261no—
195C22⋊Q8—3261no—
196C22⋊Q8—1631no—
197#121—1631no—
198C4×Q8—21no—
199C4×Q8—21no—
200C4×Q8—1631no—
201C4×Q8C2×D16311yescharacteristic
202C4×Q8C2×D16311yescharacteristic
203C4×Q8C2×D16311yescharacteristic
204C4×Q8C2×D16311yescharacteristic
205C4×D4—1631no—
206#117—21no—
207#117—21no—
208#117C2×D16311yescharacteristic
209#117C2×D16311yescharacteristic
210#107—1631no—
211#107—1631no—
212#106—1631no—
213#106—1631no—
214C4⋊C8—1631no—
215C4⋊C8—1631no—
216#103—1631no—
217#103—1631no—
218#103—1631no—
219#103—1631no—
220#103—1631no—
221#103—1631no—
222#102—1631no—
223#102—1631no—
224C22⋊C8—1631no—
225C22⋊C8—1631no—
226#97—1631no—
227C4×C8—1631no—
228C2×Q8—22no—
229C2×Q8—22no—
230C2×Q8—21no—
231C2×Q8—3261no—
232C2×Q8—1631no—
233C2×Q8C22×D16311yescharacteristic
234C2×Q8C22×D16311yescharacteristic
235C2×D4—3261no—
236C2×D4—1631no—
237C22×C4—21no—
238C22×C4—3261no—
239C22×C4C22×D16311yescharacteristic
240Q16—3262no—
241Q16—3262no—
242SD16—3262no—
243SD16—3262no—
244C2×C8—1631no—
245C2×C8—1631no—
246C2×C8—1631no—
247C2×C8—1631no—
248C4⋊C4—21no—
249C4⋊C4—21no—
250C4⋊C4—21no—
251C4⋊C4—21no—
252C4⋊C4—3261no—
253C4⋊C4—3261no—
254C4⋊C4—3261no—
255C4⋊C4—1631no—
256C4⋊C4—1631no—
257C4⋊C4C22⋊D16311yescharacteristic
258C4⋊C4C22⋊D16311yescharacteristic
259C4⋊C4C22×D16311yescharacteristic
260C4⋊C4C22×D16311yescharacteristic
261C22⋊C4—21no—
262C22⋊C4—3261no—
263C22⋊C4—3261no—
264C22⋊C4—3261no—
265C22⋊C4—3261no—
266C22⋊C4—3261no—
267C22⋊C4C22⋊D16311yescharacteristic
268C22⋊C4C22⋊D16311yescharacteristic
269C42—21no—
270C42—21no—
271C42—21no—
272C42—21no—
273C42—21no—
274C42—3261no—
275C42C22×D16311yescharacteristic
276C42C22×D16311yescharacteristic
277C23—3261no—
278C23C2×C22⋊D16311yescharacteristic
279Q8—24no—
280Q8—21no—
281Q8—21no—
282Q8—13041no—
283Q8—3261no—
284Q8—3261no—
285Q8#1457969912yes—
286Q8#1457969912yes—
287D4—13041no—
288D4—3261no—
289D4—3261no—
290C2×C4—24no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—13041no—
298C2×C4—3261no—
299C2×C4—3261no—
300C2×C4—3261no—
301C2×C4C2×C22⋊D16311yescharacteristic
302C2×C4C2×C22⋊D16311yescharacteristic
303C2×C4C2×C22⋊D16311yescharacteristic
304C2×C4C2×C22⋊D16311yescharacteristic
305C2×C4C2×C22⋊D16311yescharacteristic
306C2×C4C23×D16311yescharacteristic
307C8—3261no—
308C8—3261no—
309C8—3261no—
310C8—3261no—
311C22—21no—
312C22—21no—
313C22—21no—
314C22—6521no—
315C22—6521no—
316C22—6521no—
317C22C22×C22⋊D16311yescharacteristic
318C4—22no—
319C4—22no—
320C4—21no—
321C4—21no—
322C4—21no—
323C4—21no—
324C4—21no—
325C4—41no—
326C4—41no—
327C4—6521no—
328C4—6521no—
329C4—6521no—
330C4C2×#1457969911yescharacteristic
331C4C2×#1457969911yescharacteristic
332C2—41no—
333C2—13041no—
334C2#2378217611yescharacteristic
335C2#2378247311yescharacteristic
336C2#2378248011yescharacteristic
subgroupG/H
centerC22C22×C22⋊D163
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
Fitting#519×C163C2
socleC22×C163C22×D4
prime radical#19104709C22
hypercenter#519D163
hypocenterC163#3121
LCSG ▷ C2×C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C22 ◁ #117 ◁ #519

Complements

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

NH ≅ G/Ntype
2. #23782006C2semidirect
4. #23781960C2semidirect
5. #23781950C2semidirect
6. #23781887C2semidirect
7. #23781887C2semidirect
8. #23781821C2semidirect
10. #23781770C2semidirect
11. #23781763C2semidirect
12. C4×#14579689C2semidirect
13. C4×#14579686C2semidirect
14. #23781726C2semidirect
15. #519×C163C2semidirect
168. C163#3121semidirect
171. #519D163semidirect

Maximal subgroups

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

subgroupindex
#237822862
#237820062
#237819892
#237819602
#237819502
#237818872
#237818212
#237818042
#237817702
#237817632
C4×#145796892
C4×#145796862
#237817262
#519×C1632
#3121163