Finite Group Databasebeta

← #35227596

Subgroups — #35227596

Order 20864.

368 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23782388C211yesmaximal, characteristic, complemented
2#23782060C211yesmaximal, characteristic, complemented
3#23782052C211yesmaximal, characteristic
4#23782047C211yesmaximal, characteristic, complemented
5#23781887C211yesmaximal, characteristic, complemented
6#23781884C211yesmaximal, characteristic, complemented
7#23781880C211yesmaximal, characteristic, complemented
8#23781877C211yesmaximal, characteristic, complemented
9#23781870C211yesmaximal, characteristic, complemented
10#23781862C211yesmaximal, characteristic, complemented
11#23781796C211yesmaximal, characteristic, complemented
12#23781775C211yesmaximal, characteristic, complemented
13#23781770C211yesmaximal, characteristic, complemented
14C4×#14579687C211yesmaximal, characteristic, complemented
15#559×C163C211yesmaximal, characteristic, complemented
16C2×#14579689C2211yescharacteristic
17C2×#14579689C2211yescharacteristic
18C2×#14579687C2211yescharacteristic
19C2×#14579687C2211yescharacteristic
20C2×#14579687C2211yescharacteristic
21C2×#14579687C2211yescharacteristic
22C2×#14579686C2211yescharacteristic
23C2×#14579684C2211yescharacteristic
24C2×#14579684C2211yescharacteristic
25#19104735C2211yescharacteristic
26#19104730—21no—
27#19104730—21no—
28#19104723C2211yescharacteristic
29#19104723C2211yescharacteristic
30#19104711—21no—
31C4×Dic326C2211yescharacteristic
32C4×Dic326C2211yescharacteristic
33C4×D652C2211yescharacteristic
34#19104693—21no—
35#19104666C2211yescharacteristic
36C4⋊(C163⋊C8)C2211yescharacteristic
37C4⋊(C163⋊C8)C2211yescharacteristic
38C4⋊(C163⋊C8)C2211yescharacteristic
39#19104659C2211yescharacteristic
40#19104658C2211yescharacteristic
41#19104658C2211yescharacteristic
42#19104656C2211yescharacteristic
43#19104656C2211yescharacteristic
44#19104655C2211yescharacteristic
45#19104646C2211yescharacteristic
46#19104646C2211yescharacteristic
47C4×C163⋊C8C2211yescharacteristic
48#127×C163C2211yescharacteristic
49#124×C163—21no—
50#124×C163C2211yescharacteristic
51#124×C163C2211yescharacteristic
52#123×C163—21no—
53#123×C163—21no—
54C4×Q8×C163C2211yescharacteristic
55C4×Q8×C163C2211yescharacteristic
56C4×D4×C163—21no—
57C4×D4×C163C2211yescharacteristic
58C4×D4×C163C2211yescharacteristic
59C2×Dic326C2311yescharacteristic
60C2×Dic326C2311yescharacteristic
61C2×D652C2311yescharacteristic
62C2×C4×D163—21no—
63#14579689—22no—
64#14579689—41no—
65#14579687—22no—
66#14579687—41no—
67#14579687—41no—
68#14579687—41no—
69#14579686—41no—
70#14579684—22no—
71#14579684—41no—
72C2×C163⋊C8C2311yescharacteristic
73C2×C163⋊C8C2311yescharacteristic
74C2×C163⋊C8C2311yescharacteristic
75C2×C163⋊C8C2311yescharacteristic
76C4⋊Dic163—21no—
77C4⋊Dic163—21no—
78C4⋊Dic163C2311yescharacteristic
79Dic163⋊C4—21no—
80Dic163⋊C4—21no—
81(C2×D163)⋊C4—21no—
82(C2×D163)⋊C4—21no—
83(C2×D163)⋊C4—21no—
84(C2×D163)⋊C4—21no—
85(C2×D163)⋊C4—21no—
86C4×Dic163—21no—
87C4×Dic163—21no—
88C2×Q8×C163D411yescharacteristic
89C2×Q8×C163C2311yescharacteristic
90C2×D4×C163—21no—
91C2×D4×C163—21no—
92C2×D4×C163C2311yescharacteristic
93C2×D4×C163C2311yescharacteristic
94C22×C4×C163—21no—
95C22×C4×C163—21no—
96C4⋊C4×C163—21no—
97C4⋊C4×C163D411yescharacteristic
98C4⋊C4×C163D411yescharacteristic
99C4⋊C4×C163D411yescharacteristic
100C4⋊C4×C163C2311yescharacteristic
101C22⋊C4×C163—21no—
102C22⋊C4×C163—21no—
103C22⋊C4×C163—21no—
104C22⋊C4×C163—21no—
105C22⋊C4×C163—21no—
106C22⋊C4×C163—21no—
107C42×C163—21no—
108C42×C163C2311yescharacteristic
109C42×C163C2311yescharacteristic
110C42×C163C2311yescharacteristic
111C22×D163—21no—
112Dic326—21no—
113Dic326—21no—
114Dic326—21no—
115Dic326—21no—
116D652—21no—
117D652—21no—
118C2×Dic163—21no—
119C2×Dic163—21no—
120C2×Dic163—21no—
121C4×D163—42no—
122C163⋊C8—21no—
123C163⋊C8—21no—
124C163⋊C8—21no—
125C163⋊C8#4112yes—
126C23×C163—21no—
127C23×C163—21no—
128Q8×C163—21no—
129Q8×C163—21no—
130Q8×C163—41no—
131D4×C163—21no—
132D4×C163—21no—
133D4×C163—21no—
134D4×C163—21no—
135D4×C163—42no—
136D4×C163—81no—
137C2×C4×C163—21no—
138C2×C4×C163—21no—
139C2×C4×C163—81no—
140C2×C4×C163—81no—
141C2×C4×C163C2×D411yescharacteristic
142C2×C4×C163C2×D411yescharacteristic
143C2×C4×C163C2×D411yescharacteristic
144C2×C4×C163C2×D411yescharacteristic
145C2×C4×C163C2×D411yescharacteristic
146C2×C4×C163C2×D411yescharacteristic
147C2×C4×C163C2411yescharacteristic
148C22×C163—41no—
149C22×C163—41no—
150C22×C163—41no—
151C22×C163—41no—
152C22×C163—41no—
153C22×C163—41no—
154C22×C163C22×D411yescharacteristic
155C2×D163—41no—
156C2×D163—41no—
157C2×D163—41no—
158C4×C163—21no—
159C4×C163—21no—
160C4×C163—21no—
161C4×C163—21no—
162C4×C163—21no—
163C4×C163—21no—
164C4×C163—41no—
165C4×C163—41no—
166C4×C163C2×#4111yescharacteristic
167C4×C163H(5,𝔽2)11yescharacteristic
168Dic163—41no—
169Dic163—41no—
170Dic163—41no—
171C2×C163—81no—
172C2×C163—81no—
173C2×C163#55011yescharacteristic
174C2×C163#57511yescharacteristic
175C2×C163#57711yescharacteristic
176D163—81no—
177C163#340111yescharacteristic, complemented, Sylow
178#3401—1631nomaximal, Sylow
179#559D16311yescharacteristic, complemented
180#556—1631no—
181#497—1631no—
182#496—1631no—
183#495—1631no—
184#464—1631no—
185#464—1631no—
186#463—1631no—
187#463—1631no—
188#461—1631no—
189#460—1631no—
190#446—1631no—
191#442—1631no—
192#441—1631no—
193C4×SD16—1631no—
194C2×Q16—1631no—
195C2×Q16—1631no—
196C2×SD16—1631no—
197C2×SD16—1631no—
198C2×SD16—1631no—
199C2×SD16—1631no—
200C2×SD16—1631no—
201C2×D8—1631no—
202C2×D8—1631no—
203C4⋊Q8—1631no—
204#127C2×D16311yescharacteristic
205#126—3261no—
206#126—3261no—
207#124—21no—
208#124—1631no—
209#124—1631no—
210#124C2×D16311yescharacteristic
211#124C2×D16311yescharacteristic
212#123—21no—
213#123—21no—
214C22⋊Q8—3261no—
215C4×Q8—1631no—
216C4×Q8—1631no—
217C4×Q8C2×D16311yescharacteristic
218C4×Q8C2×D16311yescharacteristic
219C4×D4—21no—
220C4×D4—1631no—
221C4×D4C2×D16311yescharacteristic
222C4×D4C2×D16311yescharacteristic
223#117—3261no—
224#106—1631no—
225C4⋊C8—1631no—
226C4⋊C8—1631no—
227C4⋊C8—1631no—
228#103—1631no—
229#103—1631no—
230#103—1631no—
231#102—1631no—
232#102—1631no—
233#102—1631no—
234#97—1631no—
235#97—1631no—
236C4×C8—1631no—
237C2×Q8—1631no—
238C2×Q8—1631no—
239C2×Q8C22⋊D16311yescharacteristic
240C2×Q8C22×D16311yescharacteristic
241C2×D4—21no—
242C2×D4—21no—
243C2×D4—1631no—
244C2×D4C22×D16311yescharacteristic
245C2×D4C22×D16311yescharacteristic
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—3261no—
249Q16—3262no—
250Q16—6521no—
251SD16—3262no—
252SD16—6521no—
253SD16—6521no—
254SD16—6521no—
255SD16—6521no—
256D8—3262no—
257D8—6521no—
258C2×C8—1631no—
259C2×C8—1631no—
260C2×C8—1631no—
261C2×C8—1631no—
262C4⋊C4—21no—
263C4⋊C4—3261no—
264C4⋊C4—3261no—
265C4⋊C4—3261no—
266C4⋊C4—3261no—
267C4⋊C4—1631no—
268C4⋊C4C22⋊D16311yescharacteristic
269C4⋊C4C22⋊D16311yescharacteristic
270C4⋊C4C22⋊D16311yescharacteristic
271C4⋊C4C22×D16311yescharacteristic
272C22⋊C4—21no—
273C22⋊C4—21no—
274C22⋊C4—21no—
275C22⋊C4—21no—
276C22⋊C4—21no—
277C22⋊C4—21no—
278C22⋊C4—3261no—
279C22⋊C4—3261no—
280C22⋊C4—3261no—
281C22⋊C4—3261no—
282C22⋊C4—3261no—
283C42—21no—
284C42—3261no—
285C42—3261no—
286C42C22×D16311yescharacteristic
287C42C22×D16311yescharacteristic
288C42C22×D16311yescharacteristic
289C23—21no—
290C23—21no—
291C23—3261no—
292Q8—21no—
293Q8—21no—
294Q8—41no—
295Q8—3261no—
296Q8—3261no—
297Q8—3261no—
298Q8—3261no—
299D4—21no—
300D4—21no—
301D4—21no—
302D4—21no—
303D4—42no—
304D4—81no—
305D4—3261no—
306D4—3261no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—81no—
310C2×C4—81no—
311C2×C4—3261no—
312C2×C4—3261no—
313C2×C4—3261no—
314C2×C4—6522no—
315C2×C4C2×C22⋊D16311yescharacteristic
316C2×C4C2×C22⋊D16311yescharacteristic
317C2×C4C2×C22⋊D16311yescharacteristic
318C2×C4C2×C22⋊D16311yescharacteristic
319C2×C4C2×C22⋊D16311yescharacteristic
320C2×C4C2×C22⋊D16311yescharacteristic
321C2×C4C23×D16311yescharacteristic
322C8—3261no—
323C8—3261no—
324C8—3261no—
325C8—1632no—
326C22—41no—
327C22—41no—
328C22—41no—
329C22—41no—
330C22—41no—
331C22—41no—
332C22—6521no—
333C22—6521no—
334C22—6521no—
335C22C22×C22⋊D16311yescharacteristic
336C4—21no—
337C4—21no—
338C4—21no—
339C4—21no—
340C4—21no—
341C4—21no—
342C4—41no—
343C4—41no—
344C4—6521no—
345C4—6521no—
346C4—6521no—
347C4#41×D16311yescharacteristic
348C4#1910478111yescharacteristic
349C2—81no—
350C2—81no—
351C2—13041no—
352C2#2378233411yescharacteristic
353C2#2378246311yescharacteristic
354C2#2378247611yescharacteristic
subgroupG/H
centerC22C22×C22⋊D163
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
Fitting#559×C163C2
socleC22×C163C22×D4
prime radical#23782052C2
hypercenter#559D163
hypocenterC163#3401
LCSG ▷ C2×C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C22 ◁ C4×Q8 ◁ #559

Complements

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

NH ≅ G/Ntype
1. #23782388C2semidirect
2. #23782060C2semidirect
4. #23782047C2semidirect
5. #23781887C2semidirect
6. #23781884C2semidirect
7. #23781880C2semidirect
8. #23781877C2semidirect
9. #23781870C2semidirect
10. #23781862C2semidirect
11. #23781796C2semidirect
12. #23781775C2semidirect
13. #23781770C2semidirect
14. C4×#14579687C2semidirect
15. #559×C163C2semidirect
177. C163#3401semidirect
179. #559D163semidirect

Maximal subgroups

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

subgroupindex
#237823882
#237820602
#237820522
#237820472
#237818872
#237818842
#237818802
#237818772
#237818702
#237818622
#237817962
#237817752
#237817702
C4×#145796872
#559×C1632
#3401163