Finite Group Databasebeta

← #35227668

Subgroups — #35227668

Order 20864.

364 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 364 conjugacy classes of subgroups — too many to draw. Only its 113 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19104754C211yesmaximal, characteristic, complemented
2#135×D163C211yesmaximal, characteristic, complemented
3#23782108C211yesmaximal, characteristic, complemented
4#23782105C211yesmaximal, characteristic, complemented
5#23782104C212yesmaximal, complemented
6#23782101C211yesmaximal, characteristic, complemented
7#23782099C212yesmaximal, complemented
8#23782096C212yesmaximal, complemented
9#23782094C212yesmaximal, complemented
10M32×D163C211yesmaximal, characteristic, complemented
11#509×C163C211yesmaximal, characteristic, complemented
12C2×#14579698—21no—
13#41×D163—21no—
14#19104758C2211yescharacteristic, complemented
15#19104757C2211yescharacteristic
16#19104756—21no—
17#19104755—21no—
18#19104754—21no—
19#19104754—21no—
20#19104754C2212yes—
21#19104754C2211yescharacteristic, complemented
22#19104754C2212yescomplemented
23C2×Dic652C2211yescharacteristic, complemented
24Q16×D163C2211yescharacteristic, complemented
25C2×#14579687—21no—
26SD16×D163—21no—
27D8×D163C2211yescharacteristic, complemented
28C2×C8×D163C2211yescharacteristic
29Dic1304C2212yescomplemented
30#19104681C2212yescomplemented
31#19104680C2212yescomplemented
32#19104679C2212yescomplemented
33#19104679C2212yescomplemented
34#19104676C2212yescomplemented
35#19104675C2211yescharacteristic, complemented
36#19104674C2212yescomplemented
37C16×D163C2212yescomplemented
38#135×C163C2211yescharacteristic
39C2×D8×C163C2211yescharacteristic, complemented
40SD32×C163C2212yescomplemented
41D16×C163C2212yescomplemented
42M32×C163C2211yescharacteristic, complemented
43#14579700—21no—
44#14579699—21no—
45#14579699—41no—
46#14579698—21no—
47#14579698—22no—
48#14579698—22no—
49#14579698—41no—
50#14579698—41no—
51#14579698—41no—
52C2×Dic326—21no—
53Q8×D163—21no—
54C2×C22⋊D163—41no—
55D4×D163—21no—
56D4×D163—41no—
57C22×Dic163—41no—
58C2×C4×D163—41no—
59C2×C4×D163D411yescharacteristic
60Dic652—21no—
61Dic652C2311yescharacteristic
62Dic652C2312yes—
63#14579689—21no—
64#14579688—21no—
65#14579687—21no—
66#14579687—22no—
67#14579687—22no—
68#14579686—21no—
69D1304C2311yescharacteristic
70#14579684—21no—
71C2×C163⋊C8D411yescharacteristic
72C8×D163D411yescharacteristic
73C8×D163D411yescharacteristic
74C8×D163C2311yescharacteristic
75C8×D163C2311yescharacteristic
76C163⋊C16C2312yes—
77#41×C163—21no—
78C2×D4×C163—21no—
79Q16×C163C2311yescharacteristic
80SD16×C163—21no—
81D8×C163—21no—
82D8×C163C2311yescharacteristic
83D8×C163C2312yes—
84C2×C8×C163C2311yescharacteristic
85C16×C163C2312yes—
86C22×D163—41no—
87C22×D163D811yescharacteristic
88Dic326—22no—
89Dic326—21no—
90Dic326—41no—
91Dic326—41no—
92D652—21no—
93D652—41no—
94C22⋊D163—41no—
95C22⋊D163—41no—
96C22⋊D163—42no—
97C22⋊D163—42no—
98C22⋊D163—41no—
99C2×Dic163—41no—
100C2×Dic163—41no—
101C2×Dic163—42no—
102C2×Dic163—42no—
103C2×Dic163D811yescharacteristic
104C4×D163—41no—
105C4×D163—41no—
106C4×D163—41no—
107C4×D163D811yescharacteristic
108C4×D163D811yescharacteristic
109C4×D163C2×D411yescharacteristic
110C4×D163C2×D411yescharacteristic
111C163⋊C8C2×D411yescharacteristic
112C163⋊C8C2×D411yescharacteristic
113C23×C163—41no—
114Q8×C163—21no—
115D4×C163—22no—
116D4×C163—21no—
117D4×C163—41no—
118D4×C163—41no—
119C2×C4×C163—41no—
120C2×C4×C163C2×D411yescharacteristic
121C8×C163C2×D411yescharacteristic
122C8×C163C2411yescharacteristic
123C22×C163—41no—
124C22×C163—42no—
125C22×C163—82no—
126C22×C163C2×D811yescharacteristic
127C2×D163—41no—
128C2×D163—41no—
129C2×D163—82no—
130C2×D163C2×D811yescharacteristic
131C2×D163C2×D811yescharacteristic
132C4×C163—41no—
133C4×C163C2×D811yescharacteristic
134C4×C163C22×D411yescharacteristic
135Dic163—41no—
136Dic163—42no—
137Dic163C2×D811yescharacteristic
138Dic163C2×D811yescharacteristic
139C2×C163—21no—
140C2×C163—81no—
141C2×C163—82no—
142C2×C163C22×D811yescharacteristic
143D163—21no—
144D163—21no—
145D163—81no—
146C163#341411yescharacteristic, complemented, Sylow
147#3414—1631nomaximal, Sylow
148C2×#135—1631no—
149C2×#135—1631no—
150#510—1631no—
151#510—1632no—
152#510—1631no—
153#509—1631no—
154#509—1632no—
155#509D16311yescharacteristic, complemented
156#508—1632no—
157#508—1632no—
158C2×M32—1631no—
159C2×#41—3261no—
160C2×#41—3261no—
161#135—3261no—
162#135—3261no—
163#135—3261no—
164#135—3261no—
165#135—1631no—
166#135—1632no—
167#135—1632no—
168#135—1631no—
169#135—1631no—
170#135C2×D16311yescharacteristic
171C2×Q16—1631no—
172C2×Q16—1631no—
173C2×SD16—3261no—
174C2×SD16—3261no—
175C2×D8—1631no—
176C2×D8C2×D16311yescharacteristic, complemented
177C22×C8—1631no—
178Q32—1632no—
179Q32—1632no—
180SD32—1632no—
181SD32—1632no—
182SD32—1632no—
183SD32C2×D16312yescomplemented
184D16—1632no—
185D16C2×D16312yescomplemented
186M32—1631no—
187M32—1632no—
188M32C2×D16311yescharacteristic, complemented
189C2×C16—1632no—
190#41—21no—
191#41—3262no—
192#41—3262no—
193#41—3261no—
194#41—3261no—
195#41—3261no—
196#41—6521no—
197#41—6521no—
198#41—6521no—
199#41—6521no—
200C2×Q8—3261no—
201C2×Q8—3261no—
202C2×D4—21no—
203C2×D4—3261no—
204C2×D4—6521no—
205C2×D4—6521no—
206C22×C4—6521no—
207C22×C4—6521no—
208C22×C4—1631no—
209Q16—3261no—
210Q16—3261no—
211Q16—1631no—
212Q16—1632no—
213Q16C22×D16311yescharacteristic
214SD16—21no—
215SD16—3262no—
216SD16—3262no—
217SD16—3261no—
218SD16—3261no—
219SD16—3261no—
220D8—21no—
221D8—3261no—
222D8—1631no—
223D8C22×D16312yes—
224D8C22×D16311yescharacteristic
225C2×C8—1631no—
226C2×C8—1631no—
227C2×C8—1631no—
228C2×C8—1631no—
229C2×C8—1631no—
230C2×C8C22×D16311yescharacteristic
231C16—1632no—
232C16C22×D16312yes—
233C23—41no—
234C23—6521no—
235C23—1631no—
236Q8—21no—
237Q8—3262no—
238Q8—3261no—
239Q8—6521no—
240Q8—6521no—
241D4—22no—
242D4—21no—
243D4—41no—
244D4—41no—
245D4—3261no—
246D4—6521no—
247D4—6521no—
248D4—6522no—
249D4—6522no—
250D4—6521no—
251D4—6521no—
252C2×C4—41no—
253C2×C4—6521no—
254C2×C4—6521no—
255C2×C4—6521no—
256C2×C4—6521no—
257C2×C4—6522no—
258C2×C4—6522no—
259C2×C4—6521no—
260C2×C4—1631no—
261C2×C4—1631no—
262C2×C4—1631no—
263C2×C4—1631no—
264C2×C4—1631no—
265C2×C4D4×D16311yescharacteristic
266C8—1631no—
267C8—1631no—
268C8D4×D16311yescharacteristic
269C8C23×D16311yescharacteristic
270C22—41no—
271C22—42no—
272C22—82no—
273C22—13042no—
274C22—6521no—
275C22—6521no—
276C22—1631no—
277C22—1631no—
278C22D8×D16311yescharacteristic
279C4—41no—
280C4—6521no—
281C4—6522no—
282C4—1631no—
283C4—1631no—
284C4D8×D16311yescharacteristic
285C4C2×D4×D16311yescharacteristic
286C2—21no—
287C2—82no—
288C2—81no—
289C2—13041no—
290C2—3261no—
291C2—3261no—
292C2C2×D8×D16311yescharacteristic
subgroupG/H
centerC2C2×D8×D163
commutator, cosocleC8×C163C24
FrattiniC8C23×D163
Fitting#509×C163C2
socleC2×C163C22×D8
hypercenter#509D163
hypocenterC163#3414
LCSG ▷ C8×C163 ▷ C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ #509

Complements

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

NH ≅ G/Ntype
1. C2×#19104754C2semidirect
2. #135×D163C2semidirect
3. #23782108C2semidirect
4. #23782105C2semidirect
5. #23782104C2semidirect
6. #23782101C2semidirect
7. #23782099C2semidirect
8. #23782096C2semidirect
9. #23782094C2semidirect
10. M32×D163C2semidirect
11. #509×C163C2semidirect
14. #19104758C22semidirect
21. #19104754C22semidirect
22. #19104754C22semidirect
23. C2×Dic652C22semidirect
24. Q16×D163C22semidirect
27. D8×D163C22semidirect
29. Dic1304C22semidirect
30. #19104681C22semidirect
31. #19104680C22semidirect
32. #19104679C22semidirect
33. #19104679C22semidirect
34. #19104676C22semidirect
35. #19104675C22semidirect
36. #19104674C22semidirect
37. C16×D163C22semidirect
39. C2×D8×C163C22semidirect
40. SD32×C163C22semidirect
41. D16×C163C22semidirect
42. M32×C163C22semidirect
146. C163#3414semidirect
155. #509D163semidirect
176. C2×D8C2×D163semidirect
183. SD32C2×D163semidirect
185. D16C2×D163semidirect
188. M32C2×D163semidirect

Maximal subgroups

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

subgroupindex
C2×#191047542
#135×D1632
#237821082
#237821052
#237821042
#237821012
#237820992
#237820962
#237820942
M32×D1632
#509×C1632
#3414163