Finite Group Databasebeta

← #35227671

Subgroups — #35227671

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×#19104758C211yesmaximal, characteristic, complemented
2#135×D163C211yesmaximal, characteristic, complemented
3#23782107C212yesmaximal, complemented
4#23782106C211yesmaximal, characteristic, complemented
5#23782103C211yesmaximal, characteristic, complemented
6#23782102C212yesmaximal, complemented
7#23782100C211yesmaximal, characteristic, complemented
8#23782098C212yesmaximal, complemented
9#23782096C212yesmaximal, complemented
10M32×D163C211yesmaximal, characteristic, complemented
11#510×C163C211yesmaximal, characteristic, complemented
12C2×#14579700—21no—
13#41×D163—21no—
14#19104758—21no—
15#19104758—21no—
16#19104758C2212yes—
17#19104758C2211yescharacteristic, complemented
18#19104758C2212yescomplemented
19#19104757C2211yescharacteristic
20#19104756—21no—
21#19104755—21no—
22#19104754C2211yescharacteristic, complemented
23Q16×D163C2211yescharacteristic, complemented
24C2×#14579686—21no—
25SD16×D163—21no—
26C2×D1304C2211yescharacteristic, complemented
27D8×D163C2211yescharacteristic, complemented
28C2×C8×D163C2211yescharacteristic
29#19104681C2212yescomplemented
30#19104680C2212yescomplemented
31#19104678C2212yescomplemented
32#19104678C2212yescomplemented
33D2608C2212yescomplemented
34#19104676C2212yescomplemented
35#19104675C2211yescharacteristic, complemented
36#19104674C2212yescomplemented
37C16×D163C2212yescomplemented
38#135×C163C2211yescharacteristic
39C2×Q16×C163C2211yescharacteristic, complemented
40Q32×C163C2212yescomplemented
41SD32×C163C2212yescomplemented
42M32×C163C2211yescharacteristic, complemented
43#14579700—21no—
44#14579700—22no—
45#14579700—22no—
46#14579700—41no—
47#14579700—41no—
48#14579699—21no—
49#14579699—41no—
50#14579698—21no—
51#14579698—41no—
52Q8×D163—21no—
53C2×D652—21no—
54C2×D652—41no—
55D4×D163—21no—
56D4×D163—41no—
57C2×C4×D163—41no—
58C2×C4×D163—41no—
59C2×C4×D163D411yescharacteristic
60Dic652C2311yescharacteristic
61#14579689—21no—
62#14579688—21no—
63#14579687—21no—
64#14579686—21no—
65#14579686—22no—
66#14579686—22no—
67D1304—21no—
68D1304C2312yes—
69D1304C2311yescharacteristic
70#14579684—21no—
71C2×C163⋊C8D411yescharacteristic
72C8×D163D411yescharacteristic
73C8×D163D411yescharacteristic
74C8×D163C2311yescharacteristic
75C8×D163C2311yescharacteristic
76C163⋊C16C2312yes—
77#41×C163—21no—
78C2×Q8×C163—21no—
79Q16×C163—21no—
80Q16×C163C2312yes—
81Q16×C163C2311yescharacteristic
82SD16×C163—21no—
83D8×C163C2311yescharacteristic
84C2×C8×C163C2311yescharacteristic
85C16×C163C2312yes—
86C22×D163—41no—
87C22×D163—41no—
88C22×D163D811yescharacteristic
89Dic326—21no—
90Dic326—41no—
91D652—21no—
92D652—22no—
93D652—41no—
94D652—42no—
95D652—42no—
96D652—41no—
97C22⋊D163—41no—
98C22⋊D163—41no—
99C22⋊D163—41no—
100C2×Dic163—41no—
101C2×Dic163D811yescharacteristic
102C4×D163—41no—
103C4×D163—41no—
104C4×D163—41no—
105C4×D163—42no—
106C4×D163—42no—
107C4×D163D811yescharacteristic
108C4×D163D811yescharacteristic
109C4×D163C2×D411yescharacteristic
110C4×D163C2×D411yescharacteristic
111C163⋊C8C2×D411yescharacteristic
112C163⋊C8C2×D411yescharacteristic
113Q8×C163—21no—
114Q8×C163—22no—
115Q8×C163—41no—
116D4×C163—21no—
117D4×C163—41no—
118C2×C4×C163—41no—
119C2×C4×C163—41no—
120C2×C4×C163C2×D411yescharacteristic
121C8×C163C2×D411yescharacteristic
122C8×C163C2411yescharacteristic
123C22×C163—41no—
124C22×C163C2×D811yescharacteristic
125C2×D163—41no—
126C2×D163—41no—
127C2×D163—42no—
128C2×D163—82no—
129C2×D163—82no—
130C2×D163C2×D811yescharacteristic
131C2×D163C2×D811yescharacteristic
132C4×C163—41no—
133C4×C163—42no—
134C4×C163C2×D811yescharacteristic
135C4×C163C22×D411yescharacteristic
136Dic163—41no—
137Dic163C2×D811yescharacteristic
138Dic163C2×D811yescharacteristic
139C2×C163—21no—
140C2×C163—81no—
141C2×C163C22×D811yescharacteristic
142D163—21no—
143D163—21no—
144D163—81no—
145D163—82no—
146C163#341411yescharacteristic, complemented, Sylow
147#3414—1631nomaximal, Sylow
148C2×#135—1631no—
149C2×#135—1631no—
150#510—1631no—
151#510—1632no—
152#510D16311yescharacteristic, complemented
153#509—1631no—
154#509—1632no—
155#509—1631no—
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—1632no—
166#135—1631no—
167#135—1632no—
168#135—1631no—
169#135—1631no—
170#135C2×D16311yescharacteristic
171C2×Q16—1631no—
172C2×Q16C2×D16311yescharacteristic, complemented
173C2×SD16—3261no—
174C2×SD16—3261no—
175C2×D8—1631no—
176C2×D8—1631no—
177C22×C8—1631no—
178Q32—1632no—
179Q32C2×D16312yescomplemented
180SD32—1632no—
181SD32—1632no—
182SD32—1632no—
183SD32C2×D16312yescomplemented
184D16—1632no—
185D16—1632no—
186M32—1631no—
187M32—1632no—
188M32C2×D16311yescharacteristic, complemented
189C2×C16—1632no—
190#41—21no—
191#41—3261no—
192#41—3262no—
193#41—3261no—
194#41—3262no—
195#41—3261no—
196#41—6521no—
197#41—6521no—
198#41—6521no—
199#41—6521no—
200C2×Q8—21no—
201C2×Q8—3261no—
202C2×D4—3261no—
203C2×D4—3261no—
204C2×D4—6521no—
205C2×D4—6521no—
206C22×C4—6521no—
207C22×C4—6521no—
208C22×C4—1631no—
209Q16—21no—
210Q16—3261no—
211Q16—1631no—
212Q16C22×D16311yescharacteristic
213Q16C22×D16312yes—
214SD16—21no—
215SD16—3261no—
216SD16—3262no—
217SD16—3261no—
218SD16—3262no—
219SD16—3261no—
220D8—3261no—
221D8—3261no—
222D8—1632no—
223D8—1631no—
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—6521no—
234C23—6521no—
235C23—1631no—
236Q8—21no—
237Q8—22no—
238Q8—41no—
239Q8—3261no—
240Q8—6521no—
241D4—21no—
242D4—41no—
243D4—3261no—
244D4—3262no—
245D4—6521no—
246D4—6522no—
247D4—6522no—
248D4—6521no—
249D4—6521no—
250D4—6521no—
251D4—6521no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—6521no—
255C2×C4—6521no—
256C2×C4—6522no—
257C2×C4—6522no—
258C2×C4—6521no—
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—13042no—
272C22—13042no—
273C22—6521no—
274C22—6522no—
275C22—6521no—
276C22—1631no—
277C22—1631no—
278C22D8×D16311yescharacteristic
279C4—41no—
280C4—42no—
281C4—6521no—
282C4—1631no—
283C4—1631no—
284C4D8×D16311yescharacteristic
285C4C2×D4×D16311yescharacteristic
286C2—21no—
287C2—81no—
288C2—13042no—
289C2—13041no—
290C2—3261no—
291C2—3261no—
292C2C2×D8×D16311yescharacteristic
subgroupG/H
centerC2C2×D8×D163
commutator, cosocleC8×C163C24
FrattiniC8C23×D163
Fitting#510×C163C2
socleC2×C163C22×D8
hypercenter#510D163
hypocenterC163#3414
LCSG ▷ C8×C163 ▷ C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ #510

Complements

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

NH ≅ G/Ntype
1. C2×#19104758C2semidirect
2. #135×D163C2semidirect
3. #23782107C2semidirect
4. #23782106C2semidirect
5. #23782103C2semidirect
6. #23782102C2semidirect
7. #23782100C2semidirect
8. #23782098C2semidirect
9. #23782096C2semidirect
10. M32×D163C2semidirect
11. #510×C163C2semidirect
17. #19104758C22semidirect
18. #19104758C22semidirect
22. #19104754C22semidirect
23. Q16×D163C22semidirect
26. C2×D1304C22semidirect
27. D8×D163C22semidirect
29. #19104681C22semidirect
30. #19104680C22semidirect
31. #19104678C22semidirect
32. #19104678C22semidirect
33. D2608C22semidirect
34. #19104676C22semidirect
35. #19104675C22semidirect
36. #19104674C22semidirect
37. C16×D163C22semidirect
39. C2×Q16×C163C22semidirect
40. Q32×C163C22semidirect
41. SD32×C163C22semidirect
42. M32×C163C22semidirect
146. C163#3414semidirect
152. #510D163semidirect
172. C2×Q16C2×D163semidirect
179. Q32C2×D163semidirect
183. SD32C2×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×#191047582
#135×D1632
#237821072
#237821062
#237821032
#237821022
#237821002
#237820982
#237820962
M32×D1632
#510×C1632
#3414163