Finite Group Databasebeta

← #35217941

Subgroups — #35217941

Order 20864.

610 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 610 conjugacy classes of subgroups — too many to draw. Only its 165 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×#19104719C211yesmaximal, characteristic
2C2×#19104719C211yesmaximal, characteristic
3C2×#19104691C211yesmaximal, characteristic, complemented
4C22×C4⋊Dic163C211yesmaximal, characteristic, complemented
5#23781597C212yesmaximal, complemented
6#23781587C212yesmaximal, complemented
7#23781564C212yesmaximal, complemented
8#23781557C212yesmaximal, complemented
9#23781544C211yesmaximal, characteristic
10#23781526C211yesmaximal, characteristic, complemented
11#385×C163C211yesmaximal, characteristic, complemented
12C22×C22⋊D163C2211yescharacteristic
13C23×Dic163C2211yescharacteristic
14C23×Dic163C2211yescharacteristic
15#19104719—24no—
16#19104719—24no—
17#19104691—24no—
18C2×C4⋊Dic163—24no—
19C2×C4⋊Dic163C2211yescharacteristic
20C2×C4⋊Dic163C2212yes—
21C2×C4⋊Dic163C2211yescharacteristic
22C2×C4⋊Dic163C2212yes—
23C2×C4⋊Dic163C2212yes—
24C2×Dic163⋊C4C2212yes—
25C2×C22⋊Dic163C2211yescharacteristic
26C2×(C2×D163)⋊C4C2211yescharacteristic
27C2×(C2×D163)⋊C4C2212yes—
28C2×(C2×D163)⋊C4C2211yescharacteristic
29C2×(C2×D163)⋊C4C2212yes—
30C2×C4×Dic163C2212yes—
31C163⋊#95C2212yes—
32C163⋊#95C2212yes—
33C163⋊#95C2212yes—
34C23×C4×C163C2211yescharacteristic
35C2×C4⋊C4×C163C2212yes—
36C2×C22⋊C4×C163C2211yescharacteristic
37C2×C22⋊C4×C163C2211yescharacteristic
38#95×C163C2212yes—
39C23×D163C2311yescharacteristic
40C2×C22⋊D163—22no—
41C2×C22⋊D163—22no—
42C2×C22⋊D163—22no—
43C22×Dic163—22no—
44C22×Dic163—22no—
45C22×Dic163—22no—
46C22×Dic163—22no—
47C22×Dic163—22no—
48C22×Dic163D414yes—
49C22×Dic163C2312yes—
50C22×Dic163C2311yescharacteristic
51C22×Dic163C2312yes—
52C22×Dic163C2311yescharacteristic
53C22×Dic163C2311yescharacteristic
54C4⋊Dic163—24no—
55C4⋊Dic163—24no—
56C4⋊Dic163—22no—
57C4⋊Dic163—22no—
58C4⋊Dic163—24no—
59Dic163⋊C4—24no—
60C22⋊Dic163—22no—
61(C2×D163)⋊C4—22no—
62(C2×D163)⋊C4—22no—
63(C2×D163)⋊C4—24no—
64(C2×D163)⋊C4—24no—
65C4×Dic163—24no—
66C24×C163C2311yescharacteristic
67C22×C4×C163—22no—
68C22×C4×C163—24no—
69C22×C4×C163C2311yescharacteristic
70C22×C4×C163C2311yescharacteristic
71C22×C4×C163C2312yes—
72C22×C4×C163C2312yes—
73C4⋊C4×C163—24no—
74C22⋊C4×C163—22no—
75C22⋊C4×C163D414yes—
76C22×D163—21no—
77C22×D163—21no—
78C22×D163—22no—
79C22×D163—21no—
80C22×D163—22no—
81C22⋊D163—44no—
82C2×Dic163—28no—
83C2×Dic163—21no—
84C2×Dic163—21no—
85C2×Dic163—21no—
86C2×Dic163—21no—
87C2×Dic163—22no—
88C2×Dic163—22no—
89C2×Dic163—22no—
90C2×Dic163—21no—
91C2×Dic163—21no—
92C2×Dic163—22no—
93C2×Dic163—22no—
94C2×Dic163—21no—
95C2×Dic163—22no—
96C2×Dic163—44no—
97C2×Dic163C2×D412yes—
98C2×Dic163C2×D412yes—
99C23×C163—22no—
100C23×C163—22no—
101C23×C163C2×D412yes—
102C23×C163C2×D412yes—
103C23×C163#4112yes—
104C23×C163C2411yescharacteristic
105C2×C4×C163—28no—
106C2×C4×C163—22no—
107C2×C4×C163—22no—
108C2×C4×C163—22no—
109C2×C4×C163—22no—
110C2×C4×C163—21no—
111C2×C4×C163—21no—
112C2×C4×C163—22no—
113C2×C4×C163—22no—
114C2×C4×C163C2×D414yes—
115C2×C4×C163#4114yes—
116C22×C163—22no—
117C22×C163—22no—
118C22×C163—24no—
119C22×C163—24no—
120C22×C163#12114yes—
121C22×C163C22×D411yescharacteristic
122C22×C163C22×D411yescharacteristic
123C22×C163C2×#4112yes—
124C22×C163C2×#4111yescharacteristic
125C22×C1632−1+412yes—
126C2×D163—41no—
127C2×D163—41no—
128C2×D163—41no—
129C2×D163—42no—
130C2×D163—42no—
131C4×C163—24no—
132C4×C163—24no—
133C4×C163—41no—
134C4×C163—41no—
135Dic163—22no—
136Dic163—22no—
137Dic163—42no—
138Dic163—41no—
139Dic163—42no—
140C2×C163—24no—
141C2×C163C2×#12111yescharacteristic
142C2×C163#52511yescharacteristic
143C2×C163#53611yescharacteristic
144C2×C163#54712yes—
145C2×C163#55212yes—
146D163—81no—
147C163#241311yescharacteristic, complemented, Sylow
148#2413—1631nomaximal, Sylow
149C2×#123—1631no—
150C2×#123—1631no—
151C2×#117—1631no—
152C22×C4⋊C4—1631no—
153#398—1632no—
154#396—1632no—
155#389—1632no—
156#388—1632no—
157#386—1631no—
158#385D16311yescharacteristic, complemented
159#381—1631no—
160C22×D4—1631no—
161C23×C4—1631no—
162C23×C4—1631no—
163C23×C4C2×D16311yescharacteristic
164#123—3264no—
165#123—3264no—
166#117—3264no—
167C2×C4⋊C4—3264no—
168C2×C4⋊C4—1632no—
169C2×C4⋊C4—1632no—
170C2×C4⋊C4—1632no—
171C2×C4⋊C4—1631no—
172C2×C4⋊C4—1632no—
173C2×C4⋊C4—1631no—
174C2×C4⋊C4C2×D16312yes—
175C2×C22⋊C4—1631no—
176C2×C22⋊C4—1631no—
177C2×C22⋊C4—1632no—
178C2×C22⋊C4—1631no—
179C2×C22⋊C4—1632no—
180C2×C22⋊C4C2×D16311yescharacteristic
181C2×C22⋊C4C2×D16311yescharacteristic
182C2×C42—1632no—
183#95—1632no—
184#95—1632no—
185#95—1632no—
186#95C2×D16312yes—
187C24—1631no—
188C24C22×D16311yescharacteristic
189C2×D4—3262no—
190C2×D4—3262no—
191C2×D4—3262no—
192C22×C4—22no—
193C22×C4—24no—
194C22×C4—3262no—
195C22×C4—3262no—
196C22×C4—3262no—
197C22×C4—3262no—
198C22×C4—3262no—
199C22×C4—1632no—
200C22×C4—1632no—
201C22×C4—1631no—
202C22×C4—1634no—
203C22×C4—1631no—
204C22×C4—1631no—
205C22×C4C22×D16311yescharacteristic
206C22×C4C22×D16311yescharacteristic
207C22×C4C22×D16312yes—
208C22×C4C22×D16312yes—
209C4⋊C4—24no—
210C4⋊C4—3264no—
211C4⋊C4—3264no—
212C4⋊C4—3262no—
213C4⋊C4—3262no—
214C4⋊C4—3264no—
215C4⋊C4—3264no—
216C22⋊C4—22no—
217C22⋊C4—3262no—
218C22⋊C4—3262no—
219C22⋊C4—3262no—
220C22⋊C4—3264no—
221C22⋊C4—3264no—
222C22⋊C4D65214yes—
223C42—3264no—
224C23—22no—
225C23—22no—
226C23—3261no—
227C23—3261no—
228C23—3262no—
229C23—3261no—
230C23—3262no—
231C23D4×D16312yes—
232C23C2×D65212yes—
233C23#1457970012yes—
234C23C23×D16311yescharacteristic
235D4—6524no—
236C2×C4—28no—
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—22no—
240C2×C4—22no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—3261no—
246C2×C4—3261no—
247C2×C4—3261no—
248C2×C4—3261no—
249C2×C4—3268no—
250C2×C4—3262no—
251C2×C4—3261no—
252C2×C4—3262no—
253C2×C4—3262no—
254C2×C4—3261no—
255C2×C4—3262no—
256C2×C4—3262no—
257C2×C4—3261no—
258C2×C4—3262no—
259C2×C4—6524no—
260C2×C4—1632no—
261C2×C4—1632no—
262C2×C4C2×D65214yes—
263C2×C4#1457969814yes—
264C22—22no—
265C22—22no—
266C22—24no—
267C22—24no—
268C22—6521no—
269C22—6521no—
270C22—6521no—
271C22—6522no—
272C22—6522no—
273C22#1910471014yes—
274C22C2×D4×D16311yescharacteristic
275C22C22×D65211yescharacteristic
276C22C2×#1457969812yes—
277C22C2×#1457970011yescharacteristic
278C22#1910478212yes—
279C4—24no—
280C4—24no—
281C4—41no—
282C4—41no—
283C4—3262no—
284C4—3262no—
285C4—6522no—
286C4—6521no—
287C4—6522no—
288C2—24no—
289C2—13041no—
290C2C2×#1910471011yescharacteristic
291C2#2378217111yescharacteristic
292C2#2378223711yescharacteristic
293C2#2378231912yes—
294C2#2378235712yes—
subgroupG/H
center, FrattiniC23C23×D163
commutator, socle, cosocleC23×C163C24
Fitting#385×C163C2
prime radicalC22×C22⋊D163C22
hypercenter#385D163
hypocenterC163#2413
LCSG ▷ C23×C163 ▷ C163
UCSC1 ◁ C23 ◁ #385

Complements

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

NH ≅ G/Ntype
3. C2×#19104691C2semidirect
4. C22×C4⋊Dic163C2semidirect
5. #23781597C2semidirect
6. #23781587C2semidirect
7. #23781564C2semidirect
8. #23781557C2semidirect
10. #23781526C2semidirect
11. #385×C163C2semidirect
147. C163#2413semidirect
158. #385D163semidirect

Maximal subgroups

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

subgroupindex
C2×#191047192
C2×#191046912
C22×C4⋊Dic1632
#237815972
#237815872
#237815642
#237815572
#237815442
#237815262
#385×C1632
#2413163