Finite Group Databasebeta

← #35213557

Subgroups — #35213557

Order 20864.

288 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 288 conjugacy classes of subgroups — too many to draw. Only its 51 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
1#23782223C211yesmaximal, characteristic
2#23781984C211yesmaximal, characteristic, complemented
3#23781981C211yesmaximal, characteristic, complemented
4#23781602C211yesmaximal, characteristic, complemented
5#23781398C211yesmaximal, characteristic, complemented
6#23781388C211yesmaximal, characteristic, complemented
7#327×C163C211yesmaximal, characteristic, complemented
8C2×D4×D163—21no—
9#19104720—21no—
10#19104717—21no—
11#19104709C2211yescharacteristic
12#19104707—21no—
13#19104705—21no—
14#19104703—21no—
15C2×C4⋊Dic163—21no—
16C2×C4⋊Dic163C2211yescharacteristic, complemented
17C2×(C2×D163)⋊C4—21no—
18C2×(C2×D163)⋊C4C2211yescharacteristic
19#19104667—21no—
20#19104666—21no—
21#19104657—21no—
22#19104656—21no—
23C22⋊(C163⋊C8)C2211yescharacteristic, complemented
24#121×C163C2211yescharacteristic
25C22⋊C8×C163C2211yescharacteristic, complemented
26#95×C163C2211yescharacteristic, complemented
27C23×D163—21no—
28C23×D163D411yescharacteristic
29C2×D652—21no—
30C2×D652D411yescharacteristic
31C2×C22⋊D163—21no—
32C2×C22⋊D163—21no—
33C2×C22⋊D163—21no—
34D4×D163—42no—
35D4×D163—42no—
36C22×Dic163—21no—
37C22×Dic163D411yescharacteristic
38C2×C4×D163—21no—
39C2×C163⋊C8—21no—
40C4⋊Dic163—21no—
41C4⋊Dic163—21no—
42C4⋊Dic163—42no—
43C4⋊Dic163D411yescharacteristic
44Dic163⋊C4—21no—
45C22⋊Dic163—21no—
46(C2×D163)⋊C4—21no—
47(C2×D163)⋊C4—21no—
48(C2×D163)⋊C4—21no—
49(C2×D163)⋊C4—21no—
50(C2×D163)⋊C4—42no—
51C2×D4×C163—21no—
52C2×D4×C163D411yescharacteristic
53C22×C4×C163—21no—
54C22×C4×C163C2311yescharacteristic
55C2×C8×C163—21no—
56C4⋊C4×C163D411yescharacteristic
57C22⋊C4×C163—21no—
58C22×D163—21no—
59C22×D163—21no—
60C22×D163—21no—
61C22×D163—42no—
62C22×D163—41no—
63C22×D163—41no—
64C22×D163—81no—
65C22×D163—81no—
66C22×D163SD1612yes—
67D652—41no—
68D652—42no—
69C22⋊D163—42no—
70C22⋊D163—41no—
71C22⋊D163—41no—
72C22⋊D163—81no—
73C2×Dic163—21no—
74C2×Dic163—21no—
75C2×Dic163—41no—
76C2×Dic163—41no—
77C2×Dic163—41no—
78C2×Dic163—41no—
79C4×D163—42no—
80C163⋊C8—41no—
81C23×C163—21no—
82C23×C163C2×D411yescharacteristic
83D4×C163—41no—
84D4×C163—41no—
85D4×C163—41no—
86C2×C4×C163—22no—
87C2×C4×C163—21no—
88C2×C4×C163—41no—
89C2×C4×C163—41no—
90C2×C4×C163—41no—
91C2×C4×C163C2×D411yescharacteristic
92C2×C4×C163C2×D411yescharacteristic
93C8×C163—41no—
94C22×C163—21no—
95C22×C163—21no—
96C22×C163—41no—
97C22×C163—41no—
98C22×C163—41no—
99C22×C163#12011yescharacteristic
100C22×C163C2×SD1611yescharacteristic
101C22×C163AGL(1,ℤ/8)11yescharacteristic
102C2×D163—22no—
103C2×D163—41no—
104C2×D163—41no—
105C2×D163—42no—
106C2×D163—41no—
107C2×D163—41no—
108C2×D163—41no—
109C2×D163—81no—
110C2×D163—84no—
111C4×C163—21no—
112C4×C163—21no—
113C4×C163—42no—
114C4×C163—41no—
115Dic163—41no—
116Dic163—41no—
117Dic163—81no—
118C2×C163—21no—
119C2×C163—21no—
120C2×C163—81no—
121C2×C163#45011yescharacteristic
122C2×C163#45311yescharacteristic
123C2×C163#45811yescharacteristic
124D163—42no—
125D163—81no—
126C163#161311yescharacteristic, complemented, Sylow
127#1613—1631nomaximal, Sylow
128#534—1631no—
129#481—1631no—
130#481—1631no—
131#399—1631no—
132#327—1631no—
133#327D16311yescharacteristic, complemented
134#323—1631no—
135C22×D4—3261no—
136#123—3261no—
137#123—3261no—
138#121—3261no—
139#121—1631no—
140#121C2×D16311yescharacteristic
141#120—3261no—
142#120—3261no—
143C2×C4⋊C4—3261no—
144C2×C4⋊C4—1631no—
145C2×C22⋊C4—3261no—
146C2×C22⋊C4—1631no—
147#106—3261no—
148#106—3261no—
149#102—3261no—
150#102—3261no—
151C22⋊C8—1631no—
152C22⋊C8C2×D16311yescharacteristic, complemented
153#95C2×D16311yescharacteristic, complemented
154C24—3261no—
155C24—1631no—
156C2×D4—21no—
157C2×D4—3261no—
158C2×D4—3261no—
159C2×D4—3261no—
160C2×D4—3261no—
161C2×D4—6522no—
162C2×D4—6522no—
163C2×D4—1631no—
164C2×D4D65211yescharacteristic
165C22×C4—21no—
166C22×C4—3261no—
167C22×C4—3261no—
168C22×C4—1631no—
169C22×C4C22×D16311yescharacteristic
170C2×C8—21no—
171C2×C8—3261no—
172C4⋊C4—3261no—
173C4⋊C4—3261no—
174C4⋊C4—3261no—
175C4⋊C4—6522no—
176C4⋊C4—1631no—
177C4⋊C4D65211yescharacteristic
178C22⋊C4—21no—
179C22⋊C4—3261no—
180C22⋊C4—3261no—
181C22⋊C4—3261no—
182C22⋊C4—3261no—
183C22⋊C4—3261no—
184C22⋊C4—6522no—
185C23—21no—
186C23—13041no—
187C23—13041no—
188C23—3261no—
189C23—3261no—
190C23—3261no—
191C23—6521no—
192C23—6521no—
193C23—6522no—
194C23—1632no—
195C23D4×D16311yescharacteristic
196D4—41no—
197D4—41no—
198D4—41no—
199D4—13041no—
200D4—6521no—
201D4—6522no—
202D4—6522no—
203D4—6521no—
204D4—6521no—
205C2×C4—22no—
206C2×C4—21no—
207C2×C4—41no—
208C2×C4—41no—
209C2×C4—41no—
210C2×C4—3261no—
211C2×C4—3261no—
212C2×C4—6522no—
213C2×C4—6521no—
214C2×C4—6521no—
215C2×C4—6521no—
216C2×C4—6521no—
217C2×C4D4×D16311yescharacteristic
218C2×C4C2×D65211yescharacteristic
219C8—41no—
220C8—6521no—
221C22—21no—
222C22—21no—
223C22—41no—
224C22—41no—
225C22—41no—
226C22—13041no—
227C22—13044no—
228C22—3262no—
229C22—6522no—
230C22—6521no—
231C22—6521no—
232C22—6521no—
233C22—6521no—
234C22—6521no—
235C22#1910470511yescharacteristic
236C22SD16×D16311yescharacteristic
237C22#1910475911yescharacteristic
238C4—21no—
239C4—21no—
240C4—42no—
241C4—41no—
242C4—13041no—
243C4—6521no—
244C4—6521no—
245C2—21no—
246C2—21no—
247C2—81no—
248C2—13041no—
249C2—6522no—
250C2#2378181711yescharacteristic
251C2#2378183411yescharacteristic
252C2#2378185611yescharacteristic
subgroupG/H
centerC22#19104705
commutator, cosocleC22×C4×C163C23
FrattiniC22×C4C22×D163
Fitting#327×C163C2
socleC22×C163#120
prime radical#23782223C2
hypercenter#327D163
hypocenterC163#1613
LCSG ▷ C22×C4×C163 ▷ C22×C163 ▷ C163
UCSC1 ◁ C22 ◁ C22×C4 ◁ #327

Complements

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

NH ≅ G/Ntype
2. #23781984C2semidirect
3. #23781981C2semidirect
4. #23781602C2semidirect
5. #23781398C2semidirect
6. #23781388C2semidirect
7. #327×C163C2semidirect
16. C2×C4⋊Dic163C22semidirect
23. C22⋊(C163⋊C8)C22semidirect
25. C22⋊C8×C163C22semidirect
26. #95×C163C22semidirect
126. C163#1613semidirect
133. #327D163semidirect
152. C22⋊C8C2×D163semidirect
153. #95C2×D163semidirect

Maximal subgroups

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

subgroupindex
#237822232
#237819842
#237819812
#237816022
#237813982
#237813882
#327×C1632
#1613163