Finite Group Databasebeta

← #35213517

Subgroups — #35213517

Order 20864.

260 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 49 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#23782252C211yesmaximal, characteristic
2#23781823C211yesmaximal, characteristic, complemented
3#23781803C211yesmaximal, characteristic, complemented
4#23781578C211yesmaximal, characteristic, complemented
5#23781401C211yesmaximal, characteristic, complemented
6#23781388C211yesmaximal, characteristic, complemented
7#327×C163C211yesmaximal, characteristic, complemented
8C22×Dic326C2211yescharacteristic
9C2×Dic652—21no—
10C2×#14579687—21no—
11#19104730—21no—
12#19104721—21no—
13#19104719—21no—
14#19104714C2211yescharacteristic
15#19104707—21no—
16#19104703—21no—
17D4×Dic163—21no—
18C2×(C2×D163)⋊C4—21no—
19C2×(C2×D163)⋊C4C2211yescharacteristic
20#19104660—21no—
21#19104658—21no—
22C22⋊(C163⋊C8)C2211yescharacteristic
23C163⋊#95—21no—
24#121×C163C2211yescharacteristic
25C22⋊C8×C163C2211yescharacteristic
26#95×C163C2211yescharacteristic
27C23×D163D411yescharacteristic
28C2×Dic326—21no—
29C2×Dic326—21no—
30C2×Dic326—42no—
31C2×Dic326D411yescharacteristic
32C2×C22⋊D163—21no—
33C2×C22⋊D163—21no—
34C22×Dic163—21no—
35C22×Dic163—21no—
36C22×Dic163D411yescharacteristic
37Dic652—42no—
38#14579687—42no—
39C2×C163⋊C8—21no—
40C4⋊Dic163—21no—
41C4⋊Dic163D411yescharacteristic
42Dic163⋊C4—21no—
43C22⋊Dic163—21no—
44C22⋊Dic163—21no—
45(C2×D163)⋊C4—21no—
46(C2×D163)⋊C4—21no—
47(C2×D163)⋊C4—21no—
48(C2×D163)⋊C4—21no—
49(C2×D163)⋊C4—42no—
50C4×Dic163—21no—
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—41no—
62C22×D163—41no—
63Dic326—22no—
64Dic326—22no—
65Dic326—41no—
66Dic326—41no—
67Dic326—41no—
68C22⋊D163—81no—
69C22⋊D163—81no—
70C2×Dic163—22no—
71C2×Dic163—21no—
72C2×Dic163—21no—
73C2×Dic163—21no—
74C2×Dic163—41no—
75C2×Dic163—41no—
76C2×Dic163—41no—
77C2×Dic163—81no—
78C163⋊C8—41no—
79C23×C163—21no—
80C23×C163C2×D411yescharacteristic
81D4×C163—41no—
82D4×C163—41no—
83D4×C163—41no—
84C2×C4×C163—21no—
85C2×C4×C163—41no—
86C2×C4×C163—41no—
87C2×C4×C163—41no—
88C2×C4×C163—41no—
89C2×C4×C163C2×D411yescharacteristic
90C2×C4×C163C2×D411yescharacteristic
91C8×C163—41no—
92C22×C163—21no—
93C22×C163—21no—
94C22×C163—41no—
95C22×C163—41no—
96C22×C163—41no—
97C22×C163#12011yescharacteristic
98C22×C163#13511yescharacteristic
99C22×C163#13711yescharacteristic
100C2×D163—42no—
101C2×D163—41no—
102C2×D163—41no—
103C2×D163—41no—
104C2×D163—81no—
105C4×C163—21no—
106C4×C163—21no—
107C4×C163—41no—
108C4×C163—81no—
109Dic163—41no—
110Dic163—41no—
111Dic163—42no—
112Dic163—41no—
113C2×C163—21no—
114C2×C163—21no—
115C2×C163—81no—
116C2×C163#45211yescharacteristic
117C2×C163#45511yescharacteristic
118C2×C163B(4,𝔽2)11yescharacteristic
119D163—81no—
120C163#160611yescharacteristic, complemented, Sylow
121#1606—1631nomaximal, Sylow
122#538—1631no—
123#451—1631no—
124#448—1631no—
125#393—1631no—
126#328—1631no—
127#327D16311yescharacteristic, complemented
128#323—1631no—
129C22×Q8—1631no—
130C2×Q16—3261no—
131C2×SD16—3261no—
132#126—3261no—
133#124—3261no—
134#123—3261no—
135C22⋊Q8—1631no—
136#121—3261no—
137#121C2×D16311yescharacteristic
138#120—3261no—
139C4×D4—3261no—
140C2×C22⋊C4—3261no—
141C2×C22⋊C4—1631no—
142#103—3261no—
143#103—3261no—
144C22⋊C8—1631no—
145C22⋊C8C2×D16311yescharacteristic
146#95—3261no—
147#95C2×D16311yescharacteristic
148C24—1631no—
149C2×Q8—3261no—
150C2×Q8—3261no—
151C2×Q8—6522no—
152C2×Q8—1631no—
153C2×D4—21no—
154C2×D4—3261no—
155C2×D4—3261no—
156C2×D4D65211yescharacteristic
157C22×C4—21no—
158C22×C4—3261no—
159C22×C4—3261no—
160C22×C4—1631no—
161C22×C4C22×D16311yescharacteristic
162Q16—6522no—
163SD16—6522no—
164C2×C8—21no—
165C2×C8—3261no—
166C4⋊C4—3261no—
167C4⋊C4—3261no—
168C4⋊C4—1631no—
169C4⋊C4D65211yescharacteristic
170C22⋊C4—21no—
171C22⋊C4—3261no—
172C22⋊C4—3261no—
173C22⋊C4—3261no—
174C22⋊C4—3261no—
175C22⋊C4—3261no—
176C22⋊C4—3261no—
177C22⋊C4—6522no—
178C42—3261no—
179C23—21no—
180C23—3261no—
181C23—3261no—
182C23—3261no—
183C23—6521no—
184C23—6521no—
185C23D4×D16311yescharacteristic
186Q8—3262no—
187Q8—3262no—
188Q8—6521no—
189Q8—6521no—
190Q8—6521no—
191D4—41no—
192D4—41no—
193D4—41no—
194D4—13041no—
195D4—13041no—
196C2×C4—21no—
197C2×C4—41no—
198C2×C4—41no—
199C2×C4—41no—
200C2×C4—41no—
201C2×C4—13041no—
202C2×C4—3262no—
203C2×C4—3261no—
204C2×C4—3261no—
205C2×C4—3261no—
206C2×C4—6521no—
207C2×C4—6521no—
208C2×C4—6521no—
209C2×C4D4×D16311yescharacteristic
210C2×C4C2×D65211yescharacteristic
211C8—41no—
212C8—6521no—
213C22—21no—
214C22—21no—
215C22—41no—
216C22—41no—
217C22—41no—
218C22—13041no—
219C22—6522no—
220C22—6521no—
221C22—6521no—
222C22—6521no—
223C22#1910470511yescharacteristic
224C22#1910475411yescharacteristic
225C22#1910476411yescharacteristic
226C4—21no—
227C4—21no—
228C4—41no—
229C4—81no—
230C4—6521no—
231C4—6521no—
232C4—6522no—
233C4—6521no—
234C2—21no—
235C2—21no—
236C2—81no—
237C2—13041no—
238C2#2378182811yescharacteristic
239C2#2378184411yescharacteristic
240C2#2378185311yescharacteristic
subgroupG/H
centerC22#19104705
commutator, cosocleC22×C4×C163C23
FrattiniC22×C4C22×D163
Fitting#327×C163C2
socleC22×C163#120
prime radical#23782252C2
hypercenter#327D163
hypocenterC163#1606
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. #23781823C2semidirect
3. #23781803C2semidirect
4. #23781578C2semidirect
5. #23781401C2semidirect
6. #23781388C2semidirect
7. #327×C163C2semidirect
120. C163#1606semidirect
127. #327D163semidirect

Maximal subgroups

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

subgroupindex
#237822522
#237818232
#237818032
#237815782
#237814012
#237813882
#327×C1632
#1606163