Finite Group Databasebeta

← #27170061

Subgroups — #27170061

Order 12928.

272 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 272 conjugacy classes of subgroups — too many to draw. Only its 53 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#20265327C211yesmaximal, characteristic, complemented
2#20265310C211yesmaximal, characteristic
3#20265094C211yesmaximal, characteristic, complemented
4#20264914C211yesmaximal, characteristic, complemented
5#20264911C211yesmaximal, characteristic, complemented
6#20264903C211yesmaximal, characteristic, complemented
7#539×C101C211yesmaximal, characteristic, complemented
8C22×D404C2211yescharacteristic
9C2×#13108061—21no—
10C2×#13108059—21no—
11C2×#13108051—21no—
12C2×#13108051—21no—
13#15692175—21no—
14#15692175—21no—
15C22⋊#11717579C2211yescharacteristic
16C22⋊#11717579C2211yescharacteristic
17#15692162C2211yescharacteristic
18#126×C101—21no—
19#125×C101—21no—
20#123×C101—21no—
21#123×C101—21no—
22C22⋊Q8×C101—21no—
23C22⋊Q8×C101C2211yescharacteristic
24#121×C101C2211yescharacteristic
25C4×D4×C101—21no—
26C2×C4⋊C4×C101C2211yescharacteristic
27C23×D101—21no—
28C2×D404—21no—
29C2×D404—21no—
30C2×D404—42no—
31C22×Dic101—21no—
32#13108061—42no—
33#13108059—42no—
34C2×#11717579—21no—
35C2×#11717579—21no—
36#13108051—42no—
37#13108051—42no—
38C2×Q8×C101D411yescharacteristic
39C2×D4×C101—21no—
40C2×D4×C101D411yescharacteristic
41C22×C4×C101—21no—
42C22×C4×C101D411yescharacteristic
43C22×C4×C101D411yescharacteristic
44C22×C4×C101C2311yescharacteristic
45C4⋊C4×C101—21no—
46C4⋊C4×C101—21no—
47C4⋊C4×C101—21no—
48C4⋊C4×C101—21no—
49C4⋊C4×C101—21no—
50C4⋊C4×C101—21no—
51C4⋊C4×C101D411yescharacteristic
52C4⋊C4×C101D411yescharacteristic
53C22⋊C4×C101—21no—
54C22⋊C4×C101—21no—
55C22⋊C4×C101—21no—
56C22⋊C4×C101—21no—
57C22⋊C4×C101—21no—
58C42×C101—21no—
59C23×C101—21no—
60C23×C101C2×D411yescharacteristic
61C22×D101—22no—
62C22×D101—41no—
63C22×D101—41no—
64C22×D101—41no—
65C22×D101—41no—
66C22×D101—41no—
67C22×D101—41no—
68Q8×C101—41no—
69D4×C101—41no—
70D4×C101—41no—
71D4×C101—41no—
72C2×C4×C101—21no—
73C2×C4×C101—21no—
74C2×C4×C101—21no—
75C2×C4×C101—21no—
76C2×C4×C101—21no—
77C2×C4×C101—21no—
78C2×C4×C101—21no—
79C2×C4×C101—41no—
80C2×C4×C101—41no—
81C2×C4×C101—81no—
82C2×C4×C101C2×D411yescharacteristic
83C2×C4×C101C2×D411yescharacteristic
84C2×Dic101—41no—
85C2×Dic101—41no—
86C2×Dic101—41no—
87D404—22no—
88D404—22no—
89D404—41no—
90D404—41no—
91#11717579—41no—
92#11717579—41no—
93C22×C101—21no—
94C22×C101—21no—
95C22×C101—41no—
96C22×C101—41no—
97C22×C101—41no—
98C22×C101#12011yescharacteristic
99C22×C101#13511yescharacteristic
100C22×C101AGL(1,ℤ/8)11yescharacteristic
101C2×D101—42no—
102C2×D101—42no—
103C2×D101—42no—
104C2×D101—81no—
105C2×D101—81no—
106C2×D101—81no—
107C2×D101—81no—
108C4×C101—21no—
109C4×C101—21no—
110C4×C101—41no—
111C4×C101—41no—
112C4×C101—41no—
113C4×C101—41no—
114C4×C101—41no—
115Dic101—81no—
116C2×C101—21no—
117C2×C101—21no—
118C2×C101—81no—
119C2×C101#44911yescharacteristic
120C2×C101#45411yescharacteristic
121C2×C101B(4,𝔽2)11yescharacteristic
122D101—82no—
123#1597—1011nomaximal, Sylow
124C101#159711yescharacteristic, complemented, Sylow
125#539D10111yescharacteristic, complemented
126#450—1011no—
127#447—1011no—
128#394—1011no—
129#328—1011no—
130#327—1011no—
131#324—1011no—
132C22×D4—1011no—
133C2×SD16—2021no—
134C2×D8—2021no—
135#126—21no—
136#125—21no—
137#123—21no—
138#123—21no—
139C22⋊Q8—21no—
140C22⋊Q8C2×D10111yescharacteristic
141#121C2×D10111yescharacteristic
142C4×D4—21no—
143C2×C4⋊C4C2×D10111yescharacteristic
144C2×C22⋊C4—2021no—
145C2×C22⋊C4—2021no—
146#102—2021no—
147#102—2021no—
148C22⋊C8—1011no—
149C22⋊C8—1011no—
150#95—1011no—
151C24—2021no—
152C2×Q8C22⋊D10111yescharacteristic
153C2×D4—21no—
154C2×D4—4042no—
155C2×D4—2021no—
156C2×D4—2021no—
157C2×D4C22⋊D10111yescharacteristic
158C22×C4—21no—
159C22×C4—2021no—
160C22×C4C22⋊D10111yescharacteristic
161C22×C4C22⋊D10111yescharacteristic
162C22×C4C22×D10111yescharacteristic
163SD16—4042no—
164D8—4042no—
165C2×C8—2021no—
166C2×C8—2021no—
167C4⋊C4—21no—
168C4⋊C4—21no—
169C4⋊C4—21no—
170C4⋊C4—21no—
171C4⋊C4—21no—
172C4⋊C4—21no—
173C4⋊C4C22⋊D10111yescharacteristic
174C4⋊C4C22⋊D10111yescharacteristic
175C22⋊C4—21no—
176C22⋊C4—21no—
177C22⋊C4—21no—
178C22⋊C4—21no—
179C22⋊C4—21no—
180C22⋊C4—4042no—
181C22⋊C4—4042no—
182C42—21no—
183C23—21no—
184C23—4041no—
185C23—4041no—
186C23—4041no—
187C23—4041no—
188C23—4041no—
189C23—4041no—
190C23—2022no—
191C23C2×C22⋊D10111yescharacteristic
192Q8—41no—
193D4—41no—
194D4—41no—
195D4—41no—
196D4—4041no—
197D4—4041no—
198D4—2022no—
199D4—2022no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—21no—
207C2×C4—41no—
208C2×C4—41no—
209C2×C4—81no—
210C2×C4—4041no—
211C2×C4—4041no—
212C2×C4—4041no—
213C2×C4C2×C22⋊D10111yescharacteristic
214C2×C4C2×C22⋊D10111yescharacteristic
215C8—4041no—
216C8—4041no—
217C22—21no—
218C22—21no—
219C22—41no—
220C22—41no—
221C22—41no—
222C22—8081no—
223C22—8081no—
224C22—8081no—
225C22—8081no—
226C22—4042no—
227C22—4042no—
228C22—4042no—
229C22#1569222411yescharacteristic
230C22#1569227511yescharacteristic
231C22#1569228011yescharacteristic
232C4—21no—
233C4—21no—
234C4—41no—
235C4—41no—
236C4—41no—
237C4—41no—
238C4—41no—
239C4—8081no—
240C2—21no—
241C2—21no—
242C2—81no—
243C2—8082no—
244C2#2026532211yescharacteristic
245C2#2026535011yescharacteristic
246C2#2026536411yescharacteristic
subgroupG/H
centerC22#15692224
commutator, cosocleC22×C4×C101C23
FrattiniC22×C4C22×D101
Fitting#539×C101C2
socleC22×C101#120
prime radical#20265310C2
hypercenter#539D101
hypocenterC101#1597
LCSG ▷ C22×C4×C101 ▷ C22×C101 ▷ C101
UCSC1 ◁ C22 ◁ C22×C4 ◁ #539

Complements

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

NH ≅ G/Ntype
1. #20265327C2semidirect
3. #20265094C2semidirect
4. #20264914C2semidirect
5. #20264911C2semidirect
6. #20264903C2semidirect
7. #539×C101C2semidirect
124. C101#1597semidirect
125. #539D101semidirect

Maximal subgroups

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

subgroupindex
#202653272
#202653102
#202650942
#202649142
#202649112
#202649032
#539×C1012
#1597101