Finite Group Databasebeta

← #27172771

Subgroups — #27172771

Order 12928.

264 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 264 conjugacy classes of subgroups — too many to draw. Only its 65 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#20265684C211yesmaximal, characteristic
2#20265161C211yesmaximal, characteristic, complemented
3#20264966C212yesmaximal, complemented
4#20264963C212yesmaximal, complemented
5#410×C101C211yesmaximal, characteristic, complemented
6C22×C4×D101C411yescharacteristic, complemented
7C2×#13108058—21no—
8#15692236C411yescharacteristic, complemented
9#15692236C2211yescharacteristic
10#15692231—21no—
11#15692226—21no—
12#15692223C411yescharacteristic, complemented
13#15692223C2211yescharacteristic
14C4×C22⋊D101—21no—
15C4×C22⋊D101—21no—
16#15692214C411yescharacteristic, complemented
17#15692174—21no—
18#15692172C2212yes—
19C2×#41×C101C2211yescharacteristic
20#117×C101—21no—
21#99×C101—21no—
22#99×C101C2212yes—
23C23×D101C2×C411yescharacteristic
24C2×C22⋊D101—21no—
25C2×C22⋊D101—21no—
26C22×Dic101C2×C411yescharacteristic
27C2×C4×D101—21no—
28C2×C4×D101—21no—
29C2×C4×D101—41no—
30C2×C4×D101—41no—
31#13108058—22no—
32#13108058—41no—
33C2×#11717579—41no—
34C4⋊Dic101—21no—
35#13108053—21no—
36#13108053—21no—
37C22⋊Dic101—21no—
38C22⋊Dic101—21no—
39C22⋊Dic101C2×C411yescharacteristic
40C22⋊Dic101C2×C411yescharacteristic
41#13108051—21no—
42#13108051—21no—
43C4×Dic101—21no—
44#41×C101—41no—
45#41×C101—41no—
46C2×Q8×C101D411yescharacteristic
47C2×D4×C101C2×C411yescharacteristic
48C2×D4×C101D411yescharacteristic
49C2×D4×C101C2311yescharacteristic
50C22×C4×C101C2×C411yescharacteristic
51C22×C4×C101D411yescharacteristic
52C22×C4×C101D411yescharacteristic
53C4⋊C4×C101—41no—
54C22⋊C4×C101—22no—
55C42×C101—41no—
56C23×C101C22⋊C411yescharacteristic
57C23×C101C22×C411yescharacteristic
58C23×C101C2×D411yescharacteristic
59C22×D101—21no—
60C22×D101—42no—
61C22×D101—41no—
62Q8×C101—41no—
63D4×C101—41no—
64D4×C101—41no—
65D4×C101—41no—
66C2×C4×C101—21no—
67C2×C4×C101—21no—
68C2×C4×C101—21no—
69C2×C4×C101—21no—
70C2×C4×C101—21no—
71C2×C4×C101—21no—
72C2×C4×C101—42no—
73C2×C4×C101C22⋊C411yescharacteristic
74C2×C4×C101C22⋊C411yescharacteristic
75C2×C4×C101C22⋊C411yescharacteristic
76C2×C4×C101C2×D411yescharacteristic
77C22⋊D101—42no—
78C22⋊D101—42no—
79C2×Dic101—21no—
80C2×Dic101—21no—
81C2×Dic101—21no—
82C2×Dic101—41no—
83C4×D101—42no—
84C4×D101—41no—
85C4×D101—41no—
86#11717579—42no—
87C22×C101—21no—
88C22×C101—21no—
89C22×C101—21no—
90C22×C101—41no—
91C22×C101—41no—
92C22×C101—41no—
93C22×C101C2×C22⋊C411yescharacteristic
94C2×D101—22no—
95C2×D101—41no—
96C2×D101—42no—
97C2×D101—41no—
98C2×D101—42no—
99C4×C101—21no—
100C4×C101—21no—
101C4×C101—21no—
102C4×C101—82no—
103C4×C101#9911yescharacteristic
104C4×C101#9911yescharacteristic
105Dic101—41no—
106Dic101—41no—
107Dic101—41no—
108C2×C101—21no—
109C2×C101—41no—
110C2×C101—41no—
111C2×C101—41no—
112C2×C101C2×#9911yescharacteristic
113D101—42no—
114#2120—1011nomaximal, Sylow
115C101#212011yescharacteristic, complemented, Sylow
116#525—1011no—
117#413—1011no—
118#410D10111yescharacteristic, complemented
119#352—1012no—
120AΣL(1,𝔽16)—1012no—
121C2×#41C2×D10111yescharacteristic
122C23×C4—1011no—
123C2×M16—2021no—
124#123—1011no—
125#123—1011no—
126C22⋊Q8—2021no—
127#121—2021no—
128#120—1011no—
129#120—1011no—
130C4×D4—2021no—
131C4×D4—2021no—
132#117—21no—
133#117—1011no—
134#101—2021no—
135#100—1012no—
136#99—21no—
137#99C2×D10112yes—
138C24—1011no—
139#41—41no—
140#41—41no—
141C2×Q8D40411yescharacteristic
142C2×D4—2021no—
143C2×D4—2021no—
144C2×D4C4×D10111yescharacteristic
145C2×D4C22⋊D10111yescharacteristic
146C2×D4C22×D10111yescharacteristic
147C22×C4—1011no—
148C22×C4—4041no—
149C22×C4—4041no—
150C22×C4—2021no—
151C22×C4—2021no—
152C22×C4C4×D10111yescharacteristic
153C22×C4D40411yescharacteristic
154C22×C4C22⋊D10111yescharacteristic
155M16—4041no—
156M16—2022no—
157C2×C8—4041no—
158C4⋊C4—41no—
159C4⋊C4—2021no—
160C4⋊C4—2021no—
161C4⋊C4—2021no—
162C22⋊C4—22no—
163C22⋊C4—1011no—
164C22⋊C4—1011no—
165C22⋊C4—2021no—
166C22⋊C4—2021no—
167C22⋊C4—2021no—
168C22⋊C4—2021no—
169C42—41no—
170C42—2021no—
171C23—4041no—
172C23—4042no—
173C23—2021no—
174C23#1310805111yescharacteristic
175C23C2×C4×D10111yescharacteristic
176C23C2×C22⋊D10111yescharacteristic
177Q8—41no—
178D4—41no—
179D4—41no—
180D4—41no—
181D4—4042no—
182D4—4042no—
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—21no—
186C2×C4—21no—
187C2×C4—21no—
188C2×C4—21no—
189C2×C4—42no—
190C2×C4—4042no—
191C2×C4—4041no—
192C2×C4—4041no—
193C2×C4—4041no—
194C2×C4—2021no—
195C2×C4—2021no—
196C2×C4—2021no—
197C2×C4#1310805111yescharacteristic
198C2×C4#1310805111yescharacteristic
199C2×C4#1310805111yescharacteristic
200C2×C4C2×D40411yescharacteristic
201C8—4042no—
202C22—21no—
203C22—21no—
204C22—21no—
205C22—41no—
206C22—41no—
207C22—41no—
208C22—4041no—
209C22—4042no—
210C22—4042no—
211C22—4041no—
212C22—2022no—
213C22C2×#1310805111yescharacteristic
214C4—21no—
215C4—21no—
216C4—21no—
217C4—82no—
218C4—4041no—
219C4—4041no—
220C4—4041no—
221C4#1569216911yescharacteristic
222C4#1569216911yescharacteristic
223C2—21no—
224C2—41no—
225C2—41no—
226C2—41no—
227C2—4042no—
228C2C2×#1569216911yescharacteristic
subgroupG/H
centerC4#15692169
commutatorC23×C101C22×C4
FrattiniC2×D4C22×D101
Fitting#410×C101C2
socleC2×C101C2×#99
prime radical#20265684C2
cosocleC2×D4×C101C23
hypercenter#410D101
hypocenterC101#2120
LCSG ▷ C23×C101 ▷ C22×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C4 ◁ C2×C4 ◁ C2×#41 ◁ #410

Complements

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

NH ≅ G/Ntype
2. #20265161C2semidirect
3. #20264966C2semidirect
4. #20264963C2semidirect
5. #410×C101C2semidirect
6. C22×C4×D101C4semidirect
8. #15692236C4semidirect
12. #15692223C4semidirect
16. #15692214C4semidirect
115. C101#2120semidirect
118. #410D101semidirect

Maximal subgroups

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

subgroupindex
#202656842
#202651612
#202649662
#202649632
#410×C1012
#2120101