Finite Group Databasebeta

← #27186378

Subgroups — #27186378

Order 12928.

440 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 440 conjugacy classes of subgroups — too many to draw. Only its 68 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#20266148C212yesmaximal, complemented
2#20266021C212yesmaximal, complemented
3#20266020C212yesmaximal, complemented
4C2×C22⋊C4×D101C211yesmaximal, characteristic
5C2×#13108084—22no—
6#15692312—22no—
7C22⋊#11717581C2214yescomplemented
8C24×D101C411yescharacteristic
9C22×C4×D101C411yescharacteristic
10C22×C4×D101C2211yescharacteristic
11C2×C22⋊Dic101C2211yescharacteristic
12C2×#13108051C411yescharacteristic
13C2×#13108051C2211yescharacteristic
14C22⋊C4×D101—22no—
15C22⋊C4×D101—22no—
16C2×C22⋊C4×C101C411yescharacteristic
17#13108084—44no—
18C2×#11717581—24no—
19C23×D101—22no—
20C23×D101—21no—
21C23×D101—21no—
22C23×D101—22no—
23C23×D101—42no—
24C23×D101—42no—
25C23×D101C2×C411yescharacteristic
26C23×D101C2×C411yescharacteristic
27C22×Dic101C2×C411yescharacteristic
28C22×Dic101C2311yescharacteristic
29C2×C4×D101—21no—
30C2×C4×D101—21no—
31C2×C4×D101—42no—
32C2×C4×D101—42no—
33C2×C4×D101D412yes—
34C2×C4×D101D412yes—
35C22⋊Dic101—22no—
36#13108051—22no—
37#13108051—22no—
38C24×C101C2×C411yescharacteristic
39C22×C4×C101C2×C411yescharacteristic
40C22×C4×C101C2×C411yescharacteristic
41C22⋊C4×C101—22no—
42C23×C101—21no—
43C23×C101—21no—
44C23×C101—41no—
45C23×C101—41no—
46C23×C101M1612yescomplemented
47C23×C101C22×C411yescharacteristic
48C22×D101—21no—
49C22×D101—21no—
50C22×D101—22no—
51C22×D101—22no—
52C22×D101—24no—
53C22×D101—24no—
54C22×D101—21no—
55C22×D101—21no—
56C22×D101—48no—
57C22×D101—48no—
58C22×D101—42no—
59C22×D101—42no—
60C22×D101—42no—
61C22×D101—42no—
62C22×D101—41no—
63C22×D101—41no—
64C22×D101C22⋊C411yescharacteristic
65C22×D101C22⋊C411yescharacteristic
66C22×D101M1612yescomplemented
67C2×C4×C101—21no—
68C2×C4×C101—41no—
69C2×C4×C101—41no—
70C2×C4×C101C22⋊C412yes—
71C2×Dic101—21no—
72C2×Dic101—41no—
73C2×Dic101—41no—
74C2×Dic101C2×D412yes—
75C4×D101—24no—
76C4×D101—42no—
77C4×D101—42no—
78C4×D101—42no—
79#11717581—44no—
80C22×C101—21no—
81C22×C101—21no—
82C22×C101—22no—
83C22×C101—42no—
84C22×C101—42no—
85C22×C101—41no—
86C22×C101—41no—
87C22×C101C2×C22⋊C411yescharacteristic
88C22×C101C2×M1611yescharacteristic
89C22×C101C2×M1611yescharacteristic
90C2×D101—24no—
91C2×D101—24no—
92C2×D101—21no—
93C2×D101—21no—
94C2×D101—21no—
95C2×D101—21no—
96C2×D101—22no—
97C2×D101—48no—
98C2×D101—48no—
99C2×D101—42no—
100C2×D101—41no—
101C2×D101—41no—
102C2×D101—42no—
103C2×D101#9912yes—
104C2×D101#10012yes—
105C4×C101—22no—
106C4×C101—41no—
107Dic101—22no—
108Dic101—41no—
109C2×C101—21no—
110C2×C101—21no—
111C2×C101—42no—
112C2×C101#40711yescharacteristic
113C2×C101C2×#9911yescharacteristic
114C2×C101C2×#10011yescharacteristic
115D101—22no—
116D101—22no—
117D101—42no—
118#1462—1011nomaximal, Sylow
119C101#146211yescharacteristic, complemented, Sylow
120C22×C22⋊C4—1011no—
121#407—1012no—
122#323—1012no—
123#323—1012no—
124C25—1011no—
125C23×C4—1011no—
126C23×C4—1011no—
127C2×M16—2022no—
128C2×C22⋊C4—1011no—
129C2×C22⋊C4—1011no—
130C2×C22⋊C4—1011no—
131C2×C22⋊C4—2022no—
132C2×C22⋊C4—2022no—
133C2×C22⋊C4#8793811yescharacteristic
134C22⋊C8—1014no—
135C22⋊C8—2022no—
136C24—1011no—
137C24—1011no—
138C24—4042no—
139C24—4042no—
140C24—2021no—
141C24—2021no—
142C24—2022no—
143C24—2022no—
144C24C2×#8793811yescharacteristic
145C22×C4—1011no—
146C22×C4—1012no—
147C22×C4—1012no—
148C22×C4—1011no—
149C22×C4—4042no—
150C22×C4—4042no—
151C22×C4—2021no—
152C22×C4—2021no—
153C22×C4C2×#8793811yescharacteristic
154C22×C4C2×#8793811yescharacteristic
155M16—4044no—
156C2×C8—2024no—
157C22⋊C4—22no—
158C22⋊C4—2022no—
159C22⋊C4—2022no—
160C22⋊C4—2022no—
161C23—21no—
162C23—21no—
163C23—41no—
164C23—41no—
165C23—1011no—
166C23—1011no—
167C23—1012no—
168C23—4041no—
169C23—4041no—
170C23—4042no—
171C23—4042no—
172C23—4048no—
173C23—4048no—
174C23—4042no—
175C23—4042no—
176C23—2022no—
177C23—2022no—
178C23—2024no—
179C23—2024no—
180C23—2021no—
181C23—2021no—
182C23—2021no—
183C23—2021no—
184C23#1310808412yescomplemented
185C23C22×#8793811yescharacteristic
186C2×C4—21no—
187C2×C4—41no—
188C2×C4—41no—
189C2×C4—1012no—
190C2×C4—4042no—
191C2×C4—4042no—
192C2×C4—4042no—
193C2×C4—4041no—
194C2×C4—4041no—
195C2×C4—2024no—
196C2×C4—2021no—
197C2×C4C22⋊#8793812yes—
198C8—4044no—
199C22—21no—
200C22—21no—
201C22—22no—
202C22—42no—
203C22—42no—
204C22—41no—
205C22—41no—
206C22—1012no—
207C22—1012no—
208C22—4042no—
209C22—4042no—
210C22—4048no—
211C22—4048no—
212C22—4041no—
213C22—4041no—
214C22—2024no—
215C22—2024no—
216C22—2022no—
217C22—2021no—
218C22—2021no—
219C22—2021no—
220C22—2021no—
221C22C2×C22⋊#8793811yescharacteristic
222C22D101⋊M1611yescharacteristic
223C22C2×#1310808411yescharacteristic
224C4—22no—
225C4—41no—
226C4—4041no—
227C4—2022no—
228C2—21no—
229C2—21no—
230C2—42no—
231C2—4042no—
232C2—2022no—
233C2—2022no—
234C2#2026614811yescharacteristic
235C2D101⋊#9911yescharacteristic
236C2D101⋊#10011yescharacteristic
subgroupG/H
centerC22C2×C22⋊#87938
commutatorC23×C101C22×C4
FrattiniC23C22×#87938
FittingC2×C22⋊C4×C101C4
socleC22×C101C2×C22⋊C4
prime radicalC24×D101C4
cosocleC22×Dic101C23
hypercenterC2×C22⋊C4#87938
hypocenterC101#1462
LCSG ▷ C23×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C22⋊C4

Complements

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

NH ≅ G/Ntype
1. #20266148C2semidirect
2. #20266021C2semidirect
3. #20266020C2semidirect
7. C22⋊#11717581C22semidirect
46. C23×C101M16semidirect
66. C22×D101M16semidirect
119. C101#1462semidirect
184. C23#13108084semidirect

Maximal subgroups

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

subgroupindex
#202661482
#202660212
#202660202
C2×C22⋊C4×D1012
#1462101