Finite Group Databasebeta

← #27169468

Subgroups — #27169468

Order 12928.

316 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 77 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
1C2×C4×C22⋊D101C211yesmaximal, characteristic
2#20265141C211yesmaximal, characteristic, complemented
3#20264902C212yesmaximal, complemented
4#20264900C212yesmaximal, complemented
5#407×C101C211yesmaximal, characteristic, complemented
6C22×C22⋊D101C411yescharacteristic
7C22×C4×D101C411yescharacteristic
8C2×#13108058—21no—
9C4×C22⋊D101—21no—
10C4×C22⋊D101—21no—
11C4×C22⋊D101—21no—
12C4×C22⋊D101—21no—
13C2×#13108053C2211yescharacteristic
14C2×C22⋊Dic101C411yescharacteristic
15C2×#13108051C2211yescharacteristic
16C2×C4×Dic101C411yescharacteristic
17C22⋊#11717579—21no—
18C22⋊#11717579C2212yes—
19C23×C4×C101C2211yescharacteristic
20C2×M16×C101—21no—
21C22⋊C8×C101—21no—
22C22⋊C8×C101C2212yes—
23C23×D101C2×C411yescharacteristic
24C2×C22⋊D101—21no—
25C2×C22⋊D101—21no—
26C2×C22⋊D101—22no—
27C2×C22⋊D101—22no—
28C22×Dic101C2×C411yescharacteristic
29C22×Dic101C2×C411yescharacteristic
30C22×Dic101C2×C411yescharacteristic
31C2×C4×D101—21no—
32C2×C4×D101—21no—
33C2×C4×D101—42no—
34#13108058—42no—
35C2×#11717579—22no—
36#13108053—21no—
37#13108053—21no—
38C22⋊Dic101—21no—
39C22⋊Dic101—21no—
40#13108051—21no—
41#13108051—21no—
42C4×Dic101—21no—
43C4×Dic101—21no—
44C24×C101C2×C411yescharacteristic
45C22×C4×C101—42no—
46C22×C4×C101C2×C411yescharacteristic
47C22×C4×C101D411yescharacteristic
48C22×C4×C101D411yescharacteristic
49C22×C4×C101D411yescharacteristic
50C22×C4×C101D411yescharacteristic
51C22×C4×C101C2311yescharacteristic
52M16×C101—42no—
53C2×C8×C101—22no—
54C23×C101—21no—
55C23×C101—21no—
56C23×C101—41no—
57C23×C101—41no—
58C23×C101C22⋊C411yescharacteristic
59C23×C101C22⋊C411yescharacteristic
60C23×C101C22×C411yescharacteristic
61C22×D101—21no—
62C22×D101—21no—
63C22×D101—41no—
64C22×D101—41no—
65C22×D101M1612yescomplemented
66C2×C4×C101—22no—
67C2×C4×C101—22no—
68C2×C4×C101—22no—
69C2×C4×C101—41no—
70C2×C4×C101—41no—
71C2×C4×C101—41no—
72C2×C4×C101C22⋊C411yescharacteristic
73C2×C4×C101C22⋊C411yescharacteristic
74C2×C4×C101C2×D411yescharacteristic
75C2×C4×C101C2×D411yescharacteristic
76C22⋊D101—24no—
77C22⋊D101—24no—
78C2×Dic101—21no—
79C2×Dic101—21no—
80C2×Dic101—21no—
81C2×Dic101—21no—
82C2×Dic101—41no—
83C2×Dic101—41no—
84C2×Dic101M1612yescomplemented
85C4×D101—42no—
86C4×D101—42no—
87C8×C101—42no—
88#11717579—42no—
89C22×C101—21no—
90C22×C101—21no—
91C22×C101—21no—
92C22×C101—21no—
93C22×C101—21no—
94C22×C101—21no—
95C22×C101—21no—
96C22×C101—21no—
97C22×C101—42no—
98C22×C101—42no—
99C22×C101C2×C22⋊C411yescharacteristic
100C22×C101C2×M1611yescharacteristic
101C22×C101C2×M1611yescharacteristic
102C2×D101—22no—
103C2×D101—42no—
104C2×D101—41no—
105C2×D101—41no—
106C2×D101—42no—
107C4×C101—21no—
108C4×C101—21no—
109C4×C101—21no—
110C4×C101#9912yes—
111Dic101—22no—
112Dic101—41no—
113Dic101—41no—
114C2×C101—21no—
115C2×C101—21no—
116C2×C101—41no—
117C2×C101—41no—
118C2×C101#40711yescharacteristic
119C2×C101C2×#9911yescharacteristic
120C2×C101#41311yescharacteristic
121D101—42no—
122#1465—1011nomaximal, Sylow
123C101#146511yescharacteristic, complemented, Sylow
124C2×C4×D4—1011no—
125#407—1011no—
126#407D10111yescharacteristic, complemented
127#324—1012no—
128#323—1012no—
129C22×D4—1011no—
130C23×C4—1011no—
131C23×C4C2×D10111yescharacteristic
132C2×M16—21no—
133C2×M16—2021no—
134C4×D4—2021no—
135C4×D4—2021no—
136C4×D4—2021no—
137C4×D4—2021no—
138C2×C4⋊C4—1011no—
139C2×C22⋊C4—1011no—
140C2×C22⋊C4—1011no—
141C2×C42—1011no—
142C22⋊C8—21no—
143C22⋊C8—1012no—
144C22⋊C8—2021no—
145C22⋊C8C2×D10112yes—
146C24—1011no—
147C24C4×D10111yescharacteristic
148C2×D4—2022no—
149C2×D4—2022no—
150C2×D4—2021no—
151C2×D4—2021no—
152C22×C4—42no—
153C22×C4—1011no—
154C22×C4—1011no—
155C22×C4—1011no—
156C22×C4—4042no—
157C22×C4—2021no—
158C22×C4—2021no—
159C22×C4C4×D10111yescharacteristic
160C22×C4D40411yescharacteristic
161C22×C4D40411yescharacteristic
162C22×C4C22⋊D10111yescharacteristic
163C22×C4C22⋊D10111yescharacteristic
164C22×C4C22×D10111yescharacteristic
165M16—42no—
166M16—4042no—
167C2×C8—22no—
168C2×C8—2022no—
169C4⋊C4—2021no—
170C4⋊C4—2021no—
171C22⋊C4—2021no—
172C22⋊C4—2021no—
173C22⋊C4—2021no—
174C22⋊C4—2021no—
175C42—2021no—
176C42—2021no—
177C23—21no—
178C23—21no—
179C23—41no—
180C23—41no—
181C23—1012no—
182C23—4041no—
183C23—4041no—
184C23—2021no—
185C23—2021no—
186C23#1310805111yescharacteristic
187C23#1310805111yescharacteristic
188C23C2×C4×D10111yescharacteristic
189D4—2024no—
190D4—2024no—
191C2×C4—22no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—41no—
195C2×C4—41no—
196C2×C4—41no—
197C2×C4—1012no—
198C2×C4—4042no—
199C2×C4—4042no—
200C2×C4—4041no—
201C2×C4—4041no—
202C2×C4—2021no—
203C2×C4—2021no—
204C2×C4—2021no—
205C2×C4—2021no—
206C2×C4#1310805111yescharacteristic
207C2×C4#1310805111yescharacteristic
208C2×C4C2×C22⋊D10111yescharacteristic
209C2×C4C2×D40411yescharacteristic
210C8—42no—
211C8—4042no—
212C22—21no—
213C22—21no—
214C22—21no—
215C22—21no—
216C22—21no—
217C22—21no—
218C22—21no—
219C22—21no—
220C22—42no—
221C22—42no—
222C22—4042no—
223C22—4042no—
224C22—4041no—
225C22—4041no—
226C22—2022no—
227C22C2×#1310805111yescharacteristic
228C22M16×D10111yescharacteristic
229C22M16×D10111yescharacteristic
230C4—21no—
231C4—21no—
232C4—21no—
233C4—4041no—
234C4—4041no—
235C4—2022no—
236C4#1569216912yes—
237C2—21no—
238C2—21no—
239C2—41no—
240C2—41no—
241C2—4042no—
242C2#2026514011yescharacteristic
243C2C2×#1569216911yescharacteristic
244C2#2026515911yescharacteristic
subgroupG/H
centerC22C2×#13108051
commutatorC23×C101C22×C4
FrattiniC22×C4C22×D101
Fitting#407×C101C2
socleC22×C101C2×C22⋊C4
prime radicalC22×C22⋊D101C4
cosocleC22×C4×C101C23
hypercenter#407D101
hypocenterC101#1465
LCSG ▷ C23×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C23×C4 ◁ #407

Complements

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

NH ≅ G/Ntype
2. #20265141C2semidirect
3. #20264902C2semidirect
4. #20264900C2semidirect
5. #407×C101C2semidirect
65. C22×D101M16semidirect
84. C2×Dic101M16semidirect
123. C101#1465semidirect
126. #407D101semidirect

Maximal subgroups

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

subgroupindex
C2×C4×C22⋊D1012
#202651412
#202649022
#202649002
#407×C1012
#1465101