Finite Group Databasebeta

← C2×#20264964 #27172764

Subgroups — C2×#20264964 #27172764

Order 12928.

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

normal subgroups only

This group has 438 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×#15692172C211yesmaximal, characteristic, complemented
2C2×#15692170C211yesmaximal, characteristic, complemented
3#20264964C214yesmaximal, direct product factor
4C2×#120×C101C211yesmaximal, characteristic
5C2×#13108058—21no—
6C2×C22⋊Dic101—21no—
7#15692172—21no—
8#15692172C2212yescomplemented
9#15692170—21no—
10#15692170C2212yescomplemented
11C25×C101C411yescharacteristic, complemented
12C22×D4×C101—21no—
13C22×D4×C101C2211yescharacteristic
14#120×C101—21no—
15#120×C101—21no—
16#120×C101C412yescomplemented
17#120×C101C2212yes—
18C2×C22⋊C4×C101—21no—
19C2×C22⋊C4×C101C411yescharacteristic, complemented
20C22×Dic101—41no—
21#13108058—22no—
22#13108058—41no—
23C2×#11717579—41no—
24C22⋊Dic101—22no—
25C22⋊Dic101—41no—
26C24×C101—21no—
27C24×C101—21no—
28C24×C101—44no—
29C24×C101—41no—
30C24×C101—41no—
31C24×C101C2×C411yescharacteristic
32C24×C101C2×C412yescomplemented
33C24×C101D411yescharacteristic
34C2×D4×C101—22no—
35C2×D4×C101—22no—
36C2×D4×C101—41no—
37C2×D4×C101—41no—
38C2×D4×C101—44no—
39C2×D4×C101C2×C411yescharacteristic
40C2×D4×C101D411yescharacteristic
41C2×D4×C101D411yescharacteristic
42C2×D4×C101C2311yescharacteristic
43C22×C4×C101—21no—
44C22×C4×C101D411yescharacteristic
45C22⋊C4×C101—21no—
46C22⋊C4×C101—22no—
47C22⋊C4×C101—22no—
48C22⋊C4×C101C2×C412yescomplemented
49C23×C101—24no—
50C23×C101—21no—
51C23×C101—21no—
52C23×C101—21no—
53C23×C101—22no—
54C23×C101—21no—
55C23×C101—42no—
56C23×C101—44no—
57C23×C101—42no—
58C23×C101—42no—
59C23×C101—44no—
60C23×C101—42no—
61C23×C101—42no—
62C23×C101—44no—
63C23×C101—48no—
64C23×C101—44no—
65C23×C101—41no—
66C23×C101—41no—
67C23×C101C22⋊C411yescharacteristic
68C23×C101C22⋊C411yescharacteristic
69C23×C101C22⋊C411yescharacteristic
70C23×C101C22×C411yescharacteristic
71C23×C101C2×D411yescharacteristic
72D4×C101—41no—
73D4×C101—41no—
74D4×C101—41no—
75D4×C101—41no—
76D4×C101—44no—
77D4×C101—44no—
78C2×C4×C101—21no—
79C2×C4×C101—21no—
80C2×C4×C101—22no—
81C2×C4×C101—42no—
82C2×C4×C101C22⋊C411yescharacteristic
83C2×C4×C101C2×D411yescharacteristic
84C2×Dic101—42no—
85C2×Dic101—82no—
86#11717579—42no—
87C22×C101—22no—
88C22×C101—24no—
89C22×C101—21no—
90C22×C101—21no—
91C22×C101—21no—
92C22×C101—21no—
93C22×C101—41no—
94C22×C101—41no—
95C22×C101—41no—
96C22×C101—41no—
97C22×C101—42no—
98C22×C101—44no—
99C22×C101—42no—
100C22×C101—44no—
101C22×C101—44no—
102C22×C101—48no—
103C22×C101—42no—
104C22×C101—44no—
105C22×C101—42no—
106C22×C101—42no—
107C22×C101—81no—
108C22×C101#9911yescharacteristic
109C22×C101#9911yescharacteristic
110C22×C101C2×C22⋊C411yescharacteristic
111C4×C101—21no—
112C4×C101—21no—
113C4×C101—42no—
114Dic101—82no—
115C2×C101—21no—
116C2×C101—21no—
117C2×C101—41no—
118C2×C101—44no—
119C2×C101—41no—
120C2×C101—41no—
121C2×C101—41no—
122C2×C101AΣL(1,𝔽16)12yescomplemented
123C2×C101C2×#9911yescharacteristic
124C2×AΣL(1,𝔽16)—1011nomaximal, Sylow
125C101C2×AΣL(1,𝔽16)11yescharacteristic, complemented, Sylow
126C2×#120D10111yescharacteristic
127C2×#100—1011no—
128C2×#99—1011no—
129AΣL(1,𝔽16)—1014no—
130C25Dic10111yescharacteristic, complemented
131C22×D4—21no—
132C22×D4C2×D10111yescharacteristic
133C2×M16—2021no—
134#120—21no—
135#120—21no—
136#120Dic10112yescomplemented
137#120C2×D10112yes—
138C2×C22⋊C4—21no—
139C2×C22⋊C4—2021no—
140C2×C22⋊C4Dic10111yescharacteristic, complemented
141#100—1012no—
142#100—2021no—
143#99—1012no—
144#99—2021no—
145C24—21no—
146C24—21no—
147C24—44no—
148C24—41no—
149C24—41no—
150C24C2×Dic10111yescharacteristic
151C24C2×Dic10112yescomplemented
152C24C22⋊D10111yescharacteristic
153C2×D4—22no—
154C2×D4—22no—
155C2×D4—41no—
156C2×D4—41no—
157C2×D4—44no—
158C2×D4C2×Dic10111yescharacteristic
159C2×D4C22⋊D10111yescharacteristic
160C2×D4C22⋊D10111yescharacteristic
161C2×D4C22×D10111yescharacteristic
162C22×C4—21no—
163C22×C4—4041no—
164C22×C4C22⋊D10111yescharacteristic
165M16—4041no—
166M16—2022no—
167C2×C8—4041no—
168C22⋊C4—21no—
169C22⋊C4—22no—
170C22⋊C4—22no—
171C22⋊C4—4041no—
172C22⋊C4—2022no—
173C22⋊C4C2×Dic10112yescomplemented
174C23—24no—
175C23—21no—
176C23—21no—
177C23—21no—
178C23—22no—
179C23—21no—
180C23—42no—
181C23—44no—
182C23—42no—
183C23—42no—
184C23—44no—
185C23—42no—
186C23—42no—
187C23—44no—
188C23—48no—
189C23—44no—
190C23—41no—
191C23—41no—
192C23C22⋊Dic10111yescharacteristic
193C23C22⋊Dic10111yescharacteristic
194C23C22⋊Dic10111yescharacteristic
195C23C22×Dic10111yescharacteristic
196C23C2×C22⋊D10111yescharacteristic
197D4—41no—
198D4—41no—
199D4—41no—
200D4—41no—
201D4—44no—
202D4—44no—
203C2×C4—21no—
204C2×C4—21no—
205C2×C4—22no—
206C2×C4—42no—
207C2×C4—8082no—
208C2×C4—4042no—
209C2×C4C22⋊Dic10111yescharacteristic
210C2×C4C2×C22⋊D10111yescharacteristic
211C8—4042no—
212C22—22no—
213C22—24no—
214C22—21no—
215C22—21no—
216C22—21no—
217C22—21no—
218C22—41no—
219C22—41no—
220C22—41no—
221C22—41no—
222C22—42no—
223C22—44no—
224C22—42no—
225C22—44no—
226C22—44no—
227C22—48no—
228C22—42no—
229C22—44no—
230C22—42no—
231C22—42no—
232C22—81no—
233C22#1569217011yescharacteristic
234C22#1569217011yescharacteristic
235C22C2×C22⋊Dic10111yescharacteristic
236C4—21no—
237C4—21no—
238C4—42no—
239C4—8082no—
240C2—21no—
241C2—21no—
242C2—41no—
243C2—44no—
244C2—41no—
245C2—41no—
246C2—41no—
247C2#2026496412yesdirect product factor
248C2C2×#1569217011yescharacteristic
subgroupG/H
centerC22#15692170
commutatorC23×C101C22×C4
FrattiniC2×D4C22×D101
Fitting, prime radicalC2×#120×C101C2
socleC22×C101#99
cosocleC2×D4×C101C23
hypercenterC2×#120D101
hypocenterC101C2×AΣL(1,𝔽16)
LCSG ▷ C23×C101 ▷ C22×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C23 ◁ C22×D4 ◁ C2×#120

Complements

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

NH ≅ G/Ntype
1. C2×#15692172C2semidirect
2. C2×#15692170C2semidirect
3. #20264964C2direct (N × H)
8. #15692172C22semidirect
10. #15692170C22semidirect
11. C25×C101C4semidirect
16. #120×C101C4semidirect
19. C2×C22⋊C4×C101C4semidirect
32. C24×C101C2×C4semidirect
48. C22⋊C4×C101C2×C4semidirect
122. C2×C101AΣL(1,𝔽16)semidirect
125. C101C2×AΣL(1,𝔽16)semidirect
130. C25Dic101semidirect
136. #120Dic101semidirect
140. C2×C22⋊C4Dic101semidirect
151. C24C2×Dic101semidirect
173. C22⋊C4C2×Dic101semidirect
247. C2#20264964direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#156921722
C2×#156921702
#202649642
C2×#120×C1012
C2×AΣL(1,𝔽16)101