Finite Group Databasebeta

← C2×#20266078 #27187379

Subgroups — C2×#20266078 #27187379

Order 12928.

438 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 438 conjugacy classes of subgroups — too many to draw. Only its 58 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×#15692315C211yesmaximal, characteristic, complemented
2C2×#15692313C211yesmaximal, characteristic, complemented
3#20266078C214yesmaximal, direct product factor
4C2×#15692225C211yesmaximal, characteristic
5C2×#13108083—21no—
6C2×C22⋊#87938—21no—
7#15692315—21no—
8#15692315C2212yescomplemented
9#15692313—21no—
10#15692313C2212yescomplemented
11C24×D101C411yescharacteristic, complemented
12C22×D404—21no—
13C22×C22⋊D101C2211yescharacteristic
14#15692225—21no—
15#15692225—21no—
16#15692225C412yescomplemented
17#15692225C2212yes—
18C2×#13108051—21no—
19C2×C22⋊C4×C101C411yescharacteristic, complemented
20C22×#87938—41no—
21#13108083—22no—
22#13108083—41no—
23C2×#11717581—41no—
24C22⋊#87938—22no—
25C22⋊#87938—41no—
26C23×D101—21no—
27C23×D101—21no—
28C23×D101—44no—
29C23×D101—41no—
30C23×D101—41no—
31C23×D101C2×C412yescomplemented
32C23×D101D411yescharacteristic
33C2×D404—22no—
34C2×D404—22no—
35C2×D404—44no—
36C2×C22⋊D101—41no—
37C2×C22⋊D101—41no—
38C2×C22⋊D101C2×C411yescharacteristic
39C2×C22⋊D101D411yescharacteristic
40C2×C22⋊D101D411yescharacteristic
41C2×C22⋊D101C2311yescharacteristic
42C22×Dic101D411yescharacteristic
43#13108051—22no—
44#13108051—22no—
45C24×C101C2×C411yescharacteristic
46C22×C4×C101—21no—
47C22⋊C4×C101—21no—
48C22⋊C4×C101C2×C412yescomplemented
49C23×C101—21no—
50C23×C101—21no—
51C23×C101—42no—
52C23×C101C22⋊C411yescharacteristic
53C23×C101C22⋊C411yescharacteristic
54C23×C101C22×C411yescharacteristic
55C22×D101—24no—
56C22×D101—21no—
57C22×D101—22no—
58C22×D101—21no—
59C22×D101—44no—
60C22×D101—44no—
61C22×D101—42no—
62C22×D101—42no—
63C22×D101—48no—
64C22×D101—44no—
65C22×D101—41no—
66C22×D101—41no—
67C22×D101—42no—
68C22×D101—44no—
69C22×D101—42no—
70C22×D101C22⋊C411yescharacteristic
71C22×D101C2×D411yescharacteristic
72C2×#87938—42no—
73C2×#87938—82no—
74C2×C4×C101—22no—
75C2×C4×C101—42no—
76C22⋊D101—41no—
77C22⋊D101—41no—
78C22⋊D101—41no—
79C22⋊D101—41no—
80C2×Dic101—21no—
81C2×Dic101—21no—
82C2×Dic101C22⋊C411yescharacteristic
83C2×Dic101C2×D411yescharacteristic
84D404—44no—
85D404—44no—
86#11717581—42no—
87C22×C101—22no—
88C22×C101—21no—
89C22×C101—21no—
90C22×C101—41no—
91C22×C101—41no—
92C22×C101—42no—
93C22×C101—42no—
94C22×C101#9911yescharacteristic
95C22×C101#9911yescharacteristic
96C22×C101C2×C22⋊C411yescharacteristic
97C2×D101—24no—
98C2×D101—21no—
99C2×D101—21no—
100C2×D101—42no—
101C2×D101—44no—
102C2×D101—48no—
103C2×D101—44no—
104C2×D101—42no—
105C2×D101—44no—
106C2×D101—42no—
107C2×D101—41no—
108C2×D101—41no—
109C2×D101—44no—
110C2×D101—81no—
111#87938—82no—
112C4×C101—42no—
113Dic101—21no—
114Dic101—21no—
115C2×C101—21no—
116C2×C101—21no—
117C2×C101—41no—
118C2×C101—41no—
119C2×C101AΣL(1,𝔽16)12yescomplemented
120C2×C101C2×#9911yescharacteristic
121D101—41no—
122D101—44no—
123D101—41no—
124C2×AΣL(1,𝔽16)—1011nomaximal, Sylow
125C101C2×AΣL(1,𝔽16)11yescharacteristic, complemented, Sylow
126C2×#120—1011no—
127C2×#100—1011no—
128C2×#99—1011no—
129AΣL(1,𝔽16)—1014no—
130C25—1011no—
131C22×D4—1011no—
132C22×D4—2021no—
133C2×M16—2021no—
134#120—1012no—
135#120—1012no—
136#120—2021no—
137#120—2021no—
138C2×C22⋊C4—2021no—
139C2×C22⋊C4—2021no—
140C2×C22⋊C4#8793811yescharacteristic, complemented
141#100—1012no—
142#100—2021no—
143#99—1012no—
144#99—2021no—
145C24—1011no—
146C24—1012no—
147C24—4041no—
148C24—4041no—
149C24—4044no—
150C24—2021no—
151C24—2021no—
152C24C2×#8793811yescharacteristic
153C2×D4—1011no—
154C2×D4—1011no—
155C2×D4—1011no—
156C2×D4—1011no—
157C2×D4—4044no—
158C2×D4—4041no—
159C2×D4—4041no—
160C2×D4—2022no—
161C2×D4—2022no—
162C22×C4—21no—
163C22×C4—1011no—
164C22×C4—4041no—
165M16—4041no—
166M16—2022no—
167C2×C8—4041no—
168C22⋊C4—21no—
169C22⋊C4—4041no—
170C22⋊C4—2022no—
171C22⋊C4—2022no—
172C22⋊C4—2022no—
173C22⋊C4C2×#8793812yescomplemented
174C23—21no—
175C23—21no—
176C23—42no—
177C23—1011no—
178C23—1011no—
179C23—4042no—
180C23—4042no—
181C23—4042no—
182C23—4042no—
183C23—4044no—
184C23—4048no—
185C23—4044no—
186C23—4044no—
187C23—4044no—
188C23—4041no—
189C23—4041no—
190C23—2024no—
191C23—2021no—
192C23—2021no—
193C23—2022no—
194C23C22⋊#8793811yescharacteristic
195C23C22⋊#8793811yescharacteristic
196C23C22×#8793811yescharacteristic
197D4—4044no—
198D4—4044no—
199D4—4041no—
200D4—4041no—
201D4—4041no—
202D4—4041no—
203C2×C4—22no—
204C2×C4—42no—
205C2×C4—8082no—
206C2×C4—1011no—
207C2×C4—1011no—
208C2×C4—4042no—
209C2×C4—2021no—
210C2×C4—2021no—
211C8—4042no—
212C22—22no—
213C22—21no—
214C22—21no—
215C22—41no—
216C22—41no—
217C22—42no—
218C22—42no—
219C22—8081no—
220C22—4041no—
221C22—4041no—
222C22—4042no—
223C22—4044no—
224C22—4044no—
225C22—4044no—
226C22—4048no—
227C22—4042no—
228C22—4044no—
229C22—4042no—
230C22—2024no—
231C22—2021no—
232C22—2021no—
233C22#1569231411yescharacteristic
234C22#1569231411yescharacteristic
235C22C2×C22⋊#8793811yescharacteristic
236C4—42no—
237C4—8082no—
238C4—2021no—
239C4—2021no—
240C2—21no—
241C2—21no—
242C2—41no—
243C2—41no—
244C2—4041no—
245C2—4041no—
246C2—4044no—
247C2#2026607812yesdirect product factor
248C2C2×#1569231411yescharacteristic
subgroupG/H
centerC22#15692314
commutatorC23×C101C22×C4
FrattiniC23C22×#87938
FittingC2×C22⋊C4×C101C4
socleC22×C101#99
prime radicalC2×#15692225C2
cosocleC2×C22⋊D101C23
hypercenterC2×C22⋊C4#87938
hypocenterC101C2×AΣL(1,𝔽16)
LCSG ▷ C23×C101 ▷ C22×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C23 ◁ C24 ◁ C2×C22⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×#15692315C2semidirect
2. C2×#15692313C2semidirect
3. #20266078C2direct (N × H)
8. #15692315C22semidirect
10. #15692313C22semidirect
11. C24×D101C4semidirect
16. #15692225C4semidirect
19. C2×C22⋊C4×C101C4semidirect
31. C23×D101C2×C4semidirect
48. C22⋊C4×C101C2×C4semidirect
119. C2×C101AΣL(1,𝔽16)semidirect
125. C101C2×AΣL(1,𝔽16)semidirect
140. C2×C22⋊C4#87938semidirect
173. C22⋊C4C2×#87938semidirect
247. C2#20266078direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#156923152
C2×#156923132
#202660782
C2×#156922252
C2×AΣL(1,𝔽16)101