Finite Group Databasebeta

← C2×#20266077 #27187377

Subgroups — C2×#20266077 #27187377

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#20266077C214yesmaximal, direct product factor
4C2×#15692224C211yesmaximal, characteristic
5C2×#13108083—21no—
6C2×C22⋊#87938—21no—
7#15692315—21no—
8#15692315C2212yescomplemented
9#15692313—21no—
10#15692313C2212yescomplemented
11C22×C22⋊D101—21no—
12C22×C22⋊D101C2211yescharacteristic
13#15692224—21no—
14#15692224—21no—
15#15692224C412yescomplemented
16#15692224C2212yes—
17C2×C22⋊Dic101—21no—
18C2×C22⋊Dic101C411yescharacteristic, complemented
19C25×C101C411yescharacteristic, complemented
20C22×#87938—41no—
21#13108083—22no—
22#13108083—41no—
23C2×#11717581—41no—
24C22⋊#87938—22no—
25C22⋊#87938—41no—
26C23×D101D411yescharacteristic
27C2×C22⋊D101—22no—
28C2×C22⋊D101—22no—
29C2×C22⋊D101—41no—
30C2×C22⋊D101—41no—
31C2×C22⋊D101—44no—
32C2×C22⋊D101C2×C411yescharacteristic
33C2×C22⋊D101D411yescharacteristic
34C2×C22⋊D101D411yescharacteristic
35C2×C22⋊D101C2311yescharacteristic
36C22×Dic101—21no—
37C22×Dic101D411yescharacteristic
38C22⋊Dic101—21no—
39C22⋊Dic101—22no—
40C22⋊Dic101—22no—
41C22⋊Dic101C2×C412yescomplemented
42C24×C101—21no—
43C24×C101—21no—
44C24×C101—44no—
45C24×C101—41no—
46C24×C101—41no—
47C24×C101C2×C411yescharacteristic
48C24×C101C2×C412yescomplemented
49C23×C101—24no—
50C23×C101—21no—
51C23×C101—21no—
52C23×C101—22no—
53C23×C101—42no—
54C23×C101—44no—
55C23×C101—42no—
56C23×C101—42no—
57C23×C101—44no—
58C23×C101—42no—
59C23×C101—44no—
60C23×C101—48no—
61C23×C101—44no—
62C23×C101—41no—
63C23×C101—41no—
64C23×C101C22⋊C411yescharacteristic
65C23×C101C22⋊C411yescharacteristic
66C23×C101C22×C411yescharacteristic
67C22×D101—21no—
68C22×D101—21no—
69C22×D101—42no—
70C22×D101C22⋊C411yescharacteristic
71C22×D101C2×D411yescharacteristic
72C2×#87938—42no—
73C2×#87938—82no—
74C22⋊D101—41no—
75C22⋊D101—41no—
76C22⋊D101—41no—
77C22⋊D101—41no—
78C22⋊D101—44no—
79C22⋊D101—44no—
80C2×Dic101—21no—
81C2×Dic101—21no—
82C2×Dic101—22no—
83C2×Dic101—42no—
84C2×Dic101C22⋊C411yescharacteristic
85C2×Dic101C2×D411yescharacteristic
86#11717581—42no—
87C22×C101—22no—
88C22×C101—24no—
89C22×C101—21no—
90C22×C101—21no—
91C22×C101—41no—
92C22×C101—41no—
93C22×C101—42no—
94C22×C101—44no—
95C22×C101—42no—
96C22×C101—44no—
97C22×C101—44no—
98C22×C101—48no—
99C22×C101—44no—
100C22×C101—42no—
101C22×C101—42no—
102C22×C101#9911yescharacteristic
103C22×C101#9911yescharacteristic
104C22×C101C2×C22⋊C411yescharacteristic
105C2×D101—21no—
106C2×D101—21no—
107C2×D101—42no—
108C2×D101—41no—
109C2×D101—41no—
110C2×D101—81no—
111#87938—82no—
112Dic101—21no—
113Dic101—21no—
114Dic101—42no—
115C2×C101—21no—
116C2×C101—21no—
117C2×C101—44no—
118C2×C101—41no—
119C2×C101—41no—
120C2×C101AΣL(1,𝔽16)12yescomplemented
121C2×C101C2×#9911yescharacteristic
122D101—41no—
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#8793811yescharacteristic, complemented
131C22×D4—1011no—
132C22×D4—2021no—
133C2×M16—2021no—
134#120—1012no—
135#120—1012no—
136#120—2021no—
137#120—2021no—
138C2×C22⋊C4—1011no—
139C2×C22⋊C4—2021no—
140C2×C22⋊C4—2021no—
141#100—1012no—
142#100—2021no—
143#99—1012no—
144#99—2021no—
145C24—21no—
146C24—21no—
147C24—44no—
148C24—41no—
149C24—41no—
150C24—1011no—
151C24C2×#8793811yescharacteristic
152C24C2×#8793812yescomplemented
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—1011no—
163C22×C4—4041no—
164C22×C4—2021no—
165M16—4041no—
166M16—2022no—
167C2×C8—4041no—
168C22⋊C4—1012no—
169C22⋊C4—4041no—
170C22⋊C4—2022no—
171C22⋊C4—2022no—
172C22⋊C4—2022no—
173C22⋊C4—2021no—
174C23—24no—
175C23—21no—
176C23—21no—
177C23—22no—
178C23—42no—
179C23—44no—
180C23—42no—
181C23—42no—
182C23—44no—
183C23—42no—
184C23—44no—
185C23—48no—
186C23—44no—
187C23—41no—
188C23—41no—
189C23—1011no—
190C23—1011no—
191C23—4042no—
192C23—2021no—
193C23—2021no—
194C23C22⋊#8793811yescharacteristic
195C23C22⋊#8793811yescharacteristic
196C23C22×#8793811yescharacteristic
197D4—4044no—
198D4—4044no—
199D4—4041no—
200D4—4041no—
201D4—4041no—
202D4—4041no—
203C2×C4—8082no—
204C2×C4—1011no—
205C2×C4—1011no—
206C2×C4—4042no—
207C2×C4—4042no—
208C2×C4—2022no—
209C2×C4—2021no—
210C2×C4—2021no—
211C8—4042no—
212C22—22no—
213C22—24no—
214C22—21no—
215C22—21no—
216C22—41no—
217C22—41no—
218C22—42no—
219C22—44no—
220C22—42no—
221C22—44no—
222C22—44no—
223C22—48no—
224C22—44no—
225C22—42no—
226C22—42no—
227C22—8081no—
228C22—4041no—
229C22—4041no—
230C22—4042no—
231C22—2021no—
232C22—2021no—
233C22#1569231311yescharacteristic
234C22#1569231311yescharacteristic
235C22C2×C22⋊#8793811yescharacteristic
236C4—8082no—
237C4—4042no—
238C4—2021no—
239C4—2021no—
240C2—21no—
241C2—21no—
242C2—44no—
243C2—41no—
244C2—41no—
245C2—4041no—
246C2—4041no—
247C2#2026607712yesdirect product factor
248C2C2×#1569231311yescharacteristic
subgroupG/H
centerC22#15692313
commutatorC23×C101C22×C4
FrattiniC23C22×#87938
FittingC25×C101C4
socleC22×C101#99
prime radicalC2×#15692224C2
cosocleC2×C22⋊D101C23
hypercenterC25#87938
hypocenterC101C2×AΣL(1,𝔽16)
LCSG ▷ C23×C101 ▷ C22×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C23 ◁ C24 ◁ C25

Complements

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

NH ≅ G/Ntype
1. C2×#15692315C2semidirect
2. C2×#15692313C2semidirect
3. #20266077C2direct (N × H)
8. #15692315C22semidirect
10. #15692313C22semidirect
15. #15692224C4semidirect
18. C2×C22⋊Dic101C4semidirect
19. C25×C101C4semidirect
41. C22⋊Dic101C2×C4semidirect
48. C24×C101C2×C4semidirect
120. C2×C101AΣL(1,𝔽16)semidirect
125. C101C2×AΣL(1,𝔽16)semidirect
130. C25#87938semidirect
152. C24C2×#87938semidirect
247. C2#20266077direct (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
#202660772
C2×#156922242
C2×AΣL(1,𝔽16)101