Finite Group Databasebeta

← C2×#20265504 #27181019

Subgroups — C2×#20265504 #27181019

Order 12928.

468 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 468 conjugacy classes of subgroups — too many to draw. Only its 141 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×#15692230C211yesmaximal, characteristic
2C2×#15692213C211yesmaximal, characteristic, complemented
3#20265504C218yesmaximal, direct product factor
4C2×#15692186C211yesmaximal, characteristic, complemented
5C2×#15692177C211yesmaximal, characteristic, complemented
6C2×#15692175C211yesmaximal, characteristic, complemented
7C2×#15692167C211yesmaximal, characteristic, complemented
8C2×#107×C101C211yesmaximal, characteristic, complemented
9C22×D404—21no—
10C22×D404C2211yescharacteristic
11C22×C4×D101C2211yescharacteristic
12C22×#11717579C2211yescharacteristic
13#15692230—22no—
14#15692230C2214yes—
15#15692213—22no—
16#15692213C2214yescomplemented
17C2×C4⋊Dic101C2211yescharacteristic
18C2×#13108051—21no—
19C2×#13108051—21no—
20C2×C4×Dic101—21no—
21#15692186C2214yescomplemented
22#15692177C2214yescomplemented
23#15692175C2214yescomplemented
24#15692167C2214yescomplemented
25C22×C8×C101C2211yescharacteristic
26C2×C4⋊C4×C101C2211yescharacteristic
27C2×C4⋊C4×C101C2211yescharacteristic
28#107×C101C2214yescomplemented
29C23×D101—21no—
30C23×D101D411yescharacteristic
31C2×D404—22no—
32C2×D404—22no—
33C2×D404—24no—
34C2×D404—21no—
35C2×D404—44no—
36C2×D404C2312yes—
37C22×Dic101—21no—
38C22×Dic101D411yescharacteristic
39C2×C4×D101—22no—
40C2×C4×D101—22no—
41C2×C4×D101D412yes—
42C2×C4×D101C2312yes—
43C2×#11717579—22no—
44C2×#11717579C2312yes—
45C4⋊Dic101—21no—
46C4⋊Dic101C2312yes—
47#13108051—24no—
48#13108051—24no—
49C4×Dic101—24no—
50C22×C4×C101—21no—
51C22×C4×C101—21no—
52C22×C4×C101C2311yescharacteristic
53C2×C8×C101—22no—
54C2×C8×C101C2312yes—
55C4⋊C4×C101—21no—
56C4⋊C4×C101—21no—
57C4⋊C4×C101C2312yes—
58C4⋊C4×C101C2312yes—
59C23×C101C2×D411yescharacteristic
60C22×D101—22no—
61C22×D101—22no—
62C22×D101—22no—
63C22×D101—22no—
64C22×D101—42no—
65C22×D101—42no—
66C22×D101—42no—
67C22×D101C2×D412yes—
68C2×C4×C101—22no—
69C2×C4×C101—22no—
70C2×C4×C101—42no—
71C2×C4×C101—42no—
72C2×C4×C101C2×D411yescharacteristic
73C2×C4×C101#4112yes—
74C2×C4×C101#4112yes—
75C2×C4×C101C2411yescharacteristic
76C2×Dic101—22no—
77C2×Dic101—22no—
78C2×Dic101—42no—
79C2×Dic101C2×D412yes—
80D404—22no—
81D404—22no—
82D404—48no—
83D404—42no—
84C4×D101—24no—
85C4×D101—22no—
86C4×D101—22no—
87C8×C101—22no—
88#11717579—22no—
89C22×C101#12312yes—
90C22×C101#13512yes—
91C22×C101AGL(1,ℤ/8)12yes—
92C22×C101C22×D411yescharacteristic
93C2×D101—22no—
94C2×D101—22no—
95C2×D101—22no—
96C2×D101—44no—
97C2×D101—42no—
98C2×D101—42no—
99C2×D101—42no—
100C2×D101—84no—
101C4×C101—42no—
102C4×C101—42no—
103C4×C101#12312yes—
104C4×C101C2×#4111yescharacteristic
105C4×C101C2×#4111yescharacteristic
106Dic101—22no—
107Dic101—42no—
108C2×C101#48214yescomplemented
109C2×C101C2×#12311yescharacteristic
110C2×C101C2×#13511yescharacteristic
111C2×C101C2×AGL(1,ℤ/8)11yescharacteristic
112D101—42no—
113D101—82no—
114C2×#482—1011nomaximal, Sylow
115C101C2×#48211yescharacteristic, complemented, Sylow
116C2×#121—1011no—
117C2×#117—1011no—
118#482—1018no—
119C2×#107D10111yescharacteristic, complemented
120C2×#106—1011no—
121C2×#102—1011no—
122C2×#102—1011no—
123C2×C22⋊C8—1011no—
124C22×D4—1011no—
125C22×D4—2021no—
126C23×C4—1011no—
127C22×C8—1011no—
128C22×C8C2×D10111yescharacteristic
129#121—1014no—
130#121—2022no—
131#117—1014no—
132#117—2022no—
133C2×C4⋊C4—1011no—
134C2×C4⋊C4C2×D10111yescharacteristic
135C2×C4⋊C4C2×D10111yescharacteristic
136C2×C22⋊C4—2021no—
137C2×C22⋊C4—2021no—
138C2×C42—2021no—
139#107C2×D10114yescomplemented
140#106—1014no—
141#102—1014no—
142#102—1014no—
143C22⋊C8—1014no—
144C24—1011no—
145C24—2021no—
146C2×D4—1012no—
147C2×D4—4044no—
148C2×D4—2024no—
149C2×D4—2021no—
150C2×D4—2022no—
151C2×D4—2022no—
152C22×C4—21no—
153C22×C4—21no—
154C22×C4—1011no—
155C22×C4—1012no—
156C22×C4—1012no—
157C22×C4—2021no—
158C22×C4—2022no—
159C22×C4—2022no—
160C22×C4C22×D10111yescharacteristic
161C2×C8—22no—
162C2×C8—1012no—
163C2×C8—2022no—
164C2×C8C22×D10112yes—
165C4⋊C4—21no—
166C4⋊C4—21no—
167C4⋊C4—1012no—
168C4⋊C4—2021no—
169C4⋊C4C22×D10112yes—
170C4⋊C4C22×D10112yes—
171C22⋊C4—2024no—
172C22⋊C4—2024no—
173C42—2024no—
174C23—1012no—
175C23—4042no—
176C23—4042no—
177C23—4042no—
178C23—2022no—
179C23—2022no—
180C23—2022no—
181C23—2022no—
182C23D4×D10111yescharacteristic
183D4—4048no—
184D4—4042no—
185D4—2022no—
186D4—2022no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—42no—
190C2×C4—42no—
191C2×C4—1012no—
192C2×C4—4042no—
193C2×C4—2022no—
194C2×C4—2022no—
195C2×C4—2022no—
196C2×C4—2022no—
197C2×C4—2024no—
198C2×C4D4×D10111yescharacteristic
199C2×C4#1310807412yes—
200C2×C4#1310807512yes—
201C2×C4C23×D10111yescharacteristic
202C8—22no—
203C8—2022no—
204C22—8084no—
205C22—4042no—
206C22—4042no—
207C22—4042no—
208C22—4044no—
209C22—2022no—
210C22—2022no—
211C22—2022no—
212C22#1569224012yes—
213C22#1569227812yes—
214C22#1569227912yes—
215C22C2×D4×D10111yescharacteristic
216C4—42no—
217C4—42no—
218C4—4042no—
219C4—2022no—
220C4#1569224012yes—
221C4C2×#1310807411yescharacteristic
222C4C2×#1310807511yescharacteristic
223C2—8082no—
224C2—4042no—
225C2#2026550414yesdirect product factor
226C2C2×#1569224011yescharacteristic
227C2C2×#1569227811yescharacteristic
228C2C2×#1569227911yescharacteristic
subgroupG/H
centerC23D4×D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
FittingC2×#107×C101C2
socleC23×C101C2×D4
prime radicalC2×#15692230C2
hypercenterC2×#107D101
hypocenterC101C2×#482
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C23 ◁ C22×C4 ◁ C2×#107

Complements

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

NH ≅ G/Ntype
2. C2×#15692213C2semidirect
3. #20265504C2direct (N × H)
4. C2×#15692186C2semidirect
5. C2×#15692177C2semidirect
6. C2×#15692175C2semidirect
7. C2×#15692167C2semidirect
8. C2×#107×C101C2semidirect
16. #15692213C22semidirect
21. #15692186C22semidirect
22. #15692177C22semidirect
23. #15692175C22semidirect
24. #15692167C22semidirect
28. #107×C101C22semidirect
108. C2×C101#482semidirect
115. C101C2×#482semidirect
119. C2×#107D101semidirect
139. #107C2×D101semidirect
225. C2#20265504direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#156922302
C2×#156922132
#202655042
C2×#156921862
C2×#156921772
C2×#156921752
C2×#156921672
C2×#107×C1012
C2×#482101