Finite Group Databasebeta

← C2×#20265420 #27180717

Subgroups — C2×#20265420 #27180717

Order 12928.

544 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 544 conjugacy classes of subgroups — too many to draw. Only its 157 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
1C22×#13108057C211yesmaximal, characteristic, complemented
2C2×#15692231C211yesmaximal, characteristic, complemented
3C2×#15692226C211yesmaximal, characteristic
4#20265420C218yesmaximal, direct product factor
5C2×#15692189C211yesmaximal, characteristic, complemented
6C2×#15692179C211yesmaximal, characteristic, complemented
7C2×#15692176C211yesmaximal, characteristic, complemented
8C22×SD16×C101C211yesmaximal, characteristic, complemented
9C22×C22⋊D101—21no—
10C22×C4×D101C2211yescharacteristic
11C2×#13108057—24no—
12C2×#13108057C2214yescomplemented
13C22×#11717579C2211yescharacteristic
14#15692231—22no—
15#15692231C2214yescomplemented
16#15692226—22no—
17#15692226C2214yes—
18C2×C4⋊Dic101C2211yescharacteristic
19C2×C4⋊Dic101C2211yescharacteristic
20C2×#13108053—21no—
21C2×C22⋊Dic101—21no—
22C2×#13108051—21no—
23#15692189C2214yescomplemented
24#15692179C2214yescomplemented
25#15692176C2214yescomplemented
26C22×Q8×C101C2211yescharacteristic
27C22×D4×C101C2211yescharacteristic
28C2×SD16×C101—24no—
29C2×SD16×C101C2214yescomplemented
30C22×C8×C101C2211yescharacteristic
31C23×D101D411yescharacteristic
32C2×C22⋊D101—24no—
33C2×C22⋊D101—44no—
34C22×Dic101—21no—
35C22×Dic101—21no—
36C22×Dic101D411yescharacteristic
37C2×C4×D101—22no—
38C2×C4×D101—22no—
39C2×C4×D101D412yes—
40C2×C4×D101C2312yes—
41#13108057—28no—
42C2×#11717579—22no—
43C2×#11717579C2312yes—
44C4⋊Dic101—21no—
45C4⋊Dic101—21no—
46C4⋊Dic101C2312yes—
47C4⋊Dic101C2312yes—
48#13108053—24no—
49C22⋊Dic101—24no—
50#13108051—24no—
51C24×C101—21no—
52C2×Q8×C101—21no—
53C2×Q8×C101—22no—
54C2×Q8×C101—22no—
55C2×Q8×C101C2312yes—
56C2×D4×C101—22no—
57C2×D4×C101—21no—
58C2×D4×C101—22no—
59C2×D4×C101C2312yes—
60C22×C4×C101—21no—
61C22×C4×C101C2311yescharacteristic
62SD16×C101—28no—
63C2×C8×C101D414yescomplemented
64C2×C8×C101C2312yes—
65C23×C101—22no—
66C23×C101—42no—
67C23×C101—42no—
68C23×C101—42no—
69C23×C101C2×D411yescharacteristic
70C22×D101—22no—
71C22×D101—22no—
72C22×D101—22no—
73C22×D101C2×D412yes—
74Q8×C101—22no—
75Q8×C101—22no—
76Q8×C101—42no—
77D4×C101—22no—
78D4×C101—22no—
79D4×C101—42no—
80C2×C4×C101—22no—
81C2×C4×C101—42no—
82C2×C4×C101C2×D411yescharacteristic
83C2×C4×C101C2×D412yes—
84C2×C4×C101#4112yes—
85C2×C4×C101C2411yescharacteristic
86C22⋊D101—48no—
87C2×Dic101—22no—
88C2×Dic101—22no—
89C2×Dic101—22no—
90C2×Dic101—42no—
91C2×Dic101—42no—
92C2×Dic101C2×D412yes—
93C4×D101—24no—
94C4×D101—22no—
95C4×D101—22no—
96C8×C101C2×D414yescomplemented
97#11717579—22no—
98C22×C101—42no—
99C22×C101—42no—
100C22×C101—42no—
101C22×C101—84no—
102C22×C101#12112yes—
103C22×C101AGL(1,ℤ/8)12yes—
104C22×C101#13712yes—
105C22×C101C22×D411yescharacteristic
106C2×D101—22no—
107C2×D101—22no—
108C2×D101—22no—
109C2×D101—44no—
110C4×C101—42no—
111C4×C101#12112yes—
112C4×C101C22×D411yescharacteristic
113C4×C101C2×#4111yescharacteristic
114Dic101—22no—
115Dic101—42no—
116Dic101—42no—
117C2×C101—82no—
118C2×C101#46814yescomplemented
119C2×C101C2×#12111yescharacteristic
120C2×C101C2×AGL(1,ℤ/8)11yescharacteristic
121C2×C101C2×#13711yescharacteristic
122D101—42no—
123C2×#468—1011nomaximal, Sylow
124C101C2×#46811yescharacteristic, complemented, Sylow
125C22×SD16D10111yescharacteristic, complemented
126C22×M16—1011no—
127C2×C22⋊Q8—1011no—
128C2×#121—1011no—
129#468—1018no—
130C2×#107—1011no—
131C2×#103—1011no—
132C2×#102—1011no—
133C22×Q8C2×D10111yescharacteristic
134C22×D4—2021no—
135C22×D4C2×D10111yescharacteristic
136C23×C4—1011no—
137C2×SD16—24no—
138C2×SD16C2×D10114yescomplemented
139C2×M16—1014no—
140C2×M16—2024no—
141C22×C8—1011no—
142C22×C8C2×D10111yescharacteristic
143C22⋊Q8—1014no—
144C22⋊Q8—2022no—
145#121—1014no—
146#121—2022no—
147C2×C4⋊C4—1011no—
148C2×C4⋊C4—1011no—
149C2×C4⋊C4—2021no—
150C2×C22⋊C4—2021no—
151C2×C22⋊C4—2021no—
152#107—1014no—
153#103—1014no—
154#102—1014no—
155C24—21no—
156C24—1011no—
157C2×Q8—21no—
158C2×Q8—22no—
159C2×Q8—22no—
160C2×Q8C22×D10112yes—
161C2×D4—21no—
162C2×D4—22no—
163C2×D4—22no—
164C2×D4—4044no—
165C2×D4—2024no—
166C2×D4C22×D10112yes—
167C22×C4—21no—
168C22×C4—1011no—
169C22×C4—1012no—
170C22×C4—1012no—
171C22×C4—2021no—
172C22×C4—2021no—
173C22×C4—2022no—
174C22×C4—2022no—
175C22×C4C22×D10111yescharacteristic
176SD16—28no—
177M16—2028no—
178C2×C8—1012no—
179C2×C8—2022no—
180C2×C8C22⋊D10114yescomplemented
181C2×C8C22×D10112yes—
182C4⋊C4—1012no—
183C4⋊C4—1012no—
184C4⋊C4—2024no—
185C4⋊C4—2021no—
186C4⋊C4—2021no—
187C22⋊C4—2024no—
188C22⋊C4—2024no—
189C23—22no—
190C23—42no—
191C23—42no—
192C23—42no—
193C23—1012no—
194C23—2022no—
195C23—2022no—
196C23—2022no—
197C23D4×D10111yescharacteristic
198Q8—22no—
199Q8—22no—
200Q8—42no—
201D4—22no—
202D4—22no—
203D4—42no—
204D4—4048no—
205C2×C4—22no—
206C2×C4—42no—
207C2×C4—1012no—
208C2×C4—4042no—
209C2×C4—4042no—
210C2×C4—2022no—
211C2×C4—2022no—
212C2×C4—2022no—
213C2×C4—2022no—
214C2×C4—2022no—
215C2×C4—2024no—
216C2×C4D4×D10111yescharacteristic
217C2×C4C2×C22⋊D10112yes—
218C2×C4#1310807312yes—
219C2×C4C23×D10111yescharacteristic
220C8—2022no—
221C8C2×C22⋊D10114yescomplemented
222C22—42no—
223C22—42no—
224C22—42no—
225C22—84no—
226C22—4044no—
227C22—2022no—
228C22—2022no—
229C22—2022no—
230C22#1569222612yes—
231C22#1569228212yes—
232C22#1569228412yes—
233C22C2×D4×D10111yescharacteristic
234C4—42no—
235C4—4042no—
236C4—4042no—
237C4—2022no—
238C4#1569222612yes—
239C4C22×C22⋊D10111yescharacteristic
240C4C2×#1310807311yescharacteristic
241C2—82no—
242C2—4042no—
243C2#2026542014yesdirect product factor
244C2C2×#1569222611yescharacteristic
245C2C2×#1569228211yescharacteristic
246C2C2×#1569228411yescharacteristic
subgroupG/H
centerC23D4×D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
FittingC22×SD16×C101C2
socleC23×C101C2×D4
prime radicalC2×#15692226C2
hypercenterC22×SD16D101
hypocenterC101C2×#468
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C23 ◁ C22×C4 ◁ C22×SD16

Complements

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

NH ≅ G/Ntype
1. C22×#13108057C2semidirect
2. C2×#15692231C2semidirect
4. #20265420C2direct (N × H)
5. C2×#15692189C2semidirect
6. C2×#15692179C2semidirect
7. C2×#15692176C2semidirect
8. C22×SD16×C101C2semidirect
12. C2×#13108057C22semidirect
15. #15692231C22semidirect
23. #15692189C22semidirect
24. #15692179C22semidirect
25. #15692176C22semidirect
29. C2×SD16×C101C22semidirect
63. C2×C8×C101D4semidirect
96. C8×C101C2×D4semidirect
118. C2×C101#468semidirect
124. C101C2×#468semidirect
125. C22×SD16D101semidirect
138. C2×SD16C2×D101semidirect
180. C2×C8C22⋊D101semidirect
221. C8C2×C22⋊D101semidirect
243. C2#20265420direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#131080572
C2×#156922312
C2×#156922262
#202654202
C2×#156921892
C2×#156921792
C2×#156921762
C22×SD16×C1012
C2×#468101