Finite Group Databasebeta

← C2×#20265498 #27181013

Subgroups — C2×#20265498 #27181013

Order 12928.

468 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 468 conjugacy classes of subgroups — too many to draw. Only its 149 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×#15692214C211yesmaximal, characteristic, complemented
2#20265498C218yesmaximal, direct product factor
3C2×#15692188C211yesmaximal, characteristic, complemented
4C2×#15692186C211yesmaximal, characteristic, complemented
5C2×#15692176C211yesmaximal, characteristic, complemented
6C2×#15692176C211yesmaximal, characteristic, complemented
7C2×C22⋊#11717579C211yesmaximal, characteristic, complemented
8C2×#121×C101C211yesmaximal, characteristic
9C22×#11717579C2211yescharacteristic
10C22×#11717579C2211yescharacteristic
11#15692214—22no—
12#15692214C2214yescomplemented
13C2×C4⋊Dic101C2211yescharacteristic
14C2×C4⋊Dic101C2211yescharacteristic
15C2×C22⋊Dic101—21no—
16C2×C4×Dic101—21no—
17#15692188C2214yescomplemented
18#15692186C2214yescomplemented
19#15692176C2214yescomplemented
20#15692176C2214yescomplemented
21C22⋊#11717579C2214yescomplemented
22C22×D4×C101—21no—
23C22×D4×C101C2211yescharacteristic
24C23×C4×C101C2211yescharacteristic
25#121×C101—22no—
26#121×C101C2214yes—
27C2×C4⋊C4×C101C2211yescharacteristic
28C2×C22⋊C4×C101—21no—
29C22×Dic101—21no—
30C22×Dic101—21no—
31C2×#11717579—22no—
32C2×#11717579—22no—
33C2×#11717579C2312yes—
34C2×#11717579C2312yes—
35C4⋊Dic101—21no—
36C4⋊Dic101—21no—
37C4⋊Dic101C2312yes—
38C4⋊Dic101C2312yes—
39C22⋊Dic101—24no—
40C4×Dic101—24no—
41C24×C101—21no—
42C24×C101D411yescharacteristic
43C2×D4×C101—24no—
44C2×D4×C101—21no—
45C2×D4×C101—22no—
46C2×D4×C101—22no—
47C2×D4×C101—44no—
48C2×D4×C101C2312yes—
49C22×C4×C101—22no—
50C22×C4×C101—22no—
51C22×C4×C101—21no—
52C22×C4×C101D411yescharacteristic
53C22×C4×C101D412yes—
54C22×C4×C101C2312yes—
55C22×C4×C101C2311yescharacteristic
56C4⋊C4×C101—21no—
57C4⋊C4×C101C2312yes—
58C22⋊C4×C101—24no—
59C23×C101—22no—
60C23×C101—22no—
61C23×C101—22no—
62C23×C101—22no—
63C23×C101—42no—
64C23×C101—42no—
65C23×C101—42no—
66C23×C101C2×D412yes—
67C23×C101C2×D411yescharacteristic
68D4×C101—22no—
69D4×C101—22no—
70D4×C101—48no—
71D4×C101—42no—
72C2×C4×C101—22no—
73C2×C4×C101—24no—
74C2×C4×C101—22no—
75C2×C4×C101—22no—
76C2×C4×C101—22no—
77C2×C4×C101—42no—
78C2×C4×C101C2×D412yes—
79C2×C4×C101C2×D411yescharacteristic
80C2×C4×C101#4112yes—
81C2×C4×C101#4112yes—
82C2×C4×C101C2411yescharacteristic
83C2×Dic101—22no—
84C2×Dic101—22no—
85C2×Dic101—42no—
86C2×Dic101—42no—
87#11717579—22no—
88#11717579—22no—
89C22×C101—22no—
90C22×C101—22no—
91C22×C101—22no—
92C22×C101—44no—
93C22×C101—42no—
94C22×C101—42no—
95C22×C101—42no—
96C22×C101—84no—
97C22×C101#12312yes—
98C22×C101#13512yes—
99C22×C101AGL(1,ℤ/8)12yes—
100C22×C101C22×D411yescharacteristic
101C4×C101—22no—
102C4×C101—42no—
103C4×C101#12312yes—
104C4×C101C2×#4111yescharacteristic
105C4×C101C2×#4111yescharacteristic
106Dic101—42no—
107Dic101—42no—
108C2×C101—42no—
109C2×C101—82no—
110C2×C101#48214yescomplemented
111C2×C101C2×#12311yescharacteristic
112C2×C101C2×#13511yescharacteristic
113C2×C101C2×AGL(1,ℤ/8)11yescharacteristic
114C2×#482—1011nomaximal, Sylow
115C101C2×#48211yescharacteristic, complemented, Sylow
116C2×#121D10111yescharacteristic
117C2×#117—1011no—
118#482—1018no—
119C2×#107—1011no—
120C2×#106—1011no—
121C2×#102—1011no—
122C2×#102—1011no—
123C2×C22⋊C8—1011no—
124C22×D4—21no—
125C22×D4C2×D10111yescharacteristic
126C23×C4C2×D10111yescharacteristic
127C22×C8—1011no—
128C22×C8—1011no—
129#121—22no—
130#121C2×D10114yes—
131#117—1014no—
132#117—2022no—
133C2×C4⋊C4—1011no—
134C2×C4⋊C4—1011no—
135C2×C4⋊C4C2×D10111yescharacteristic
136C2×C22⋊C4—21no—
137C2×C22⋊C4—2021no—
138C2×C42—2021no—
139#107—1014no—
140#106—1014no—
141#102—1014no—
142#102—1014no—
143C22⋊C8—1014no—
144C24—21no—
145C24C22⋊D10111yescharacteristic
146C2×D4—24no—
147C2×D4—21no—
148C2×D4—22no—
149C2×D4—22no—
150C2×D4—44no—
151C2×D4C22×D10112yes—
152C22×C4—22no—
153C22×C4—22no—
154C22×C4—21no—
155C22×C4—2021no—
156C22×C4—2021no—
157C22×C4C22⋊D10112yes—
158C22×C4C22⋊D10111yescharacteristic
159C22×C4C22×D10112yes—
160C22×C4C22×D10111yescharacteristic
161C2×C8—1012no—
162C2×C8—1012no—
163C2×C8—2022no—
164C2×C8—2022no—
165C4⋊C4—21no—
166C4⋊C4—1012no—
167C4⋊C4—1012no—
168C4⋊C4—2021no—
169C4⋊C4—2021no—
170C4⋊C4C22×D10112yes—
171C22⋊C4—24no—
172C22⋊C4—2024no—
173C42—2024no—
174C23—22no—
175C23—22no—
176C23—22no—
177C23—22no—
178C23—42no—
179C23—42no—
180C23—42no—
181C23C2×C22⋊D10112yes—
182C23C2×C22⋊D10111yescharacteristic
183D4—22no—
184D4—22no—
185D4—48no—
186D4—42no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—24no—
191C2×C4—22no—
192C2×C4—42no—
193C2×C4—4042no—
194C2×C4—4042no—
195C2×C4—2022no—
196C2×C4—2022no—
197C2×C4C2×C22⋊D10112yes—
198C2×C4C2×C22⋊D10111yescharacteristic
199C2×C4#1310807312yes—
200C2×C4#1310807312yes—
201C2×C4C23×D10111yescharacteristic
202C8—2022no—
203C8—2022no—
204C22—22no—
205C22—22no—
206C22—22no—
207C22—44no—
208C22—42no—
209C22—42no—
210C22—42no—
211C22—84no—
212C22#1569223612yes—
213C22#1569227512yes—
214C22#1569228112yes—
215C22C22×C22⋊D10111yescharacteristic
216C4—22no—
217C4—42no—
218C4—4042no—
219C4—4042no—
220C4#1569223612yes—
221C4C2×#1310807311yescharacteristic
222C4C2×#1310807311yescharacteristic
223C2—42no—
224C2—82no—
225C2#2026549814yesdirect product factor
226C2C2×#1569223611yescharacteristic
227C2C2×#1569227511yescharacteristic
228C2C2×#1569228111yescharacteristic
subgroupG/H
centerC23C2×C22⋊D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
Fitting, prime radicalC2×#121×C101C2
socleC23×C101C2×D4
hypercenterC2×#121D101
hypocenterC101C2×#482
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#121

Complements

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

NH ≅ G/Ntype
1. C2×#15692214C2semidirect
2. #20265498C2direct (N × H)
3. C2×#15692188C2semidirect
4. C2×#15692186C2semidirect
5. C2×#15692176C2semidirect
6. C2×#15692176C2semidirect
7. C2×C22⋊#11717579C2semidirect
12. #15692214C22semidirect
17. #15692188C22semidirect
18. #15692186C22semidirect
19. #15692176C22semidirect
20. #15692176C22semidirect
21. C22⋊#11717579C22semidirect
110. C2×C101#482semidirect
115. C101C2×#482semidirect
225. C2#20265498direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#156922142
#202654982
C2×#156921882
C2×#156921862
C2×#156921762
C2×C22⋊#117175792
C2×#121×C1012
C2×#482101