Finite Group Databasebeta

← C2×#466×C227 #44397004

Subgroups — C2×#466×C227 #44397004

Order 29056.

616 conjugacy classes of subgroups; only the 232 normal subgroups are shown.

normal subgroups only

This group has 616 conjugacy classes of subgroups — too many to draw. Only its 232 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×D8×C227C211yesmaximal, characteristic, complemented
2C23×C8×C227C211yesmaximal, characteristic, complemented
3C2×#121×C227C212yesmaximal, complemented
4#466×C227C218yesmaximal, direct product factor
5C2×#107×C227C211yesmaximal, characteristic, complemented
6C2×#102×C227C212yesmaximal, complemented
7C22×D4×C227—22no—
8C22×D4×C227C2212yes—
9C23×C4×C227C2211yescharacteristic
10C2×D8×C227—24no—
11C2×D8×C227C2214yescomplemented
12C22×C8×C227—24no—
13C22×C8×C227C2211yescharacteristic
14C22×C8×C227C2211yescharacteristic
15C22×C8×C227C2214yescomplemented
16#121×C227—24no—
17#121×C227C2218yescomplemented
18C2×C4⋊C4×C227C2212yes—
19C2×C22⋊C4×C227—22no—
20#107×C227C2214yescomplemented
21#102×C227C2218yescomplemented
22C24×C227—22no—
23C24×C227D411yescharacteristic
24C2×D4×C227—28no—
25C2×D4×C227—24no—
26C2×D4×C227—22no—
27C2×D4×C227—24no—
28C2×D4×C227—48no—
29C2×D4×C227C2314yes—
30C22×C4×C227—22no—
31C22×C4×C227—22no—
32C22×C4×C227—22no—
33C22×C4×C227D412yes—
34C22×C4×C227D411yescharacteristic
35C22×C4×C227C2312yes—
36C22×C4×C227C2311yescharacteristic
37D8×C227—28no—
38C2×C8×C227—22no—
39C2×C8×C227—28no—
40C2×C8×C227D414yescomplemented
41C2×C8×C227C2312yes—
42C2×C8×C227C2312yes—
43C4⋊C4×C227—22no—
44C4⋊C4×C227C2314yes—
45C22⋊C4×C227—28no—
46C23×C227—22no—
47C23×C227—22no—
48C23×C227—24no—
49C23×C227—44no—
50C23×C227—44no—
51C23×C227—44no—
52C23×C227D814yescomplemented
53C23×C227C2×D412yes—
54C23×C227C2×D411yescharacteristic
55D4×C227—24no—
56D4×C227—24no—
57D4×C227—416no—
58D4×C227—44no—
59C2×C4×C227—22no—
60C2×C4×C227—24no—
61C2×C4×C227—22no—
62C2×C4×C227—22no—
63C2×C4×C227—24no—
64C2×C4×C227—44no—
65C2×C4×C227C2×D412yes—
66C2×C4×C227C2×D411yescharacteristic
67C2×C4×C227C2×D412yes—
68C2×C4×C227#4112yes—
69C2×C4×C227C2411yescharacteristic
70C8×C227—22no—
71C8×C227C2×D414yescomplemented
72C22×C227—22no—
73C22×C227—22no—
74C22×C227—28no—
75C22×C227—44no—
76C22×C227—44no—
77C22×C227—44no—
78C22×C227—88no—
79C22×C227#12112yes—
80C22×C227C2×D812yes—
81C22×C227C2×D814yescomplemented
82C22×C227#13512yes—
83C22×C227C22×D411yescharacteristic
84C4×C227—22no—
85C4×C227—44no—
86C4×C227#12112yes—
87C4×C227C22×D411yescharacteristic
88C4×C227C2×#4111yescharacteristic
89C2×C227—24no—
90C2×C227—84no—
91C2×C227#46614yesdirect product factor
92C2×C227C2×#12111yescharacteristic
93C2×C227C22×D811yescharacteristic
94C2×C227C2×#13511yescharacteristic
95C227C2×#46611yescharacteristic, direct product factor, Sylow
96C2×#466C22711yesmaximal, characteristic, direct product factor, Sylow
97C22×D8C2×C22711yescharacteristic, complemented
98C23×C8C2×C22711yescharacteristic, complemented
99C2×#121C2×C22712yescomplemented
100#466C2×C22718yesdirect product factor
101C2×#107C2×C22711yescharacteristic, complemented
102C2×#102C2×C22712yescomplemented
103C22×D4—22no—
104C22×D4C22×C22712yes—
105C23×C4C22×C22711yescharacteristic
106C2×D8—24no—
107C2×D8C22×C22714yescomplemented
108C22×C8—24no—
109C22×C8C22×C22711yescharacteristic
110C22×C8C22×C22711yescharacteristic
111C22×C8C22×C22714yescomplemented
112#121—24no—
113#121C22×C22718yescomplemented
114C2×C4⋊C4C22×C22712yes—
115C2×C22⋊C4—22no—
116#107C22×C22714yescomplemented
117#102C22×C22718yescomplemented
118C24—22no—
119C24D4×C22711yescharacteristic
120C2×D4—28no—
121C2×D4—24no—
122C2×D4—22no—
123C2×D4—24no—
124C2×D4—48no—
125C2×D4C23×C22714yes—
126C22×C4—22no—
127C22×C4—22no—
128C22×C4—22no—
129C22×C4D4×C22711yescharacteristic
130C22×C4D4×C22712yes—
131C22×C4C23×C22712yes—
132C22×C4C23×C22711yescharacteristic
133D8—28no—
134C2×C8—22no—
135C2×C8—28no—
136C2×C8D4×C22714yescomplemented
137C2×C8C23×C22712yes—
138C2×C8C23×C22712yes—
139C4⋊C4—22no—
140C4⋊C4C23×C22714yes—
141C22⋊C4—28no—
142C23—22no—
143C23—22no—
144C23—24no—
145C23—44no—
146C23—44no—
147C23—44no—
148C23D8×C22714yescomplemented
149C23C2×D4×C22712yes—
150C23C2×D4×C22711yescharacteristic
151D4—24no—
152D4—24no—
153D4—416no—
154D4—44no—
155C2×C4—22no—
156C2×C4—24no—
157C2×C4—22no—
158C2×C4—22no—
159C2×C4—24no—
160C2×C4—44no—
161C2×C4C2×D4×C22712yes—
162C2×C4C2×D4×C22711yescharacteristic
163C2×C4C2×D4×C22712yes—
164C2×C4#41×C22712yes—
165C2×C4C24×C22711yescharacteristic
166C8—22no—
167C8C2×D4×C22714yescomplemented
168C22—28no—
169C22—22no—
170C22—22no—
171C22—44no—
172C22—44no—
173C22—44no—
174C22—88no—
175C22#121×C22712yes—
176C22C2×D8×C22712yes—
177C22C2×D8×C22714yescomplemented
178C22#135×C22712yes—
179C22C22×D4×C22711yescharacteristic
180C4—22no—
181C4—44no—
182C4#121×C22712yes—
183C4C22×D4×C22711yescharacteristic
184C4C2×#41×C22711yescharacteristic
185C2—24no—
186C2—84no—
187C2#466×C22714yesdirect product factor
188C2C2×#121×C22711yescharacteristic
189C2C22×D8×C22711yescharacteristic
190C2C2×#135×C22711yescharacteristic
subgroupG/H
center, socleC23×C227C2×D4
commutator, Frattini, cosocleC2×C4C24×C227
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C227 ◁ C23×C4×C227 ◁ G

Complements

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

NH ≅ G/Ntype
1. C22×D8×C227C2semidirect
2. C23×C8×C227C2semidirect
3. C2×#121×C227C2semidirect
4. #466×C227C2direct (N × H)
5. C2×#107×C227C2semidirect
6. C2×#102×C227C2semidirect
11. C2×D8×C227C22semidirect
15. C22×C8×C227C22semidirect
17. #121×C227C22semidirect
20. #107×C227C22semidirect
21. #102×C227C22semidirect
40. C2×C8×C227D4semidirect
52. C23×C227D8semidirect
71. C8×C227C2×D4semidirect
81. C22×C227C2×D8semidirect
91. C2×C227#466direct (N × H)
95. C227C2×#466direct (N × H)
96. C2×#466C227direct (N × H)
97. C22×D8C2×C227semidirect
98. C23×C8C2×C227semidirect
99. C2×#121C2×C227semidirect
100. #466C2×C227direct (N × H)
101. C2×#107C2×C227semidirect
102. C2×#102C2×C227semidirect
107. C2×D8C22×C227semidirect
111. C22×C8C22×C227semidirect
113. #121C22×C227semidirect
116. #107C22×C227semidirect
117. #102C22×C227semidirect
136. C2×C8D4×C227semidirect
148. C23D8×C227semidirect
167. C8C2×D4×C227semidirect
177. C22C2×D8×C227semidirect
187. C2#466×C227direct (N × H)

Maximal subgroups

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

subgroupindex
C22×D8×C2272
C23×C8×C2272
C2×#121×C2272
#466×C2272
C2×#107×C2272
C2×#102×C2272
C2×#466227