Finite Group Databasebeta

← C2×C53×#466 #20271985

Subgroups — C2×C53×#466 #20271985

Order 6784.

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×C53C211yesmaximal, characteristic, complemented
2C23×C8×C53C211yesmaximal, characteristic, complemented
3C2×#121×C53C212yesmaximal, complemented
4C53×#466C218yesmaximal, direct product factor
5C2×#107×C53C211yesmaximal, characteristic, complemented
6C2×#102×C53C212yesmaximal, complemented
7C22×D4×C53—22no—
8C22×D4×C53C2212yes—
9C23×C4×C53C2211yescharacteristic
10C2×D8×C53—24no—
11C2×D8×C53C2214yescomplemented
12C22×C8×C53—24no—
13C22×C8×C53C2211yescharacteristic
14C22×C8×C53C2211yescharacteristic
15C22×C8×C53C2214yescomplemented
16#121×C53—24no—
17#121×C53C2218yescomplemented
18C2×C4⋊C4×C53C2212yes—
19C2×C22⋊C4×C53—22no—
20#107×C53C2214yescomplemented
21#102×C53C2218yescomplemented
22C24×C53—22no—
23C24×C53D411yescharacteristic
24C2×D4×C53—28no—
25C2×D4×C53—24no—
26C2×D4×C53—22no—
27C2×D4×C53—24no—
28C2×D4×C53—48no—
29C2×D4×C53C2314yes—
30C22×C4×C53—22no—
31C22×C4×C53—22no—
32C22×C4×C53—22no—
33C22×C4×C53D412yes—
34C22×C4×C53D411yescharacteristic
35C22×C4×C53C2312yes—
36C22×C4×C53C2311yescharacteristic
37D8×C53—28no—
38C2×C8×C53—22no—
39C2×C8×C53—28no—
40C2×C8×C53D414yescomplemented
41C2×C8×C53C2312yes—
42C2×C8×C53C2312yes—
43C4⋊C4×C53—22no—
44C4⋊C4×C53C2314yes—
45C22⋊C4×C53—28no—
46C23×C53—22no—
47C23×C53—22no—
48C23×C53—24no—
49C23×C53—44no—
50C23×C53—44no—
51C23×C53—44no—
52C23×C53D814yescomplemented
53C23×C53C2×D412yes—
54C23×C53C2×D411yescharacteristic
55D4×C53—24no—
56D4×C53—24no—
57D4×C53—416no—
58D4×C53—44no—
59C2×C4×C53—22no—
60C2×C4×C53—24no—
61C2×C4×C53—22no—
62C2×C4×C53—22no—
63C2×C4×C53—24no—
64C2×C4×C53—44no—
65C2×C4×C53C2×D412yes—
66C2×C4×C53C2×D411yescharacteristic
67C2×C4×C53C2×D412yes—
68C2×C4×C53#4112yes—
69C2×C4×C53C2411yescharacteristic
70C8×C53—22no—
71C8×C53C2×D414yescomplemented
72C22×C53—22no—
73C22×C53—22no—
74C22×C53—28no—
75C22×C53—44no—
76C22×C53—44no—
77C22×C53—44no—
78C22×C53—88no—
79C22×C53#12112yes—
80C22×C53C2×D812yes—
81C22×C53C2×D814yescomplemented
82C22×C53#13512yes—
83C22×C53C22×D411yescharacteristic
84C4×C53—22no—
85C4×C53—44no—
86C4×C53#12112yes—
87C4×C53C22×D411yescharacteristic
88C4×C53C2×#4111yescharacteristic
89C2×#466C5311yesmaximal, characteristic, direct product factor, Sylow
90C2×C53—24no—
91C2×C53—84no—
92C2×C53#46614yesdirect product factor
93C2×C53C2×#12111yescharacteristic
94C2×C53C22×D811yescharacteristic
95C2×C53C2×#13511yescharacteristic
96C22×D8C2×C5311yescharacteristic, complemented
97C23×C8C2×C5311yescharacteristic, complemented
98C2×#121C2×C5312yescomplemented
99#466C2×C5318yesdirect product factor
100C2×#107C2×C5311yescharacteristic, complemented
101C2×#102C2×C5312yescomplemented
102C53C2×#46611yescharacteristic, direct product factor, Sylow
103C22×D4—22no—
104C22×D4C22×C5312yes—
105C23×C4C22×C5311yescharacteristic
106C2×D8—24no—
107C2×D8C22×C5314yescomplemented
108C22×C8—24no—
109C22×C8C22×C5311yescharacteristic
110C22×C8C22×C5311yescharacteristic
111C22×C8C22×C5314yescomplemented
112#121—24no—
113#121C22×C5318yescomplemented
114C2×C4⋊C4C22×C5312yes—
115C2×C22⋊C4—22no—
116#107C22×C5314yescomplemented
117#102C22×C5318yescomplemented
118C24—22no—
119C24D4×C5311yescharacteristic
120C2×D4—28no—
121C2×D4—24no—
122C2×D4—22no—
123C2×D4—24no—
124C2×D4—48no—
125C2×D4C23×C5314yes—
126C22×C4—22no—
127C22×C4—22no—
128C22×C4—22no—
129C22×C4D4×C5311yescharacteristic
130C22×C4D4×C5312yes—
131C22×C4C23×C5312yes—
132C22×C4C23×C5311yescharacteristic
133D8—28no—
134C2×C8—22no—
135C2×C8—28no—
136C2×C8D4×C5314yescomplemented
137C2×C8C23×C5312yes—
138C2×C8C23×C5312yes—
139C4⋊C4—22no—
140C4⋊C4C23×C5314yes—
141C22⋊C4—28no—
142C23—22no—
143C23—22no—
144C23—24no—
145C23—44no—
146C23—44no—
147C23—44no—
148C23D8×C5314yescomplemented
149C23C2×D4×C5312yes—
150C23C2×D4×C5311yescharacteristic
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×C5312yes—
162C2×C4C2×D4×C5311yescharacteristic
163C2×C4C2×D4×C5312yes—
164C2×C4#41×C5312yes—
165C2×C4C24×C5311yescharacteristic
166C8—22no—
167C8C2×D4×C5314yescomplemented
168C22—22no—
169C22—22no—
170C22—28no—
171C22—44no—
172C22—44no—
173C22—44no—
174C22—88no—
175C22#121×C5312yes—
176C22C2×D8×C5312yes—
177C22C2×D8×C5314yescomplemented
178C22#135×C5312yes—
179C22C22×D4×C5311yescharacteristic
180C4—22no—
181C4—44no—
182C4#121×C5312yes—
183C4C22×D4×C5311yescharacteristic
184C4C2×#41×C5311yescharacteristic
185C2—24no—
186C2—84no—
187C2C53×#46614yesdirect product factor
188C2C2×#121×C5311yescharacteristic
189C2C22×D8×C5311yescharacteristic
190C2C2×#135×C5311yescharacteristic
subgroupG/H
center, socleC23×C53C2×D4
commutator, Frattini, cosocleC2×C4C24×C53
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C53 ◁ C23×C4×C53 ◁ G

Complements

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

NH ≅ G/Ntype
1. C22×D8×C53C2semidirect
2. C23×C8×C53C2semidirect
3. C2×#121×C53C2semidirect
4. C53×#466C2direct (N × H)
5. C2×#107×C53C2semidirect
6. C2×#102×C53C2semidirect
11. C2×D8×C53C22semidirect
15. C22×C8×C53C22semidirect
17. #121×C53C22semidirect
20. #107×C53C22semidirect
21. #102×C53C22semidirect
40. C2×C8×C53D4semidirect
52. C23×C53D8semidirect
71. C8×C53C2×D4semidirect
81. C22×C53C2×D8semidirect
89. C2×#466C53direct (N × H)
92. C2×C53#466direct (N × H)
96. C22×D8C2×C53semidirect
97. C23×C8C2×C53semidirect
98. C2×#121C2×C53semidirect
99. #466C2×C53direct (N × H)
100. C2×#107C2×C53semidirect
101. C2×#102C2×C53semidirect
102. C53C2×#466direct (N × H)
107. C2×D8C22×C53semidirect
111. C22×C8C22×C53semidirect
113. #121C22×C53semidirect
116. #107C22×C53semidirect
117. #102C22×C53semidirect
136. C2×C8D4×C53semidirect
148. C23D8×C53semidirect
167. C8C2×D4×C53semidirect
177. C22C2×D8×C53semidirect
187. C2C53×#466direct (N × H)

Maximal subgroups

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

subgroupindex
C22×D8×C532
C23×C8×C532
C2×#121×C532
C53×#4662
C2×#107×C532
C2×#102×C532
C2×#46653