Finite Group Databasebeta

← C2×#20261341 #27155097

Subgroups — C2×#20261341 #27155097

Order 12416.

1260 conjugacy classes of subgroups; only the 485 normal subgroups are shown.

normal subgroups only

This group has 1260 conjugacy classes of subgroups — too many to draw. Only its 485 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
1#20261341C2116yesmaximal, direct product factor
2C2×#15690807C211yesmaximal, characteristic, complemented
3C2×#15690795C214yesmaximal, complemented
4C2×#15690785C214yesmaximal, complemented
5C2×#15690785C212yesmaximal, complemented
6C2×#15690772C212yesmaximal, complemented
7C2×#15690767C211yesmaximal, characteristic, complemented
8C2×#125×C97C211yesmaximal, characteristic, complemented
9C22×D388C2212yes—
10C22×D388C2214yes—
11C22×D388C2214yes—
12C22×C4×D97C2211yescharacteristic
13C22×C4×D97C2214yes—
14#15690807C2218yescomplemented
15#15690795C22132yescomplemented
16#15690785C22116yescomplemented
17#15690785C22132yescomplemented
18#15690772C22116yescomplemented
19#15690767C2218yescomplemented
20C2×#13095490C2212yes—
21C2×#13095488C2212yes—
22C2×#13095488C2214yes—
23C2×#13095488C2214yes—
24C2×C4×Dic97C2211yescharacteristic
25#125×C97C2218yescomplemented
26C2×C4⋊C4×C97C2212yes—
27C2×C4⋊C4×C97C2214yes—
28C2×C42×C97C2211yescharacteristic
29C23×D97C2311yescharacteristic
30C23×D97C2314yes—
31C2×D388—28no—
32C2×D388—48no—
33C2×D388—48no—
34C2×D388C2318yes—
35C2×D388C23116yes—
36C2×D388C23116yes—
37C22×Dic97C2312yes—
38C22×Dic97C2311yescharacteristic
39C2×C4×D97—24no—
40C2×C4×D97—48no—
41C2×C4×D97C2314yes—
42C2×C4×D97C23116yes—
43#13095490C2318yes—
44#13095488C2318yes—
45#13095488C23116yes—
46#13095488C23116yes—
47C4×Dic97C2314yes—
48C22×C4×C97C2314yes—
49C22×C4×C97C2312yes—
50C22×C4×C97C2311yescharacteristic
51C4⋊C4×C97C2318yes—
52C4⋊C4×C97C23116yes—
53C42×C97C2314yes—
54C23×C97C2411yescharacteristic
55C22×D97—22no—
56C22×D97—28no—
57C22×D97—28no—
58C22×D97—24no—
59C22×D97—28no—
60C22×D97C2412yes—
61C22×D97C2418yes—
62C2×C4×C97—28no—
63C2×C4×C97—24no—
64C2×C4×C97#4114yes—
65C2×C4×C97C2414yes—
66C2×C4×C97C2418yes—
67C2×C4×C97C2412yes—
68C2×Dic97—22no—
69C2×Dic97—24no—
70C2×Dic97C2414yes—
71C2×Dic97C2412yes—
72D388—216no—
73D388—416no—
74D388—416no—
75C4×D97—28no—
76C4×D97—416no—
77C22×C97C2×#4112yes—
78C22×C97H(5,𝔽2)14yes—
79C22×C97C2511yescharacteristic
80C2×D97—22no—
81C2×D97—28no—
82C2×D97—28no—
83C2×D97—24no—
84C2×D97—28no—
85C2×D97—44no—
86C2×D97—416no—
87C4×C97—28no—
88C4×C97—24no—
89C4×C97C2×#4114yes—
90Dic97—22no—
91Dic97—24no—
92C2×C97#54014yescomplemented
93C2×C97C22×#4111yescharacteristic
94C2×C97C2×H(5,𝔽2)12yes—
95D97—42no—
96D97—48no—
97C2×#540—971nomaximal, Sylow
98C97C2×#54011yescharacteristic, complemented, Sylow
99#540—9716no—
100C2×#127—971no—
101C2×#125D9711yescharacteristic, complemented
102C2×#123—974no—
103C2×#121—974no—
104C2×#121—972no—
105C2×C4×D4—972no—
106C2×#117—971no—
107C22×D4—974no—
108C22×D4—974no—
109C22×D4—972no—
110C23×C4—974no—
111C23×C4—971no—
112#127—978no—
113#125C2×D9718yescomplemented
114#123—9732no—
115#121—9732no—
116#121—9716no—
117C4×D4—9716no—
118#117—978no—
119C2×C4⋊C4—972no—
120C2×C4⋊C4C2×D9712yes—
121C2×C4⋊C4C2×D9714yes—
122C2×C22⋊C4—972no—
123C2×C22⋊C4—974no—
124C2×C22⋊C4—974no—
125C2×C42—971no—
126C2×C42C2×D9711yescharacteristic
127C24—971no—
128C24—974no—
129C2×D4—9716no—
130C2×D4—9716no—
131C2×D4—978no—
132C2×D4—3888no—
133C2×D4—3888no—
134C2×D4—1948no—
135C22×C4—972no—
136C22×C4—971no—
137C22×C4—9716no—
138C22×C4—974no—
139C22×C4—3888no—
140C22×C4—1944no—
141C22×C4C22×D9712yes—
142C22×C4C22×D9714yes—
143C22×C4C22×D9711yescharacteristic
144C4⋊C4—978no—
145C4⋊C4C22×D9718yes—
146C4⋊C4C22×D97116yes—
147C22⋊C4—978no—
148C22⋊C4—9716no—
149C22⋊C4—9716no—
150C42—974no—
151C42C22×D9714yes—
152C23—972no—
153C23—978no—
154C23—1942no—
155C23—1948no—
156C23—1948no—
157C23—1944no—
158C23—1948no—
159C23C23×D9711yescharacteristic
160D4—38816no—
161D4—38816no—
162D4—19416no—
163C2×C4—28no—
164C2×C4—24no—
165C2×C4—974no—
166C2×C4—972no—
167C2×C4—38816no—
168C2×C4—1942no—
169C2×C4—1944no—
170C2×C4—1948no—
171C2×C4#1309551214yes—
172C2×C4C23×D9714yes—
173C2×C4C23×D9718yes—
174C2×C4C23×D9712yes—
175C22—3884no—
176C22—38816no—
177C22—1942no—
178C22—1948no—
179C22—1948no—
180C22—1944no—
181C22—1948no—
182C22C2×#1309551212yes—
183C22#1569085614yes—
184C22C24×D9711yescharacteristic
185C4—28no—
186C4—24no—
187C4—1942no—
188C4—1944no—
189C4C2×#1309551214yes—
190C2—3882no—
191C2—3888no—
192C2#2026134114yesdirect product factor
193C2C22×#1309551211yescharacteristic
194C2C2×#1569085612yes—
subgroupG/H
centerC23C23×D97
commutator, cosocleC22×C97C25
FrattiniC22C24×D97
FittingC2×#125×C97C2
socleC23×C97C24
hypercenterC2×#125D97
hypocenterC97C2×#540
LCSG ▷ C22×C97 ▷ C97
UCSC1 ◁ C23 ◁ C2×#125

Complements

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

NH ≅ G/Ntype
1. #20261341C2direct (N × H)
2. C2×#15690807C2semidirect
3. C2×#15690795C2semidirect
4. C2×#15690785C2semidirect
5. C2×#15690785C2semidirect
6. C2×#15690772C2semidirect
7. C2×#15690767C2semidirect
8. C2×#125×C97C2semidirect
14. #15690807C22semidirect
15. #15690795C22semidirect
16. #15690785C22semidirect
17. #15690785C22semidirect
18. #15690772C22semidirect
19. #15690767C22semidirect
25. #125×C97C22semidirect
92. C2×C97#540semidirect
98. C97C2×#540semidirect
101. C2×#125D97semidirect
113. #125C2×D97semidirect
192. C2#20261341direct (N × H)

Maximal subgroups

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

subgroupindex
#202613412
C2×#156908072
C2×#156907952
C2×#156907852
C2×#156907722
C2×#156907672
C2×#125×C972
C2×#54097