Finite Group Databasebeta

← C2×#27161641 #38756015

Subgroups — C2×#27161641 #38756015

Order 25216.

1984 conjugacy classes of subgroups; only the 565 normal subgroups are shown.

normal subgroups only

This group has 1984 conjugacy classes of subgroups — too many to draw. Only its 565 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
1C23×C22⋊D197C212yesmaximal, complemented
2#27161641C2116yesmaximal, direct product factor
3C2×#20263310C214yesmaximal, complemented
4C2×#20263300C214yesmaximal, complemented
5C2×#20263296C214yesmaximal, complemented
6C22×C22⋊C4×C197C211yesmaximal, characteristic, complemented
7C24×D197C2212yescomplemented
8C22×D788C2214yescomplemented
9C22×C22⋊D197—216no—
10C22×C22⋊D197C2214yes—
11C22×C22⋊D197C2214yescomplemented
12C22×C22⋊D197C22116yescomplemented
13C23×Dic197C2212yescomplemented
14#20263310C22132yescomplemented
15#20263300C22132yescomplemented
16#20263296C22132yescomplemented
17C2×C4⋊Dic197C2214yescomplemented
18C2×#15691335C2218yescomplemented
19C25×C197C2211yescharacteristic
20C23×C4×C197C2212yescomplemented
21C2×C22⋊C4×C197C2214yescomplemented
22C2×C22⋊C4×C197C2218yescomplemented
23C23×D197—28no—
24C23×D197—48no—
25C23×D197C2314yes—
26C23×D197C2318yescomplemented
27C2×D788—48no—
28C2×D788C23116yescomplemented
29C2×C22⋊D197—216no—
30C2×C22⋊D197—216no—
31C2×C22⋊D197—264no—
32C2×C22⋊D197C23116yes—
33C2×C22⋊D197C23116yescomplemented
34C22×Dic197—28no—
35C22×Dic197C2314yes—
36C22×Dic197C2318yescomplemented
37C4⋊Dic197C23116yescomplemented
38#15691335C23132yescomplemented
39C24×C197—44no—
40C24×C197D418yescomplemented
41C24×C197C2312yes—
42C24×C197C2311yescharacteristic
43C24×C197C2314yes—
44C22×C4×C197—44no—
45C22×C4×C197C2314yes—
46C22×C4×C197C2318yescomplemented
47C22⋊C4×C197C23116yescomplemented
48C22×D197—28no—
49C22×D197—216no—
50C22×D197—216no—
51C22×D197—416no—
52C22×D197—432no—
53C22×D197C2418yes—
54D788—416no—
55C22⋊D197—232no—
56C22⋊D197—232no—
57C2×Dic197—28no—
58C2×Dic197—216no—
59C2×Dic197C2418yes—
60C23×C197—24no—
61C23×C197—28no—
62C23×C197—416no—
63C23×C197—48no—
64C23×C197C2×D418yes—
65C23×C197C2×D414yes—
66C23×C197C2×D4116yescomplemented
67C23×C197C2414yes—
68C23×C197C2412yes—
69C23×C197C2411yescharacteristic
70C2×C4×C197—28no—
71C2×C4×C197—48no—
72C2×C4×C197C2418yes—
73C22×C197—28no—
74C22×C197—216no—
75C22×C197—24no—
76C22×C197—28no—
77C22×C197—416no—
78C22×C197C22×D418yes—
79C22×C197C22×D414yes—
80C22×C197C22×D412yes—
81C22×C197H(5,𝔽2)14yes—
82C22×C197C2511yescharacteristic
83C2×D197—28no—
84C2×D197—216no—
85C2×D197—432no—
86C2×D197—416no—
87C4×C197—28no—
88Dic197—28no—
89C2×C197—28no—
90C2×C197—24no—
91C2×C197#53414yescomplemented
92C2×C197C23×D411yescharacteristic
93C2×C197C2×H(5,𝔽2)12yes—
94D197—48no—
95C197C2×#53411yescharacteristic, complemented, Sylow
96C2×#534—1971nomaximal, Sylow
97C23×D4—1972no—
98#534—19716no—
99C2×#123—1974no—
100C2×#121—1974no—
101C2×#120—1974no—
102C22×C22⋊C4D19711yescharacteristic, complemented
103C25—1972no—
104C25C2×D19711yescharacteristic
105C22×D4—39416no—
106C22×D4—1974no—
107C22×D4—1974no—
108C22×D4—19716no—
109C22×D4—1974no—
110C23×C4—1972no—
111C23×C4C2×D19712yescomplemented
112#123—19732no—
113#121—19732no—
114#120—19732no—
115C2×C4⋊C4—1974no—
116C2×C22⋊C4—1978no—
117C2×C22⋊C4C2×D19714yescomplemented
118C2×C22⋊C4C2×D19718yescomplemented
119C24—44no—
120C24—7888no—
121C24—3948no—
122C24—1978no—
123C24—1974no—
124C24D78818yescomplemented
125C24C22×D19712yes—
126C24C22×D19711yescharacteristic
127C24C22×D19714yes—
128C2×D4—7888no—
129C2×D4—39416no—
130C2×D4—39416no—
131C2×D4—39464no—
132C2×D4—19716no—
133C2×D4—19716no—
134C2×D4—19716no—
135C22×C4—44no—
136C22×C4—3948no—
137C22×C4—1978no—
138C22×C4—1974no—
139C22×C4C22×D19714yes—
140C22×C4C22×D19718yescomplemented
141C4⋊C4—19716no—
142C22⋊C4—19732no—
143C22⋊C4C22×D197116yescomplemented
144C23—24no—
145C23—28no—
146C23—416no—
147C23—48no—
148C23—78832no—
149C23—78816no—
150C23—39416no—
151C23—3948no—
152C23—39416no—
153C23—1978no—
154C23C2×D78818yes—
155C23C2×D78814yes—
156C23C2×D788116yescomplemented
157C23C23×D19714yes—
158C23C23×D19712yes—
159C23C23×D19711yescharacteristic
160D4—78816no—
161D4—39432no—
162D4—39432no—
163C2×C4—28no—
164C2×C4—48no—
165C2×C4—39416no—
166C2×C4—3948no—
167C2×C4—1978no—
168C2×C4C23×D19718yes—
169C22—216no—
170C22—28no—
171C22—24no—
172C22—28no—
173C22—416no—
174C22—78832no—
175C22—78816no—
176C22—3948no—
177C22—39416no—
178C22C22×D78818yes—
179C22C22×D78814yes—
180C22C22×D78812yes—
181C22#2026337114yes—
182C22C24×D19711yescharacteristic
183C4—28no—
184C4—3948no—
185C2—28no—
186C2—24no—
187C2—7888no—
188C2#2716164114yesdirect product factor
189C2C23×D78811yescharacteristic
190C2C2×#2026337112yes—
subgroupG/H
centerC23C23×D197
commutator, cosocleC22×C197C25
FrattiniC22C24×D197
FittingC22×C22⋊C4×C197C2
socleC23×C197C24
hypercenterC22×C22⋊C4D197
hypocenterC197C2×#534
LCSG ▷ C22×C197 ▷ C197
UCSC1 ◁ C23 ◁ C22×C22⋊C4

Complements

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

NH ≅ G/Ntype
1. C23×C22⋊D197C2semidirect
2. #27161641C2direct (N × H)
3. C2×#20263310C2semidirect
4. C2×#20263300C2semidirect
5. C2×#20263296C2semidirect
6. C22×C22⋊C4×C197C2semidirect
7. C24×D197C22semidirect
8. C22×D788C22semidirect
11. C22×C22⋊D197C22semidirect
12. C22×C22⋊D197C22semidirect
13. C23×Dic197C22semidirect
14. #20263310C22semidirect
15. #20263300C22semidirect
16. #20263296C22semidirect
17. C2×C4⋊Dic197C22semidirect
18. C2×#15691335C22semidirect
20. C23×C4×C197C22semidirect
21. C2×C22⋊C4×C197C22semidirect
22. C2×C22⋊C4×C197C22semidirect
26. C23×D197C23semidirect
28. C2×D788C23semidirect
33. C2×C22⋊D197C23semidirect
36. C22×Dic197C23semidirect
37. C4⋊Dic197C23semidirect
38. #15691335C23semidirect
40. C24×C197D4semidirect
46. C22×C4×C197C23semidirect
47. C22⋊C4×C197C23semidirect
66. C23×C197C2×D4semidirect
91. C2×C197#534semidirect
95. C197C2×#534semidirect
102. C22×C22⋊C4D197semidirect
111. C23×C4C2×D197semidirect
117. C2×C22⋊C4C2×D197semidirect
118. C2×C22⋊C4C2×D197semidirect
124. C24D788semidirect
140. C22×C4C22×D197semidirect
143. C22⋊C4C22×D197semidirect
156. C23C2×D788semidirect
188. C2#27161641direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C22⋊D1972
#271616412
C2×#202633102
C2×#202633002
C2×#202632962
C22×C22⋊C4×C1972
C2×#534197