Finite Group Databasebeta

← C22×#15690781 #27154955

Subgroups — C22×#15690781 #27154955

Order 12416.

2528 conjugacy classes of subgroups; only the 685 normal subgroups are shown.

normal subgroups only

This group has 2528 conjugacy classes of subgroups — too many to draw. Only its 685 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⋊D97C212yesmaximal, complemented
2C23×C4×D97C211yesmaximal, characteristic, complemented
3C2×#15690781C2124yesmaximal, direct product factor
4C22×C4⋊Dic97C211yesmaximal, characteristic, complemented
5C22×C22⋊Dic97C212yesmaximal, complemented
6C23×D4×C97C211yesmaximal, characteristic, complemented
7C24×D97C2211yescharacteristic
8C22×C22⋊D97—216no—
9C22×C22⋊D97C22124yescomplemented
10C23×Dic97C2211yescharacteristic
11C23×Dic97C2212yes—
12C22×C4×D97—28no—
13C22×C4×D97C22112yescomplemented
14#15690781C22164yesdirect product factor
15C2×C4⋊Dic97C22112yescomplemented
16C2×C22⋊Dic97C22124yescomplemented
17C25×C97C2212yes—
18C22×D4×C97—28no—
19C22×D4×C97C22112yescomplemented
20C23×C4×C97C2211yescharacteristic
21C23×D97—24no—
22C23×D97—24no—
23C23×D97D418yescomplemented
24C23×D97C2316yes—
25C2×C22⋊D97—296no—
26C2×C22⋊D97C23132yescomplemented
27C22×Dic97—24no—
28C22×Dic97—28no—
29C22×Dic97C2316yes—
30C22×Dic97C23112yes—
31C2×C4×D97—248no—
32C2×C4×D97C23116yescomplemented
33C4⋊Dic97C23116yescomplemented
34C22⋊Dic97C23132yescomplemented
35C24×C97—28no—
36C24×C97—28no—
37C24×C97—28no—
38C24×C97C23112yes—
39C24×C97C2311yescharacteristic
40C2×D4×C97—248no—
41C2×D4×C97C23116yescomplemented
42C22×C4×C97D418yescomplemented
43C22×C4×C97C2316yes—
44C23×C97—224no—
45C23×C97—224no—
46C23×C97—224no—
47C23×C97—432no—
48C23×C97C2×D414yes—
49C23×C97C2×D414yes—
50C23×C97#4114yes—
51C23×C97C2418yes—
52C23×C97C2413yes—
53C22×D97—232no—
54C22×D97—212no—
55C22×D97—212no—
56C22×D97C2×D4124yescomplemented
57C22×D97C2414yes—
58D4×C97—232no—
59C2×C4×C97C2×D4124yescomplemented
60C2×C4×C97C2414yes—
61C22⋊D97—264no—
62C2×Dic97—212no—
63C2×Dic97—224no—
64C2×Dic97C2414yes—
65C2×Dic97C2418yes—
66C4×D97—232no—
67C22×C97—28no—
68C22×C97—28no—
69C22×C97—28no—
70C22×C97—448no—
71C22×C97#121116yescomplemented
72C22×C97C22×D416yes—
73C22×C97C22×D416yes—
74C22×C97C2×#4116yes—
75C22×C97C2511yescharacteristic
76C2×D97—248no—
77C2×D97—24no—
78C2×D97—24no—
79C2×D97C22×D418yescomplemented
80C4×C97C22×D418yescomplemented
81Dic97—24no—
82Dic97—28no—
83C2×C97—48no—
84C2×C97C2×#121112yescomplemented
85C2×C97C23×D411yescharacteristic
86C2×C97C23×D411yescharacteristic
87C2×C97C22×#4111yescharacteristic
88D97—28no—
89C22×#121—971nomaximal, Sylow
90C97C22×#12111yescharacteristic, complemented, Sylow
91C23×D4—972no—
92C23×D4D9711yescharacteristic, complemented
93C24×C4—971no—
94C2×#121—9724no—
95C22×C4⋊C4—971no—
96C22×C22⋊C4—972no—
97C25—971no—
98C25C2×D9712yes—
99C22×D4—28no—
100C22×D4—9724no—
101C22×D4—19416no—
102C22×D4C2×D97112yescomplemented
103C23×C4—971no—
104C23×C4—972no—
105C23×C4—9712no—
106C23×C4—1948no—
107C23×C4C2×D9711yescharacteristic
108#121—9764no—
109C2×C4⋊C4—9712no—
110C2×C22⋊C4—9724no—
111C24—28no—
112C24—28no—
113C24—28no—
114C24—978no—
115C24—976no—
116C24—1944no—
117C24—1944no—
118C24C22×D97112yes—
119C24C22×D9711yescharacteristic
120C2×D4—248no—
121C2×D4—9732no—
122C2×D4—19496no—
123C2×D4C22×D97116yescomplemented
124C22×C4—976no—
125C22×C4—9712no—
126C22×C4—9716no—
127C22×C4—1948no—
128C22×C4—1944no—
129C22×C4—19448no—
130C22×C4C22⋊D9718yescomplemented
131C22×C4C22×D9716yes—
132C4⋊C4—9716no—
133C22⋊C4—9732no—
134C23—224no—
135C23—224no—
136C23—224no—
137C23—432no—
138C23—9724no—
139C23—974no—
140C23—19432no—
141C23—19412no—
142C23—19412no—
143C23D4×D9714yes—
144C23C2×C22⋊D9714yes—
145C23#1309551014yes—
146C23C23×D9718yes—
147C23C23×D9713yes—
148D4—232no—
149D4—19464no—
150C2×C4—974no—
151C2×C4—978no—
152C2×C4—19424no—
153C2×C4—19412no—
154C2×C4—19432no—
155C2×C4C2×C22⋊D97124yescomplemented
156C2×C4C23×D9714yes—
157C22—28no—
158C22—28no—
159C22—28no—
160C22—448no—
161C22—978no—
162C22—19448no—
163C22—1944no—
164C22—1944no—
165C22#15690781116yesdirect product factor
166C22C2×D4×D9716yes—
167C22C22×C22⋊D9716yes—
168C22C2×#1309551016yes—
169C22C24×D9711yescharacteristic
170C4—1948no—
171C4—1944no—
172C4C22×C22⋊D9718yescomplemented
173C2—48no—
174C2—1948no—
175C2C2×#15690781112yesdirect product factor
176C2C22×D4×D9711yescharacteristic
177C2C23×C22⋊D9711yescharacteristic
178C2C22×#1309551011yescharacteristic
subgroupG/H
centerC24C22×D97
commutator, cosocleC22×C97C25
FrattiniC22C24×D97
FittingC23×D4×C97C2
socleC24×C97C23
hypercenterC23×D4D97
hypocenterC97C22×#121
LCSG ▷ C22×C97 ▷ C97
UCSC1 ◁ C24 ◁ C23×D4

Complements

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

NH ≅ G/Ntype
1. C23×C22⋊D97C2semidirect
2. C23×C4×D97C2semidirect
3. C2×#15690781C2direct (N × H)
4. C22×C4⋊Dic97C2semidirect
5. C22×C22⋊Dic97C2semidirect
6. C23×D4×C97C2semidirect
9. C22×C22⋊D97C22semidirect
13. C22×C4×D97C22semidirect
14. #15690781C22direct (N × H)
15. C2×C4⋊Dic97C22semidirect
16. C2×C22⋊Dic97C22semidirect
19. C22×D4×C97C22semidirect
23. C23×D97D4semidirect
26. C2×C22⋊D97C23semidirect
32. C2×C4×D97C23semidirect
33. C4⋊Dic97C23semidirect
34. C22⋊Dic97C23semidirect
41. C2×D4×C97C23semidirect
42. C22×C4×C97D4semidirect
56. C22×D97C2×D4semidirect
59. C2×C4×C97C2×D4semidirect
71. C22×C97#121semidirect
79. C2×D97C22×D4semidirect
80. C4×C97C22×D4semidirect
84. C2×C97C2×#121semidirect
90. C97C22×#121semidirect
92. C23×D4D97semidirect
102. C22×D4C2×D97semidirect
123. C2×D4C22×D97semidirect
130. C22×C4C22⋊D97semidirect
155. C2×C4C2×C22⋊D97semidirect
165. C22#15690781direct (N × H)
172. C4C22×C22⋊D97semidirect
175. C2C2×#15690781direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C22⋊D972
C23×C4×D972
C2×#156907812
C22×C4⋊Dic972
C22×C22⋊Dic972
C23×D4×C972
C22×#12197