Finite Group Databasebeta

← #27152125

Subgroups — #27152125

Order 12416.

400 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 400 conjugacy classes of subgroups — too many to draw. Only its 133 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
1C2×#15690798C211yesmaximal, characteristic
2#20261097C214yesmaximal, complemented
3#20261092C214yesmaximal, complemented
4#20260743C212yesmaximal, complemented
5#20260739C212yesmaximal
6C4×#13095495C211yesmaximal, characteristic, complemented
7#533×C97C211yesmaximal, characteristic, complemented
8C22×Dic194C2211yescharacteristic
9C22×D388C2211yescharacteristic
10C2×#13095495C2212yes—
11#15690798—22no—
12#15690798C2212yes—
13#15690798C2212yes—
14C2×#13095488—22no—
15#15690733C2218yescomplemented
16#15690730C2218yes—
17#15690721C2212yescomplemented
18C4×#11715895C2212yescomplemented
19C4⋊Q8×C97C2212yescomplemented
20#125×C97C2212yescomplemented
21C22⋊Q8×C97—22no—
22C4×Q8×C97—22no—
23#117×C97C2212yes—
24C2×C42×C97C2211yescharacteristic
25C23×D97—21no—
26C2×Dic194—24no—
27C2×Dic194—21no—
28C2×Dic194C2312yes—
29C2×D388—24no—
30C2×D388—21no—
31C2×D388C2312yes—
32C22×Dic97—21no—
33#13095495—24no—
34C2×#11715895C2314yes—
35#13095488—24no—
36#13095488—24no—
37C2×Q8×C97—22no—
38C22×C4×C97D412yes—
39C22×C4×C97C2311yescharacteristic
40C4⋊C4×C97—24no—
41C4⋊C4×C97—22no—
42C4⋊C4×C97C2314yes—
43C22⋊C4×C97—22no—
44C42×C97—22no—
45C42×C97D412yes—
46C42×C97C2311yescharacteristic
47C42×C97C2311yescharacteristic
48C23×C97C2×D411yescharacteristic
49C22×D97—22no—
50C22×D97—44no—
51C22×D97—42no—
52Q8×C97—44no—
53C2×C4×C97—24no—
54C2×C4×C97—42no—
55C2×C4×C97C2×D412yes—
56C2×C4×C97C2×D412yes—
57C2×C4×C97C2×D411yescharacteristic
58C2×C4×C97#4114yes—
59C2×C4×C97C2411yescharacteristic
60C2×Dic97—22no—
61C2×Dic97—42no—
62D388—24no—
63D388—42no—
64Dic194—24no—
65Dic194—42no—
66#11715895—24no—
67C22×C97—22no—
68C22×C97#12412yes—
69C22×C97C22×D411yescharacteristic
70C2×D97—42no—
71C2×D97—44no—
72C2×D97—84no—
73C4×C97—22no—
74C4×C97—22no—
75C4×C97—44no—
76C4×C97#12412yes—
77C4×C97C2×#4112yes—
78Dic97—42no—
79C2×C97—22no—
80C2×C97C2×#12411yescharacteristic
81C2×C97#57512yes—
82D97—82no—
83#3140—971nomaximal, Sylow
84C97#314011yescharacteristic, complemented, Sylow
85#533D9711yescharacteristic, complemented
86C2×#124—971no—
87#489—974no—
88#488—974no—
89#419—972no—
90#418—972no—
91C4×M16—971no—
92C22×Q8—971no—
93C22×D4—971no—
94C2×M16—972no—
95C4⋊Q8C2×D9712yescomplemented
96#125C2×D9712yescomplemented
97#124—972no—
98#124—972no—
99#124—1942no—
100C22⋊Q8—22no—
101C4×Q8—22no—
102#117C2×D9712yes—
103C2×C22⋊C4—1942no—
104C2×C42C2×D9711yescharacteristic
105#103—978no—
106#102—978no—
107#97—972no—
108C4×C8—972no—
109C24—1941no—
110C2×Q8—22no—
111C2×Q8—972no—
112C2×Q8—1944no—
113C2×Q8—1941no—
114C2×D4—972no—
115C2×D4—1941no—
116C2×D4—1944no—
117C22×C4—1941no—
118C22×C4C22⋊D9712yes—
119C22×C4C22×D9711yescharacteristic
120M16—1944no—
121C2×C8—974no—
122C4⋊C4—24no—
123C4⋊C4—22no—
124C4⋊C4C22×D9714yes—
125C22⋊C4—22no—
126C22⋊C4—1944no—
127C22⋊C4—1944no—
128C42—22no—
129C42C22⋊D9712yes—
130C42C22×D9711yescharacteristic
131C42C22×D9711yescharacteristic
132C23—3884no—
133C23—3882no—
134C23—1942no—
135C23C2×C22⋊D9711yescharacteristic
136Q8—44no—
137Q8—3882no—
138Q8—1944no—
139D4—3882no—
140D4—1944no—
141C2×C4—24no—
142C2×C4—42no—
143C2×C4—3882no—
144C2×C4—1942no—
145C2×C4C2×C22⋊D9712yes—
146C2×C4C2×C22⋊D9712yes—
147C2×C4C2×C22⋊D9711yescharacteristic
148C2×C4#1309551214yes—
149C2×C4C23×D9711yescharacteristic
150C8—1944no—
151C22—22no—
152C22—7764no—
153C22—3882no—
154C22—3884no—
155C22#1569079912yes—
156C22C22×C22⋊D9711yescharacteristic
157C4—22no—
158C4—22no—
159C4—44no—
160C4—3882no—
161C4#1569079912yes—
162C4C2×#1309551212yes—
163C2—22no—
164C2—7762no—
165C2C2×#1569079911yescharacteristic
166C2#2026153312yes—
subgroupG/H
centerC22C22×C22⋊D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#533×C97C2
socleC22×C97C22×D4
prime radicalC22×D388C22
hypercenter#533D97
hypocenterC97#3140
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C2×C42 ◁ #533

Complements

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

NH ≅ G/Ntype
2. #20261097C2semidirect
3. #20261092C2semidirect
4. #20260743C2semidirect
6. C4×#13095495C2semidirect
7. #533×C97C2semidirect
15. #15690733C22semidirect
17. #15690721C22semidirect
18. C4×#11715895C22semidirect
19. C4⋊Q8×C97C22semidirect
20. #125×C97C22semidirect
84. C97#3140semidirect
85. #533D97semidirect
95. C4⋊Q8C2×D97semidirect
96. #125C2×D97semidirect

Maximal subgroups

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

subgroupindex
C2×#156907982
#202610972
#202610922
#202607432
#202607392
C4×#130954952
#533×C972
#314097