Finite Group Databasebeta

← #27150055

Subgroups — #27150055

Order 12416.

582 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 582 conjugacy classes of subgroups — too many to draw. Only its 157 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×C4×Dic97C211yesmaximal, characteristic, complemented
2#20260623C214yesmaximal, complemented
3#20260616C214yesmaximal, complemented
4#20260615C212yesmaximal
5#20260607C212yesmaximal, complemented
6#20260598C211yesmaximal, characteristic, complemented
7C2×#123×C97C211yesmaximal, characteristic
8C23×Dic97C2212yes—
9C2×#13095490C2214yes—
10C2×C22⋊Dic97C2214yes—
11C2×C22⋊Dic97C2212yes—
12C2×C4×Dic97—24no—
13C2×C4×Dic97C2214yes—
14#15690717C2214yes—
15#15690717C2214yes—
16#15690717C2214yes—
17C22×D4×C97C2211yescharacteristic
18C23×C4×C97C2211yescharacteristic
19#123×C97—24no—
20C2×C4⋊C4×C97C2212yes—
21C2×C22⋊C4×C97C2212yes—
22C2×C22⋊C4×C97C2211yescharacteristic
23C22×Dic97—24no—
24C22×Dic97—28no—
25C22×Dic97C2314yes—
26C22×Dic97C2314yes—
27#13095490—28no—
28C22⋊Dic97—24no—
29C22⋊Dic97—28no—
30C4×Dic97—28no—
31C24×C97C2311yescharacteristic
32C24×C97C2311yescharacteristic
33C2×D4×C97—22no—
34C2×D4×C97—24no—
35C22×C4×C97—24no—
36C22×C4×C97D414yes—
37C22×C4×C97C2312yes—
38C22×C4×C97C2311yescharacteristic
39C22×C4×C97C2312yes—
40C4⋊C4×C97—24no—
41C22⋊C4×C97—22no—
42C22⋊C4×C97—24no—
43C23×C97—24no—
44C23×C97—21no—
45C23×C97—22no—
46C23×C97—24no—
47C23×C97C2×D412yes—
48C23×C97#4114yes—
49C23×C97C2411yescharacteristic
50D4×C97—44no—
51C2×C4×C97—28no—
52C2×C4×C97—24no—
53C2×C4×C97—22no—
54C2×C4×C97—21no—
55C2×C4×C97—22no—
56C2×C4×C97—24no—
57C2×C4×C97C2×D414yes—
58C2×Dic97—216no—
59C2×Dic97—24no—
60C2×Dic97—24no—
61C2×Dic97—24no—
62C2×Dic97—24no—
63C2×Dic97—24no—
64C2×Dic97#4118yes—
65C22×C97—24no—
66C22×C97—28no—
67C22×C97—41no—
68C22×C97—42no—
69C22×C97—44no—
70C22×C97#12314yes—
71C22×C97C22×D411yescharacteristic
72C22×C97C2×#4114yes—
73C22×C97C2×#4112yes—
74C4×C97—24no—
75C4×C97—41no—
76C4×C97—42no—
77Dic97—28no—
78Dic97—44no—
79C2×C97—24no—
80C2×C97—81no—
81C2×C97C2×#12311yescharacteristic
82C2×C97#52512yes—
83C2×C97#52914yes—
84#2853—971nomaximal, Sylow
85C97#285311yescharacteristic, complemented, Sylow
86C2×#123D9711yescharacteristic
87C22×C42—971no—
88#388—974no—
89#387—974no—
90#386—972no—
91#385—972no—
92#381—971no—
93C22×D4C2×D9711yescharacteristic
94C23×C4—972no—
95C23×C4C2×D9711yescharacteristic
96#123—24no—
97C2×C4⋊C4—974no—
98C2×C4⋊C4C2×D9712yes—
99C2×C22⋊C4—972no—
100C2×C22⋊C4—974no—
101C2×C22⋊C4C2×D9711yescharacteristic
102C2×C22⋊C4C2×D9712yes—
103C2×C42—974no—
104C2×C42—1944no—
105#95—974no—
106#95—974no—
107#95—974no—
108C24C22×D9711yescharacteristic
109C24C22×D9711yescharacteristic
110C2×D4—22no—
111C2×D4—24no—
112C22×C4—24no—
113C22×C4—974no—
114C22×C4—974no—
115C22×C4—1948no—
116C22×C4—1944no—
117C22×C4C22⋊D9714yes—
118C22×C4C22×D9711yescharacteristic
119C22×C4C22×D9712yes—
120C22×C4C22×D9712yes—
121C4⋊C4—24no—
122C4⋊C4—1948no—
123C22⋊C4—22no—
124C22⋊C4—24no—
125C22⋊C4—1948no—
126C22⋊C4—1944no—
127C42—1948no—
128C23—24no—
129C23—21no—
130C23—24no—
131C23—22no—
132C23C2×C22⋊D9712yes—
133C23#1309551014yes—
134C23C23×D9711yescharacteristic
135D4—44no—
136C2×C4—28no—
137C2×C4—24no—
138C2×C4—22no—
139C2×C4—21no—
140C2×C4—24no—
141C2×C4—22no—
142C2×C4—978no—
143C2×C4—1944no—
144C2×C4—1944no—
145C2×C4—1944no—
146C2×C4—19416no—
147C2×C4—1944no—
148C2×C4—1944no—
149C2×C4C2×C22⋊D9714yes—
150C22—24no—
151C22—28no—
152C22—41no—
153C22—42no—
154C22—44no—
155C22#1569079114yes—
156C22C22×C22⋊D9711yescharacteristic
157C22#41×D9714yes—
158C22C2×#1309551012yes—
159C4—24no—
160C4—41no—
161C4—42no—
162C4—3884no—
163C4—1948no—
164C2—24no—
165C2—81no—
166C2C2×#1569079111yescharacteristic
167C2#2026124212yes—
168C2#2026126214yes—
subgroupG/H
center, FrattiniC23C23×D97
commutator, socle, cosocleC23×C97C24
FittingC2×#123×C97C2
prime radicalC22×D4×C97C22
hypercenterC2×#123D97
hypocenterC97#2853
LCSG ▷ C23×C97 ▷ C97
UCSC1 ◁ C23 ◁ C2×#123

Complements

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

NH ≅ G/Ntype
1. C22×C4×Dic97C2semidirect
2. #20260623C2semidirect
3. #20260616C2semidirect
5. #20260607C2semidirect
6. #20260598C2semidirect
85. C97#2853semidirect

Maximal subgroups

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

subgroupindex
C22×C4×Dic972
#202606232
#202606162
#202606152
#202606072
#202605982
C2×#123×C972
#285397