Finite Group Databasebeta

← #27150058

Subgroups — #27150058

Order 12416.

582 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 582 conjugacy classes of subgroups — too many to draw. Only its 165 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×#15690793C211yesmaximal, characteristic
2#20260626C214yesmaximal, complemented
3#20260619C214yesmaximal, complemented
4#20260614C212yesmaximal
5#20260608C212yesmaximal, complemented
6#20260598C211yesmaximal, characteristic, complemented
7C22×C42×C97C211yesmaximal, characteristic, complemented
8C22×C22⋊D97C2211yescharacteristic
9#15690793—24no—
10C2×#13095490C2214yes—
11C2×#13095490C2212yes—
12C2×C22⋊Dic97C2211yescharacteristic
13C2×C22⋊Dic97C2212yes—
14C2×#13095488C2214yes—
15C2×#13095488C2212yes—
16#15690717C2214yes—
17#15690717C2214yes—
18#15690717C2214yes—
19C23×C4×C97C2212yes—
20C23×C4×C97C2211yescharacteristic
21C2×C42×C97—24no—
22C2×C42×C97C2214yes—
23C23×D97C2311yescharacteristic
24C2×C22⋊D97—22no—
25C2×C22⋊D97—24no—
26C22×Dic97C2312yes—
27C22×Dic97C2314yes—
28C22×Dic97C2311yescharacteristic
29#13095490—24no—
30#13095490—28no—
31C22⋊Dic97—22no—
32C22⋊Dic97—24no—
33#13095488—24no—
34#13095488—28no—
35C24×C97C2311yescharacteristic
36C22×C4×C97—24no—
37C22×C4×C97—24no—
38C22×C4×C97—28no—
39C22×C4×C97D414yes—
40C22×C4×C97C2312yes—
41C22×C4×C97C2314yes—
42C42×C97—28no—
43C23×C97—24no—
44C23×C97C2×D412yes—
45C23×C97#4114yes—
46C23×C97C2411yescharacteristic
47C22×D97—21no—
48C22×D97—24no—
49C22×D97—22no—
50C2×C4×C97—216no—
51C2×C4×C97—28no—
52C2×C4×C97—24no—
53C2×C4×C97—24no—
54C2×C4×C97—24no—
55C2×C4×C97C2×D414yes—
56C2×C4×C97#4118yes—
57C22⋊D97—44no—
58C2×Dic97—22no—
59C2×Dic97—21no—
60C2×Dic97—24no—
61C2×Dic97—24no—
62C2×Dic97—24no—
63C2×Dic97—24no—
64C2×Dic97—22no—
65C22×C97—24no—
66C22×C97—28no—
67C22×C97#12314yes—
68C22×C97C22×D411yescharacteristic
69C22×C97C2×#4112yes—
70C22×C97C2×#4114yes—
71C2×D97—41no—
72C2×D97—42no—
73C2×D97—44no—
74C4×C97—24no—
75C4×C97—28no—
76Dic97—41no—
77Dic97—42no—
78Dic97—44no—
79C2×C97—24no—
80C2×C97C2×#12311yescharacteristic
81C2×C97#52512yes—
82C2×C97#52914yes—
83D97—81no—
84#2853—971nomaximal, Sylow
85C97#285311yescharacteristic, complemented, Sylow
86C2×#123—971no—
87C22×C42D9711yescharacteristic, complemented
88#388—974no—
89#387—974no—
90#386—972no—
91#385—972no—
92#381—971no—
93C22×D4—971no—
94C23×C4C2×D9712yes—
95C23×C4C2×D9711yescharacteristic
96#123—1944no—
97C2×C4⋊C4—972no—
98C2×C4⋊C4—974no—
99C2×C22⋊C4—971no—
100C2×C22⋊C4—972no—
101C2×C22⋊C4—972no—
102C2×C22⋊C4—974no—
103C2×C42—24no—
104C2×C42C2×D9714yes—
105#95—974no—
106#95—974no—
107#95—974no—
108C24—971no—
109C24C22×D9711yescharacteristic
110C2×D4—1942no—
111C2×D4—1944no—
112C22×C4—24no—
113C22×C4—24no—
114C22×C4—28no—
115C22×C4—974no—
116C22×C4—971no—
117C22×C4—972no—
118C22×C4C22⋊D9714yes—
119C22×C4C22×D9712yes—
120C22×C4C22×D9714yes—
121C4⋊C4—1944no—
122C4⋊C4—1948no—
123C22⋊C4—1942no—
124C22⋊C4—1944no—
125C22⋊C4—1944no—
126C22⋊C4—1948no—
127C42—28no—
128C23—24no—
129C23—1941no—
130C23—1944no—
131C23—1942no—
132C23C2×C22⋊D9712yes—
133C23#1309551114yes—
134C23C23×D9711yescharacteristic
135D4—3884no—
136C2×C4—216no—
137C2×C4—28no—
138C2×C4—24no—
139C2×C4—24no—
140C2×C4—24no—
141C2×C4—1942no—
142C2×C4—1941no—
143C2×C4—1944no—
144C2×C4—1944no—
145C2×C4—1944no—
146C2×C4—1944no—
147C2×C4—1942no—
148C2×C4C2×C22⋊D9714yes—
149C2×C4#1309551118yes—
150C22—24no—
151C22—28no—
152C22—3881no—
153C22—3882no—
154C22—3884no—
155C22#1569079314yes—
156C22C22×C22⋊D9711yescharacteristic
157C22C2×#1309551112yes—
158C22C2×#1309551114yes—
159C4—24no—
160C4—28no—
161C4—3881no—
162C4—3882no—
163C4—3884no—
164C2—24no—
165C2—7761no—
166C2C2×#1569079311yescharacteristic
167C2#2026124612yes—
168C2#2026126914yes—
subgroupG/H
center, FrattiniC23C23×D97
commutator, socle, cosocleC23×C97C24
FittingC22×C42×C97C2
prime radicalC22×C22⋊D97C22
hypercenterC22×C42D97
hypocenterC97#2853
LCSG ▷ C23×C97 ▷ C97
UCSC1 ◁ C23 ◁ C22×C42

Complements

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

NH ≅ G/Ntype
2. #20260626C2semidirect
3. #20260619C2semidirect
5. #20260608C2semidirect
6. #20260598C2semidirect
7. C22×C42×C97C2semidirect
85. C97#2853semidirect
87. C22×C42D97semidirect

Maximal subgroups

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

subgroupindex
C2×#156907932
#202606262
#202606192
#202606142
#202606082
#202605982
C22×C42×C972
#285397