Finite Group Databasebeta

← #27157199

Subgroups — #27157199

Order 12416.

212 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 212 conjugacy classes of subgroups — too many to draw. Only its 58 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
1#20261758C211yesmaximal, characteristic
2C4×#13095521C211yesmaximal, characteristic
3#20261592C212yesmaximal, complemented
4#20261592C212yesmaximal, complemented
5#125×D97C211yesmaximal, characteristic
6C2×#13095521—21no—
7#15690877—21no—
8#15690867—21no—
9#15690865C2212yes—
10#15690865C2212yes—
11C4×#11715897—21no—
12#15690802C411yescharacteristic
13#15690801C2211yescharacteristic
14#15690801C2211yescharacteristic
15#15690800—21no—
16#15690800—21no—
17C4⋊C4×D97—21no—
18C4⋊C4×D97—21no—
19C4⋊C4×D97C411yescharacteristic
20C4⋊C4×D97C411yescharacteristic
21C42×D97C2211yescharacteristic
22#125×C97C411yescharacteristic
23#13095521—24no—
24C2×#11715897—22no—
25C2×#11715897—22no—
26C2×C4×D97—21no—
27C2×C4×D97—21no—
28C2×C4×D97C2×C411yescharacteristic
29C2×C4×D97D411yescharacteristic
30C2×C4×D97D411yescharacteristic
31C4⋊Dic97—21no—
32C4⋊Dic97—21no—
33C4⋊Dic97C2×C411yescharacteristic
34C4⋊Dic97C2×C411yescharacteristic
35#13095490—21no—
36#13095490—21no—
37#13095490—21no—
38#13095490—21no—
39#13095490—21no—
40#13095490—21no—
41C4×Dic97D411yescharacteristic
42C4×Dic97D411yescharacteristic
43C4×Dic97C2311yescharacteristic
44C4⋊C4×C97—21no—
45C4⋊C4×C97—21no—
46C4⋊C4×C97C2×C411yescharacteristic
47C4⋊C4×C97C2×C411yescharacteristic
48C42×C97C2×C411yescharacteristic
49C22×D97C22⋊C411yescharacteristic
50C2×C4×C97—21no—
51C2×C4×C97—21no—
52C2×C4×C97C22⋊C411yescharacteristic
53C2×C4×C97C22⋊C411yescharacteristic
54C2×C4×C97C22×C411yescharacteristic
55C2×Dic97—21no—
56C2×Dic97—21no—
57C2×Dic97C22⋊C411yescharacteristic
58C2×Dic97C2×D411yescharacteristic
59C2×Dic97C2×D411yescharacteristic
60C4×D97—22no—
61C4×D97—22no—
62C4×D97—41no—
63C4×D97—42no—
64C4×D97—42no—
65#11715897—24no—
66#11715897—42no—
67C22×C97C2×C22⋊C411yescharacteristic
68C2×D97—21no—
69C2×D97—21no—
70C2×D97#10412yes—
71C4×C97—21no—
72C4×C97—21no—
73C4×C97—21no—
74C4×C97—41no—
75C4×C97—41no—
76Dic97—21no—
77Dic97—21no—
78Dic97—41no—
79Dic97—41no—
80Dic97#10112yes—
81C2×C97C2×#10111yescharacteristic
82C2×C97C2×#10411yescharacteristic
83C2×C97#42111yescharacteristic
84D97—22no—
85#1528—971nomaximal, Sylow
86C97#152811yescharacteristic, complemented, Sylow
87C2×#125—971no—
88#432—971no—
89C4×M16—971no—
90#330—972no—
91#330—972no—
92C2×M16—1941no—
93#125—971no—
94#125—971no—
95#125—971no—
96#125—1941no—
97#125—1941no—
98#125#8760411yescharacteristic
99C2×C4⋊C4—971no—
100C2×C4⋊C4—971no—
101C2×C4⋊C4—1941no—
102C2×C4⋊C4—1941no—
103C2×C42—971no—
104C4⋊C8—1941no—
105C22⋊C8—1941no—
106#97—972no—
107#97—972no—
108C4×C8—1941no—
109C22×C4—971no—
110C22×C4—971no—
111C22×C4—971no—
112C22×C4—1941no—
113C22×C4—1941no—
114M16—1944no—
115C2×C8—1942no—
116C2×C8—1942no—
117C4⋊C4—21no—
118C4⋊C4—21no—
119C4⋊C4—971no—
120C4⋊C4—971no—
121C4⋊C4—1941no—
122C4⋊C4—1941no—
123C4⋊C4—1941no—
124C4⋊C4—1941no—
125C4⋊C4—1941no—
126C4⋊C4—1941no—
127C4⋊C4—1941no—
128C4⋊C4—1941no—
129C4⋊C4C2×#8760411yescharacteristic
130C4⋊C4C2×#8760411yescharacteristic
131C42—971no—
132C42—971no—
133C42—971no—
134C42C2×#8760411yescharacteristic
135C23—971no—
136C2×C4—21no—
137C2×C4—21no—
138C2×C4—971no—
139C2×C4—971no—
140C2×C4—971no—
141C2×C4—3881no—
142C2×C4—3882no—
143C2×C4—3882no—
144C2×C4—1942no—
145C2×C4—1942no—
146C2×C4—1941no—
147C2×C4—1941no—
148C2×C4C22⋊#8760411yescharacteristic
149C2×C4C22⋊#8760411yescharacteristic
150C2×C4C22×#8760411yescharacteristic
151C8—3882no—
152C8—1944no—
153C22—972no—
154C22—1941no—
155C22—1941no—
156C22C2×C22⋊#8760411yescharacteristic
157C4—21no—
158C4—21no—
159C4—21no—
160C4—41no—
161C4—41no—
162C4—972no—
163C4—3881no—
164C4—3881no—
165C4—1941no—
166C4—1941no—
167C2—1942no—
168C2D97⋊#10111yescharacteristic
169C2D97⋊#10411yescharacteristic
170C2#2026173211yescharacteristic
subgroupG/H
centerC22C2×C22⋊#87604
commutatorC2×C4×C97C22×C4
FrattiniC2×C4C22×#87604
Fitting#125×C97C4
socleC22×C97C2×C22⋊C4
prime radicalC22×D97C22⋊C4
cosocleC4×Dic97C23
hypercenter#125#87604
hypocenterC97#1528
LCSG ▷ C2×C4×C97 ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ C42 ◁ #125

Complements

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

NH ≅ G/Ntype
3. #20261592C2semidirect
4. #20261592C2semidirect
86. C97#1528semidirect

Maximal subgroups

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

subgroupindex
#202617582
C4×#130955212
#202615922
#125×D972
#152897