Finite Group Databasebeta

← #27157694

Subgroups — #27157694

Order 12416.

252 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 252 conjugacy classes of subgroups — too many to draw. Only its 66 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#20261815C211yesmaximal, characteristic
2#20261750C212yesmaximal, complemented
3#20261710C211yesmaximal, characteristic
4#20261610C212yesmaximal, complemented
5#20260718C211yesmaximal, characteristic
6#15690898—22no—
7#15690898—22no—
8C2×#13095521C2211yescharacteristic
9D97⋊M16—22no—
10D97⋊M16C2212yes—
11C2×#13095519C2211yescharacteristic
12#15690881—22no—
13#15690880—22no—
14#15690867—22no—
15C2×#13095512C2211yescharacteristic
16M16×D97C2212yes—
17#15690729C411yescharacteristic
18#15690726C412yes—
19#101×C97C411yescharacteristic
20#13095521—22no—
21#13095521—22no—
22#13095521—22no—
23#13095521D412yes—
24#13095521D412yes—
25#13095520—22no—
26#13095519—24no—
27#13095519—22no—
28#13095519—22no—
29C2×#11715897—21no—
30C2×#11715897—21no—
31#13095512—22no—
32#13095512—22no—
33C2×D388C2×C411yescharacteristic
34C2×D388Q812yes—
35C2×C4×D97D412yes—
36C2×C4×D97C2311yescharacteristic
37#13095495C2×C412yes—
38#13095494—22no—
39C8×D97—22no—
40C2×Q8×C97C2×C411yescharacteristic
41M16×C97C2×C412yes—
42C22×D97C4⋊C412yes—
43C22×D97C2×Q811yescharacteristic
44Q8×C97—22no—
45C2×C4×C97C4⋊C412yes—
46C2×C4×C97C22×C411yescharacteristic
47C2×Dic97C2×D411yescharacteristic
48D388—24no—
49D388—22no—
50C4×D97—22no—
51C4×D97—22no—
52C4×D97C2×D412yes—
53C4×D97#4112yes—
54#11715897—22no—
55#11715897—22no—
56#11715897—41no—
57#11715897—41no—
58C8×C97—22no—
59#11715895—22no—
60C22×C97C2×C4⋊C411yescharacteristic
61C2×D97—22no—
62C2×D97—41no—
63C2×D97—42no—
64C2×D97C22⋊Q812yes—
65C4×C97—22no—
66C4×C97C4×D412yes—
67Dic97#12011yescharacteristic
68Dic97#12311yescharacteristic
69C2×C97—21no—
70C2×C97#38511yescharacteristic
71D97—22no—
72D97—42no—
73#1858—971nomaximal, Sylow
74C97#185811yescharacteristic, complemented, Sylow
75#568—971no—
76#430—972no—
77#413—971no—
78#408—971no—
79#341—972no—
80C2×#41—971no—
81#131—1942no—
82#131—1942no—
83C2×M16—972no—
84C2×M16—972no—
85C2×M16—971no—
86C2×M16—1942no—
87C22×C8—971no—
88#108—1942no—
89#108—1942no—
90#101—971no—
91#101#8760411yescharacteristic
92#100—972no—
93C22⋊C8—1942no—
94#41—1942no—
95#41—1942no—
96C2×Q8C2×#8760411yescharacteristic
97C2×D4—972no—
98C2×D4—971no—
99C22×C4—971no—
100C22×C4—972no—
101M16—972no—
102M16—972no—
103M16—972no—
104M16—1942no—
105M16—1942no—
106M16—1942no—
107M16—1942no—
108M16—1942no—
109M16C2×#8760412yes—
110C2×C8—1942no—
111C2×C8—1941no—
112C2×C8—1941no—
113C2×C8—1942no—
114C2×C8—1942no—
115C2×C8—1944no—
116C23—971no—
117C23—972no—
118Q8—22no—
119D4—1944no—
120D4—1942no—
121C2×C4—971no—
122C2×C4—972no—
123C2×C4—972no—
124C2×C4—1942no—
125C2×C4—1942no—
126C2×C4C4⋊#8760412yes—
127C2×C4C22×#8760411yescharacteristic
128C8—22no—
129C8—3881no—
130C8—3881no—
131C8—1942no—
132C8—1942no—
133C8—1942no—
134C22—972no—
135C22—3882no—
136C22—3881no—
137C22—1942no—
138C22C2×C4⋊#8760411yescharacteristic
139C4—22no—
140C4—971no—
141C4—971no—
142C4D4×#8760412yes—
143C2—21no—
144C2—3882no—
145C2—1942no—
146C2#2026168211yescharacteristic
subgroupG/H
centerC2#20261682
commutatorC2×C4×C97C22×C4
FrattiniC2×C4C22×#87604
Fitting#101×C97C4
socleC2×C97#385
prime radicalC2×#13095512C22
cosocleC2×C4×D97C23
hypercenter#101#87604
hypocenterC97#1858
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C2 ◁ C2×C4 ◁ #101

Complements

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

NH ≅ G/Ntype
2. #20261750C2semidirect
4. #20261610C2semidirect
74. C97#1858semidirect

Maximal subgroups

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

subgroupindex
#202618152
#202617502
#202617102
#202616102
#202607182
#185897