Finite Group Databasebeta

← #27158009

Subgroups — #27158009

Order 12416.

258 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 258 conjugacy classes of subgroups — too many to draw. Only its 54 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#20261714C212yesmaximal, complemented
2#20261642C212yesmaximal, complemented
3#20261640C212yesmaximal, complemented
4#20261285C211yesmaximal, characteristic
5#15690889—22no—
6#15690869C2214yes—
7#15690868—22no—
8C2×#13095511C411yescharacteristic, complemented
9C2×#13095511C2211yescharacteristic
10#15690810C411yescharacteristic, complemented
11#15690807C2211yescharacteristic
12#15690798C411yescharacteristic, complemented
13#15690798C2211yescharacteristic
14#15690784—22no—
15C4×D388—22no—
16C2×C42×C97C411yescharacteristic, complemented
17C4⋊#87604—42no—
18C22⋊#87604—24no—
19C4×#87604—42no—
20#13095511—41no—
21#13095511—41no—
22#13095511—41no—
23#13095511—41no—
24C2×Dic194C2×C411yescharacteristic
25C2×Dic194C2×C411yescharacteristic
26C2×D388—22no—
27C2×D388C2×C411yescharacteristic
28C2×D388C2311yescharacteristic
29C2×C22⋊D97—21no—
30C2×C22⋊D97D412yes—
31C2×C4×D97—21no—
32C2×C4×D97D412yes—
33C4⋊Dic97—22no—
34#13095488—22no—
35#13095488—22no—
36C22×C4×C97—21no—
37C22×C4×C97C2×C411yescharacteristic
38C42×C97—21no—
39C42×C97C2×C411yescharacteristic
40C42×C97C2×C411yescharacteristic
41C23×C97C22⋊C411yescharacteristic
42C22×D97—21no—
43C22×D97C2×D412yes—
44C2×#87604—44no—
45C2×C4×C97—21no—
46C2×C4×C97—21no—
47C2×C4×C97—21no—
48C2×C4×C97—21no—
49C2×C4×C97—42no—
50C2×C4×C97C22⋊C411yescharacteristic
51C2×C4×C97C22×C411yescharacteristic
52C22⋊D97—42no—
53C22⋊D97—42no—
54C2×Dic97—21no—
55C2×Dic97C22⋊C412yes—
56D388—41no—
57D388—41no—
58D388—44no—
59C4×D97—22no—
60C4×D97—22no—
61C4×D97—41no—
62C4×D97—41no—
63Dic194—41no—
64Dic194—41no—
65C22×C97—21no—
66C22×C97—41no—
67C22×C97C2×C22⋊C411yescharacteristic
68C2×D97—22no—
69C2×D97—42no—
70C2×D97—42no—
71C2×D97—41no—
72#87604—84no—
73C4×C97—22no—
74C4×C97—21no—
75C4×C97—41no—
76C4×C97#9911yescharacteristic
77C4×C97#9911yescharacteristic
78Dic97—22no—
79Dic97—41no—
80C2×C97—21no—
81C2×C97—41no—
82C2×C97C2×#9911yescharacteristic
83D97—42no—
84D97—81no—
85#2126—971nomaximal, Sylow
86C97#212611yescharacteristic, complemented, Sylow
87#532—971no—
88#410—972no—
89#354—972no—
90#353—972no—
91C2×#41—971no—
92C2×#41—971no—
93C4⋊Q8—971no—
94#127—971no—
95#124—971no—
96#124—971no—
97#121—1942no—
98C4×D4—1942no—
99#117—1942no—
100C2×C42#8760411yescharacteristic, complemented
101#99—974no—
102#99—1942no—
103#41—3881no—
104#41—3881no—
105#41—3881no—
106#41—3881no—
107C2×Q8—971no—
108C2×Q8—971no—
109C2×D4—972no—
110C2×D4—971no—
111C2×D4—971no—
112C2×D4—1942no—
113C2×D4—1941no—
114C22×C4—21no—
115C22×C4—972no—
116C22×C4—1941no—
117C22×C4C2×#8760411yescharacteristic
118C4⋊C4—3882no—
119C4⋊C4—1942no—
120C22⋊C4—1942no—
121C22⋊C4—1942no—
122C22⋊C4—1944no—
123C42—21no—
124C42—3882no—
125C42C2×#8760411yescharacteristic
126C42C2×#8760411yescharacteristic
127C23—972no—
128C23—1941no—
129C23C22⋊#8760411yescharacteristic
130Q8—3881no—
131Q8—3881no—
132D4—3881no—
133D4—3884no—
134D4—3881no—
135D4—3882no—
136D4—3882no—
137C2×C4—21no—
138C2×C4—21no—
139C2×C4—21no—
140C2×C4—21no—
141C2×C4—42no—
142C2×C4—972no—
143C2×C4—3884no—
144C2×C4—3881no—
145C2×C4—3881no—
146C2×C4—1941no—
147C2×C4—1942no—
148C2×C4—1942no—
149C2×C4C22⋊#8760411yescharacteristic
150C2×C4C22×#8760411yescharacteristic
151C22—21no—
152C22—41no—
153C22—3882no—
154C22—3882no—
155C22—3881no—
156C22—1942no—
157C22C2×C22⋊#8760411yescharacteristic
158C4—22no—
159C4—21no—
160C4—41no—
161C4—7764no—
162C4—3881no—
163C4—1942no—
164C4#1569086911yescharacteristic
165C4#1569086911yescharacteristic
166C2—21no—
167C2—41no—
168C2—7761no—
169C2—3882no—
170C2C2×#1569086911yescharacteristic
subgroupG/H
centerC4#15690869
commutatorC2×C4×C97C22×C4
FrattiniC2×C4C22×#87604
FittingC2×C42×C97C4
socleC2×C97C2×#99
prime radical#20261285C2
cosocleC2×D388C23
hypercenterC2×C42#87604
hypocenterC97#2126
LCSG ▷ C2×C4×C97 ▷ C22×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C4 ◁ C2×C4 ◁ C22×C4 ◁ C2×C42

Complements

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

NH ≅ G/Ntype
1. #20261714C2semidirect
2. #20261642C2semidirect
3. #20261640C2semidirect
8. C2×#13095511C4semidirect
10. #15690810C4semidirect
12. #15690798C4semidirect
16. C2×C42×C97C4semidirect
86. C97#2126semidirect
100. C2×C42#87604semidirect

Maximal subgroups

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

subgroupindex
#202617142
#202616422
#202616402
#202612852
#212697