Finite Group Databasebeta

← #27157997

Subgroups — #27157997

Order 12416.

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

normal subgroups only

This group has 266 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#20261721C211yesmaximal, characteristic, complemented
2#20261714C211yesmaximal, characteristic, complemented
3#20261636C214yesmaximal, complemented
4#20261304C211yesmaximal, characteristic
5D97⋊M16—21no—
6#15690889—21no—
7#15690871—21no—
8#15690870C2212yes—
9#15690869—21no—
10#15690868C2212yes—
11C2×#13095511C2211yescharacteristic
12C22×D388C411yescharacteristic, complemented
13#15690807—21no—
14#15690798—21no—
15#15690785—21no—
16#15690785—21no—
17#15690780C412yescomplemented
18#15690780C2212yes—
19#117×C97C411yescharacteristic, complemented
20#13095521—41no—
21#13095520—22no—
22#13095519—41no—
23C4⋊#87604—41no—
24C22⋊#87604—22no—
25C4×#87604—41no—
26C23×D97C2×C412yes—
27#13095511—41no—
28#13095511—41no—
29C2×Dic194D411yescharacteristic
30C2×D388—21no—
31C2×D388—21no—
32C2×D388—22no—
33C2×D388—22no—
34C2×D388—41no—
35C2×D388—41no—
36C2×D388D411yescharacteristic
37C2×C22⋊D97C2×C411yescharacteristic
38C2×C22⋊D97C2311yescharacteristic
39C2×C4×D97D411yescharacteristic
40C2×C4×D97D411yescharacteristic
41#13095488—22no—
42#13095488—22no—
43C22×C4×C97C2×C411yescharacteristic
44C4⋊C4×C97—21no—
45C22⋊C4×C97C2×C412yes—
46C42×C97—21no—
47C23×C97C22×C411yescharacteristic
48C22×D97—22no—
49C22×D97—42no—
50C22×D97—82no—
51C22×D97C22⋊C411yescharacteristic
52C22×D97C2×D411yescharacteristic
53C2×#87604—42no—
54C2×C4×C97—21no—
55C2×C4×C97—21no—
56C2×C4×C97—22no—
57C2×C4×C97C22⋊C411yescharacteristic
58C2×C4×C97C22⋊C411yescharacteristic
59C22⋊D97—41no—
60C22⋊D97—41no—
61C2×Dic97C22⋊C411yescharacteristic
62C2×Dic97C2×D411yescharacteristic
63D388—41no—
64D388—42no—
65D388—41no—
66D388—41no—
67D388—82no—
68D388—82no—
69C4×D97—21no—
70C4×D97—21no—
71C4×D97—21no—
72C4×D97—21no—
73Dic194—41no—
74#11715897—42no—
75C22×C97—21no—
76C22×C97—41no—
77C22×C97C2×C22⋊C411yescharacteristic
78C2×D97—21no—
79C2×D97—21no—
80C2×D97—41no—
81C2×D97—41no—
82C2×D97—42no—
83C2×D97—42no—
84C2×D97—42no—
85C2×D97—84no—
86#87604—82no—
87C4×C97—21no—
88C4×C97—42no—
89C4×C97#9911yescharacteristic
90C4×C97#9911yescharacteristic
91Dic97—21no—
92Dic97—21no—
93C2×C97—21no—
94C2×C97—41no—
95C2×C97C2×#9911yescharacteristic
96D97—41no—
97D97—41no—
98D97—82no—
99#2122—971nomaximal, Sylow
100C97#212211yescharacteristic, complemented, Sylow
101#535—971no—
102#413—971no—
103#410—971no—
104AΣL(1,𝔽16)—974no—
105C2×#41—971no—
106C22×D4—971no—
107C2×M16—1941no—
108#127—1941no—
109#124—1941no—
110#121—1941no—
111#121—1941no—
112#120—972no—
113#120—972no—
114#117—1941no—
115#117#8760411yescharacteristic, complemented
116#101—1941no—
117#100—972no—
118#99—972no—
119#99—1941no—
120C24—972no—
121#41—3881no—
122#41—3881no—
123C2×Q8—971no—
124C2×D4—971no—
125C2×D4—971no—
126C2×D4—971no—
127C2×D4—3881no—
128C2×D4—3881no—
129C2×D4—1942no—
130C2×D4—1942no—
131C2×D4—1941no—
132C2×D4—1941no—
133C22×C4—971no—
134C22×C4—971no—
135C22×C4C2×#8760411yescharacteristic
136M16—3881no—
137M16—1942no—
138C2×C8—3881no—
139C4⋊C4—21no—
140C4⋊C4—3881no—
141C22⋊C4—1942no—
142C22⋊C4—1942no—
143C22⋊C4—1942no—
144C22⋊C4C2×#8760412yes—
145C42—21no—
146C42—3881no—
147C23—7762no—
148C23—971no—
149C23—971no—
150C23—3882no—
151C23—1942no—
152C23C22×#8760411yescharacteristic
153Q8—3881no—
154D4—7762no—
155D4—7762no—
156D4—3882no—
157D4—3881no—
158D4—3881no—
159D4—3881no—
160D4—3881no—
161D4—3881no—
162C2×C4—21no—
163C2×C4—21no—
164C2×C4—22no—
165C2×C4—971no—
166C2×C4—971no—
167C2×C4—3882no—
168C2×C4—1941no—
169C2×C4—1941no—
170C2×C4—1941no—
171C2×C4—1941no—
172C2×C4C22⋊#8760411yescharacteristic
173C2×C4C22⋊#8760411yescharacteristic
174C8—3882no—
175C22—21no—
176C22—41no—
177C22—7764no—
178C22—3882no—
179C22—3882no—
180C22—3881no—
181C22—3881no—
182C22—3882no—
183C22—1941no—
184C22—1941no—
185C22C2×C22⋊#8760411yescharacteristic
186C4—21no—
187C4—42no—
188C4—7762no—
189C4—1941no—
190C4—1941no—
191C4#1569086911yescharacteristic
192C4#1569086911yescharacteristic
193C2—21no—
194C2—41no—
195C2—7762no—
196C2—3881no—
197C2—3881no—
198C2C2×#1569086911yescharacteristic
subgroupG/H
centerC2C2×#15690869
commutatorC23×C97C22×C4
FrattiniC23C22×#87604
Fitting#117×C97C4
socleC2×C97C2×#99
prime radical#20261304C2
cosocleC2×C22⋊D97C23
hypercenter#117#87604
hypocenterC97#2122
LCSG ▷ C23×C97 ▷ C22×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C2 ◁ C2×C4 ◁ C22×C4 ◁ #117

Complements

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

NH ≅ G/Ntype
1. #20261721C2semidirect
2. #20261714C2semidirect
3. #20261636C2semidirect
12. C22×D388C4semidirect
17. #15690780C4semidirect
19. #117×C97C4semidirect
100. C97#2122semidirect
115. #117#87604semidirect

Maximal subgroups

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

subgroupindex
#202617212
#202617142
#202616362
#202613042
#212297