Finite Group Databasebeta

← #27157987

Subgroups — #27157987

Order 12416.

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

normal subgroups only

This group has 264 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#20261637C212yesmaximal, complemented
4#20261635C212yesmaximal, complemented
5#20261246C211yesmaximal, characteristic
6D97⋊M16—21no—
7#15690889—21no—
8#15690871—21no—
9#15690870C2212yes—
10#15690869—21no—
11#15690868C2212yes—
12C2×#13095511C2211yescharacteristic
13#15690793C411yescharacteristic, complemented
14#15690793C2211yescharacteristic
15#15690789—21no—
16#15690784—21no—
17#15690779C411yescharacteristic, complemented
18#15690779C2211yescharacteristic
19C4×C22⋊D97—21no—
20C4×C22⋊D97—21no—
21#15690769C411yescharacteristic, complemented
22C23×C4×C97C411yescharacteristic, complemented
23#13095521—41no—
24#13095520—22no—
25#13095519—41no—
26C4⋊#87604—41no—
27C22⋊#87604—22no—
28C4×#87604—41no—
29#13095511—41no—
30#13095511—41no—
31C2×Dic194D411yescharacteristic
32C2×D388D411yescharacteristic
33C2×C22⋊D97—21no—
34C2×C22⋊D97—21no—
35C2×C22⋊D97C2×C411yescharacteristic
36C2×C22⋊D97C2311yescharacteristic
37C2×C4×D97D411yescharacteristic
38C2×C4×D97D411yescharacteristic
39C4⋊Dic97—21no—
40#13095490—21no—
41#13095490—21no—
42C22⋊Dic97—21no—
43C22⋊Dic97—21no—
44C22⋊Dic97C2×C411yescharacteristic
45C22⋊Dic97C2×C411yescharacteristic
46#13095488—21no—
47#13095488—21no—
48C4×Dic97—21no—
49C24×C97C2×C411yescharacteristic
50C22×C4×C97—21no—
51C22×C4×C97—21no—
52C22×C4×C97—41no—
53C22×C4×C97—41no—
54C22×C4×C97C2×C411yescharacteristic
55C22×C4×C97C2×C411yescharacteristic
56C23×C97—21no—
57C23×C97—41no—
58C23×C97—42no—
59C23×C97C22×C411yescharacteristic
60C22×D97C22⋊C411yescharacteristic
61C22×D97C2×D411yescharacteristic
62C2×#87604—42no—
63C2×C4×C97—21no—
64C2×C4×C97—21no—
65C2×C4×C97—21no—
66C2×C4×C97—42no—
67C2×C4×C97—41no—
68C2×C4×C97—41no—
69C2×C4×C97—41no—
70C2×C4×C97C22⋊C411yescharacteristic
71C2×C4×C97C22⋊C411yescharacteristic
72C22⋊D97—41no—
73C22⋊D97—41no—
74C22⋊D97—42no—
75C22⋊D97—42no—
76C2×Dic97—21no—
77C2×Dic97—21no—
78C2×Dic97C22⋊C411yescharacteristic
79C2×Dic97C2×D411yescharacteristic
80D388—41no—
81C4×D97—21no—
82C4×D97—21no—
83C4×D97—21no—
84C4×D97—21no—
85Dic194—41no—
86#11715897—42no—
87C22×C97—22no—
88C22×C97—21no—
89C22×C97—41no—
90C22×C97—41no—
91C22×C97—42no—
92C22×C97—42no—
93C22×C97—41no—
94C22×C97C2×C22⋊C411yescharacteristic
95C2×D97—21no—
96C2×D97—21no—
97C2×D97—41no—
98C2×D97—41no—
99#87604—82no—
100C4×C97—21no—
101C4×C97—41no—
102C4×C97#9911yescharacteristic
103C4×C97#9911yescharacteristic
104Dic97—21no—
105Dic97—21no—
106Dic97—41no—
107Dic97—41no—
108C2×C97—21no—
109C2×C97—42no—
110C2×C97—41no—
111C2×C97C2×#9911yescharacteristic
112D97—41no—
113D97—41no—
114#2120—971nomaximal, Sylow
115C97#212011yescharacteristic, complemented, Sylow
116#525—971no—
117#413—971no—
118#410—971no—
119#352—972no—
120AΣL(1,𝔽16)—972no—
121C2×#41—971no—
122C23×C4#8760411yescharacteristic, complemented
123C2×M16—1941no—
124#123—971no—
125#123—971no—
126C22⋊Q8—1941no—
127#121—1941no—
128#120—971no—
129#120—971no—
130C4×D4—1941no—
131C4×D4—1941no—
132#117—971no—
133#117—1941no—
134#101—1941no—
135#100—972no—
136#99—972no—
137#99—1941no—
138C24C2×#8760411yescharacteristic
139#41—3881no—
140#41—3881no—
141C2×Q8—971no—
142C2×D4—971no—
143C2×D4—971no—
144C2×D4—971no—
145C2×D4—1941no—
146C2×D4—1941no—
147C22×C4—21no—
148C22×C4—21no—
149C22×C4—41no—
150C22×C4—41no—
151C22×C4—971no—
152C22×C4—971no—
153C22×C4C2×#8760411yescharacteristic
154C22×C4C2×#8760411yescharacteristic
155M16—3881no—
156M16—1942no—
157C2×C8—3881no—
158C4⋊C4—3881no—
159C4⋊C4—1941no—
160C4⋊C4—1941no—
161C4⋊C4—1941no—
162C22⋊C4—971no—
163C22⋊C4—971no—
164C22⋊C4—1941no—
165C22⋊C4—1941no—
166C22⋊C4—1941no—
167C22⋊C4—1941no—
168C22⋊C4—1942no—
169C42—3881no—
170C42—1941no—
171C23—21no—
172C23—41no—
173C23—42no—
174C23—971no—
175C23—971no—
176C23C22×#8760411yescharacteristic
177Q8—3881no—
178D4—3881no—
179D4—3882no—
180D4—3882no—
181D4—3881no—
182D4—3881no—
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—21no—
186C2×C4—42no—
187C2×C4—41no—
188C2×C4—41no—
189C2×C4—41no—
190C2×C4—971no—
191C2×C4—971no—
192C2×C4—3882no—
193C2×C4—1941no—
194C2×C4—1941no—
195C2×C4—1941no—
196C2×C4—1941no—
197C2×C4—1941no—
198C2×C4—1941no—
199C2×C4C22⋊#8760411yescharacteristic
200C2×C4C22⋊#8760411yescharacteristic
201C8—3882no—
202C22—22no—
203C22—21no—
204C22—41no—
205C22—41no—
206C22—42no—
207C22—42no—
208C22—41no—
209C22—3881no—
210C22—3881no—
211C22—1941no—
212C22—1941no—
213C22C2×C22⋊#8760411yescharacteristic
214C4—21no—
215C4—41no—
216C4—7762no—
217C4—3881no—
218C4—3881no—
219C4—1941no—
220C4—1941no—
221C4#1569086811yescharacteristic
222C4#1569086811yescharacteristic
223C2—21no—
224C2—42no—
225C2—41no—
226C2—3881no—
227C2—3881no—
228C2C2×#1569086811yescharacteristic
subgroupG/H
centerC4#15690868
commutatorC23×C97C22×C4
FrattiniC23C22×#87604
FittingC23×C4×C97C4
socleC2×C97C2×#99
prime radical#20261246C2
cosocleC2×C22⋊D97C23
hypercenterC23×C4#87604
hypocenterC97#2120
LCSG ▷ C23×C97 ▷ C22×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C4 ◁ C2×C4 ◁ C22×C4 ◁ C23×C4

Complements

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

NH ≅ G/Ntype
1. #20261721C2semidirect
2. #20261714C2semidirect
3. #20261637C2semidirect
4. #20261635C2semidirect
13. #15690793C4semidirect
17. #15690779C4semidirect
21. #15690769C4semidirect
22. C23×C4×C97C4semidirect
115. C97#2120semidirect
122. C23×C4#87604semidirect

Maximal subgroups

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

subgroupindex
#202617212
#202617142
#202616372
#202616352
#202612462
#212097