Finite Group Databasebeta

← D97⋊#568 #27158728

Subgroups — D97⋊#568 #27158728

Order 12416.

1460 conjugacy classes of subgroups.

The subgroup lattice has 230 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20261815C212yesmaximal, complemented
2#20261814C218yesmaximal, complemented
3#20261813C216yesmaximal, complemented
4D97⋊#131C218yesmaximal, complemented
5C2×D97⋊M16C216yesmaximal, complemented
6C2×#41×D97C211yesmaximal, characteristic
7#15690898C22116yescomplemented
8#15690897C22148yescomplemented
9C2×#13095521C22112yescomplemented
10D97⋊M16C2214yes—
11D97⋊M16C22112yescomplemented
12D97⋊M16C22112yescomplemented
13D97⋊M16C22112yescomplemented
14C2×#13095520C22112yescomplemented
15C2×#13095519C22112yescomplemented
16C2×#13095512C2211yescharacteristic
17C2×#13095511C413yes—
18C2×#13095510C2213yes—
19#41×D97C414yes—
20#41×D97C414yes—
21#41×D97C2214yes—
22#41×D97C2214yes—
23C2×Q8×D97C411yescharacteristic
24C2×D4×D97C413yes—
25C22×C4×D97C2213yes—
26C2×#41×C97C411yescharacteristic
27#13095521C2318yes—
28#13095521C23124yescomplemented
29#13095520C2318yes—
30#13095520C23124yescomplemented
31#13095519C2318yes—
32#13095519C23124yescomplemented
33C2×#11715897C23124yescomplemented
34C23×D97C2×C413yes—
35#13095512C2×C414yes—
36#13095512C2314yes—
37#13095511C2×C4112yes—
38#13095511C2×C4112yes—
39#13095510C2×C4112yes—
40#13095510C23112yes—
41C2×Dic194C2×C413yes—
42Q8×D97C2×C414yes—
43Q8×D97C2×C414yes—
44C2×D388C2×C413yes—
45C2×C22⋊D97C2×C416yes—
46D4×D97C2×C4112yes—
47D4×D97C2×C4112yes—
48C22×Dic97C2313yes—
49C2×C4×D97C2×C413yes—
50C2×C4×D97C2×C416yes—
51C2×C4×D97C2×C416yes—
52C2×C4×D97C2×C413yes—
53C2×C4×D97C2311yescharacteristic
54C2×C4×D97C2316yes—
55C2×C4×D97C2316yes—
56C2×C4×D97C2313yes—
57#41×C97C2×C414yes—
58#41×C97C2×C414yes—
59C2×Q8×C97C2×C411yescharacteristic
60C2×D4×C97C2×C413yes—
61C22×C4×C97C2×C413yes—
62C23×C97C22×C413yes—
63C22×D97—46no—
64C22×D97C22×C411yescharacteristic
65C22×D97C22×C416yes—
66C22×D97C22×C416yes—
67C22×D97C22×C413yes—
68Q8×C97C22×C414yes—
69D4×C97C22×C4112yes—
70C2×C4×C97C22×C411yescharacteristic
71C2×C4×C97C22×C413yes—
72C2×C4×C97C22×C416yes—
73C2×C4×C97C22×C416yes—
74C22⋊D97C22×C4124yes—
75C2×Dic97C22×C413yes—
76C2×Dic97C22×C416yes—
77C2×Dic97C2411yescharacteristic
78C2×Dic97C2416yes—
79D388C22×C4112yes—
80C4×D97C22×C411yescharacteristic
81C4×D97C22×C411yescharacteristic
82C4×D97C22×C416yes—
83C4×D97C22×C416yes—
84C4×D97C22×C416yes—
85C4×D97C22×C416yes—
86C4×D97C22×C416yes—
87C4×D97C2411yescharacteristic
88C4×D97C2411yescharacteristic
89C4×D97C2416yes—
90Dic194C22×C4112yes—
91#11715897C24116yes—
92C22×C97—43no—
93C22×C97C23×C411yescharacteristic
94C22×C97C23×C416yes—
95C2×D97—22no—
96C2×D97—212no—
97C2×D97—46no—
98C2×D97—43no—
99C2×D97C23×C411yescharacteristic
100C2×D97C23×C416yes—
101C2×D97C23×C411yescharacteristic
102C4×C97C23×C416yes—
103C4×C97C23×C411yescharacteristic
104C4×C97C23×C411yescharacteristic
105Dic97C23×C416yes—
106Dic97C23×C411yescharacteristic
107Dic97C2511yescharacteristic
108C2×C97—21no—
109C2×C97—26no—
110C2×C97C24×C411yescharacteristic
111D97—21no—
112D97—26no—
113D97#56812yescomplemented
114C2×#568—971nomaximal, Sylow
115C97C2×#56811yescharacteristic, complemented, Sylow
116C22×#41—971no—
117#568—976no—
118#568—978no—
119#568—972no—
120C2×#131—978no—
121C22×M16—976no—
122C2×#41—974no—
123C2×#41—974no—
124C2×#41—974no—
125C2×#41—974no—
126C2×#41—971no—
127C2×#41—973no—
128C2×#41—973no—
129C2×#41#8760411yescharacteristic
130C22×Q8—971no—
131C22×D4—973no—
132C23×C4—973no—
133#131—9748no—
134#131—9716no—
135C2×M16—974no—
136C2×M16—9712no—
137C2×M16—9712no—
138C2×M16—9712no—
139C2×M16—9712no—
140C2×M16—9712no—
141C22×C8—9712no—
142C24—973no—
143#41—974no—
144#41—974no—
145#41—9712no—
146#41—9712no—
147#41—9712no—
148#41—9712no—
149#41C2×#8760414yes—
150#41C2×#8760414yes—
151C2×Q8—974no—
152C2×Q8—974no—
153C2×Q8—973no—
154C2×Q8C2×#8760411yescharacteristic
155C2×D4—976no—
156C2×D4—9712no—
157C2×D4—9712no—
158C2×D4—973no—
159C2×D4C2×#8760413yes—
160C22×C4—973no—
161C22×C4—973no—
162C22×C4—973no—
163C22×C4—976no—
164C22×C4—976no—
165C22×C4—976no—
166C22×C4—971no—
167C22×C4—976no—
168C22×C4—973no—
169C22×C4C2×#8760413yes—
170M16—978no—
171M16—9724no—
172M16—978no—
173M16—9724no—
174C2×C8—9724no—
175C2×C8—978no—
176C2×C8—9724no—
177C23—976no—
178C23—973no—
179C23—971no—
180C23—976no—
181C23—3886no—
182C23C22×#8760413yes—
183Q8—9712no—
184Q8C22×#8760414yes—
185D4—9724no—
186D4—9712no—
187D4C22×#87604112yes—
188C2×C4—976no—
189C2×C4—971no—
190C2×C4—973no—
191C2×C4—976no—
192C2×C4—976no—
193C2×C4—971no—
194C2×C4—971no—
195C2×C4—976no—
196C2×C4—976no—
197C2×C4—976no—
198C2×C4—976no—
199C2×C4—971no—
200C2×C4—971no—
201C2×C4—976no—
202C2×C4C22×#8760413yes—
203C2×C4C22×#8760416yes—
204C2×C4C22×#8760411yescharacteristic
205C2×C4C22×#8760416yes—
206C8—9716no—
207C22—43no—
208C22—971no—
209C22—971no—
210C22—976no—
211C22—3883no—
212C22—3886no—
213C22—1942no—
214C22—19412no—
215C22C23×#8760411yescharacteristic
216C22C23×#8760416yes—
217C4—971no—
218C4—971no—
219C4—976no—
220C4C23×#8760411yescharacteristic
221C4C23×#8760416yes—
222C4C23×#8760411yescharacteristic
223C2—21no—
224C2—26no—
225C2—972no—
226C2—1941no—
227C2—1946no—
228C2C24×#8760411yescharacteristic
subgroupG/H
centerC4C23×#87604
commutator, socleC2×C97C24×C4
FrattiniC2C24×#87604
FittingC2×#41×C97C4
prime radicalC2×#41×D97C2
cosocleDic97C25
hypercenterC2×#41#87604
hypocenterC97C2×#568
LCSG ▷ C2×C97 ▷ C97
UCSC1 ◁ C4 ◁ C2×#41

Complements

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

NH ≅ G/Ntype
1. #20261815C2semidirect
2. #20261814C2semidirect
3. #20261813C2semidirect
4. D97⋊#131C2semidirect
5. C2×D97⋊M16C2semidirect
7. #15690898C22semidirect
8. #15690897C22semidirect
9. C2×#13095521C22semidirect
11. D97⋊M16C22semidirect
12. D97⋊M16C22semidirect
13. D97⋊M16C22semidirect
14. C2×#13095520C22semidirect
15. C2×#13095519C22semidirect
28. #13095521C23semidirect
30. #13095520C23semidirect
32. #13095519C23semidirect
33. C2×#11715897C23semidirect
113. D97#568semidirect
115. C97C2×#568semidirect

Maximal subgroups

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

subgroupindex
#202618152
#202618142
#202618132
D97⋊#1312
C2×D97⋊M162
C2×#41×D972
C2×#56897