Finite Group Databasebeta

← D97⋊#352 #27157985

Subgroups — D97⋊#352 #27157985

Order 12416.

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

normal subgroups only

This group has 278 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
1D97⋊#100C211yesmaximal, characteristic
2D97⋊#99C211yesmaximal, characteristic
3#20261638C212yesmaximal, complemented
4#20261637C212yesmaximal, complemented
5#123×D97C211yesmaximal, characteristic
6D97⋊M16—21no—
7C2×C22⋊#87604—21no—
8#15690870—21no—
9#15690870C2212yes—
10#15690869—21no—
11#15690868C2212yes—
12C2×D4×D97C2211yescharacteristic
13C22×C4×D97C411yescharacteristic, complemented
14#15690795—21no—
15#15690794C2211yescharacteristic
16#15690793C2211yescharacteristic
17#15690792—21no—
18#15690791C411yescharacteristic, complemented
19C4⋊C4×D97—21no—
20C22⋊C4×D97—21no—
21C22⋊C4×D97C411yescharacteristic, complemented
22#123×C97C411yescharacteristic, complemented
23C22×#87604—41no—
24#13095521—41no—
25#13095520—22no—
26#13095519—41no—
27C22⋊#87604—22no—
28C22⋊#87604—41no—
29C23×D97C2×C411yescharacteristic
30C23×D97D411yescharacteristic
31C2×D388D411yescharacteristic
32C2×C22⋊D97D411yescharacteristic
33C2×C22⋊D97C2311yescharacteristic
34D4×D97—41no—
35D4×D97—41no—
36C22×Dic97C2×C411yescharacteristic, complemented
37C2×C4×D97—21no—
38C2×C4×D97—21no—
39C2×C4×D97—21no—
40C2×C4×D97—41no—
41C2×C4×D97—41no—
42C2×C4×D97D411yescharacteristic
43C4⋊Dic97—21no—
44#13095490—21no—
45#13095490—21no—
46C22⋊Dic97—21no—
47C22⋊Dic97C2×C411yescharacteristic, complemented
48#13095488—21no—
49#13095488—21no—
50#13095488—21no—
51C2×D4×C97C2×C411yescharacteristic
52C22×C4×C97C2×C411yescharacteristic, complemented
53C4⋊C4×C97—21no—
54C22⋊C4×C97—21no—
55C22⋊C4×C97C2×C411yescharacteristic, complemented
56C23×C97C22⋊C411yescharacteristic
57C23×C97C22×C411yescharacteristic
58C22×D97—21no—
59C22×D97—21no—
60C22×D97—21no—
61C22×D97—21no—
62C22×D97—42no—
63C22×D97—42no—
64C22×D97C22⋊C411yescharacteristic
65C22×D97C22⋊C411yescharacteristic
66C22×D97C2×D411yescharacteristic
67C2×#87604—42no—
68C2×#87604—82no—
69D4×C97—41no—
70C2×C4×C97—21no—
71C2×C4×C97—21no—
72C2×C4×C97—41no—
73C2×C4×C97C22⋊C411yescharacteristic
74C22⋊D97—41no—
75C22⋊D97—41no—
76C2×Dic97—21no—
77C2×Dic97—21no—
78C2×Dic97—41no—
79C2×Dic97C2×D411yescharacteristic
80D388—41no—
81C4×D97—21no—
82C4×D97—21no—
83C4×D97—42no—
84C4×D97—42no—
85C4×D97—41no—
86C4×D97—41no—
87#11715897—42no—
88C22×C97—21no—
89C22×C97—21no—
90C22×C97—41no—
91C22×C97—41no—
92C22×C97C2×C22⋊C411yescharacteristic
93C2×D97—22no—
94C2×D97—21no—
95C2×D97—21no—
96C2×D97—42no—
97C2×D97—42no—
98C2×D97—42no—
99C2×D97—41no—
100C2×D97—41no—
101C2×D97—81no—
102C2×D97#9911yescharacteristic
103C2×D97#9911yescharacteristic
104#87604—82no—
105C4×C97—21no—
106C4×C97—41no—
107C4×C97—41no—
108Dic97—21no—
109Dic97—41no—
110Dic97—41no—
111C2×C97—21no—
112C2×C97—41no—
113C2×C97—41no—
114C2×C97C2×#9911yescharacteristic
115D97—21no—
116D97—41no—
117D97—41no—
118D97#35212yescomplemented
119C2×#352—971nomaximal, Sylow
120C97C2×#35211yescharacteristic, complemented, Sylow
121C2×#123—971no—
122C2×#100—971no—
123C2×#99—971no—
124#352—972no—
125#352—972no—
126C22×D4—971no—
127C23×C4—971no—
128C2×M16—1941no—
129#123—971no—
130#123—971no—
131#123—971no—
132#123—1941no—
133#123—1941no—
134#123#8760411yescharacteristic, complemented
135C2×C4⋊C4—1941no—
136C2×C22⋊C4—971no—
137C2×C22⋊C4—1941no—
138C2×C22⋊C4—1941no—
139#100—972no—
140#100—1941no—
141#99—972no—
142#99—1941no—
143C24—971no—
144C24—971no—
145C2×D4—971no—
146C2×D4—971no—
147C2×D4—971no—
148C2×D4—3881no—
149C2×D4—3881no—
150C2×D4C2×#8760411yescharacteristic
151C22×C4—971no—
152C22×C4—971no—
153C22×C4—3881no—
154C22×C4—3881no—
155C22×C4—3881no—
156C22×C4—1941no—
157C22×C4—1941no—
158C22×C4—1941no—
159C22×C4C2×#8760411yescharacteristic, complemented
160M16—3881no—
161M16—1942no—
162C2×C8—3881no—
163C4⋊C4—21no—
164C4⋊C4—1941no—
165C4⋊C4—1941no—
166C4⋊C4—1941no—
167C22⋊C4—21no—
168C22⋊C4—971no—
169C22⋊C4—3881no—
170C22⋊C4—1941no—
171C22⋊C4—1941no—
172C22⋊C4—1941no—
173C22⋊C4—1942no—
174C22⋊C4—1941no—
175C22⋊C4C2×#8760411yescharacteristic, complemented
176C23—971no—
177C23—971no—
178C23—971no—
179C23—3882no—
180C23—3882no—
181C23—1941no—
182C23—1941no—
183C23—1941no—
184C23—1941no—
185C23C22⋊#8760411yescharacteristic
186C23C22×#8760411yescharacteristic
187D4—41no—
188D4—3881no—
189D4—3881no—
190D4—3881no—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—41no—
194C2×C4—7762no—
195C2×C4—971no—
196C2×C4—3881no—
197C2×C4—3881no—
198C2×C4—3882no—
199C2×C4—3882no—
200C2×C4—3882no—
201C2×C4—3881no—
202C2×C4—1941no—
203C2×C4—1941no—
204C2×C4—1941no—
205C2×C4—1941no—
206C2×C4C22⋊#8760411yescharacteristic
207C8—3882no—
208C22—21no—
209C22—21no—
210C22—41no—
211C22—41no—
212C22—7761no—
213C22—971no—
214C22—971no—
215C22—3881no—
216C22—3881no—
217C22—3882no—
218C22—3882no—
219C22—3882no—
220C22—1942no—
221C22—1941no—
222C22—1941no—
223C22C2×C22⋊#8760411yescharacteristic
224C4—21no—
225C4—41no—
226C4—41no—
227C4—7762no—
228C4—3881no—
229C4—3881no—
230C4—1941no—
231C2—21no—
232C2—41no—
233C2—41no—
234C2—972no—
235C2—3881no—
236C2—3881no—
237C2—1941no—
238C2D97⋊#9911yescharacteristic
subgroupG/H
centerC2D97⋊#99
commutatorC23×C97C22×C4
FrattiniC23C22×#87604
Fitting#123×C97C4
socleC2×C97C2×#99
prime radicalC2×D4×D97C22
cosocleC2×C22⋊D97C23
hypercenter#123#87604
hypocenterC97C2×#352
LCSG ▷ C23×C97 ▷ C22×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #123

Complements

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

NH ≅ G/Ntype
3. #20261638C2semidirect
4. #20261637C2semidirect
13. C22×C4×D97C4semidirect
18. #15690791C4semidirect
21. C22⋊C4×D97C4semidirect
22. #123×C97C4semidirect
36. C22×Dic97C2×C4semidirect
47. C22⋊Dic97C2×C4semidirect
52. C22×C4×C97C2×C4semidirect
55. C22⋊C4×C97C2×C4semidirect
118. D97#352semidirect
120. C97C2×#352semidirect
134. #123#87604semidirect
159. C22×C4C2×#87604semidirect
175. C22⋊C4C2×#87604semidirect

Maximal subgroups

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

subgroupindex
D97⋊#1002
D97⋊#992
#202616382
#202616372
#123×D972
C2×#35297