Finite Group Databasebeta

← D97⋊#327 #27157102

Subgroups — D97⋊#327 #27157102

Order 12416.

388 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 388 conjugacy classes of subgroups — too many to draw. Only its 74 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#20261703C211yesmaximal, characteristic, complemented
2C2×(C2×#87604)⋊C4C211yesmaximal, characteristic, complemented
3#20261587C212yesmaximal, complemented
4#20261586C212yesmaximal, complemented
5#121×D97C211yesmaximal, characteristic
6C23×#87604—21no—
7C2×#13095519—21no—
8#15690867C2212yescomplemented
9C22⋊#11715897—21no—
10(C2×#87604)⋊C4—21no—
11(C2×#87604)⋊C4C2212yescomplemented
12C2×D4×D97—21no—
13C2×D4×D97C411yescharacteristic, complemented
14C22×C4×D97C2211yescharacteristic
15#15690785C2211yescharacteristic
16#15690784C411yescharacteristic, complemented
17#15690783—21no—
18#15690782—21no—
19#15690781C2211yescharacteristic
20C4⋊C4×D97C411yescharacteristic, complemented
21C22⋊C4×D97—21no—
22#121×C97C411yescharacteristic, complemented
23C22×#87604—24no—
24C22×#87604—22no—
25C22×#87604—42no—
26C22×#87604—42no—
27#13095519—42no—
28C2×#11715897—22no—
29C23×D97—21no—
30C23×D97D411yescharacteristic
31C2×D388—21no—
32C2×D388C2×C411yescharacteristic, complemented
33C2×C22⋊D97—21no—
34C2×C22⋊D97—21no—
35C2×C22⋊D97—21no—
36D4×D97—42no—
37D4×D97—42no—
38D4×D97—42no—
39C22×Dic97D411yescharacteristic
40C2×C4×D97—21no—
41C2×C4×D97—42no—
42C2×C4×D97C2×C411yescharacteristic
43C2×C4×D97D411yescharacteristic
44C2×C4×D97D411yescharacteristic
45C2×C4×D97C2311yescharacteristic
46C4⋊Dic97C2×C411yescharacteristic, complemented
47#13095490—21no—
48C22⋊Dic97—21no—
49#13095488—21no—
50#13095488—21no—
51C2×D4×C97—21no—
52C2×D4×C97C2×C411yescharacteristic, complemented
53C22×C4×C97C2×C411yescharacteristic
54C4⋊C4×C97C2×C411yescharacteristic, complemented
55C22⋊C4×C97—21no—
56C23×C97—21no—
57C23×C97C22⋊C411yescharacteristic
58C22×D97—22no—
59C22×D97—22no—
60C22×D97—21no—
61C22×D97—42no—
62C22×D97—42no—
63C22×D97—42no—
64C22×D97C22⋊C411yescharacteristic
65C22×D97D812yes—
66C22×D97SD1612yes—
67C22×D97C2×D411yescharacteristic
68C2×#87604—24no—
69C2×#87604—42no—
70C2×#87604—48no—
71C2×#87604—42no—
72D4×C97—41no—
73D4×C97—42no—
74C2×C4×C97—21no—
75C2×C4×C97—41no—
76C2×C4×C97C22⋊C411yescharacteristic
77C2×C4×C97C22×C411yescharacteristic
78C22⋊D97—42no—
79C22⋊D97—42no—
80C22⋊D97—41no—
81C22⋊D97—41no—
82C2×Dic97—21no—
83C2×Dic97—41no—
84C2×Dic97C22⋊C411yescharacteristic
85C2×Dic97C2×D411yescharacteristic
86D388—41no—
87D388—41no—
88D388—41no—
89C4×D97—22no—
90C4×D97—22no—
91C4×D97—41no—
92C4×D97—41no—
93C4×D97—42no—
94#11715897—42no—
95C22×C97—21no—
96C22×C97—21no—
97C22×C97—41no—
98C22×C97—41no—
99C22×C97—41no—
100C22×C97#10212yes—
101C22×C97C2×C22⋊C411yescharacteristic
102C2×D97—21no—
103C2×D97—21no—
104C2×D97—24no—
105C2×D97—24no—
106C2×D97—41no—
107C2×D97—41no—
108C2×D97—41no—
109C2×D97—84no—
110C2×D97#9912yes—
111C2×D97#10212yes—
112C2×D97#10412yes—
113C2×D97C2×D811yescharacteristic
114C2×D97C2×SD1611yescharacteristic
115#87604—44no—
116C4×C97—21no—
117C4×C97—21no—
118C4×C97—41no—
119Dic97—21no—
120Dic97—21no—
121Dic97—41no—
122C2×C97—22no—
123C2×C97—81no—
124C2×C97C2×#9911yescharacteristic
125C2×C97C2×#10211yescharacteristic
126C2×C97C2×#10411yescharacteristic
127D97—21no—
128D97—21no—
129D97—81no—
130D97#32714yescomplemented
131C2×#327—971nomaximal, Sylow
132C97C2×#32711yescharacteristic, complemented, Sylow
133C2×#121—971no—
134C2×C22⋊C8—971no—
135C2×#95—971no—
136#327—972no—
137#327—972no—
138C22×D4—971no—
139C22×D4—1941no—
140C23×C4—971no—
141C23×C4—1941no—
142C22×C8—1941no—
143#121—971no—
144#121—971no—
145#121—971no—
146#121—1941no—
147#121—1941no—
148#121#8760411yescharacteristic, complemented
149C2×C4⋊C4—971no—
150C2×C22⋊C4—1941no—
151C22⋊C8—972no—
152C22⋊C8—1941no—
153#95—972no—
154#95—1941no—
155C24—971no—
156C24—1941no—
157C2×D4—21no—
158C2×D4—971no—
159C2×D4—3882no—
160C2×D4—3882no—
161C2×D4—3882no—
162C2×D4—1941no—
163C2×D4—1941no—
164C2×D4—1941no—
165C2×D4—1941no—
166C2×D4C2×#8760411yescharacteristic, complemented
167C22×C4—971no—
168C22×C4—971no—
169C22×C4—971no—
170C22×C4—971no—
171C22×C4—971no—
172C22×C4—3882no—
173C22×C4—3882no—
174C22×C4—3882no—
175C22×C4—1941no—
176C22×C4—1942no—
177C22×C4—1944no—
178C22×C4C2×#8760411yescharacteristic
179C2×C8—3882no—
180C2×C8—1942no—
181C4⋊C4—971no—
182C4⋊C4—1941no—
183C4⋊C4C2×#8760411yescharacteristic, complemented
184C22⋊C4—21no—
185C22⋊C4—1941no—
186C22⋊C4—1941no—
187C22⋊C4—1941no—
188C23—21no—
189C23—971no—
190C23—972no—
191C23—972no—
192C23—971no—
193C23—3882no—
194C23—3882no—
195C23—3882no—
196C23—1942no—
197C23—1942no—
198C23—1941no—
199C23C22⋊#8760411yescharacteristic
200D4—41no—
201D4—42no—
202D4—3881no—
203D4—3881no—
204D4—3881no—
205D4—3881no—
206D4—3881no—
207D4—3882no—
208D4—3882no—
209C2×C4—21no—
210C2×C4—41no—
211C2×C4—971no—
212C2×C4—971no—
213C2×C4—3881no—
214C2×C4—3881no—
215C2×C4—3882no—
216C2×C4—3882no—
217C2×C4—3888no—
218C2×C4—3882no—
219C2×C4—3881no—
220C2×C4—1942no—
221C2×C4—1942no—
222C2×C4—1944no—
223C2×C4—1941no—
224C2×C4C22⋊#8760411yescharacteristic
225C2×C4C22×#8760411yescharacteristic
226C8—3882no—
227C22—21no—
228C22—21no—
229C22—41no—
230C22—41no—
231C22—41no—
232C22—7764no—
233C22—972no—
234C22—972no—
235C22—972no—
236C22—971no—
237C22—971no—
238C22—3881no—
239C22—3881no—
240C22—3881no—
241C22—1944no—
242C22—1944no—
243C22—1941no—
244C22—1941no—
245C22#1569087212yes—
246C22C2×C22⋊#8760411yescharacteristic
247C4—21no—
248C4—21no—
249C4—41no—
250C4—3884no—
251C4—3881no—
252C4—1941no—
253C4—1941no—
254C2—22no—
255C2—81no—
256C2—7761no—
257C2—974no—
258C2—1941no—
259C2—1941no—
260C2D97⋊#9911yescharacteristic
261C2C2×#1569087211yescharacteristic
262C2D97⋊#10411yescharacteristic
subgroupG/H
centerC22C2×C22⋊#87604
commutatorC2×C4×C97C22×C4
FrattiniC2×C4C22×#87604
Fitting#121×C97C4
socleC22×C97C2×C22⋊C4
prime radical#121×D97C2
cosocleC2×C4×D97C23
hypercenter#121#87604
hypocenterC97C2×#327
LCSG ▷ C2×C4×C97 ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ #121

Complements

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

NH ≅ G/Ntype
1. #20261703C2semidirect
2. C2×(C2×#87604)⋊C4C2semidirect
3. #20261587C2semidirect
4. #20261586C2semidirect
8. #15690867C22semidirect
11. (C2×#87604)⋊C4C22semidirect
13. C2×D4×D97C4semidirect
16. #15690784C4semidirect
20. C4⋊C4×D97C4semidirect
22. #121×C97C4semidirect
32. C2×D388C2×C4semidirect
46. C4⋊Dic97C2×C4semidirect
52. C2×D4×C97C2×C4semidirect
54. C4⋊C4×C97C2×C4semidirect
130. D97#327semidirect
132. C97C2×#327semidirect
148. #121#87604semidirect
166. C2×D4C2×#87604semidirect
183. C4⋊C4C2×#87604semidirect

Maximal subgroups

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

subgroupindex
#202617032
C2×(C2×#87604)⋊C42
#202615872
#202615862
#121×D972
C2×#32797