Finite Group Databasebeta

← #352×D197 #38744346

Subgroups — #352×D197 #38744346

Order 25216.

278 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 278 conjugacy classes of subgroups — too many to draw. Only its 57 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#123×D197C211yesmaximal, characteristic
2#100×D197C211yesmaximal, characteristic
3#99×D197C211yesmaximal, characteristic
4#27160870C211yesmaximal, characteristic, complemented
5#27160869C211yesmaximal, characteristic, complemented
6#27160868C211yesmaximal, characteristic, complemented
7#352×C197C211yesmaximal, characteristic, complemented
8C2×D4×D197C2211yescharacteristic
9C22×C4×D197C411yescharacteristic, complemented
10M16×D197—21no—
11#20263311—21no—
12#20263310C411yescharacteristic, complemented
13#20263309C411yescharacteristic, complemented
14#20263308—21no—
15#20263307C2211yescharacteristic
16C4⋊C4×D197—21no—
17C22⋊C4×D197—21no—
18C22⋊C4×D197—21no—
19C22⋊C4×D197C411yescharacteristic, complemented
20#20263243C2211yescharacteristic
21#20263242—21no—
22#20263241C2211yescharacteristic
23#20263240—21no—
24#123×C197C2211yescharacteristic
25#100×C197C2211yescharacteristic
26#99×C197C2211yescharacteristic
27C23×D197C2×C411yescharacteristic
28C23×D197D411yescharacteristic
29C2×D788D411yescharacteristic
30C2×C22⋊D197C2×C411yescharacteristic
31C2×C22⋊D197D411yescharacteristic
32D4×D197—41no—
33D4×D197—41no—
34C22×Dic197C2×C411yescharacteristic, complemented
35C2×C4×D197—21no—
36C2×C4×D197—21no—
37C2×C4×D197—21no—
38C2×C4×D197—41no—
39C2×C4×D197—41no—
40C2×C4×D197—41no—
41C2×C4×D197D411yescharacteristic
42#15691342—21no—
43#15691341—41no—
44C8×D197—41no—
45C4⋊Dic197—21no—
46#15691337—21no—
47#15691337—21no—
48C22⋊Dic197—21no—
49C22⋊Dic197—21no—
50C22⋊Dic197C2×C411yescharacteristic, complemented
51#15691335—21no—
52#15691335—21no—
53#15691335—21no—
54#15691335—41no—
55C2×D4×C197C2311yescharacteristic
56C22×C4×C197C2×C411yescharacteristic, complemented
57M16×C197—21no—
58C4⋊C4×C197—21no—
59C22⋊C4×C197—21no—
60C22⋊C4×C197—21no—
61C22⋊C4×C197C2×C411yescharacteristic, complemented
62C22×D197—21no—
63C22×D197—21no—
64C22×D197—21no—
65C22×D197—21no—
66C22×D197—42no—
67C22×D197—42no—
68C22×D197C22⋊C411yescharacteristic
69C22×D197C22⋊C411yescharacteristic
70C22×D197C22⋊C411yescharacteristic
71D788—41no—
72C22⋊D197—41no—
73C22⋊D197—41no—
74C2×Dic197—21no—
75C2×Dic197—21no—
76C2×Dic197—41no—
77C2×Dic197—41no—
78C2×Dic197C22⋊C411yescharacteristic
79C4×D197—21no—
80C4×D197—21no—
81C4×D197—42no—
82C4×D197—42no—
83C4×D197—41no—
84C4×D197—41no—
85C4×D197—82no—
86#13096802—41no—
87C23×C197C22×C411yescharacteristic
88C23×C197C2×D411yescharacteristic
89D4×C197—41no—
90C2×C4×C197—21no—
91C2×C4×C197—21no—
92C2×C4×C197—41no—
93C2×C4×C197—41no—
94C2×C4×C197C2×D411yescharacteristic
95C8×C197—41no—
96C22×C197—21no—
97C22×C197—21no—
98C22×C197—41no—
99C22×C197—41no—
100C22×C197C2×C22⋊C411yescharacteristic
101C2×D197—22no—
102C2×D197—21no—
103C2×D197—21no—
104C2×D197—42no—
105C2×D197—42no—
106C2×D197—42no—
107C2×D197—41no—
108C2×D197—41no—
109C2×D197—81no—
110C2×D197#9911yescharacteristic
111C2×D197#9911yescharacteristic
112C4×C197—21no—
113C4×C197—41no—
114C4×C197—41no—
115C4×C197—81no—
116Dic197—21no—
117Dic197—41no—
118Dic197—41no—
119Dic197—81no—
120C2×C197—21no—
121C2×C197—41no—
122C2×C197—41no—
123C2×C197C2×#9911yescharacteristic
124D197—21no—
125D197—41no—
126D197—41no—
127D197#35212yesdirect product factor
128C197C2×#35211yescharacteristic, complemented, Sylow
129C2×#352—1971nomaximal, Sylow
130C2×#123—1971no—
131C2×#100—1971no—
132C2×#99—1971no—
133#352—1971no—
134#352—1971no—
135#352—1971no—
136#352D19711yescharacteristic, direct product factor
137C22×D4—1971no—
138C23×C4—1971no—
139C2×M16—3941no—
140#123—3941no—
141#123—3941no—
142#123—1971no—
143#123—1971no—
144#123—1971no—
145#123C2×D19711yescharacteristic
146C2×C4⋊C4—3941no—
147C2×C22⋊C4—3941no—
148C2×C22⋊C4—3941no—
149C2×C22⋊C4—1971no—
150#100—3941no—
151#100—1971no—
152#100C2×D19711yescharacteristic
153#99—3941no—
154#99—1971no—
155#99C2×D19711yescharacteristic
156C24—1971no—
157C24—1971no—
158C2×D4—7881no—
159C2×D4—7881no—
160C2×D4—1971no—
161C2×D4—1971no—
162C2×D4—1971no—
163C2×D4C22×D19711yescharacteristic
164C22×C4—7881no—
165C22×C4—7881no—
166C22×C4—7881no—
167C22×C4—3941no—
168C22×C4—3941no—
169C22×C4—3941no—
170C22×C4—1971no—
171C22×C4—1971no—
172C22×C4C4×D19711yescharacteristic, complemented
173M16—21no—
174M16—7881no—
175M16—3941no—
176C2×C8—7881no—
177C4⋊C4—21no—
178C4⋊C4—3941no—
179C4⋊C4—3941no—
180C4⋊C4—3941no—
181C22⋊C4—21no—
182C22⋊C4—21no—
183C22⋊C4—7881no—
184C22⋊C4—3941no—
185C22⋊C4—3941no—
186C22⋊C4—3941no—
187C22⋊C4—3941no—
188C22⋊C4—3941no—
189C22⋊C4—1971no—
190C22⋊C4C4×D19711yescharacteristic, complemented
191C23—7882no—
192C23—7882no—
193C23—3941no—
194C23—3941no—
195C23—3941no—
196C23—3941no—
197C23—1971no—
198C23—1971no—
199C23—1971no—
200C23C2×C4×D19711yescharacteristic
201C23D4×D19711yescharacteristic
202D4—41no—
203D4—7881no—
204D4—7881no—
205D4—7881no—
206C2×C4—21no—
207C2×C4—21no—
208C2×C4—41no—
209C2×C4—41no—
210C2×C4—7882no—
211C2×C4—7882no—
212C2×C4—7881no—
213C2×C4—7881no—
214C2×C4—7881no—
215C2×C4—7881no—
216C2×C4—15762no—
217C2×C4—3941no—
218C2×C4—3941no—
219C2×C4—3941no—
220C2×C4—3941no—
221C2×C4—1971no—
222C2×C4D4×D19711yescharacteristic
223C8—41no—
224C8—7881no—
225C22—21no—
226C22—21no—
227C22—41no—
228C22—41no—
229C22—7881no—
230C22—7881no—
231C22—7882no—
232C22—7882no—
233C22—7882no—
234C22—15761no—
235C22—3942no—
236C22—3941no—
237C22—3941no—
238C22—1971no—
239C22—1971no—
240C22C22⋊C4×D19711yescharacteristic
241C4—21no—
242C4—41no—
243C4—41no—
244C4—81no—
245C4—7881no—
246C4—7881no—
247C4—15761no—
248C4—3941no—
249C2—21no—
250C2—41no—
251C2—41no—
252C2—7881no—
253C2—7881no—
254C2—3941no—
255C2—1972no—
256C2#99×D19711yescharacteristic
subgroupG/H
centerC2#99×D197
commutatorC23×C197C22×C4
FrattiniC2×D4C22×D197
Fitting#352×C197C2
socleC2×C197C2×#99
prime radicalC2×D4×D197C22
cosocleC2×D4×C197C23
hypercenter#352D197
hypocenterC197C2×#352
LCSG ▷ C23×C197 ▷ C22×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #352

Complements

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

NH ≅ G/Ntype
4. #27160870C2semidirect
5. #27160869C2semidirect
6. #27160868C2semidirect
7. #352×C197C2semidirect
9. C22×C4×D197C4semidirect
12. #20263310C4semidirect
13. #20263309C4semidirect
19. C22⋊C4×D197C4semidirect
34. C22×Dic197C2×C4semidirect
50. C22⋊Dic197C2×C4semidirect
56. C22×C4×C197C2×C4semidirect
61. C22⋊C4×C197C2×C4semidirect
127. D197#352direct (N × H)
128. C197C2×#352semidirect
136. #352D197direct (N × H)
172. C22×C4C4×D197semidirect
190. C22⋊C4C4×D197semidirect

Maximal subgroups

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

subgroupindex
#123×D1972
#100×D1972
#99×D1972
#271608702
#271608692
#271608682
#352×C1972
C2×#352197