Finite Group Databasebeta

← #38744364

Subgroups — #38744364

Order 25216.

234 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 55 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#27161651C211yesmaximal, characteristic
2#27161060C211yesmaximal, characteristic
3#27161051C211yesmaximal, characteristic
4#27160870C211yesmaximal, characteristic, complemented
5#27160869C211yesmaximal, characteristic, complemented
6#27160868C211yesmaximal, characteristic, complemented
7#352×C197C211yesmaximal, characteristic, complemented
8C2×#15691357C2211yescharacteristic
9C22×Dic394C411yescharacteristic, complemented
10M16×D197—21no—
11#20263325—21no—
12#20263312—21no—
13#20263310C411yescharacteristic, complemented
14#20263309C411yescharacteristic, complemented
15#20263307C2211yescharacteristic
16#20263304—21no—
17#20263303—21no—
18#20263282—21no—
19#20263282C411yescharacteristic, complemented
20#20263244—21no—
21#20263243C2211yescharacteristic
22#20263241C2211yescharacteristic
23#20263240—21no—
24#123×C197C2211yescharacteristic
25#100×C197C2211yescharacteristic
26#99×C197C2211yescharacteristic
27#15691357—41no—
28#15691357—41no—
29C2×Dic394—21no—
30C2×Dic394—21no—
31C2×Dic394—41no—
32C2×Dic394—41no—
33C2×Dic394D411yescharacteristic
34C2×C22⋊D197C2×C411yescharacteristic
35C2×C22⋊D197D411yescharacteristic
36C22×Dic197C2×C411yescharacteristic
37C22×Dic197C2×C411yescharacteristic
38C22×Dic197D411yescharacteristic
39C2×C4×D197D411yescharacteristic
40#15691342—21no—
41#15691341—41no—
42C8×D197—41no—
43C4⋊Dic197—21no—
44#15691337—21no—
45#15691337—21no—
46#15691337—21no—
47#15691337—41no—
48C22⋊Dic197—21no—
49C22⋊Dic197—21no—
50C22⋊Dic197C2×C411yescharacteristic
51#15691335—21no—
52#15691335—21no—
53C4×Dic197—21no—
54C4×Dic197—41no—
55C2×D4×C197C2311yescharacteristic
56C22×C4×C197C2×C411yescharacteristic
57M16×C197—21no—
58C4⋊C4×C197—21no—
59C22⋊C4×C197—21no—
60C22⋊C4×C197—21no—
61C22⋊C4×C197C2×C411yescharacteristic
62C22×D197C22⋊C411yescharacteristic
63Dic394—41no—
64Dic394—42no—
65Dic394—41no—
66Dic394—41no—
67C22⋊D197—41no—
68C22⋊D197—41no—
69C2×Dic197—21no—
70C2×Dic197—21no—
71C2×Dic197—21no—
72C2×Dic197—21no—
73C2×Dic197—21no—
74C2×Dic197—21no—
75C2×Dic197—41no—
76C2×Dic197—41no—
77C2×Dic197C22⋊C411yescharacteristic
78C2×Dic197C22⋊C411yescharacteristic
79C2×Dic197C22⋊C411yescharacteristic
80C4×D197—21no—
81C4×D197—21no—
82#13096802—41no—
83C23×C197C22×C411yescharacteristic
84C23×C197C2×D411yescharacteristic
85D4×C197—41no—
86C2×C4×C197—21no—
87C2×C4×C197—21no—
88C2×C4×C197—41no—
89C2×C4×C197—41no—
90C2×C4×C197C2×D411yescharacteristic
91C8×C197—41no—
92C22×C197—21no—
93C22×C197—21no—
94C22×C197—41no—
95C22×C197—41no—
96C22×C197C2×C22⋊C411yescharacteristic
97C2×D197—21no—
98C2×D197—41no—
99C4×C197—21no—
100C4×C197—41no—
101C4×C197—41no—
102C4×C197—81no—
103Dic197—21no—
104Dic197—21no—
105Dic197—41no—
106Dic197—41no—
107Dic197—81no—
108Dic197#9911yescharacteristic
109Dic197#9911yescharacteristic
110C2×C197—21no—
111C2×C197—41no—
112C2×C197—41no—
113C2×C197C2×#9911yescharacteristic
114D197—41no—
115C197#212311yescharacteristic, complemented, Sylow
116#2123—1971nomaximal, Sylow
117#536—1971no—
118#413—1971no—
119#410—1971no—
120#352—1971no—
121#352—1971no—
122#352—1971no—
123#352D19711yescharacteristic, complemented
124C2×#41—1971no—
125C22×Q8—1971no—
126C2×M16—3941no—
127C4⋊Q8—3941no—
128#124—3941no—
129#123—1971no—
130#123—1971no—
131#123—1971no—
132#123C2×D19711yescharacteristic
133C22⋊Q8—3941no—
134C22⋊Q8—3941no—
135#117—3941no—
136#117—1971no—
137#101—3941no—
138#100—1971no—
139#100C2×D19711yescharacteristic
140#99—3941no—
141#99—1971no—
142#99C2×D19711yescharacteristic
143#41—7881no—
144#41—7881no—
145C2×Q8—7881no—
146C2×Q8—7881no—
147C2×Q8—3941no—
148C2×Q8—3941no—
149C2×Q8—1971no—
150C2×D4—1971no—
151C2×D4—1971no—
152C2×D4C22×D19711yescharacteristic
153C22×C4—1971no—
154C22×C4—1971no—
155C22×C4—1971no—
156C22×C4—1971no—
157C22×C4C4×D19711yescharacteristic
158M16—21no—
159M16—7881no—
160M16—3941no—
161C2×C8—7881no—
162C4⋊C4—21no—
163C4⋊C4—7881no—
164C4⋊C4—3941no—
165C4⋊C4—3941no—
166C4⋊C4—3941no—
167C4⋊C4—3941no—
168C22⋊C4—21no—
169C22⋊C4—21no—
170C22⋊C4—3941no—
171C22⋊C4—3941no—
172C22⋊C4—3941no—
173C22⋊C4—3941no—
174C22⋊C4—1971no—
175C22⋊C4C4×D19711yescharacteristic
176C42—7881no—
177C42—3941no—
178C23—1971no—
179C23C2×C4×D19711yescharacteristic
180C23D4×D19711yescharacteristic
181Q8—7881no—
182Q8—7882no—
183Q8—7881no—
184Q8—7881no—
185D4—41no—
186D4—7881no—
187D4—7881no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—41no—
191C2×C4—41no—
192C2×C4—7881no—
193C2×C4—7881no—
194C2×C4—3941no—
195C2×C4—3941no—
196C2×C4—3941no—
197C2×C4—3941no—
198C2×C4—3941no—
199C2×C4—3941no—
200C2×C4—3941no—
201C2×C4—3941no—
202C2×C4—1971no—
203C2×C4—1971no—
204C2×C4—1971no—
205C2×C4D4×D19711yescharacteristic
206C8—41no—
207C8—7881no—
208C22—21no—
209C22—21no—
210C22—41no—
211C22—41no—
212C22—7881no—
213C22—3941no—
214C22C22⋊C4×D19711yescharacteristic
215C4—21no—
216C4—41no—
217C4—41no—
218C4—81no—
219C4—7881no—
220C4—7881no—
221C4—15761no—
222C4—3941no—
223C4—3941no—
224C4—1971no—
225C4—1971no—
226C2—21no—
227C2—41no—
228C2—41no—
229C2—7881no—
230C2#99×D19711yescharacteristic
subgroupG/H
centerC2#99×D197
commutatorC23×C197C22×C4
FrattiniC2×D4C22×D197
Fitting#352×C197C2
socleC2×C197C2×#99
prime radicalC2×#15691357C22
cosocleC2×D4×C197C23
hypercenter#352D197
hypocenterC197#2123
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×Dic394C4semidirect
13. #20263310C4semidirect
14. #20263309C4semidirect
19. #20263282C4semidirect
115. C197#2123semidirect
123. #352D197semidirect

Maximal subgroups

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

subgroupindex
#271616512
#271610602
#271610512
#271608702
#271608692
#271608682
#352×C1972
#2123197