Finite Group Databasebeta

← #38743836

Subgroups — #38743836

Order 25216.

638 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 638 conjugacy classes of subgroups — too many to draw. Only its 75 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
1C2×#20263294C212yesmaximal, complemented
2C2×#20263241C212yesmaximal, complemented
3#27160934C211yesmaximal, characteristic, complemented
4#27160846C211yesmaximal, characteristic, complemented
5#534×C197C211yesmaximal, characteristic, complemented
6C24×D197C2211yescharacteristic
7C22×C22⋊D197—22no—
8#20263294—24no—
9#20263294—24no—
10C2×C22⋊Dic197C2211yescharacteristic
11C2×C22⋊Dic197C2212yes—
12C2×#15691335—22no—
13C2×#15691335—22no—
14#20263241—24no—
15C22×D4×C197C2212yes—
16#123×C197—22no—
17#121×C197—22no—
18#120×C197—22no—
19C2×C22⋊C4×C197C2211yescharacteristic
20C23×D197—21no—
21C23×D197—22no—
22C23×D197—22no—
23C23×D197—21no—
24C23×D197—44no—
25C23×D197D412yescomplemented
26C2×C22⋊D197—28no—
27C2×C22⋊D197—48no—
28C22×Dic197—22no—
29C22×Dic197—21no—
30C22⋊Dic197—22no—
31C22⋊Dic197—22no—
32C22⋊Dic197D414yescomplemented
33#15691335—28no—
34#15691335—44no—
35C24×C197D412yescomplemented
36C24×C197C2311yescharacteristic
37C2×D4×C197—22no—
38C2×D4×C197—22no—
39C2×D4×C197—22no—
40C2×D4×C197—22no—
41C2×D4×C197—22no—
42C2×D4×C197—22no—
43C2×D4×C197—22no—
44C22×C4×C197D412yescomplemented
45C22×C4×C197D412yescomplemented
46C4⋊C4×C197—22no—
47C22⋊C4×C197—24no—
48C22⋊C4×C197—22no—
49C22×D197—24no—
50C22×D197—28no—
51C22×D197—24no—
52C22×D197—22no—
53C22×D197—44no—
54C22×D197—48no—
55C22×D197—44no—
56C22×D197—42no—
57C22×D197—44no—
58C22×D197—41no—
59C22×D197—42no—
60C22×D197—41no—
61C22⋊D197—416no—
62C2×Dic197—24no—
63C2×Dic197—44no—
64C2×Dic197—42no—
65C2×Dic197—41no—
66C23×C197—22no—
67C23×C197—22no—
68C23×C197—22no—
69C23×C197—42no—
70C23×C197—44no—
71C23×C197C2×D411yescharacteristic
72C23×C197C2×D412yes—
73C23×C197C2×D412yes—
74C23×C197C2×D411yescharacteristic
75C23×C197#4111yescharacteristic
76D4×C197—42no—
77D4×C197—42no—
78D4×C197—42no—
79D4×C197—42no—
80D4×C197—82no—
81C2×C4×C197—22no—
82C2×C4×C197—22no—
83C2×C4×C197—22no—
84C2×C4×C197—22no—
85C2×C4×C197—42no—
86C22×C197—22no—
87C22×C197—24no—
88C22×C197—41no—
89C22×C197—42no—
90C22×C197—41no—
91C22×C197—44no—
92C22×C197—42no—
93C22×C197—44no—
94C22×C197—82no—
95C22×C197#12012yes—
96C22×C197#12011yescharacteristic
97C22×C197#12111yescharacteristic
98C22×C197#12112yes—
99C22×C197#12711yescharacteristic
100C2×D197—28no—
101C2×D197—24no—
102C2×D197—44no—
103C2×D197—44no—
104C2×D197—42no—
105C2×D197—48no—
106C2×D197—42no—
107C2×D197—44no—
108C4×C197—42no—
109C4×C197—42no—
110Dic197—44no—
111Dic197—81no—
112C2×C197—22no—
113C2×C197—21no—
114C2×C197—21no—
115C2×C197—22no—
116C2×C197—82no—
117C2×C197#39211yescharacteristic
118C2×C197B(4,𝔽2)12yes—
119D197—44no—
120C197#202111yescharacteristic, complemented, Sylow
121#2021—1971nomaximal, Sylow
122#534D19711yescharacteristic, complemented
123C2×#120—1972no—
124C2×#99—1972no—
125#379—1971no—
126#342—1971no—
127C25—1971no—
128C22×D4—3942no—
129C22×D4C2×D19712yes—
130#123—22no—
131#121—22no—
132#120—22no—
133#120—3944no—
134#120—3944no—
135C2×C22⋊C4—3942no—
136C2×C22⋊C4—3942no—
137C2×C22⋊C4—1971no—
138C2×C22⋊C4—1972no—
139C2×C22⋊C4C2×D19711yescharacteristic
140#99—3944no—
141C24—7884no—
142C24—3942no—
143C24—3941no—
144C24—3941no—
145C24—3942no—
146C24—1972no—
147C24C22⋊D19712yescomplemented
148C24C22×D19711yescharacteristic
149C2×D4—22no—
150C2×D4—22no—
151C2×D4—22no—
152C2×D4—22no—
153C2×D4—22no—
154C2×D4—22no—
155C2×D4—22no—
156C2×D4—7888no—
157C2×D4—3948no—
158C22×C4—3941no—
159C22×C4—3942no—
160C22×C4C22⋊D19712yescomplemented
161C22×C4C22⋊D19712yescomplemented
162C4⋊C4—22no—
163C22⋊C4—24no—
164C22⋊C4—22no—
165C22⋊C4—7884no—
166C22⋊C4—3948no—
167C22⋊C4—3942no—
168C22⋊C4—3942no—
169C22⋊C4—1974no—
170C23—22no—
171C23—22no—
172C23—22no—
173C23—42no—
174C23—44no—
175C23—7882no—
176C23—7884no—
177C23—7888no—
178C23—7884no—
179C23—7884no—
180C23—7881no—
181C23—7882no—
182C23—7881no—
183C23—3944no—
184C23—3948no—
185C23—3944no—
186C23—3942no—
187C23D4×D19711yescharacteristic
188C23D4×D19712yes—
189C23C2×C22⋊D19712yes—
190C23C2×C22⋊D19711yescharacteristic
191C23#1569135711yescharacteristic
192D4—42no—
193D4—42no—
194D4—42no—
195D4—42no—
196D4—82no—
197D4—78816no—
198C2×C4—22no—
199C2×C4—22no—
200C2×C4—22no—
201C2×C4—22no—
202C2×C4—42no—
203C2×C4—7884no—
204C2×C4—7882no—
205C2×C4—7881no—
206C2×C4—3944no—
207C22—22no—
208C22—24no—
209C22—41no—
210C22—42no—
211C22—41no—
212C22—44no—
213C22—42no—
214C22—44no—
215C22—82no—
216C22—7884no—
217C22—7884no—
218C22—7888no—
219C22—7882no—
220C22—7882no—
221C22—7884no—
222C22—3948no—
223C22—3944no—
224C22#2026329412yes—
225C22#2026329511yescharacteristic
226C22#2026329711yescharacteristic
227C22#2026329812yes—
228C22#2026332211yescharacteristic
229C4—42no—
230C4—42no—
231C4—7884no—
232C4—15761no—
233C2—22no—
234C2—21no—
235C2—22no—
236C2—21no—
237C2—82no—
238C2—7884no—
239C2#2716098711yescharacteristic
240C2#2716126612yes—
subgroupG/H
centerC22#20263295
commutator, cosocleC24×C197C23
FrattiniC24C22×D197
Fitting#534×C197C2
socleC22×C197#120
hypercenter#534D197
hypocenterC197#2021
LCSG ▷ C24×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ C24 ◁ #534

Complements

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

NH ≅ G/Ntype
1. C2×#20263294C2semidirect
2. C2×#20263241C2semidirect
3. #27160934C2semidirect
4. #27160846C2semidirect
5. #534×C197C2semidirect
25. C23×D197D4semidirect
32. C22⋊Dic197D4semidirect
35. C24×C197D4semidirect
44. C22×C4×C197D4semidirect
45. C22×C4×C197D4semidirect
120. C197#2021semidirect
122. #534D197semidirect
147. C24C22⋊D197semidirect
160. C22×C4C22⋊D197semidirect
161. C22×C4C22⋊D197semidirect

Maximal subgroups

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

subgroupindex
C2×#202632942
C2×#202632412
#271609342
#271608462
#534×C1972
#2021197