Finite Group Databasebeta

← #38744702

Subgroups — #38744702

Order 25216.

258 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 258 conjugacy classes of subgroups — too many to draw. Only its 45 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#27161268C211yesmaximal, characteristic, complemented
2#27161265C211yesmaximal, characteristic
3#27161256C211yesmaximal, characteristic, complemented
4#27160874C211yesmaximal, characteristic, complemented
5#27160869C211yesmaximal, characteristic, complemented
6#27160866C211yesmaximal, characteristic, complemented
7#551×C197C211yesmaximal, characteristic, complemented
8#20263371C2211yescharacteristic
9#20263357—21no—
10#20263309—21no—
11#20263295—21no—
12#20263254—21no—
13#20263243C2211yescharacteristic
14#20263241C2211yescharacteristic
15#20263241C2211yescharacteristic
16#20263240—21no—
17#20263240—21no—
18#126×C197—21no—
19#124×C197C2211yescharacteristic
20#123×C197—21no—
21#123×C197C2211yescharacteristic
22C22⋊Q8×C197—21no—
23#120×C197C2211yescharacteristic
24C4×D4×C197—21no—
25#117×C197—21no—
26C2×C22⋊C4×C197—21no—
27#15691358—21no—
28#15691357—41no—
29C2×C22⋊D197—21no—
30C2×C22⋊D197—21no—
31C2×C22⋊D197—41no—
32D4×D197—41no—
33#15691348—41no—
34#15691345—41no—
35#15691342—21no—
36#15691337—41no—
37C22⋊Dic197—21no—
38C22⋊Dic197—21no—
39C22⋊Dic197—41no—
40#15691335—41no—
41C24×C197D411yescharacteristic
42C2×Q8×C197D411yescharacteristic, complemented
43C2×D4×C197—21no—
44C2×D4×C197C2311yescharacteristic
45C22×C4×C197—21no—
46C22×C4×C197—21no—
47C22×C4×C197D411yescharacteristic, complemented
48C4⋊C4×C197—21no—
49C4⋊C4×C197—21no—
50C4⋊C4×C197—21no—
51C4⋊C4×C197—21no—
52C22⋊C4×C197—21no—
53C22⋊C4×C197—21no—
54C22⋊C4×C197—21no—
55C22⋊C4×C197—21no—
56C22⋊C4×C197—21no—
57C22⋊C4×C197—21no—
58C22⋊C4×C197D411yescharacteristic
59C22⋊C4×C197D411yescharacteristic, complemented
60C42×C197—21no—
61C42×C197D411yescharacteristic, complemented
62C22×D197—21no—
63C22×D197—21no—
64Dic394—21no—
65D788—21no—
66C22⋊D197—41no—
67C22⋊D197—41no—
68C22⋊D197—41no—
69C22⋊D197—82no—
70C2×Dic197—21no—
71C2×Dic197—21no—
72C2×Dic197—41no—
73C2×Dic197—41no—
74C4×D197—41no—
75#13096802—41no—
76C23×C197—21no—
77C23×C197—42no—
78C23×C197—41no—
79C23×C197C2×D411yescharacteristic
80C23×C197C2×D411yescharacteristic
81Q8×C197—41no—
82D4×C197—41no—
83D4×C197—42no—
84C2×C4×C197—21no—
85C2×C4×C197—21no—
86C2×C4×C197—21no—
87C2×C4×C197—21no—
88C2×C4×C197—21no—
89C2×C4×C197—42no—
90C2×C4×C197—41no—
91C2×C4×C197—81no—
92C2×C4×C197C2×D411yescharacteristic
93C22×C197—22no—
94C22×C197—21no—
95C22×C197—21no—
96C22×C197—41no—
97C22×C197—41no—
98C22×C197—41no—
99C22×C197—42no—
100C22×C197—41no—
101C22×C197—81no—
102C22×C197#12011yescharacteristic
103C2×D197—41no—
104C2×D197—81no—
105C2×D197—81no—
106C4×C197—21no—
107C4×C197—41no—
108C4×C197—41no—
109C4×C197—41no—
110C4×C197—41no—
111C4×C197—41no—
112Dic197—41no—
113Dic197—81no—
114Dic197—81no—
115C2×C197—21no—
116C2×C197—41no—
117C2×C197—41no—
118C2×C197—42no—
119C2×C197B(4,𝔽2)11yescharacteristic
120D197—81no—
121C197#219711yescharacteristic, complemented, Sylow
122#2197—1971nomaximal, Sylow
123#551D19711yescharacteristic, complemented
124#458—1971no—
125B(4,𝔽2)—1971no—
126#455—1971no—
127#354—1971no—
128#352—1971no—
129AΣL(1,𝔽16)—1971no—
130H(5,𝔽2)—1971no—
131#137—3941no—
132#126—21no—
133#124C2×D19711yescharacteristic
134#123—21no—
135#123—3941no—
136#123C2×D19711yescharacteristic
137C22⋊Q8—21no—
138#120—3941no—
139#120C2×D19711yescharacteristic
140C4×D4—21no—
141#117—21no—
142C2×C22⋊C4—21no—
143#104—3941no—
144#100—1971no—
145#99—3941no—
146#99—3941no—
147#99—1971no—
148#99—1971no—
149C24C22⋊D19711yescharacteristic
150#41—7881no—
151#41—3941no—
152C2×Q8C22⋊D19711yescharacteristic, complemented
153C2×D4—21no—
154C2×D4—7881no—
155C2×D4—7881no—
156C2×D4—3941no—
157C2×D4—3941no—
158C2×D4C22×D19711yescharacteristic
159C22×C4—21no—
160C22×C4—21no—
161C22×C4C22⋊D19711yescharacteristic, complemented
162Q16—7881no—
163SD16—7881no—
164M16—3941no—
165C4⋊C4—21no—
166C4⋊C4—21no—
167C4⋊C4—21no—
168C4⋊C4—21no—
169C4⋊C4—7881no—
170C22⋊C4—21no—
171C22⋊C4—21no—
172C22⋊C4—21no—
173C22⋊C4—21no—
174C22⋊C4—21no—
175C22⋊C4—21no—
176C22⋊C4—7881no—
177C22⋊C4—7881no—
178C22⋊C4—3941no—
179C22⋊C4—3941no—
180C22⋊C4C22⋊D19711yescharacteristic
181C22⋊C4C22⋊D19711yescharacteristic, complemented
182C42—21no—
183C42C22⋊D19711yescharacteristic, complemented
184C23—21no—
185C23—41no—
186C23—42no—
187C23—3941no—
188C23—3941no—
189C23C2×C22⋊D19711yescharacteristic
190C23C2×C22⋊D19711yescharacteristic
191Q8—41no—
192Q8—3941no—
193D4—41no—
194D4—42no—
195D4—7881no—
196D4—7881no—
197D4—7881no—
198D4—15762no—
199D4—3941no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—21no—
205C2×C4—42no—
206C2×C4—41no—
207C2×C4—81no—
208C2×C4—7881no—
209C2×C4—7881no—
210C2×C4—7881no—
211C2×C4—3941no—
212C2×C4—3941no—
213C2×C4C2×C22⋊D19711yescharacteristic
214C8—7881no—
215C22—22no—
216C22—21no—
217C22—21no—
218C22—41no—
219C22—41no—
220C22—41no—
221C22—42no—
222C22—41no—
223C22—81no—
224C22—7881no—
225C22—15761no—
226C22—15761no—
227C22#2026329511yescharacteristic
228C4—21no—
229C4—41no—
230C4—41no—
231C4—41no—
232C4—41no—
233C4—41no—
234C4—7881no—
235C4—15761no—
236C4—15761no—
237C2—21no—
238C2—41no—
239C2—41no—
240C2—42no—
241C2—15761no—
242C2#2716126611yescharacteristic
subgroupG/H
centerC2#27161266
commutator, cosocleC2×D4×C197C23
FrattiniC2×D4C22×D197
Fitting#551×C197C2
socleC2×C197B(4,𝔽2)
prime radical#27161265C2
hypercenter#551D197
hypocenterC197#2197
LCSG ▷ C2×D4×C197 ▷ C22×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #551

Complements

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

NH ≅ G/Ntype
1. #27161268C2semidirect
3. #27161256C2semidirect
4. #27160874C2semidirect
5. #27160869C2semidirect
6. #27160866C2semidirect
7. #551×C197C2semidirect
42. C2×Q8×C197D4semidirect
47. C22×C4×C197D4semidirect
59. C22⋊C4×C197D4semidirect
61. C42×C197D4semidirect
121. C197#2197semidirect
123. #551D197semidirect
152. C2×Q8C22⋊D197semidirect
161. C22×C4C22⋊D197semidirect
181. C22⋊C4C22⋊D197semidirect
183. C42C22⋊D197semidirect

Maximal subgroups

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

subgroupindex
#271612682
#271612652
#271612562
#271608742
#271608692
#271608662
#551×C1972
#2197197