Finite Group Databasebeta

← #38741255

Subgroups — #38741255

Order 25216.

286 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 286 conjugacy classes of subgroups — too many to draw. Only its 81 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#27161043C211yesmaximal, characteristic, complemented
2#27160939C211yesmaximal, characteristic, complemented
3#27160816C212yesmaximal, complemented
4#27160813C212yesmaximal, complemented
5#525×C197C211yesmaximal, characteristic
6C2×#15691342—21no—
7C2×C4⋊Dic197—21no—
8C2×C22⋊Dic197—21no—
9C22⋊#13096802—21no—
10C22⋊#13096802C2212yes—
11#20263233C2212yes—
12C2×#41×C197C411yescharacteristic, complemented
13C23×C4×C197C2211yescharacteristic
14#123×C197—21no—
15C22⋊Q8×C197C411yescharacteristic, complemented
16C22⋊Q8×C197C2211yescharacteristic
17#121×C197C411yescharacteristic, complemented
18#121×C197C2211yescharacteristic
19#120×C197—21no—
20C4×D4×C197—21no—
21C4×D4×C197—21no—
22#117×C197C411yescharacteristic, complemented
23C22×Dic197—22no—
24#15691342—42no—
25C2×#13096802—22no—
26C4⋊Dic197—42no—
27C22⋊Dic197—42no—
28C24×C197D411yescharacteristic
29#41×C197—42no—
30C2×Q8×C197C2×C411yescharacteristic
31C2×D4×C197—21no—
32C2×D4×C197—21no—
33C2×D4×C197—21no—
34C2×D4×C197C2×C411yescharacteristic
35C22×C4×C197—21no—
36C22×C4×C197—42no—
37C22×C4×C197C2×C411yescharacteristic
38C22×C4×C197C2×C411yescharacteristic
39C22×C4×C197D411yescharacteristic
40C22×C4×C197D411yescharacteristic
41C22×C4×C197D411yescharacteristic
42C22×C4×C197C2311yescharacteristic
43C4⋊C4×C197—21no—
44C4⋊C4×C197—21no—
45C4⋊C4×C197C2×C411yescharacteristic
46C4⋊C4×C197C2×C411yescharacteristic
47C22⋊C4×C197—21no—
48C22⋊C4×C197—21no—
49C22⋊C4×C197—21no—
50C22⋊C4×C197—21no—
51C22⋊C4×C197—21no—
52C42×C197—21no—
53C2×Dic197—42no—
54C2×Dic197—42no—
55C2×Dic197—42no—
56#13096802—42no—
57C23×C197—21no—
58C23×C197—21no—
59C23×C197—21no—
60C23×C197—41no—
61C23×C197—41no—
62C23×C197C22⋊C411yescharacteristic
63C23×C197C22⋊C411yescharacteristic
64C23×C197C2×D411yescharacteristic
65Q8×C197—41no—
66D4×C197—41no—
67D4×C197—42no—
68D4×C197—42no—
69D4×C197—41no—
70D4×C197—41no—
71C2×C4×C197—22no—
72C2×C4×C197—22no—
73C2×C4×C197—22no—
74C2×C4×C197—21no—
75C2×C4×C197—21no—
76C2×C4×C197—21no—
77C2×C4×C197—41no—
78C2×C4×C197—41no—
79C2×C4×C197—41no—
80C2×C4×C197—42no—
81C2×C4×C197C22⋊C411yescharacteristic
82C2×C4×C197C22⋊C411yescharacteristic
83C2×C4×C197C22×C411yescharacteristic
84C2×C4×C197C2×D411yescharacteristic
85C22×C197—21no—
86C22×C197—21no—
87C22×C197—21no—
88C22×C197—21no—
89C22×C197—21no—
90C22×C197—21no—
91C22×C197—21no—
92C22×C197—21no—
93C22×C197—42no—
94C22×C197—42no—
95C22×C197—41no—
96C22×C197—41no—
97C22×C197—41no—
98C22×C197C2×C22⋊C411yescharacteristic
99C22×C197AGL(1,ℤ/8)11yescharacteristic
100C22×C197#13711yescharacteristic
101C4×C197—21no—
102C4×C197—21no—
103C4×C197—21no—
104C4×C197—41no—
105C4×C197—41no—
106C4×C197—41no—
107C4×C197#9912yes—
108Dic197—82no—
109C2×C197—21no—
110C2×C197—21no—
111C2×C197—41no—
112C2×C197—41no—
113C2×C197—81no—
114C2×C197C2×#9911yescharacteristic
115C2×C197#41711yescharacteristic
116C2×C197#42111yescharacteristic
117C197#151311yescharacteristic, complemented, Sylow
118#1513—1971nomaximal, Sylow
119#525D19711yescharacteristic
120#407—1971no—
121#380—1971no—
122#328—1972no—
123#327—1972no—
124C2×#41Dic19711yescharacteristic, complemented
125C23×C4C2×D19711yescharacteristic
126C2×M16—3941no—
127#123—21no—
128C22⋊Q8Dic19711yescharacteristic, complemented
129C22⋊Q8C2×D19711yescharacteristic
130#121Dic19711yescharacteristic, complemented
131#121C2×D19711yescharacteristic
132#120—21no—
133C4×D4—21no—
134C4×D4—21no—
135#117Dic19711yescharacteristic, complemented
136C2×C4⋊C4—3941no—
137C2×C22⋊C4—3941no—
138C22⋊C8—3941no—
139C22⋊C8—1972no—
140#95—1972no—
141C24C22⋊D19711yescharacteristic
142#41—42no—
143C2×Q8C2×Dic19711yescharacteristic
144C2×D4—21no—
145C2×D4—21no—
146C2×D4—21no—
147C2×D4C2×Dic19711yescharacteristic
148C22×C4—21no—
149C22×C4—42no—
150C22×C4—3942no—
151C22×C4C2×Dic19711yescharacteristic
152C22×C4C2×Dic19711yescharacteristic
153C22×C4C22⋊D19711yescharacteristic
154C22×C4C22⋊D19711yescharacteristic
155C22×C4C22⋊D19711yescharacteristic
156C22×C4C22×D19711yescharacteristic
157M16—7882no—
158C2×C8—3942no—
159C4⋊C4—21no—
160C4⋊C4—21no—
161C4⋊C4—7882no—
162C4⋊C4C2×Dic19711yescharacteristic
163C4⋊C4C2×Dic19711yescharacteristic
164C22⋊C4—21no—
165C22⋊C4—21no—
166C22⋊C4—21no—
167C22⋊C4—21no—
168C22⋊C4—21no—
169C22⋊C4—7882no—
170C42—21no—
171C23—21no—
172C23—21no—
173C23—21no—
174C23—41no—
175C23—41no—
176C23C22⋊Dic19711yescharacteristic
177C23C22⋊Dic19711yescharacteristic
178C23C2×C22⋊D19711yescharacteristic
179Q8—41no—
180D4—41no—
181D4—42no—
182D4—42no—
183D4—41no—
184D4—41no—
185C2×C4—22no—
186C2×C4—22no—
187C2×C4—22no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—21no—
191C2×C4—41no—
192C2×C4—41no—
193C2×C4—41no—
194C2×C4—42no—
195C2×C4—7882no—
196C2×C4—7882no—
197C2×C4—7882no—
198C2×C4C22⋊Dic19711yescharacteristic
199C2×C4C22⋊Dic19711yescharacteristic
200C2×C4C22×Dic19711yescharacteristic
201C2×C4C2×C22⋊D19711yescharacteristic
202C8—7882no—
203C22—21no—
204C22—21no—
205C22—21no—
206C22—21no—
207C22—21no—
208C22—21no—
209C22—21no—
210C22—21no—
211C22—42no—
212C22—42no—
213C22—41no—
214C22—41no—
215C22—41no—
216C22C2×C22⋊Dic19711yescharacteristic
217C22#2026335111yescharacteristic
218C22#2026335611yescharacteristic
219C4—21no—
220C4—21no—
221C4—21no—
222C4—41no—
223C4—41no—
224C4—41no—
225C4—15762no—
226C4#2026324112yes—
227C2—21no—
228C2—21no—
229C2—41no—
230C2—41no—
231C2—81no—
232C2C2×#2026324111yescharacteristic
233C2#2716108011yescharacteristic
234C2#2716109611yescharacteristic
subgroupG/H
centerC22C2×C22⋊Dic197
commutatorC2×C4×C197C22×C4
FrattiniC22×C4C22×D197
Fitting, prime radical#525×C197C2
socleC22×C197C2×C22⋊C4
cosocleC22×C4×C197C23
hypercenter#525D197
hypocenterC197#1513
LCSG ▷ C2×C4×C197 ▷ C22×C197 ▷ C197
UCSC1 ◁ C22 ◁ C23×C4 ◁ #525

Complements

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

NH ≅ G/Ntype
1. #27161043C2semidirect
2. #27160939C2semidirect
3. #27160816C2semidirect
4. #27160813C2semidirect
12. C2×#41×C197C4semidirect
15. C22⋊Q8×C197C4semidirect
17. #121×C197C4semidirect
22. #117×C197C4semidirect
117. C197#1513semidirect
124. C2×#41Dic197semidirect
128. C22⋊Q8Dic197semidirect
130. #121Dic197semidirect
135. #117Dic197semidirect

Maximal subgroups

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

subgroupindex
#271610432
#271609392
#271608162
#271608132
#525×C1972
#1513197