Finite Group Databasebeta

← #38752119

Subgroups — #38752119

Order 25216.

716 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 716 conjugacy classes of subgroups — too many to draw. Only its 145 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#27161915C211yesmaximal, characteristic
2#27161655C212yesmaximal, complemented
3#27161650C211yesmaximal, characteristic, complemented
4#27161270C218yesmaximal, complemented
5#27161052C212yesmaximal, complemented
6#410×C197C211yesmaximal, characteristic, complemented
7#20263374—21no—
8#20263373—22no—
9#20263372—22no—
10#20263371—21no—
11#20263371C2214yes—
12C2×#15691358—22no—
13C2×#15691358C2212yes—
14#41×D197—24no—
15C22×Dic394C2211yescharacteristic, complemented
16#20263326—22no—
17#20263324—21no—
18#20263314—22no—
19#20263313—21no—
20#20263310C2218yescomplemented
21#20263307C2214yescomplemented
22#20263306—24no—
23#20263305—22no—
24#20263285C2211yescharacteristic, complemented
25#20263240C2218yescomplemented
26C2×#41×C197C2211yescharacteristic
27#117×C197C2212yescomplemented
28#99×C197C2214yescomplemented
29#15691359—24no—
30#15691358—24no—
31#15691358—24no—
32#15691358—22no—
33#15691358—22no—
34#15691358—24no—
35#15691358—22no—
36#15691358—22no—
37#15691357—24no—
38#15691357—28no—
39C2×Dic394—22no—
40C2×Dic394—21no—
41C2×Dic394—22no—
42C2×Dic394—21no—
43C2×Dic394—22no—
44C2×Dic394D412yes—
45Q8×D197—24no—
46C2×D788—22no—
47C2×D788D412yes—
48C2×C22⋊D197—24no—
49C2×C22⋊D197C2314yes—
50D4×D197—24no—
51D4×D197—28no—
52C22×Dic197C2312yes—
53C2×C4×D197—24no—
54C2×C4×D197D414yes—
55C4⋊Dic197—28no—
56C4⋊Dic197—21no—
57#15691337—24no—
58C22⋊Dic197—24no—
59C22⋊Dic197C2312yes—
60#15691335—28no—
61C4×Dic197—21no—
62#41×C197—22no—
63#41×C197—22no—
64C2×Q8×C197D411yescharacteristic
65C2×D4×C197D411yescharacteristic
66C2×D4×C197C2312yes—
67C22×C4×C197D412yes—
68C22×C4×C197C2311yescharacteristic
69C4⋊C4×C197—22no—
70C22⋊C4×C197C2314yes—
71C42×C197—22no—
72C22×D197—24no—
73C22×D197C2×D414yes—
74Dic394—24no—
75Dic394—22no—
76Dic394—22no—
77Dic394—41no—
78Dic394—41no—
79Dic394—42no—
80D788—24no—
81D788—22no—
82D788—22no—
83C22⋊D197—28no—
84C22⋊D197—24no—
85C22⋊D197—24no—
86C2×Dic197—24no—
87C2×Dic197—22no—
88C2×Dic197—24no—
89C2×Dic197—22no—
90C2×Dic197C2×D414yes—
91C4×D197—28no—
92C4×D197—24no—
93C4×D197—24no—
94C23×C197C2×D412yes—
95C23×C197C2411yescharacteristic
96Q8×C197—22no—
97D4×C197—24no—
98D4×C197—22no—
99C2×C4×C197—22no—
100C2×C4×C197—22no—
101C2×C4×C197—24no—
102C2×C4×C197C2×D412yes—
103C2×C4×C197C2×D412yes—
104C2×C4×C197C2×D412yes—
105C2×C4×C197C2×D411yescharacteristic
106C2×C4×C197C2×D411yescharacteristic
107C22×C197—22no—
108C22×C197—41no—
109C22×C197—42no—
110C22×C197C22×D412yes—
111C22×C197C22×D411yescharacteristic
112C2×D197—24no—
113C2×D197—44no—
114C2×D197—84no—
115C4×C197—22no—
116C4×C197—44no—
117C4×C197#12011yescharacteristic
118C4×C197#12011yescharacteristic
119C4×C197#12012yes—
120Dic197—24no—
121Dic197—42no—
122Dic197—42no—
123C2×C197—21no—
124C2×C197—22no—
125C2×C197—42no—
126C2×C197C2×#12011yescharacteristic
127D197—44no—
128C197#302811yescharacteristic, complemented, Sylow
129#3028—1971nomaximal, Sylow
130#585—1971no—
131#536—1971no—
132#536—1972no—
133#458—1978no—
134#410—1972no—
135#410D19711yescharacteristic, complemented
1362−1+4—3942no—
1372−1+4—3941no—
138H(5,𝔽2)—3942no—
139H(5,𝔽2)—3941no—
140H(5,𝔽2)—1974no—
141C2×#41—3944no—
142C2×#41—3942no—
143C2×#41—1972no—
144C2×#41C2×D19711yescharacteristic
145C22×Q8—1971no—
146C4⋊Q8—3942no—
147C4⋊Q8—3941no—
148#124—3942no—
149#124—3941no—
150#123—1974no—
151#123—1978no—
152C22⋊Q8—3944no—
153C22⋊Q8—3942no—
154#117—1971no—
155#117C2×D19712yescomplemented
156#99—1978no—
157#99C2×D19714yescomplemented
158#41—22no—
159#41—22no—
160#41—3944no—
161#41—3944no—
162#41—3944no—
163#41—3944no—
164#41—3948no—
165#41—3942no—
166#41—3942no—
167#41—3942no—
168#41—3944no—
169#41—3942no—
170C2×Q8—3942no—
171C2×Q8—3944no—
172C2×Q8—3942no—
173C2×Q8—3942no—
174C2×Q8—3941no—
175C2×Q8—3941no—
176C2×Q8—1972no—
177C2×Q8D78811yescharacteristic
178C2×D4—3942no—
179C2×D4—3944no—
180C2×D4—3948no—
181C2×D4—3944no—
182C2×D4—1974no—
183C2×D4—1972no—
184C2×D4D78811yescharacteristic
185C2×D4C22×D19712yes—
186C22×C4—3944no—
187C22×C4—1972no—
188C22×C4—1974no—
189C22×C4D78812yes—
190C22×C4C22×D19711yescharacteristic
191C4⋊C4—22no—
192C4⋊C4—3948no—
193C4⋊C4—3944no—
194C4⋊C4—3941no—
195C22⋊C4—3948no—
196C22⋊C4—3944no—
197C22⋊C4—1972no—
198C22⋊C4C22×D19714yes—
199C42—22no—
200C42—3941no—
201C23—3944no—
202C23—1974no—
203C23C2×D78812yes—
204C23C23×D19711yescharacteristic
205Q8—22no—
206Q8—7881no—
207Q8—7881no—
208Q8—7882no—
209Q8—3944no—
210Q8—3942no—
211Q8—3942no—
212D4—24no—
213D4—22no—
214D4—3944no—
215D4—3948no—
216D4—3944no—
217D4—3944no—
218D4—3942no—
219D4—3942no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—24no—
223C2×C4—3948no—
224C2×C4—3944no—
225C2×C4—3942no—
226C2×C4—3942no—
227C2×C4—3944no—
228C2×C4—3944no—
229C2×C4—3944no—
230C2×C4—1974no—
231C2×C4D4×D19712yes—
232C2×C4D4×D19712yes—
233C2×C4C2×D78812yes—
234C2×C4C2×D78811yescharacteristic
235C2×C4C2×D78811yescharacteristic
236C22—22no—
237C22—41no—
238C22—42no—
239C22—7884no—
240C22—15764no—
241C22—3944no—
242C22C2×D4×D19712yes—
243C22C22×D78811yescharacteristic
244C4—22no—
245C4—44no—
246C4—7882no—
247C4—7882no—
248C4—3944no—
249C4#2026329611yescharacteristic
250C4#2026329611yescharacteristic
251C4#2026329612yes—
252C2—22no—
253C2—21no—
254C2—42no—
255C2—7884no—
256C2C2×#2026329611yescharacteristic
subgroupG/H
centerC2C2×#20263296
commutator, cosocleC23×C197C24
FrattiniC23C23×D197
Fitting#410×C197C2
socleC2×C197C2×#120
prime radical#27161915C2
hypercenter#410D197
hypocenterC197#3028
LCSG ▷ C23×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C2 ◁ C22×C4 ◁ #410

Complements

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

NH ≅ G/Ntype
2. #27161655C2semidirect
3. #27161650C2semidirect
4. #27161270C2semidirect
5. #27161052C2semidirect
6. #410×C197C2semidirect
15. C22×Dic394C22semidirect
20. #20263310C22semidirect
21. #20263307C22semidirect
24. #20263285C22semidirect
25. #20263240C22semidirect
27. #117×C197C22semidirect
28. #99×C197C22semidirect
128. C197#3028semidirect
135. #410D197semidirect
155. #117C2×D197semidirect
157. #99C2×D197semidirect

Maximal subgroups

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

subgroupindex
#271619152
#271616552
#271616502
#271612702
#271610522
#410×C1972
#3028197