Finite Group Databasebeta

← #38756124

Subgroups — #38756124

Order 25216.

1160 conjugacy classes of subgroups; only the 471 normal subgroups are shown.

normal subgroups only

This group has 1160 conjugacy classes of subgroups — too many to draw. Only its 471 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#27161678C216yesmaximal, complemented
2#27161669C216yesmaximal, complemented
3#27161641C213yesmaximal, complemented
4#27161615C216yesmaximal, complemented
5#27161592C216yesmaximal, complemented
6C2×#20263286C213yesmaximal, complemented
7#543×C197C211yesmaximal, characteristic, complemented
8C2×#15691359C2212yescomplemented
9C22×C22⋊D197C2213yes—
10C23×Dic197C2211yescharacteristic
11C22×C4×D197C2213yescomplemented
12#20263321C22112yescomplemented
13#20263317C2216yescomplemented
14#20263315C2216yescomplemented
15#20263311C22112yescomplemented
16#20263310C22112yescomplemented
17#20263302C2216yescomplemented
18#20263301C2216yescomplemented
19#20263300C2216yes—
20#20263300C2216yes—
21#20263296C22112yescomplemented
22Q8×Dic197C2212yescomplemented
23#20263288C2216yescomplemented
24#20263286C22112yescomplemented
25#20263286C22112yescomplemented
26#20263284C2216yescomplemented
27C2×#15691337C2213yescomplemented
28C2×#15691335C2213yescomplemented
29C2×C4×Dic197C2213yescomplemented
30C22⋊Q8×C197C2216yescomplemented
31C22⋊Q8×C197C2216yescomplemented
32C2×C22⋊C4×C197C2213yes—
33C23×D197C2313yes—
34#15691359—44no—
35C2×D788C2316yes—
36C2×D788C2316yes—
37C2×C22⋊D197—212no—
38C2×C22⋊D197C23112yes—
39C22×Dic197—42no—
40C22×Dic197C2313yes—
41C22×Dic197C2313yes—
42C22×Dic197C2313yes—
43C2×C4×D197—46no—
44C2×C4×D197C2316yes—
45C2×C4×D197C2316yes—
46C4⋊Dic197C2316yes—
47C4⋊Dic197C2316yes—
48#15691337C23112yes—
49#15691335C23112yes—
50#15691335C23112yes—
51#15691335C23112yes—
52C4×Dic197C2316yes—
53C4×Dic197C2316yes—
54C24×C197C2311yescharacteristic
55C2×Q8×C197C2312yes—
56C2×Q8×C197C2312yes—
57C22×C4×C197C2313yes—
58C22×C4×C197C2313yes—
59C4⋊C4×C197C2316yes—
60C4⋊C4×C197C2316yes—
61C22⋊C4×C197C23112yes—
62C22×D197—26no—
63C22×D197—43no—
64C22×D197—43no—
65C22×D197C2416yes—
66D788—46no—
67D788—46no—
68C22⋊D197—224no—
69C2×Dic197—26no—
70C2×Dic197—26no—
71C2×Dic197—43no—
72C2×Dic197C2413yes—
73C2×Dic197C2416yes—
74C2×Dic197C2411yescharacteristic
75C4×D197—212no—
76C4×D197—46no—
77C23×C197—41no—
78C23×C197—41no—
79C23×C197#4114yescomplemented
80C23×C197C2413yes—
81Q8×C197—42no—
82Q8×C197—42no—
83C2×C4×C197—43no—
84C2×C4×C197—43no—
85C2×C4×C197C2416yes—
86C2×C4×C197C2416yes—
87C22×C197—23no—
88C22×C197—23no—
89C22×C197—44no—
90C22×C197C2×#4116yes—
91C22×C197C2511yescharacteristic
92C2×D197—26no—
93C2×D197—26no—
94C2×D197—26no—
95C2×D197—412no—
96C4×C197—26no—
97C4×C197—26no—
98Dic197—26no—
99Dic197—23no—
100Dic197H(5,𝔽2)12yes—
101C2×C197—26no—
102C2×C197C22×#4111yescharacteristic
103C2×C197C2×H(5,𝔽2)11yescharacteristic
104C2×C197#58511yescharacteristic
105D197—46no—
106C197#345911yescharacteristic, complemented, Sylow
107#3459—1971nomaximal, Sylow
108#543D19711yescharacteristic, complemented
109#539—1976no—
110#538—1976no—
111#534—1973no—
112#529—1976no—
113#525—1976no—
114C2×C4×D4—1973no—
115C2×#41—1972no—
116C22×D4—1973no—
117C23×C4—1971no—
118C23×C4—1973no—
119#126—19712no—
120#125—1976no—
121#124—1976no—
122#123—19712no—
123#123—19712no—
124C22⋊Q8—1976no—
125C22⋊Q8C2×D19716yescomplemented
126C22⋊Q8C2×D19716yescomplemented
127#121—1976no—
128#121—1976no—
129#121—1976no—
130#120—19712no—
131C4×Q8—1972no—
132C4×D4—1976no—
133C4×D4—19712no—
134C4×D4—19712no—
135#117—1976no—
136C2×C4⋊C4—1973no—
137C2×C22⋊C4—1973no—
138C2×C22⋊C4C2×D19713yes—
139C2×C42—1973no—
140C24—1973no—
141C24C22×D19711yescharacteristic
142#41—7884no—
143C2×Q8C22×D19712yes—
144C2×Q8C22×D19712yes—
145C2×D4—39412no—
146C2×D4—19712no—
147C2×D4—1976no—
148C2×D4—1976no—
149C22×C4—7886no—
150C22×C4—7882no—
151C22×C4—1973no—
152C22×C4—1976no—
153C22×C4—1973no—
154C22×C4—1973no—
155C22×C4—1976no—
156C22×C4C22×D19713yes—
157C22×C4C22×D19713yes—
158C4⋊C4—19712no—
159C4⋊C4—1976no—
160C4⋊C4—1976no—
161C4⋊C4C22×D19716yes—
162C4⋊C4C22×D19716yes—
163C22⋊C4—19712no—
164C22⋊C4—19712no—
165C22⋊C4—19712no—
166C22⋊C4C22×D197112yes—
167C42—1976no—
168C42—1976no—
169C23—41no—
170C23—41no—
171C23—7883no—
172C23—7883no—
173C23—3946no—
174C23—1976no—
175C23#1569135914yescomplemented
176C23C23×D19713yes—
177Q8—42no—
178Q8—42no—
179D4—7886no—
180D4—7886no—
181D4—39424no—
182C2×C4—43no—
183C2×C4—43no—
184C2×C4—7886no—
185C2×C4—7883no—
186C2×C4—39412no—
187C2×C4—3946no—
188C2×C4—3946no—
189C2×C4—1976no—
190C2×C4—1973no—
191C2×C4—1971no—
192C2×C4C23×D19716yes—
193C2×C4C23×D19716yes—
194C22—23no—
195C22—23no—
196C22—44no—
197C22—78812no—
198C22—3946no—
199C22—3946no—
200C22—3946no—
201C22C2×#1569135916yes—
202C22C24×D19711yescharacteristic
203C4—26no—
204C4—26no—
205C4—3946no—
206C4—3943no—
207C4—1972no—
208C2—26no—
209C2—7886no—
210C2C22×#1569135911yescharacteristic
211C2H(5,𝔽2)×D19711yescharacteristic
212C2#2716191411yescharacteristic
subgroupG/H
center, FrattiniC22C24×D197
commutator, socle, cosocleC22×C197C25
Fitting#543×C197C2
hypercenter#543D197
hypocenterC197#3459
LCSG ▷ C22×C197 ▷ C197
UCSC1 ◁ C22 ◁ #543

Complements

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

NH ≅ G/Ntype
1. #27161678C2semidirect
2. #27161669C2semidirect
3. #27161641C2semidirect
4. #27161615C2semidirect
5. #27161592C2semidirect
6. C2×#20263286C2semidirect
7. #543×C197C2semidirect
8. C2×#15691359C22semidirect
11. C22×C4×D197C22semidirect
12. #20263321C22semidirect
13. #20263317C22semidirect
14. #20263315C22semidirect
15. #20263311C22semidirect
16. #20263310C22semidirect
17. #20263302C22semidirect
18. #20263301C22semidirect
21. #20263296C22semidirect
22. Q8×Dic197C22semidirect
23. #20263288C22semidirect
24. #20263286C22semidirect
25. #20263286C22semidirect
26. #20263284C22semidirect
27. C2×#15691337C22semidirect
28. C2×#15691335C22semidirect
29. C2×C4×Dic197C22semidirect
30. C22⋊Q8×C197C22semidirect
31. C22⋊Q8×C197C22semidirect
79. C23×C197#41semidirect
106. C197#3459semidirect
108. #543D197semidirect
125. C22⋊Q8C2×D197semidirect
126. C22⋊Q8C2×D197semidirect
175. C23#15691359semidirect

Maximal subgroups

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

subgroupindex
#271616782
#271616692
#271616412
#271616152
#271615922
C2×#202632862
#543×C1972
#3459197