Finite Group Databasebeta

← #38755845

Subgroups — #38755845

Order 25216.

364 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 364 conjugacy classes of subgroups — too many to draw. Only its 113 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#135×D197C212yesmaximal, complemented
2#27161521C212yesmaximal, complemented
3#27161518C212yesmaximal, complemented
4#27161516C212yesmaximal, complemented
5#27161513C212yesmaximal, complemented
6#27161511C211yesmaximal, characteristic, complemented
7#27161509C212yesmaximal, complemented
8C2×#20263265C211yesmaximal, characteristic, complemented
9#508×C197C211yesmaximal, characteristic, complemented
10#41×D197—22no—
11#20263349C2212yes—
12#20263348C2212yescomplemented
13#20263347—22no—
14#20263346—22no—
15#20263345C2212yes—
16Q16×D197C2212yescomplemented
17SD16×D197—22no—
18D8×D197C2212yescomplemented
19C2×C8×D197C2211yescharacteristic
20Dic1576C2211yescharacteristic, complemented
21#20263272C2212yescomplemented
22#20263271C2212yescomplemented
23#20263270C2212yescomplemented
24#20263269C2212yescomplemented
25D3152C2211yescharacteristic, complemented
26#20263267C2212yescomplemented
27#20263265C2212yescomplemented
28#20263265C2212yescomplemented
29C2×#15691333C2211yescharacteristic, complemented
30#135×C197C2212yescomplemented
31Q32×C197C2211yescharacteristic, complemented
32SD32×C197C2212yescomplemented
33D16×C197C2211yescharacteristic, complemented
34C2×C16×C197C2211yescharacteristic, complemented
35#15691359—22no—
36#15691358—22no—
37#15691358—42no—
38#15691357—22no—
39#15691357—42no—
40Q8×D197—22no—
41D4×D197—22no—
42D4×D197—42no—
43C2×C4×D197—42no—
44C2×C4×D197D411yescharacteristic
45Dic788C2312yes—
46#15691348—22no—
47#15691347—22no—
48#15691346—22no—
49#15691345—22no—
50D1576C2312yes—
51#15691343—22no—
52C2×#13096802D411yescharacteristic
53C8×D197D412yes—
54C8×D197C2312yes—
55#15691333C2312yes—
56#41×C197—22no—
57Q16×C197C2312yes—
58SD16×C197—22no—
59D8×C197C2312yes—
60C2×C8×C197C2311yescharacteristic
61C16×C197C2311yescharacteristic
62C16×C197C2311yescharacteristic
63C22×D197—42no—
64C22×D197D811yescharacteristic
65Dic394—22no—
66Dic394—42no—
67D788—22no—
68D788—42no—
69C22⋊D197—42no—
70C22⋊D197—42no—
71C22⋊D197—42no—
72C2×Dic197—42no—
73C2×Dic197D811yescharacteristic
74C4×D197—42no—
75C4×D197—42no—
76C4×D197—42no—
77C4×D197D812yes—
78C4×D197C2×D412yes—
79#13096802C2×D412yes—
80Q8×C197—22no—
81D4×C197—22no—
82D4×C197—42no—
83C2×C4×C197—42no—
84C2×C4×C197C2×D411yescharacteristic
85C8×C197C2×D411yescharacteristic
86C8×C197C2411yescharacteristic
87C22×C197—42no—
88C22×C197C2×D811yescharacteristic
89C2×D197—41no—
90C2×D197—42no—
91C2×D197—84no—
92C2×D197C2×D812yes—
93C4×C197—42no—
94C4×C197C2×D811yescharacteristic
95C4×C197C22×D411yescharacteristic
96Dic197—42no—
97Dic197C2×D812yes—
98C2×C197—21no—
99C2×C197—82no—
100C2×C197C22×D811yescharacteristic
101D197—22no—
102D197—82no—
103C197#341411yescharacteristic, complemented, Sylow
104#3414—1971nomaximal, Sylow
105C2×#135—1972no—
106#510—1972no—
107#510—1972no—
108#509—1972no—
109#509—1972no—
110#508—1972no—
111#508—1971no—
112#508D19711yescharacteristic, complemented
113C2×M32—1971no—
114C2×#41—3942no—
115#135—3942no—
116#135—3942no—
117#135—1972no—
118#135—1972no—
119#135—1972no—
120#135C2×D19712yescomplemented
121C2×Q16—1972no—
122C2×SD16—3942no—
123C2×D8—1972no—
124C22×C8—1971no—
125Q32—1972no—
126Q32—1971no—
127Q32C2×D19711yescharacteristic, complemented
128SD32—1972no—
129SD32—1972no—
130SD32—1972no—
131SD32C2×D19712yescomplemented
132D16—1972no—
133D16—1971no—
134D16C2×D19711yescharacteristic, complemented
135M32—1972no—
136M32—1972no—
137C2×C16—1971no—
138C2×C16C2×D19711yescharacteristic, complemented
139#41—22no—
140#41—7882no—
141#41—7882no—
142#41—3942no—
143#41—3942no—
144#41—3942no—
145C2×Q8—3942no—
146C2×D4—7882no—
147C2×D4—3942no—
148C22×C4—7882no—
149C22×C4—1971no—
150Q16—3942no—
151Q16—1972no—
152Q16C22×D19712yes—
153SD16—22no—
154SD16—3942no—
155SD16—3942no—
156SD16—3942no—
157D8—3942no—
158D8—1972no—
159D8C22×D19712yes—
160C2×C8—1971no—
161C2×C8—1972no—
162C2×C8—1972no—
163C2×C8C22×D19711yescharacteristic
164C16—1972no—
165C16C22×D19711yescharacteristic
166C16C22×D19711yescharacteristic
167C23—7882no—
168C23—1971no—
169Q8—22no—
170Q8—7882no—
171Q8—3942no—
172D4—22no—
173D4—42no—
174D4—7882no—
175D4—7882no—
176D4—7882no—
177D4—7882no—
178D4—3942no—
179C2×C4—42no—
180C2×C4—7882no—
181C2×C4—7882no—
182C2×C4—7882no—
183C2×C4—7882no—
184C2×C4—1971no—
185C2×C4—1972no—
186C2×C4—1972no—
187C2×C4D4×D19711yescharacteristic
188C8—1972no—
189C8D4×D19711yescharacteristic
190C8C23×D19711yescharacteristic
191C22—42no—
192C22—7881no—
193C22—7882no—
194C22—15764no—
195C22—1972no—
196C22D8×D19711yescharacteristic
197C4—42no—
198C4—7882no—
199C4—1972no—
200C4D8×D19711yescharacteristic
201C4C2×D4×D19711yescharacteristic
202C2—21no—
203C2—82no—
204C2—15762no—
205C2—3942no—
206C2C2×D8×D19711yescharacteristic
subgroupG/H
centerC4D8×D197
commutator, cosocleC8×C197C24
FrattiniC8C23×D197
Fitting#508×C197C2
socleC2×C197C22×D8
hypercenter#508D197
hypocenterC197#3414
LCSG ▷ C8×C197 ▷ C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C4 ◁ C2×C4 ◁ C2×C8 ◁ #508

Complements

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

NH ≅ G/Ntype
1. #135×D197C2semidirect
2. #27161521C2semidirect
3. #27161518C2semidirect
4. #27161516C2semidirect
5. #27161513C2semidirect
6. #27161511C2semidirect
7. #27161509C2semidirect
8. C2×#20263265C2semidirect
9. #508×C197C2semidirect
12. #20263348C22semidirect
16. Q16×D197C22semidirect
18. D8×D197C22semidirect
20. Dic1576C22semidirect
21. #20263272C22semidirect
22. #20263271C22semidirect
23. #20263270C22semidirect
24. #20263269C22semidirect
25. D3152C22semidirect
26. #20263267C22semidirect
27. #20263265C22semidirect
28. #20263265C22semidirect
29. C2×#15691333C22semidirect
30. #135×C197C22semidirect
31. Q32×C197C22semidirect
32. SD32×C197C22semidirect
33. D16×C197C22semidirect
34. C2×C16×C197C22semidirect
103. C197#3414semidirect
112. #508D197semidirect
120. #135C2×D197semidirect
127. Q32C2×D197semidirect
131. SD32C2×D197semidirect
134. D16C2×D197semidirect
138. C2×C16C2×D197semidirect

Maximal subgroups

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

subgroupindex
#135×D1972
#271615212
#271615182
#271615162
#271615132
#271615112
#271615092
C2×#202632652
#508×C1972
#3414197