Finite Group Databasebeta

← #38763321

Subgroups — #38763321

Order 25408.

262 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 262 conjugacy classes of subgroups — too many to draw. Only its 99 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
1C2×#20263732C211yesmaximal, characteristic, complemented
2#41×D397C211yesmaximal, characteristic, complemented
3#27163128C212yesmaximal, complemented
4#27163127C211yesmaximal, characteristic, complemented
5#27163124C211yesmaximal, characteristic, complemented
6#27163123C212yesmaximal, complemented
7#27163121C211yesmaximal, characteristic, complemented
8#27163119C212yesmaximal, complemented
9#27163117C212yesmaximal, complemented
10M16×D397C211yesmaximal, characteristic, complemented
11#137×C397C211yesmaximal, characteristic, complemented
12#20263732—21no—
13#20263732—21no—
14#20263732C2212yes—
15#20263732C2211yescharacteristic, complemented
16#20263732C2212yescomplemented
17#20263731—21no—
18#20263731C2211yescharacteristic
19#20263730—21no—
20#20263730C2211yescharacteristic, complemented
21Q8×D397C2211yescharacteristic, complemented
22C2×D1588—21no—
23C2×D1588C2211yescharacteristic, complemented
24D4×D397—21no—
25D4×D397C2211yescharacteristic, complemented
26C2×C4×D397—21no—
27C2×C4×D397—21no—
28C2×C4×D397C2211yescharacteristic
29#20263721C2212yescomplemented
30#20263720C2212yescomplemented
31#20263718C2212yescomplemented
32#20263718C2212yescomplemented
33D3176C2212yescomplemented
34#20263716C2212yescomplemented
35#20263715C2211yescharacteristic, complemented
36#20263714C2212yescomplemented
37C8×D397C2212yescomplemented
38#41×C397C2211yescharacteristic
39C2×Q8×C397C2211yescharacteristic, complemented
40Q16×C397C2212yescomplemented
41SD16×C397C2212yescomplemented
42M16×C397C2211yescharacteristic, complemented
43C22×D397—21no—
44C22×D397—21no—
45C22×D397D411yescharacteristic
46Dic794—21no—
47Dic794C2311yescharacteristic
48D1588—21no—
49D1588—22no—
50D1588—22no—
51D1588—21no—
52D1588C2312yes—
53D1588C2311yescharacteristic
54C22⋊D397—21no—
55C22⋊D397—21no—
56C22⋊D397—21no—
57C2×Dic397—21no—
58C2×Dic397D411yescharacteristic
59C4×D397—21no—
60C4×D397—21no—
61C4×D397—21no—
62C4×D397—22no—
63C4×D397—22no—
64C4×D397D411yescharacteristic
65C4×D397D411yescharacteristic
66C4×D397C2311yescharacteristic
67C4×D397C2311yescharacteristic
68#15691787C2312yes—
69Q8×C397—21no—
70Q8×C397C2312yes—
71Q8×C397C2311yescharacteristic
72D4×C397—21no—
73D4×C397C2311yescharacteristic
74C2×C4×C397—21no—
75C2×C4×C397—21no—
76C2×C4×C397C2311yescharacteristic
77C8×C397C2312yes—
78C22×C397—21no—
79C22×C397C2×D411yescharacteristic
80C2×D397—21no—
81C2×D397—22no—
82C2×D397—41no—
83C2×D397—42no—
84C2×D397—42no—
85C2×D397C2×D411yescharacteristic
86C2×D397C2×D411yescharacteristic
87C4×C397—21no—
88C4×C397—22no—
89C4×C397C2×D411yescharacteristic
90C4×C397C2411yescharacteristic
91Dic397—21no—
92Dic397C2×D411yescharacteristic
93Dic397C2×D411yescharacteristic
94C2×C397—21no—
95C2×C397—41no—
96C2×C397C22×D411yescharacteristic
97D397—21no—
98D397—21no—
99D397—41no—
100D397—42no—
101C397#57511yescharacteristic, complemented, Sylow
102#575—3971nomaximal, Sylow
103C2×#41—3971no—
104C2×#41—3971no—
105#137—3971no—
106#137—3972no—
107#137D39711yescharacteristic, complemented
108AGL(1,ℤ/8)—3971no—
109AGL(1,ℤ/8)—3972no—
110AGL(1,ℤ/8)—3971no—
111#135—3972no—
112#135—3972no—
113C2×M16—3971no—
114#41—7941no—
115#41—7941no—
116#41—7941no—
117#41—7941no—
118#41—3972no—
119#41—3971no—
120#41—3972no—
121#41—3971no—
122#41—3971no—
123#41C2×D39711yescharacteristic
124C2×Q8—3971no—
125C2×Q8C2×D39711yescharacteristic, complemented
126C2×D4—7941no—
127C2×D4—7941no—
128C2×D4—3971no—
129C2×D4—3971no—
130C22×C4—7941no—
131C22×C4—7941no—
132C22×C4—3971no—
133Q16—3972no—
134Q16C2×D39712yescomplemented
135SD16—3972no—
136SD16—3972no—
137SD16—3972no—
138SD16C2×D39712yescomplemented
139D8—3972no—
140D8—3972no—
141M16—3971no—
142M16—3972no—
143M16C2×D39711yescharacteristic, complemented
144C2×C8—3972no—
145C23—7941no—
146C23—7941no—
147C23—3971no—
148Q8—21no—
149Q8—7941no—
150Q8—3971no—
151Q8C22×D39711yescharacteristic
152Q8C22×D39712yes—
153D4—21no—
154D4—7941no—
155D4—7942no—
156D4—7942no—
157D4—7941no—
158D4—7941no—
159D4—7941no—
160D4—7941no—
161D4—3972no—
162D4—3971no—
163D4C22×D39711yescharacteristic
164C2×C4—21no—
165C2×C4—21no—
166C2×C4—7941no—
167C2×C4—7941no—
168C2×C4—7942no—
169C2×C4—7942no—
170C2×C4—7941no—
171C2×C4—7941no—
172C2×C4—3971no—
173C2×C4—3971no—
174C2×C4—3971no—
175C2×C4—3971no—
176C2×C4—3971no—
177C2×C4C22×D39711yescharacteristic
178C8—3972no—
179C8C22×D39712yes—
180C22—21no—
181C22—7942no—
182C22—7941no—
183C22—15881no—
184C22—15882no—
185C22—15882no—
186C22—3971no—
187C22—3971no—
188C22D4×D39711yescharacteristic
189C4—21no—
190C4—22no—
191C4—7941no—
192C4—3971no—
193C4—3971no—
194C4D4×D39711yescharacteristic
195C4C23×D39711yescharacteristic
196C2—21no—
197C2—41no—
198C2—7941no—
199C2—7941no—
200C2—15882no—
201C2—15881no—
202C2C2×D4×D39711yescharacteristic
subgroupG/H
centerC2C2×D4×D397
commutator, cosocleC4×C397C24
FrattiniC4C23×D397
Fitting#137×C397C2
socleC2×C397C22×D4
hypercenter#137D397
hypocenterC397#575
LCSG ▷ C4×C397 ▷ C2×C397 ▷ C397
UCSC1 ◁ C2 ◁ C2×C4 ◁ #137

Complements

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

NH ≅ G/Ntype
1. C2×#20263732C2semidirect
2. #41×D397C2semidirect
3. #27163128C2semidirect
4. #27163127C2semidirect
5. #27163124C2semidirect
6. #27163123C2semidirect
7. #27163121C2semidirect
8. #27163119C2semidirect
9. #27163117C2semidirect
10. M16×D397C2semidirect
11. #137×C397C2semidirect
15. #20263732C22semidirect
16. #20263732C22semidirect
20. #20263730C22semidirect
21. Q8×D397C22semidirect
23. C2×D1588C22semidirect
25. D4×D397C22semidirect
29. #20263721C22semidirect
30. #20263720C22semidirect
31. #20263718C22semidirect
32. #20263718C22semidirect
33. D3176C22semidirect
34. #20263716C22semidirect
35. #20263715C22semidirect
36. #20263714C22semidirect
37. C8×D397C22semidirect
39. C2×Q8×C397C22semidirect
40. Q16×C397C22semidirect
41. SD16×C397C22semidirect
42. M16×C397C22semidirect
101. C397#575semidirect
107. #137D397semidirect
125. C2×Q8C2×D397semidirect
134. Q16C2×D397semidirect
138. SD16C2×D397semidirect
143. M16C2×D397semidirect

Maximal subgroups

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

subgroupindex
C2×#202637322
#41×D3972
#271631282
#271631272
#271631242
#271631232
#271631212
#271631192
#271631172
M16×D3972
#137×C3972
#575397