Finite Group Databasebeta

← #37554993

Subgroups — #37554993

Order 23488.

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

normal subgroups only

This group has 258 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
1#26044308C211yesmaximal, characteristic, complemented
2#26044307C211yesmaximal, characteristic, complemented
3#26044293C212yesmaximal, complemented
4#26044291C211yesmaximal, characteristic, complemented
5#26044286C211yesmaximal, characteristic, complemented
6#26044285C212yesmaximal, complemented
7#26044282C212yesmaximal, complemented
8C2×#19154566C211yesmaximal, characteristic, complemented
9SD16×D367C212yesmaximal, complemented
10#26044267C211yesmaximal, characteristic, complemented
11#135×C367C211yesmaximal, characteristic, complemented
12#19154578C2212yes—
13#19154577—21no—
14#19154577—21no—
15#19154577C2211yescharacteristic
16#19154576—22no—
17#19154576C2212yescomplemented
18C2×Dic734—21no—
19C2×Dic734C2211yescharacteristic, complemented
20Q8×D367C2212yescomplemented
21C2×D1468—21no—
22C2×D1468C2211yescharacteristic
23D4×D367—22no—
24D4×D367C2212yes—
25#19154567C2212yescomplemented
26#19154566C2212yescomplemented
27#19154566C2212yescomplemented
28#19154565C2212yescomplemented
29#19154564C2212yescomplemented
30#19154562C2212yescomplemented
31#19154561C2211yescharacteristic, complemented
32#19154560C2212yescomplemented
33C8×D367C2212yescomplemented
34#41×C367C2211yescharacteristic
35#41×C367C2211yescharacteristic, complemented
36Q16×C367C2211yescharacteristic, complemented
37SD16×C367C2211yescharacteristic, complemented
38SD16×C367C2211yescharacteristic, complemented
39D8×C367C2211yescharacteristic, complemented
40C2×C8×C367C2211yescharacteristic, complemented
41C22×D367—21no—
42C22×D367—22no—
43Dic734—21no—
44Dic734—22no—
45Dic734D411yescharacteristic
46Dic734C2312yes—
47D1468—22no—
48D1468—21no—
49D1468D411yescharacteristic
50D1468C2312yes—
51C22⋊D367—22no—
52C22⋊D367—22no—
53C22⋊D367D412yes—
54C2×Dic367—21no—
55C2×Dic367—22no—
56C4×D367—22no—
57C4×D367—22no—
58C4×D367C2312yes—
59C367⋊C8C2312yes—
60Q8×C367C2311yescharacteristic
61Q8×C367C2311yescharacteristic
62D4×C367—21no—
63D4×C367—21no—
64D4×C367C2311yescharacteristic
65D4×C367C2311yescharacteristic
66C2×C4×C367—21no—
67C2×C4×C367—21no—
68C2×C4×C367C2311yescharacteristic
69C8×C367C2311yescharacteristic
70C8×C367C2311yescharacteristic
71C22×C367—21no—
72C22×C367—21no—
73C22×C367C2×D411yescharacteristic
74C2×D367—22no—
75C2×D367—82no—
76C2×D367—81no—
77C2×D367C2×D412yes—
78C4×C367—21no—
79C4×C367—21no—
80C4×C367C2×D411yescharacteristic
81C4×C367C2411yescharacteristic
82Dic367—22no—
83Dic367C2×D412yes—
84C2×C367—21no—
85C2×C367—41no—
86C2×C367—41no—
87C2×C367C22×D411yescharacteristic
88D367—22no—
89D367—42no—
90C367#57711yescharacteristic, complemented, Sylow
91#577—3671nomaximal, Sylow
922−1+4—3671no—
93H(5,𝔽2)—3671no—
94#137—3671no—
95#137—3672no—
96AGL(1,ℤ/8)—3671no—
97AGL(1,ℤ/8)—3672no—
98#135—3672no—
99#135D36711yescharacteristic, complemented
100C2×SD16—3672no—
101C2×SD16—3671no—
102#131—3671no—
103#41—3671no—
104#41—3672no—
105#41—3672no—
106#41—7341no—
107#41—7341no—
108#41—7342no—
109#41C2×D36711yescharacteristic
110#41C2×D36711yescharacteristic, complemented
111C2×Q8—3672no—
112C2×Q8—3671no—
113C2×Q8—7341no—
114C2×D4—3672no—
115C2×D4—3671no—
116C2×D4—7341no—
117C2×D4—7342no—
118Q16—3672no—
119Q16C2×D36711yescharacteristic, complemented
120SD16—3672no—
121SD16—3672no—
122SD16—3672no—
123SD16—3672no—
124SD16C2×D36711yescharacteristic, complemented
125SD16C2×D36711yescharacteristic, complemented
126D8—3672no—
127D8C2×D36711yescharacteristic, complemented
128M16—3671no—
129M16—3672no—
130C2×C8—3672no—
131C2×C8C2×D36711yescharacteristic, complemented
132C23—7342no—
133C23—7341no—
134Q8—3671no—
135Q8—3672no—
136Q8—7342no—
137Q8—7341no—
138Q8C22×D36711yescharacteristic
139Q8C22×D36711yescharacteristic
140D4—21no—
141D4—21no—
142D4—3672no—
143D4—3671no—
144D4—3672no—
145D4—7342no—
146D4—7342no—
147D4—7342no—
148D4—7341no—
149D4C22×D36711yescharacteristic
150D4C22×D36711yescharacteristic
151C2×C4—21no—
152C2×C4—21no—
153C2×C4—3672no—
154C2×C4—7342no—
155C2×C4—7342no—
156C2×C4—7342no—
157C2×C4—7341no—
158C2×C4C22×D36711yescharacteristic
159C8—3672no—
160C8C22×D36711yescharacteristic
161C8C22×D36711yescharacteristic
162C22—21no—
163C22—21no—
164C22—3672no—
165C22—29361no—
166C22—29362no—
167C22—7342no—
168C22D4×D36711yescharacteristic
169C4—21no—
170C4—21no—
171C4—3672no—
172C4—7342no—
173C4D4×D36711yescharacteristic
174C4C23×D36711yescharacteristic
175C2—21no—
176C2—41no—
177C2—41no—
178C2—14682no—
179C2—7342no—
180C2C2×D4×D36711yescharacteristic
subgroupG/H
centerC2C2×D4×D367
commutator, cosocleC4×C367C24
FrattiniC4C23×D367
Fitting#135×C367C2
socleC2×C367C22×D4
hypercenter#135D367
hypocenterC367#577
LCSG ▷ C4×C367 ▷ C2×C367 ▷ C367
UCSC1 ◁ C2 ◁ C2×C4 ◁ #135

Complements

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

NH ≅ G/Ntype
1. #26044308C2semidirect
2. #26044307C2semidirect
3. #26044293C2semidirect
4. #26044291C2semidirect
5. #26044286C2semidirect
6. #26044285C2semidirect
7. #26044282C2semidirect
8. C2×#19154566C2semidirect
9. SD16×D367C2semidirect
10. #26044267C2semidirect
11. #135×C367C2semidirect
17. #19154576C22semidirect
19. C2×Dic734C22semidirect
20. Q8×D367C22semidirect
25. #19154567C22semidirect
26. #19154566C22semidirect
27. #19154566C22semidirect
28. #19154565C22semidirect
29. #19154564C22semidirect
30. #19154562C22semidirect
31. #19154561C22semidirect
32. #19154560C22semidirect
33. C8×D367C22semidirect
35. #41×C367C22semidirect
36. Q16×C367C22semidirect
37. SD16×C367C22semidirect
38. SD16×C367C22semidirect
39. D8×C367C22semidirect
40. C2×C8×C367C22semidirect
90. C367#577semidirect
99. #135D367semidirect
110. #41C2×D367semidirect
119. Q16C2×D367semidirect
124. SD16C2×D367semidirect
125. SD16C2×D367semidirect
127. D8C2×D367semidirect
131. C2×C8C2×D367semidirect

Maximal subgroups

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

subgroupindex
#260443082
#260443072
#260442932
#260442912
#260442862
#260442852
#260442822
C2×#191545662
SD16×D3672
#260442672
#135×C3672
#577367