Finite Group Databasebeta

← #37554996

Subgroups — #37554996

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#26044306C211yesmaximal, characteristic, complemented
3#26044294C211yesmaximal, characteristic, complemented
4#26044290C212yesmaximal, complemented
5#26044285C212yesmaximal, complemented
6#26044281C211yesmaximal, characteristic, complemented
7#26044280C212yesmaximal, complemented
8C2×#19154565C211yesmaximal, characteristic, complemented
9SD16×D367C212yesmaximal, complemented
10#26044269C211yesmaximal, characteristic, complemented
11AGL(1,ℤ/8)×C367C211yesmaximal, characteristic, complemented
12#19154577—21no—
13#19154577C2211yescharacteristic
14#19154577C2211yescharacteristic
15#19154576—21no—
16#19154576—22no—
17#19154576C2212yes—
18#19154576C2212yescomplemented
19C2×Dic734—21no—
20C2×Dic734C2211yescharacteristic, complemented
21Q8×D367C2212yescomplemented
22C2×C22⋊D367—22no—
23D4×D367—21no—
24D4×D367C2212yes—
25Dic1468C2212yescomplemented
26#19154567C2211yescharacteristic, complemented
27#19154566C2212yescomplemented
28#19154565C2212yescomplemented
29#19154565C2212yescomplemented
30#19154565C2211yescharacteristic, complemented
31#19154564C2211yescharacteristic, complemented
32#19154562C2211yescharacteristic, complemented
33#19154560C2212yescomplemented
34C2×C367⋊C8C2211yescharacteristic, complemented
35C8×D367C2212yescomplemented
36#41×C367C2211yescharacteristic, complemented
37C2×D4×C367C2211yescharacteristic
38SD16×C367C2212yescomplemented
39D8×C367C2212yescomplemented
40M16×C367C2211yescharacteristic, complemented
41C22×D367—22no—
42Dic734—21no—
43Dic734—22no—
44Dic734D411yescharacteristic
45Dic734C2312yes—
46Dic734C2311yescharacteristic
47D1468D411yescharacteristic
48D1468C2311yescharacteristic
49C22⋊D367—21no—
50C22⋊D367—22no—
51C22⋊D367—22no—
52C22⋊D367—22no—
53C22⋊D367D412yes—
54C2×Dic367—21no—
55C2×Dic367—22no—
56C2×Dic367—22no—
57C4×D367—22no—
58C4×D367—21no—
59C4×D367C2312yes—
60C367⋊C8C2311yescharacteristic
61C367⋊C8C2311yescharacteristic
62C23×C367—21no—
63Q8×C367C2311yescharacteristic
64D4×C367—21no—
65D4×C367—21no—
66D4×C367C2311yescharacteristic
67D4×C367C2312yes—
68C2×C4×C367—21no—
69C2×C4×C367C2311yescharacteristic
70C8×C367C2312yes—
71C22×C367—21no—
72C22×C367—22no—
73C22×C367—81no—
74C22×C367C2×D411yescharacteristic
75C2×D367—21no—
76C2×D367—82no—
77C2×D367C2×D412yes—
78C4×C367—21no—
79C4×C367C2×D411yescharacteristic
80C4×C367C2411yescharacteristic
81Dic367—21no—
82Dic367—22no—
83Dic367C2×D412yes—
84C2×C367—21no—
85C2×C367—42no—
86C2×C367—41no—
87C2×C367C22×D411yescharacteristic
88D367—22no—
89D367—41no—
90C367#57711yescharacteristic, complemented, Sylow
91#577—3671nomaximal, Sylow
922−1+4—3671no—
93H(5,𝔽2)—3671no—
94#137—3672no—
95#137—3671no—
96AGL(1,ℤ/8)—3672no—
97AGL(1,ℤ/8)D36711yescharacteristic, complemented
98#135—3671no—
99#135—3672no—
100C2×SD16—3671no—
101C2×SD16—3672no—
102#131—3671no—
103#41—3671no—
104#41—3672no—
105#41—3672no—
106#41—3671no—
107#41—7341no—
108#41—7342no—
109#41—7341no—
110#41C2×D36711yescharacteristic, complemented
111C2×Q8—3672no—
112C2×Q8—3671no—
113C2×Q8—7341no—
114C2×D4—3672no—
115C2×D4—7341no—
116C2×D4—7342no—
117C2×D4C2×D36711yescharacteristic
118Q16—3671no—
119Q16—3672no—
120SD16—3671no—
121SD16—3672no—
122SD16—3672no—
123SD16—3671no—
124SD16—3672no—
125SD16C2×D36712yescomplemented
126D8—3671no—
127D8C2×D36712yescomplemented
128M16—3672no—
129M16C2×D36711yescharacteristic, complemented
130C2×C8—3671no—
131C2×C8—3672no—
132C23—21no—
133C23—7342no—
134Q8—3671no—
135Q8—3672no—
136Q8—3671no—
137Q8—7342no—
138Q8—7341no—
139Q8C22×D36711yescharacteristic
140D4—21no—
141D4—21no—
142D4—3672no—
143D4—3671no—
144D4—3671no—
145D4—7342no—
146D4—7342no—
147D4—7342no—
148D4—7341no—
149D4C22×D36711yescharacteristic
150D4C22×D36712yes—
151C2×C4—21no—
152C2×C4—3672no—
153C2×C4—7342no—
154C2×C4—7342no—
155C2×C4—7342no—
156C2×C4—7341no—
157C2×C4—7341no—
158C2×C4C22×D36711yescharacteristic
159C8—3671no—
160C8—3671no—
161C8C22×D36712yes—
162C22—21no—
163C22—22no—
164C22—81no—
165C22—3672no—
166C22—29362no—
167C22—7341no—
168C22D4×D36711yescharacteristic
169C4—21no—
170C4—3672no—
171C4—7341no—
172C4—7342no—
173C4D4×D36711yescharacteristic
174C4C23×D36711yescharacteristic
175C2—21no—
176C2—42no—
177C2—41no—
178C2—14681no—
179C2—7342no—
180C2C2×D4×D36711yescharacteristic
subgroupG/H
centerC2C2×D4×D367
commutator, cosocleC4×C367C24
FrattiniC4C23×D367
FittingAGL(1,ℤ/8)×C367C2
socleC2×C367C22×D4
hypercenterAGL(1,ℤ/8)D367
hypocenterC367#577
LCSG ▷ C4×C367 ▷ C2×C367 ▷ C367
UCSC1 ◁ C2 ◁ C2×C4 ◁ AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. #26044308C2semidirect
2. #26044306C2semidirect
3. #26044294C2semidirect
4. #26044290C2semidirect
5. #26044285C2semidirect
6. #26044281C2semidirect
7. #26044280C2semidirect
8. C2×#19154565C2semidirect
9. SD16×D367C2semidirect
10. #26044269C2semidirect
11. AGL(1,ℤ/8)×C367C2semidirect
18. #19154576C22semidirect
20. C2×Dic734C22semidirect
21. Q8×D367C22semidirect
25. Dic1468C22semidirect
26. #19154567C22semidirect
27. #19154566C22semidirect
28. #19154565C22semidirect
29. #19154565C22semidirect
30. #19154565C22semidirect
31. #19154564C22semidirect
32. #19154562C22semidirect
33. #19154560C22semidirect
34. C2×C367⋊C8C22semidirect
35. C8×D367C22semidirect
36. #41×C367C22semidirect
38. SD16×C367C22semidirect
39. D8×C367C22semidirect
40. M16×C367C22semidirect
90. C367#577semidirect
97. AGL(1,ℤ/8)D367semidirect
110. #41C2×D367semidirect
125. SD16C2×D367semidirect
127. D8C2×D367semidirect
129. M16C2×D367semidirect

Maximal subgroups

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

subgroupindex
#260443082
#260443062
#260442942
#260442902
#260442852
#260442812
#260442802
C2×#191545652
SD16×D3672
#260442692
AGL(1,ℤ/8)×C3672
#577367