Finite Group Databasebeta

← #37554995

Subgroups — #37554995

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#26044309C211yesmaximal, characteristic, complemented
2#26044307C211yesmaximal, characteristic, complemented
3#26044293C212yesmaximal, complemented
4#26044289C211yesmaximal, characteristic, complemented
5#26044288C212yesmaximal, complemented
6#26044284C212yesmaximal, complemented
7#26044281C211yesmaximal, characteristic, complemented
8C2×#19154564C211yesmaximal, characteristic, complemented
9SD16×D367C212yesmaximal, complemented
10#26044269C211yesmaximal, characteristic, complemented
11#137×C367C211yesmaximal, characteristic, complemented
12#19154578—21no—
13#19154578C2212yes—
14#19154578C2212yescomplemented
15#19154577—21no—
16#19154577—22no—
17#19154577C2211yescharacteristic
18#19154577C2211yescharacteristic, complemented
19Q8×D367—21no—
20Q8×D367C2212yescomplemented
21C2×D1468—21no—
22C2×D1468C2211yescharacteristic
23D4×D367—22no—
24D4×D367C2212yes—
25#19154567C2211yescharacteristic, complemented
26#19154566C2212yescomplemented
27#19154565C2211yescharacteristic, complemented
28#19154564C2212yescomplemented
29#19154564C2212yescomplemented
30#19154564C2211yescharacteristic, complemented
31D2936C2212yescomplemented
32#19154562C2211yescharacteristic, complemented
33#19154560C2212yescomplemented
34C2×C367⋊C8C2211yescharacteristic, complemented
35C8×D367C2212yescomplemented
36#41×C367C2211yescharacteristic
37C2×Q8×C367C2211yescharacteristic, complemented
38Q16×C367C2212yescomplemented
39SD16×C367C2212yescomplemented
40M16×C367C2211yescharacteristic, complemented
41C22×D367—21no—
42C22×D367—22no—
43Dic734—22no—
44Dic734D411yescharacteristic
45Dic734C2311yescharacteristic
46D1468—22no—
47D1468—22no—
48D1468—21no—
49D1468D411yescharacteristic
50D1468C2311yescharacteristic
51D1468C2312yes—
52C22⋊D367—21no—
53C22⋊D367—22no—
54C22⋊D367D412yes—
55C4×D367—22no—
56C4×D367—21no—
57C4×D367—22no—
58C4×D367—22no—
59C4×D367C2312yes—
60C367⋊C8C2311yescharacteristic
61C367⋊C8C2311yescharacteristic
62Q8×C367—21no—
63Q8×C367C2312yes—
64Q8×C367C2311yescharacteristic
65D4×C367—21no—
66D4×C367C2311yescharacteristic
67C2×C4×C367—21no—
68C2×C4×C367—21no—
69C2×C4×C367C2311yescharacteristic
70C8×C367C2312yes—
71C22×C367—21no—
72C22×C367C2×D411yescharacteristic
73C2×D367—21no—
74C2×D367—22no—
75C2×D367—82no—
76C2×D367—81no—
77C2×D367C2×D412yes—
78C4×C367—21no—
79C4×C367—22no—
80C4×C367C2×D411yescharacteristic
81C4×C367C2411yescharacteristic
82Dic367—21no—
83Dic367C2×D412yes—
84C2×C367—21no—
85C2×C367—41no—
86C2×C367C22×D411yescharacteristic
87D367—22no—
88D367—41no—
89D367—42no—
90C367#57711yescharacteristic, complemented, Sylow
91#577—3671nomaximal, Sylow
922−1+4—3671no—
93H(5,𝔽2)—3671no—
94#137—3672no—
95#137D36711yescharacteristic, complemented
96AGL(1,ℤ/8)—3672no—
97AGL(1,ℤ/8)—3671no—
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—7342no—
108#41—7341no—
109#41—7341no—
110#41C2×D36711yescharacteristic
111C2×Q8—3672no—
112C2×Q8—7341no—
113C2×Q8C2×D36711yescharacteristic, complemented
114C2×D4—3672no—
115C2×D4—3671no—
116C2×D4—7341no—
117C2×D4—7342no—
118Q16—3671no—
119Q16C2×D36712yescomplemented
120SD16—3672no—
121SD16—3672no—
122SD16—3671no—
123SD16—3671no—
124SD16—3672no—
125SD16C2×D36712yescomplemented
126D8—3671no—
127D8—3672no—
128M16—3672no—
129M16C2×D36711yescharacteristic, complemented
130C2×C8—3671no—
131C2×C8—3672no—
132C23—7342no—
133C23—7341no—
134Q8—21no—
135Q8—3671no—
136Q8—3671no—
137Q8—7342no—
138Q8C22×D36712yes—
139Q8C22×D36711yescharacteristic
140D4—21no—
141D4—3672no—
142D4—3671no—
143D4—3671no—
144D4—3672no—
145D4—7342no—
146D4—7342no—
147D4—7342no—
148D4—7341no—
149D4—7341no—
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—3671no—
160C8—3671no—
161C8C22×D36712yes—
162C22—21no—
163C22—3672no—
164C22—29361no—
165C22—29362no—
166C22—7341no—
167C22—7342no—
168C22D4×D36711yescharacteristic
169C4—21no—
170C4—22no—
171C4—3672no—
172C4—7341no—
173C4D4×D36711yescharacteristic
174C4C23×D36711yescharacteristic
175C2—21no—
176C2—41no—
177C2—14682no—
178C2—14681no—
179C2—7342no—
180C2C2×D4×D36711yescharacteristic
subgroupG/H
centerC2C2×D4×D367
commutator, cosocleC4×C367C24
FrattiniC4C23×D367
Fitting#137×C367C2
socleC2×C367C22×D4
hypercenter#137D367
hypocenterC367#577
LCSG ▷ C4×C367 ▷ C2×C367 ▷ C367
UCSC1 ◁ C2 ◁ C2×C4 ◁ #137

Complements

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

NH ≅ G/Ntype
1. #26044309C2semidirect
2. #26044307C2semidirect
3. #26044293C2semidirect
4. #26044289C2semidirect
5. #26044288C2semidirect
6. #26044284C2semidirect
7. #26044281C2semidirect
8. C2×#19154564C2semidirect
9. SD16×D367C2semidirect
10. #26044269C2semidirect
11. #137×C367C2semidirect
14. #19154578C22semidirect
18. #19154577C22semidirect
20. Q8×D367C22semidirect
25. #19154567C22semidirect
26. #19154566C22semidirect
27. #19154565C22semidirect
28. #19154564C22semidirect
29. #19154564C22semidirect
30. #19154564C22semidirect
31. D2936C22semidirect
32. #19154562C22semidirect
33. #19154560C22semidirect
34. C2×C367⋊C8C22semidirect
35. C8×D367C22semidirect
37. C2×Q8×C367C22semidirect
38. Q16×C367C22semidirect
39. SD16×C367C22semidirect
40. M16×C367C22semidirect
90. C367#577semidirect
95. #137D367semidirect
113. C2×Q8C2×D367semidirect
119. Q16C2×D367semidirect
125. SD16C2×D367semidirect
129. M16C2×D367semidirect

Maximal subgroups

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

subgroupindex
#260443092
#260443072
#260442932
#260442892
#260442882
#260442842
#260442812
C2×#191545642
SD16×D3672
#260442692
#137×C3672
#577367