Finite Group Databasebeta

← #37554992

Subgroups — #37554992

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#26044306C211yesmaximal, characteristic, complemented
3#26044292C211yesmaximal, characteristic, complemented
4#26044290C212yesmaximal, complemented
5#26044288C212yesmaximal, complemented
6#26044287C211yesmaximal, characteristic, complemented
7#26044283C211yesmaximal, characteristic, complemented
8#26044282C212yesmaximal, complemented
9SD16×D367C212yesmaximal, complemented
10#26044267C211yesmaximal, characteristic, complemented
11C2×SD16×C367C211yesmaximal, characteristic, complemented
12#19154578—21no—
13#19154578C2212yescomplemented
14#19154577—22no—
15#19154577C2211yescharacteristic
16#19154577C2211yescharacteristic
17#19154577C2211yescharacteristic, complemented
18#19154576—21no—
19#19154576C2212yes—
20Q8×D367—21no—
21Q8×D367C2212yescomplemented
22C2×C22⋊D367—22no—
23D4×D367—21no—
24D4×D367C2212yes—
25Dic1468C2211yescharacteristic, complemented
26#19154567C2212yescomplemented
27#19154566C2211yescharacteristic, complemented
28#19154566C2211yescharacteristic, complemented
29#19154565C2212yescomplemented
30#19154564C2212yescomplemented
31D2936C2211yescharacteristic, complemented
32#19154562C2212yescomplemented
33#19154561C2211yescharacteristic, complemented
34#19154560C2212yescomplemented
35C8×D367C2212yescomplemented
36C2×Q8×C367C2211yescharacteristic, complemented
37C2×D4×C367C2211yescharacteristic
38SD16×C367C2212yescomplemented
39SD16×C367C2212yescomplemented
40C2×C8×C367C2211yescharacteristic, complemented
41C22×D367—22no—
42Dic734—22no—
43Dic734D411yescharacteristic
44Dic734C2311yescharacteristic
45Dic734C2311yescharacteristic
46D1468—22no—
47D1468D411yescharacteristic
48D1468C2311yescharacteristic
49D1468C2311yescharacteristic
50C22⋊D367—21no—
51C22⋊D367—21no—
52C22⋊D367—22no—
53C22⋊D367—22no—
54C22⋊D367D412yes—
55C2×Dic367—22no—
56C4×D367—22no—
57C4×D367—21no—
58C4×D367—21no—
59C4×D367—22no—
60C4×D367C2312yes—
61C367⋊C8C2312yes—
62C23×C367—21no—
63Q8×C367—21no—
64Q8×C367C2312yes—
65D4×C367—21no—
66D4×C367C2312yes—
67C2×C4×C367—21no—
68C2×C4×C367C2311yescharacteristic
69C8×C367C2311yescharacteristic
70C8×C367C2311yescharacteristic
71C22×C367—22no—
72C22×C367—81no—
73C22×C367C2×D411yescharacteristic
74C2×D367—21no—
75C2×D367—21no—
76C2×D367—82no—
77C2×D367C2×D412yes—
78C4×C367—22no—
79C4×C367C2×D411yescharacteristic
80C4×C367C2411yescharacteristic
81Dic367—21no—
82Dic367—21no—
83Dic367C2×D412yes—
84C2×C367—21no—
85C2×C367—42no—
86C2×C367C22×D411yescharacteristic
87D367—22no—
88D367—41no—
89D367—41no—
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#135—3671no—
100C2×SD16—3672no—
101C2×SD16D36711yescharacteristic, complemented
102#131—3671no—
103#41—3671no—
104#41—3672no—
105#41—3672no—
106#41—3671no—
107#41—3671no—
108#41—7342no—
109#41—7341no—
110#41—7341no—
111C2×Q8—3672no—
112C2×Q8—7341no—
113C2×Q8C2×D36711yescharacteristic, complemented
114C2×D4—3672no—
115C2×D4—7341no—
116C2×D4—7342no—
117C2×D4C2×D36711yescharacteristic
118Q16—3672no—
119Q16—3671no—
120SD16—3672no—
121SD16—3672no—
122SD16—3671no—
123SD16—3671no—
124SD16C2×D36712yescomplemented
125SD16C2×D36712yescomplemented
126D8—3672no—
127D8—3671no—
128M16—3671no—
129M16—3672no—
130C2×C8—3672no—
131C2×C8C2×D36711yescharacteristic, complemented
132C23—21no—
133C23—7342no—
134Q8—21no—
135Q8—3671no—
136Q8—3671no—
137Q8—3671no—
138Q8—7342no—
139Q8C22×D36712yes—
140D4—21no—
141D4—3672no—
142D4—3671no—
143D4—3671no—
144D4—3671no—
145D4—7342no—
146D4—7342no—
147D4—7342no—
148D4—7341no—
149D4—7341no—
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—3672no—
160C8C22×D36711yescharacteristic
161C8C22×D36711yescharacteristic
162C22—22no—
163C22—81no—
164C22—3672no—
165C22—29362no—
166C22—7341no—
167C22—7341no—
168C22D4×D36711yescharacteristic
169C4—22no—
170C4—3672no—
171C4—7341no—
172C4—7341no—
173C4D4×D36711yescharacteristic
174C4C23×D36711yescharacteristic
175C2—21no—
176C2—42no—
177C2—14681no—
178C2—14681no—
179C2—7342no—
180C2C2×D4×D36711yescharacteristic
subgroupG/H
centerC2C2×D4×D367
commutator, cosocleC4×C367C24
FrattiniC4C23×D367
FittingC2×SD16×C367C2
socleC2×C367C22×D4
hypercenterC2×SD16D367
hypocenterC367#577
LCSG ▷ C4×C367 ▷ C2×C367 ▷ C367
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×SD16

Complements

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

NH ≅ G/Ntype
1. #26044309C2semidirect
2. #26044306C2semidirect
3. #26044292C2semidirect
4. #26044290C2semidirect
5. #26044288C2semidirect
6. #26044287C2semidirect
7. #26044283C2semidirect
8. #26044282C2semidirect
9. SD16×D367C2semidirect
10. #26044267C2semidirect
11. C2×SD16×C367C2semidirect
13. #19154578C22semidirect
17. #19154577C22semidirect
21. Q8×D367C22semidirect
25. Dic1468C22semidirect
26. #19154567C22semidirect
27. #19154566C22semidirect
28. #19154566C22semidirect
29. #19154565C22semidirect
30. #19154564C22semidirect
31. D2936C22semidirect
32. #19154562C22semidirect
33. #19154561C22semidirect
34. #19154560C22semidirect
35. C8×D367C22semidirect
36. C2×Q8×C367C22semidirect
38. SD16×C367C22semidirect
39. SD16×C367C22semidirect
40. C2×C8×C367C22semidirect
90. C367#577semidirect
101. C2×SD16D367semidirect
113. C2×Q8C2×D367semidirect
124. SD16C2×D367semidirect
125. SD16C2×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
#260443092
#260443062
#260442922
#260442902
#260442882
#260442872
#260442832
#260442822
SD16×D3672
#260442672
C2×SD16×C3672
#577367