Finite Group Databasebeta

← #36374953

Subgroups — #36374953

Order 22336.

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#26013687C211yesmaximal, characteristic, complemented
2#26013685C211yesmaximal, characteristic, complemented
3#26013671C212yesmaximal, complemented
4#26013667C211yesmaximal, characteristic, complemented
5#26013666C212yesmaximal, complemented
6#26013662C212yesmaximal, complemented
7#26013659C211yesmaximal, characteristic, complemented
8C2×#19150220C211yesmaximal, characteristic, complemented
9SD16×D349C212yesmaximal, complemented
10#26013647C211yesmaximal, characteristic, complemented
11#137×C349C211yesmaximal, characteristic, complemented
12#19150234—21no—
13#19150234C2212yes—
14#19150234C2212yescomplemented
15#19150233—21no—
16#19150233—22no—
17#19150233C2211yescharacteristic
18#19150233C2211yescharacteristic, complemented
19Q8×D349—21no—
20Q8×D349C2212yescomplemented
21C2×D1396—21no—
22C2×D1396C2211yescharacteristic
23D4×D349—22no—
24D4×D349C2212yes—
25#19150223C2211yescharacteristic, complemented
26#19150222C2212yescomplemented
27#19150221C2211yescharacteristic, complemented
28#19150220C2212yescomplemented
29#19150220C2212yescomplemented
30#19150220C2211yescharacteristic, complemented
31D2792C2212yescomplemented
32#19150218C2211yescharacteristic, complemented
33#19150216C2212yescomplemented
34C2×#14584095C2211yescharacteristic, complemented
35C8×D349C2212yescomplemented
36#41×C349C2211yescharacteristic
37C2×Q8×C349C2211yescharacteristic, complemented
38Q16×C349C2212yescomplemented
39SD16×C349C2212yescomplemented
40M16×C349C2211yescharacteristic, complemented
41C22×D349—21no—
42C22×D349—22no—
43Dic698—22no—
44Dic698D411yescharacteristic
45Dic698C2311yescharacteristic
46D1396—22no—
47D1396—22no—
48D1396—21no—
49D1396D411yescharacteristic
50D1396C2311yescharacteristic
51D1396C2312yes—
52C22⋊D349—21no—
53C22⋊D349—22no—
54C22⋊D349D412yes—
55C4×D349—22no—
56C4×D349—21no—
57C4×D349—22no—
58C4×D349—22no—
59C4×D349C2312yes—
60#14584095C2311yescharacteristic
61#14584095C2311yescharacteristic
62Q8×C349—21no—
63Q8×C349C2312yes—
64Q8×C349C2311yescharacteristic
65D4×C349—21no—
66D4×C349C2311yescharacteristic
67C2×C4×C349—21no—
68C2×C4×C349—21no—
69C2×C4×C349C2311yescharacteristic
70C8×C349C2312yes—
71C22×C349—21no—
72C22×C349C2×D411yescharacteristic
73C2×D349—21no—
74C2×D349—22no—
75C2×D349—82no—
76C2×D349—81no—
77C2×D349C2×D412yes—
78C4×C349—21no—
79C4×C349—22no—
80C4×C349C2×D411yescharacteristic
81C4×C349C2411yescharacteristic
82Dic349—21no—
83Dic349C2×D412yes—
84C2×C349—21no—
85C2×C349—41no—
86C2×C349C22×D411yescharacteristic
87D349—22no—
88D349—41no—
89D349—42no—
90C349#57711yescharacteristic, complemented, Sylow
91#577—3491nomaximal, Sylow
922−1+4—3491no—
93H(5,𝔽2)—3491no—
94#137—3492no—
95#137D34911yescharacteristic, complemented
96AGL(1,ℤ/8)—3492no—
97AGL(1,ℤ/8)—3491no—
98#135—3491no—
99#135—3492no—
100C2×SD16—3491no—
101C2×SD16—3492no—
102#131—3491no—
103#41—3491no—
104#41—3492no—
105#41—3492no—
106#41—3491no—
107#41—6982no—
108#41—6981no—
109#41—6981no—
110#41C2×D34911yescharacteristic
111C2×Q8—3492no—
112C2×Q8—6981no—
113C2×Q8C2×D34911yescharacteristic, complemented
114C2×D4—3492no—
115C2×D4—3491no—
116C2×D4—6981no—
117C2×D4—6982no—
118Q16—3491no—
119Q16C2×D34912yescomplemented
120SD16—3492no—
121SD16—3492no—
122SD16—3491no—
123SD16—3491no—
124SD16—3492no—
125SD16C2×D34912yescomplemented
126D8—3491no—
127D8—3492no—
128M16—3492no—
129M16C2×D34911yescharacteristic, complemented
130C2×C8—3491no—
131C2×C8—3492no—
132C23—6982no—
133C23—6981no—
134Q8—21no—
135Q8—3491no—
136Q8—3491no—
137Q8—6982no—
138Q8C22×D34912yes—
139Q8C22×D34911yescharacteristic
140D4—21no—
141D4—3492no—
142D4—3491no—
143D4—3491no—
144D4—3492no—
145D4—6982no—
146D4—6982no—
147D4—6982no—
148D4—6981no—
149D4—6981no—
150D4C22×D34911yescharacteristic
151C2×C4—21no—
152C2×C4—21no—
153C2×C4—3492no—
154C2×C4—6982no—
155C2×C4—6982no—
156C2×C4—6982no—
157C2×C4—6981no—
158C2×C4C22×D34911yescharacteristic
159C8—3491no—
160C8—3491no—
161C8C22×D34912yes—
162C22—21no—
163C22—3492no—
164C22—6981no—
165C22—6982no—
166C22—27921no—
167C22—27922no—
168C22D4×D34911yescharacteristic
169C4—21no—
170C4—22no—
171C4—3492no—
172C4—6981no—
173C4D4×D34911yescharacteristic
174C4C23×D34911yescharacteristic
175C2—21no—
176C2—41no—
177C2—13962no—
178C2—13961no—
179C2—6982no—
180C2C2×D4×D34911yescharacteristic
subgroupG/H
centerC2C2×D4×D349
commutator, cosocleC4×C349C24
FrattiniC4C23×D349
Fitting#137×C349C2
socleC2×C349C22×D4
hypercenter#137D349
hypocenterC349#577
LCSG ▷ C4×C349 ▷ C2×C349 ▷ C349
UCSC1 ◁ C2 ◁ C2×C4 ◁ #137

Complements

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

NH ≅ G/Ntype
1. #26013687C2semidirect
2. #26013685C2semidirect
3. #26013671C2semidirect
4. #26013667C2semidirect
5. #26013666C2semidirect
6. #26013662C2semidirect
7. #26013659C2semidirect
8. C2×#19150220C2semidirect
9. SD16×D349C2semidirect
10. #26013647C2semidirect
11. #137×C349C2semidirect
14. #19150234C22semidirect
18. #19150233C22semidirect
20. Q8×D349C22semidirect
25. #19150223C22semidirect
26. #19150222C22semidirect
27. #19150221C22semidirect
28. #19150220C22semidirect
29. #19150220C22semidirect
30. #19150220C22semidirect
31. D2792C22semidirect
32. #19150218C22semidirect
33. #19150216C22semidirect
34. C2×#14584095C22semidirect
35. C8×D349C22semidirect
37. C2×Q8×C349C22semidirect
38. Q16×C349C22semidirect
39. SD16×C349C22semidirect
40. M16×C349C22semidirect
90. C349#577semidirect
95. #137D349semidirect
113. C2×Q8C2×D349semidirect
119. Q16C2×D349semidirect
125. SD16C2×D349semidirect
129. M16C2×D349semidirect

Maximal subgroups

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

subgroupindex
#260136872
#260136852
#260136712
#260136672
#260136662
#260136622
#260136592
C2×#191502202
SD16×D3492
#260136472
#137×C3492
#577349