Finite Group Databasebeta

← #36374951

Subgroups — #36374951

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#26013686C211yesmaximal, characteristic, complemented
2#26013685C211yesmaximal, characteristic, complemented
3#26013671C212yesmaximal, complemented
4#26013669C211yesmaximal, characteristic, complemented
5#26013664C211yesmaximal, characteristic, complemented
6#26013663C212yesmaximal, complemented
7#26013660C212yesmaximal, complemented
8C2×#19150222C211yesmaximal, characteristic, complemented
9SD16×D349C212yesmaximal, complemented
10#26013645C211yesmaximal, characteristic, complemented
11#135×C349C211yesmaximal, characteristic, complemented
12#19150234C2212yes—
13#19150233—21no—
14#19150233—21no—
15#19150233C2211yescharacteristic
16#19150232—22no—
17#19150232C2212yescomplemented
18C2×Dic698—21no—
19C2×Dic698C2211yescharacteristic, complemented
20Q8×D349C2212yescomplemented
21C2×D1396—21no—
22C2×D1396C2211yescharacteristic
23D4×D349—22no—
24D4×D349C2212yes—
25#19150223C2212yescomplemented
26#19150222C2212yescomplemented
27#19150222C2212yescomplemented
28#19150221C2212yescomplemented
29#19150220C2212yescomplemented
30#19150218C2212yescomplemented
31#19150217C2211yescharacteristic, complemented
32#19150216C2212yescomplemented
33C8×D349C2212yescomplemented
34#41×C349C2211yescharacteristic
35#41×C349C2211yescharacteristic, complemented
36Q16×C349C2211yescharacteristic, complemented
37SD16×C349C2211yescharacteristic, complemented
38SD16×C349C2211yescharacteristic, complemented
39D8×C349C2211yescharacteristic, complemented
40C2×C8×C349C2211yescharacteristic, complemented
41C22×D349—21no—
42C22×D349—22no—
43Dic698—21no—
44Dic698—22no—
45Dic698D411yescharacteristic
46Dic698C2312yes—
47D1396—22no—
48D1396—21no—
49D1396D411yescharacteristic
50D1396C2312yes—
51C22⋊D349—22no—
52C22⋊D349—22no—
53C22⋊D349D412yes—
54C2×Dic349—21no—
55C2×Dic349—22no—
56C4×D349—22no—
57C4×D349—22no—
58C4×D349C2312yes—
59#14584095C2312yes—
60Q8×C349C2311yescharacteristic
61Q8×C349C2311yescharacteristic
62D4×C349—21no—
63D4×C349—21no—
64D4×C349C2311yescharacteristic
65D4×C349C2311yescharacteristic
66C2×C4×C349—21no—
67C2×C4×C349—21no—
68C2×C4×C349C2311yescharacteristic
69C8×C349C2311yescharacteristic
70C8×C349C2311yescharacteristic
71C22×C349—21no—
72C22×C349—21no—
73C22×C349C2×D411yescharacteristic
74C2×D349—22no—
75C2×D349—82no—
76C2×D349—81no—
77C2×D349C2×D412yes—
78C4×C349—21no—
79C4×C349—21no—
80C4×C349C2×D411yescharacteristic
81C4×C349C2411yescharacteristic
82Dic349—22no—
83Dic349C2×D412yes—
84C2×C349—21no—
85C2×C349—41no—
86C2×C349—41no—
87C2×C349C22×D411yescharacteristic
88D349—22no—
89D349—42no—
90C349#57711yescharacteristic, complemented, Sylow
91#577—3491nomaximal, Sylow
922−1+4—3491no—
93H(5,𝔽2)—3491no—
94#137—3491no—
95#137—3492no—
96AGL(1,ℤ/8)—3491no—
97AGL(1,ℤ/8)—3492no—
98#135—3492no—
99#135D34911yescharacteristic, complemented
100C2×SD16—3492no—
101C2×SD16—3491no—
102#131—3491no—
103#41—3491no—
104#41—3492no—
105#41—3492no—
106#41—6981no—
107#41—6981no—
108#41—6982no—
109#41C2×D34911yescharacteristic
110#41C2×D34911yescharacteristic, complemented
111C2×Q8—3492no—
112C2×Q8—3491no—
113C2×Q8—6981no—
114C2×D4—3492no—
115C2×D4—3491no—
116C2×D4—6981no—
117C2×D4—6982no—
118Q16—3492no—
119Q16C2×D34911yescharacteristic, complemented
120SD16—3492no—
121SD16—3492no—
122SD16—3492no—
123SD16—3492no—
124SD16C2×D34911yescharacteristic, complemented
125SD16C2×D34911yescharacteristic, complemented
126D8—3492no—
127D8C2×D34911yescharacteristic, complemented
128M16—3491no—
129M16—3492no—
130C2×C8—3492no—
131C2×C8C2×D34911yescharacteristic, complemented
132C23—6982no—
133C23—6981no—
134Q8—3491no—
135Q8—3492no—
136Q8—6982no—
137Q8—6981no—
138Q8C22×D34911yescharacteristic
139Q8C22×D34911yescharacteristic
140D4—21no—
141D4—21no—
142D4—3492no—
143D4—3491no—
144D4—3492no—
145D4—6982no—
146D4—6982no—
147D4—6982no—
148D4—6981no—
149D4C22×D34911yescharacteristic
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—3492no—
160C8C22×D34911yescharacteristic
161C8C22×D34911yescharacteristic
162C22—21no—
163C22—21no—
164C22—3492no—
165C22—6982no—
166C22—27921no—
167C22—27922no—
168C22D4×D34911yescharacteristic
169C4—21no—
170C4—21no—
171C4—3492no—
172C4—6982no—
173C4D4×D34911yescharacteristic
174C4C23×D34911yescharacteristic
175C2—21no—
176C2—41no—
177C2—41no—
178C2—13962no—
179C2—6982no—
180C2C2×D4×D34911yescharacteristic
subgroupG/H
centerC2C2×D4×D349
commutator, cosocleC4×C349C24
FrattiniC4C23×D349
Fitting#135×C349C2
socleC2×C349C22×D4
hypercenter#135D349
hypocenterC349#577
LCSG ▷ C4×C349 ▷ C2×C349 ▷ C349
UCSC1 ◁ C2 ◁ C2×C4 ◁ #135

Complements

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

NH ≅ G/Ntype
1. #26013686C2semidirect
2. #26013685C2semidirect
3. #26013671C2semidirect
4. #26013669C2semidirect
5. #26013664C2semidirect
6. #26013663C2semidirect
7. #26013660C2semidirect
8. C2×#19150222C2semidirect
9. SD16×D349C2semidirect
10. #26013645C2semidirect
11. #135×C349C2semidirect
17. #19150232C22semidirect
19. C2×Dic698C22semidirect
20. Q8×D349C22semidirect
25. #19150223C22semidirect
26. #19150222C22semidirect
27. #19150222C22semidirect
28. #19150221C22semidirect
29. #19150220C22semidirect
30. #19150218C22semidirect
31. #19150217C22semidirect
32. #19150216C22semidirect
33. C8×D349C22semidirect
35. #41×C349C22semidirect
36. Q16×C349C22semidirect
37. SD16×C349C22semidirect
38. SD16×C349C22semidirect
39. D8×C349C22semidirect
40. C2×C8×C349C22semidirect
90. C349#577semidirect
99. #135D349semidirect
110. #41C2×D349semidirect
119. Q16C2×D349semidirect
124. SD16C2×D349semidirect
125. SD16C2×D349semidirect
127. D8C2×D349semidirect
131. C2×C8C2×D349semidirect

Maximal subgroups

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

subgroupindex
#260136862
#260136852
#260136712
#260136692
#260136642
#260136632
#260136602
C2×#191502222
SD16×D3492
#260136452
#135×C3492
#577349