Finite Group Databasebeta

← #135×D349 #36374922

Subgroups — #135×D349 #36374922

Order 22336.

266 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 266 conjugacy classes of subgroups — too many to draw. Only its 101 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#41×D349C212yesmaximal, complemented
2#26013662C211yesmaximal, characteristic, complemented
3#26013661C211yesmaximal, characteristic, complemented
4#26013660C212yesmaximal, complemented
5#26013659C212yesmaximal, complemented
6#26013658C211yesmaximal, characteristic, complemented
7Q16×D349C211yesmaximal, characteristic, complemented
8SD16×D349C212yesmaximal, complemented
9D8×D349C211yesmaximal, characteristic, complemented
10C2×C8×D349C211yesmaximal, characteristic, complemented
11#135×C349C211yesmaximal, characteristic, complemented
12#19150234C2212yescomplemented
13#19150233—22no—
14#19150233C2212yescomplemented
15#19150232—22no—
16#19150232C2212yescomplemented
17Q8×D349C2212yes—
18D4×D349—22no—
19D4×D349C2212yes—
20C2×C4×D349—22no—
21C2×C4×D349C2211yescharacteristic
22Dic1396C2211yescharacteristic, complemented
23#19150223C2212yescomplemented
24#19150222C2212yescomplemented
25#19150221C2212yescomplemented
26#19150220C2212yescomplemented
27D2792C2211yescharacteristic, complemented
28#19150218C2212yescomplemented
29C2×#14584095C2211yescharacteristic, complemented
30C8×D349C2211yescharacteristic
31C8×D349C2211yescharacteristic
32C8×D349C2211yescharacteristic, complemented
33C8×D349C2211yescharacteristic, complemented
34#41×C349C2212yescomplemented
35Q16×C349C2211yescharacteristic, complemented
36SD16×C349C2212yescomplemented
37D8×C349C2211yescharacteristic, complemented
38C2×C8×C349C2211yescharacteristic, complemented
39C22×D349—22no—
40C22×D349D411yescharacteristic
41Dic698—22no—
42Dic698C2312yes—
43D1396—22no—
44D1396C2312yes—
45C22⋊D349—22no—
46C22⋊D349—22no—
47C22⋊D349—22no—
48C2×Dic349—22no—
49C2×Dic349D411yescharacteristic
50C4×D349—22no—
51C4×D349—22no—
52C4×D349—22no—
53C4×D349D411yescharacteristic
54C4×D349D411yescharacteristic
55C4×D349C2311yescharacteristic
56C4×D349C2311yescharacteristic
57#14584095C2311yescharacteristic
58#14584095C2311yescharacteristic
59Q8×C349C2312yes—
60D4×C349—22no—
61D4×C349C2312yes—
62C2×C4×C349—22no—
63C2×C4×C349C2311yescharacteristic
64C8×C349C2311yescharacteristic
65C8×C349C2311yescharacteristic
66C22×C349—22no—
67C22×C349C2×D411yescharacteristic
68C2×D349—22no—
69C2×D349—22no—
70C2×D349—44no—
71C2×D349C2×D411yescharacteristic
72C2×D349C2×D411yescharacteristic
73C4×C349—22no—
74C4×C349C2×D411yescharacteristic
75C4×C349C2411yescharacteristic
76Dic349—22no—
77Dic349C2×D411yescharacteristic
78Dic349C2×D411yescharacteristic
79C2×C349—21no—
80C2×C349—42no—
81C2×C349C22×D411yescharacteristic
82D349—21no—
83D349—42no—
84D349#13512yesdirect product factor
85C349C2×#13511yescharacteristic, complemented, Sylow
86C2×#135—3491nomaximal, Sylow
87C2×#41—3492no—
88#135—3492no—
89#135—3491no—
90#135—3492no—
91#135—3491no—
92#135—3491no—
93#135D34911yescharacteristic, direct product factor
94C2×Q16—3491no—
95C2×SD16—3492no—
96C2×D8—3491no—
97C22×C8—3491no—
98#41—3492no—
99#41—3492no—
100#41—3492no—
101#41—6982no—
102#41—6982no—
103#41C2×D34912yescomplemented
104C2×Q8—3492no—
105C2×D4—3492no—
106C2×D4—6982no—
107C22×C4—3491no—
108C22×C4—6982no—
109Q16—3492no—
110Q16—3491no—
111Q16C2×D34911yescharacteristic, complemented
112SD16—3492no—
113SD16—3492no—
114SD16—3492no—
115SD16C2×D34912yescomplemented
116D8—3492no—
117D8—3491no—
118D8C2×D34911yescharacteristic, complemented
119C2×C8—3491no—
120C2×C8—3491no—
121C2×C8—3491no—
122C2×C8—3491no—
123C2×C8—3491no—
124C2×C8C2×D34911yescharacteristic, complemented
125C23—3491no—
126C23—6982no—
127Q8—3492no—
128Q8—6982no—
129Q8C22×D34912yes—
130D4—22no—
131D4—3492no—
132D4—6982no—
133D4—6982no—
134D4—6982no—
135D4—6982no—
136D4C22×D34912yes—
137C2×C4—22no—
138C2×C4—3491no—
139C2×C4—3491no—
140C2×C4—3491no—
141C2×C4—3491no—
142C2×C4—3491no—
143C2×C4—6982no—
144C2×C4—6982no—
145C2×C4—6982no—
146C2×C4—6982no—
147C2×C4C22×D34911yescharacteristic
148C8—3491no—
149C8—3491no—
150C8C22×D34911yescharacteristic
151C8C22×D34911yescharacteristic
152C22—22no—
153C22—3491no—
154C22—3491no—
155C22—13964no—
156C22—6982no—
157C22—6982no—
158C22D4×D34911yescharacteristic
159C4—22no—
160C4—3491no—
161C4—3491no—
162C4—6982no—
163C4D4×D34911yescharacteristic
164C4C23×D34911yescharacteristic
165C2—21no—
166C2—42no—
167C2—3492no—
168C2—13962no—
169C2—6981no—
170C2C2×D4×D34911yescharacteristic
subgroupG/H
centerC4D4×D349
commutator, cosocleC4×C349C24
FrattiniC4C23×D349
Fitting#135×C349C2
socleC2×C349C22×D4
hypercenter#135D349
hypocenterC349C2×#135
LCSG ▷ C4×C349 ▷ C2×C349 ▷ C349
UCSC1 ◁ C4 ◁ C2×C4 ◁ #135

Complements

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

NH ≅ G/Ntype
1. #41×D349C2semidirect
2. #26013662C2semidirect
3. #26013661C2semidirect
4. #26013660C2semidirect
5. #26013659C2semidirect
6. #26013658C2semidirect
7. Q16×D349C2semidirect
8. SD16×D349C2semidirect
9. D8×D349C2semidirect
10. C2×C8×D349C2semidirect
11. #135×C349C2semidirect
12. #19150234C22semidirect
14. #19150233C22semidirect
16. #19150232C22semidirect
22. Dic1396C22semidirect
23. #19150223C22semidirect
24. #19150222C22semidirect
25. #19150221C22semidirect
26. #19150220C22semidirect
27. D2792C22semidirect
28. #19150218C22semidirect
29. C2×#14584095C22semidirect
32. C8×D349C22semidirect
33. C8×D349C22semidirect
34. #41×C349C22semidirect
35. Q16×C349C22semidirect
36. SD16×C349C22semidirect
37. D8×C349C22semidirect
38. C2×C8×C349C22semidirect
84. D349#135direct (N × H)
85. C349C2×#135semidirect
93. #135D349direct (N × H)
103. #41C2×D349semidirect
111. Q16C2×D349semidirect
115. SD16C2×D349semidirect
118. D8C2×D349semidirect
124. C2×C8C2×D349semidirect

Maximal subgroups

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

subgroupindex
#41×D3492
#260136622
#260136612
#260136602
#260136592
#260136582
Q16×D3492
SD16×D3492
D8×D3492
C2×C8×D3492
#135×C3492
C2×#135349