Finite Group Databasebeta

← #36374702

Subgroups — #36374702

Order 22336.

346 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 346 conjugacy classes of subgroups — too many to draw. Only its 103 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
1C2×D4×D349C212yesmaximal, complemented
2#26013624C212yesmaximal, complemented
3#26013621C212yesmaximal, complemented
4#26013613C211yesmaximal, characteristic, complemented
5#26013611C212yesmaximal, complemented
6#26013609C212yesmaximal, complemented
7#26013607C212yesmaximal, complemented
8C2×#19150210C211yesmaximal, characteristic, complemented
9#121×C349C211yesmaximal, characteristic, complemented
10C23×D349C2211yescharacteristic
11C23×D349C2212yescomplemented
12C2×D1396C2212yes—
13C2×D1396C2211yescharacteristic, complemented
14C2×C22⋊D349C2212yes—
15C2×C22⋊D349C2212yescomplemented
16C2×C22⋊D349C2212yescomplemented
17D4×D349—28no—
18C22×Dic349C2211yescharacteristic, complemented
19C2×C4×D349C2212yescomplemented
20C4⋊Dic349C2211yescharacteristic, complemented
21#19150212C2212yescomplemented
22C22⋊Dic349C2212yescomplemented
23#19150210C2212yescomplemented
24#19150210C2212yescomplemented
25#19150210C2212yescomplemented
26#19150210C2212yescomplemented
27C2×D4×C349C2212yescomplemented
28C2×D4×C349C2211yescharacteristic, complemented
29C22×C4×C349C2211yescharacteristic, complemented
30C4⋊C4×C349C2211yescharacteristic, complemented
31C22⋊C4×C349C2212yescomplemented
32C22×D349—24no—
33C22×D349—41no—
34C22×D349—41no—
35C22×D349—42no—
36C22×D349—42no—
37C22×D349D414yescomplemented
38C22×D349C2312yes—
39C22×D349C2312yes—
40D1396—24no—
41D1396—41no—
42C22⋊D349—24no—
43C22⋊D349—24no—
44C22⋊D349—42no—
45C2×Dic349—41no—
46C2×Dic349C2312yes—
47C2×Dic349C2312yes—
48C4×D349—24no—
49C23×C349C2312yes—
50C23×C349C2311yescharacteristic
51D4×C349—24no—
52D4×C349—41no—
53C2×C4×C349—41no—
54C2×C4×C349C2311yescharacteristic
55C2×C4×C349C2312yes—
56C2×C4×C349C2311yescharacteristic
57C22×C349—21no—
58C22×C349—21no—
59C22×C349—22no—
60C22×C349—22no—
61C22×C349—22no—
62C22×C349C2×D412yes—
63C22×C349C2411yescharacteristic
64C2×D349—22no—
65C2×D349—22no—
66C2×D349—22no—
67C2×D349—22no—
68C2×D349—22no—
69C2×D349—44no—
70C2×D349—48no—
71C2×D349C2×D414yes—
72C4×C349—22no—
73C4×C349—21no—
74C4×C349—21no—
75Dic349—22no—
76Dic349—22no—
77C2×C349—22no—
78C2×C349—42no—
79C2×C349C22×D411yescharacteristic
80C2×C349H(5,𝔽2)11yescharacteristic
81C2×C349H(5,𝔽2)11yescharacteristic
82D349—24no—
83D349—42no—
84C349#53411yescharacteristic, complemented, Sylow
85#534—3491nomaximal, Sylow
86C22×D4—3492no—
87#123—3492no—
88#123—3492no—
89#121—3492no—
90#121—3491no—
91#121D34911yescharacteristic, complemented
92#120—3492no—
93#120—3492no—
94C2×C22⋊C4—3491no—
95C24—3491no—
96C24—3492no—
97C2×D4—3492no—
98C2×D4—3492no—
99C2×D4—3492no—
100C2×D4—3491no—
101C2×D4—3492no—
102C2×D4—6988no—
103C2×D4C2×D34912yescomplemented
104C2×D4C2×D34911yescharacteristic, complemented
105C22×C4—3491no—
106C22×C4—3492no—
107C22×C4C2×D34911yescharacteristic, complemented
108C4⋊C4—3492no—
109C4⋊C4—3491no—
110C4⋊C4C2×D34911yescharacteristic, complemented
111C22⋊C4—3492no—
112C22⋊C4—3492no—
113C22⋊C4—3492no—
114C22⋊C4—3492no—
115C22⋊C4—3492no—
116C22⋊C4C2×D34912yescomplemented
117C23—3494no—
118C23—3492no—
119C23—3492no—
120C23—13961no—
121C23—13961no—
122C23—13962no—
123C23—13962no—
124C23—6984no—
125C23C22×D34912yes—
126C23C22×D34911yescharacteristic
127D4—24no—
128D4—41no—
129D4—13961no—
130D4—13962no—
131D4—6984no—
132D4—6984no—
133D4—6984no—
134C2×C4—41no—
135C2×C4—3492no—
136C2×C4—3492no—
137C2×C4—13961no—
138C2×C4—6984no—
139C2×C4C22×D34911yescharacteristic
140C2×C4C22×D34911yescharacteristic
141C2×C4C22×D34912yes—
142C22—21no—
143C22—21no—
144C22—22no—
145C22—22no—
146C22—22no—
147C22—3494no—
148C22—13964no—
149C22—13968no—
150C22—6982no—
151C22—6982no—
152C22—6982no—
153C22—6982no—
154C22—6982no—
155C22D4×D34912yes—
156C22C23×D34911yescharacteristic
157C4—22no—
158C4—21no—
159C4—21no—
160C4—6982no—
161C4—6982no—
162C2—22no—
163C2—42no—
164C2—13962no—
165C2—6984no—
166C2C2×D4×D34911yescharacteristic
167C2#2601368411yescharacteristic
168C2#2601368511yescharacteristic
subgroupG/H
center, FrattiniC22C23×D349
commutator, socle, cosocleC22×C349C24
Fitting#121×C349C2
hypercenter#121D349
hypocenterC349#534
LCSG ▷ C22×C349 ▷ C349
UCSC1 ◁ C22 ◁ #121

Complements

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

NH ≅ G/Ntype
1. C2×D4×D349C2semidirect
2. #26013624C2semidirect
3. #26013621C2semidirect
4. #26013613C2semidirect
5. #26013611C2semidirect
6. #26013609C2semidirect
7. #26013607C2semidirect
8. C2×#19150210C2semidirect
9. #121×C349C2semidirect
11. C23×D349C22semidirect
13. C2×D1396C22semidirect
15. C2×C22⋊D349C22semidirect
16. C2×C22⋊D349C22semidirect
18. C22×Dic349C22semidirect
19. C2×C4×D349C22semidirect
20. C4⋊Dic349C22semidirect
21. #19150212C22semidirect
22. C22⋊Dic349C22semidirect
23. #19150210C22semidirect
24. #19150210C22semidirect
25. #19150210C22semidirect
26. #19150210C22semidirect
27. C2×D4×C349C22semidirect
28. C2×D4×C349C22semidirect
29. C22×C4×C349C22semidirect
30. C4⋊C4×C349C22semidirect
31. C22⋊C4×C349C22semidirect
37. C22×D349D4semidirect
84. C349#534semidirect
91. #121D349semidirect
103. C2×D4C2×D349semidirect
104. C2×D4C2×D349semidirect
107. C22×C4C2×D349semidirect
110. C4⋊C4C2×D349semidirect
116. C22⋊C4C2×D349semidirect

Maximal subgroups

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

subgroupindex
C2×D4×D3492
#260136242
#260136212
#260136132
#260136112
#260136092
#260136072
C2×#191502102
#121×C3492
#534349