Finite Group Databasebeta

← #36374650

Subgroups — #36374650

Order 22336.

330 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 330 conjugacy classes of subgroups — too many to draw. Only its 111 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×#19150232C211yesmaximal, characteristic, complemented
2C23×Dic349C211yesmaximal, characteristic, complemented
3#26013623C211yesmaximal, characteristic, complemented
4#26013620C211yesmaximal, characteristic, complemented
5#26013616C212yesmaximal, complemented
6#26013611C212yesmaximal, complemented
7#26013608C211yesmaximal, characteristic, complemented
8D4×Dic349C212yesmaximal, complemented
9#26013599C212yesmaximal, complemented
10#26013595C211yesmaximal, characteristic, complemented
11#120×C349C211yesmaximal, characteristic, complemented
12#19150232—24no—
13C2×Dic698C2211yescharacteristic, complemented
14C2×C22⋊D349C2212yes—
15C2×C22⋊D349C2211yescharacteristic
16C2×C22⋊D349C2211yescharacteristic, complemented
17C22×Dic349—24no—
18C22×Dic349C2212yes—
19C22×Dic349C2212yes—
20C22×Dic349C2211yescharacteristic
21C22×Dic349C2211yescharacteristic
22C22×Dic349C2211yescharacteristic, complemented
23C2×C4×D349C2211yescharacteristic, complemented
24C4⋊Dic349C2212yescomplemented
25#19150212C2212yescomplemented
26#19150212C2212yescomplemented
27C22⋊Dic349C2211yescharacteristic
28C22⋊Dic349C2212yes—
29C22⋊Dic349C2212yescomplemented
30#19150210C2212yescomplemented
31C4×Dic349C2212yescomplemented
32C24×C349C2211yescharacteristic
33C2×D4×C349C2212yes—
34C2×D4×C349C2211yescharacteristic
35C22⋊C4×C349C2211yescharacteristic
36C22⋊C4×C349C2212yes—
37C22×D349C2311yescharacteristic
38Dic698—22no—
39C22⋊D349—22no—
40C22⋊D349—24no—
41C22⋊D349—22no—
42C2×Dic349—22no—
43C2×Dic349—24no—
44C2×Dic349—24no—
45C2×Dic349—22no—
46C2×Dic349D414yescomplemented
47C2×Dic349C2312yes—
48C2×Dic349C2312yes—
49C2×Dic349C2311yescharacteristic
50C2×Dic349C2311yescharacteristic
51C2×Dic349C2311yescharacteristic
52C4×D349—22no—
53C23×C349—22no—
54C23×C349—24no—
55C23×C349C2311yescharacteristic
56C23×C349C2311yescharacteristic
57C23×C349C2312yes—
58D4×C349—22no—
59D4×C349—24no—
60C2×C4×C349C2312yes—
61C2×C4×C349C2311yescharacteristic
62C22×C349—28no—
63C22×C349—22no—
64C22×C349—22no—
65C22×C349—22no—
66C22×C349—21no—
67C22×C349—22no—
68C22×C349C2×D412yes—
69C22×C349#4114yescomplemented
70C22×C349C2411yescharacteristic
71C2×D349—21no—
72C2×D349—22no—
73C4×C349—21no—
74C4×C349—22no—
75Dic349—22no—
76Dic349—21no—
77Dic349—22no—
78Dic349C2×D412yes—
79Dic349C2×D412yes—
80C2×C349—22no—
81C2×C349—24no—
82C2×C349—41no—
83C2×C349C22×D411yescharacteristic
84C2×C349C2×#4112yes—
85D349—41no—
86C349#52511yescharacteristic, complemented, Sylow
87#525—3491nomaximal, Sylow
88C2×#41—3491no—
89C23×C4—3491no—
90#123—3491no—
91#123—3491no—
92C22⋊Q8—3492no—
93#121—3492no—
94#120—3491no—
95#120D34911yescharacteristic, complemented
96C4×D4—3492no—
97C4×D4—3492no—
98#117—3491no—
99C24C2×D34911yescharacteristic
100#41—6984no—
101C2×Q8—3491no—
102C2×D4—3492no—
103C2×D4—3491no—
104C2×D4—3491no—
105C2×D4C2×D34912yes—
106C2×D4C2×D34911yescharacteristic
107C22×C4—3491no—
108C22×C4—3492no—
109C22×C4—3491no—
110C22×C4—3492no—
111C22×C4—3491no—
112C22×C4—3491no—
113C22×C4—6984no—
114C4⋊C4—3492no—
115C4⋊C4—3492no—
116C4⋊C4—3492no—
117C22⋊C4—3491no—
118C22⋊C4—3492no—
119C22⋊C4—3492no—
120C22⋊C4—3492no—
121C22⋊C4C2×D34911yescharacteristic
122C22⋊C4C2×D34912yes—
123C42—3492no—
124C23—22no—
125C23—24no—
126C23—3491no—
127C23C22×D34911yescharacteristic
128C23C22×D34911yescharacteristic
129C23C22×D34912yes—
130Q8—6982no—
131D4—22no—
132D4—24no—
133D4—6982no—
134D4—6982no—
135D4—6984no—
136C2×C4—3494no—
137C2×C4—3491no—
138C2×C4—3491no—
139C2×C4—3492no—
140C2×C4—3492no—
141C2×C4—3491no—
142C2×C4—6982no—
143C2×C4—6984no—
144C2×C4—6984no—
145C2×C4—6982no—
146C2×C4—6982no—
147C2×C4C22×D34912yes—
148C2×C4C22×D34911yescharacteristic
149C22—28no—
150C22—22no—
151C22—22no—
152C22—22no—
153C22—21no—
154C22—22no—
155C22—6981no—
156C22—6982no—
157C22D4×D34912yes—
158C22#1915023214yescomplemented
159C22C23×D34911yescharacteristic
160C4—21no—
161C4—22no—
162C4—3492no—
163C4—3492no—
164C4—6982no—
165C4—6981no—
166C4—6982no—
167C2—22no—
168C2—24no—
169C2—41no—
170C2—13961no—
171C2C2×D4×D34911yescharacteristic
172C2C2×#1915023212yes—
subgroupG/H
center, FrattiniC22C23×D349
commutator, socle, cosocleC22×C349C24
Fitting#120×C349C2
hypercenter#120D349
hypocenterC349#525
LCSG ▷ C22×C349 ▷ C349
UCSC1 ◁ C22 ◁ #120

Complements

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

NH ≅ G/Ntype
1. C2×#19150232C2semidirect
2. C23×Dic349C2semidirect
3. #26013623C2semidirect
4. #26013620C2semidirect
5. #26013616C2semidirect
6. #26013611C2semidirect
7. #26013608C2semidirect
8. D4×Dic349C2semidirect
9. #26013599C2semidirect
10. #26013595C2semidirect
11. #120×C349C2semidirect
13. C2×Dic698C22semidirect
16. C2×C22⋊D349C22semidirect
22. C22×Dic349C22semidirect
23. C2×C4×D349C22semidirect
24. C4⋊Dic349C22semidirect
25. #19150212C22semidirect
26. #19150212C22semidirect
29. C22⋊Dic349C22semidirect
30. #19150210C22semidirect
31. C4×Dic349C22semidirect
46. C2×Dic349D4semidirect
69. C22×C349#41semidirect
86. C349#525semidirect
95. #120D349semidirect
158. C22#19150232semidirect

Maximal subgroups

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

subgroupindex
C2×#191502322
C23×Dic3492
#260136232
#260136202
#260136162
#260136112
#260136082
D4×Dic3492
#260135992
#260135952
#120×C3492
#525349