Finite Group Databasebeta

← #36374652

Subgroups — #36374652

Order 22336.

330 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 330 conjugacy classes of subgroups — too many to draw. Only its 107 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
2C22×C4×D349C211yesmaximal, characteristic, complemented
3#26013622C211yesmaximal, characteristic, complemented
4#26013617C212yesmaximal, complemented
5#26013612C212yesmaximal, complemented
6#26013609C211yesmaximal, characteristic, complemented
7#26013607C211yesmaximal, characteristic, complemented
8#26013601C212yesmaximal, complemented
9#26013599C212yesmaximal, complemented
10#26013595C211yesmaximal, characteristic, complemented
11#123×C349C211yesmaximal, characteristic, complemented
12C23×D349C2211yescharacteristic
13#19150232—24no—
14C2×Dic698C2211yescharacteristic, complemented
15C2×D1396C2212yes—
16C2×C22⋊D349C2212yes—
17C2×C22⋊D349C2211yescharacteristic
18C2×C22⋊D349C2211yescharacteristic, complemented
19C22×Dic349C2211yescharacteristic
20C22×Dic349C2211yescharacteristic, complemented
21C2×C4×D349—24no—
22C2×C4×D349C2212yes—
23C2×C4×D349C2212yes—
24C2×C4×D349C2211yescharacteristic, complemented
25#19150212C2212yescomplemented
26#19150212C2212yescomplemented
27C22⋊Dic349C2211yescharacteristic
28#19150210C2212yes—
29#19150210C2212yes—
30#19150210C2212yescomplemented
31C4×Dic349C2212yescomplemented
32C2×D4×C349C2211yescharacteristic
33C22×C4×C349C2211yescharacteristic
34C4⋊C4×C349C2212yescomplemented
35C22⋊C4×C349C2211yescharacteristic
36C22⋊C4×C349C2212yescomplemented
37C22×D349—22no—
38C22×D349—24no—
39C22×D349C2311yescharacteristic
40C22×D349C2312yes—
41Dic698—22no—
42D1396—24no—
43C22⋊D349—22no—
44C22⋊D349—24no—
45C22⋊D349—22no—
46C2×Dic349—22no—
47C2×Dic349D414yescomplemented
48C2×Dic349C2312yes—
49C2×Dic349C2311yescharacteristic
50C2×Dic349C2311yescharacteristic
51C2×Dic349C2311yescharacteristic
52C4×D349—24no—
53C4×D349—24no—
54C4×D349—22no—
55C23×C349C2311yescharacteristic
56C23×C349C2311yescharacteristic
57D4×C349—22no—
58C2×C4×C349—22no—
59C2×C4×C349C2312yes—
60C2×C4×C349C2312yes—
61C2×C4×C349C2311yescharacteristic
62C22×C349—22no—
63C22×C349—21no—
64C22×C349—22no—
65C22×C349C2×D412yes—
66C22×C349C2411yescharacteristic
67C2×D349—28no—
68C2×D349—22no—
69C2×D349—22no—
70C2×D349—21no—
71C2×D349—22no—
72C2×D349#4114yescomplemented
73C4×C349—22no—
74C4×C349—21no—
75C4×C349—22no—
76Dic349—21no—
77Dic349—22no—
78Dic349C2×D412yes—
79Dic349C2×D412yes—
80C2×C349—22no—
81C2×C349—41no—
82C2×C349C22×D411yescharacteristic
83C2×C349C2×#4112yes—
84D349—24no—
85D349—41no—
86C349#52511yescharacteristic, complemented, Sylow
87#525—3491nomaximal, Sylow
88C2×#41—3491no—
89C23×C4—3491no—
90#123—3491no—
91#123D34911yescharacteristic, complemented
92C22⋊Q8—3492no—
93#121—3492no—
94#120—3491no—
95#120—3491no—
96C4×D4—3492no—
97C4×D4—3492no—
98#117—3491no—
99C24—3491no—
100#41—6984no—
101C2×Q8—3491no—
102C2×D4—3492no—
103C2×D4—3491no—
104C2×D4—3491no—
105C2×D4—3492no—
106C2×D4C2×D34911yescharacteristic
107C22×C4—3491no—
108C22×C4—3491no—
109C22×C4—3492no—
110C22×C4—3491no—
111C22×C4—3492no—
112C22×C4—6984no—
113C22×C4C2×D34911yescharacteristic
114C4⋊C4—3492no—
115C4⋊C4—3492no—
116C4⋊C4C2×D34912yescomplemented
117C22⋊C4—3491no—
118C22⋊C4—3492no—
119C22⋊C4—3492no—
120C22⋊C4—3492no—
121C22⋊C4C2×D34911yescharacteristic
122C22⋊C4C2×D34912yescomplemented
123C42—3492no—
124C23—3492no—
125C23—3491no—
126C23—6984no—
127C23—6982no—
128C23C22×D34911yescharacteristic
129C23C22×D34911yescharacteristic
130Q8—6982no—
131D4—22no—
132D4—6984no—
133D4—6982no—
134D4—6984no—
135D4—6982no—
136C2×C4—22no—
137C2×C4—3494no—
138C2×C4—3491no—
139C2×C4—3491no—
140C2×C4—3492no—
141C2×C4—3491no—
142C2×C4—6984no—
143C2×C4—6984no—
144C2×C4—6982no—
145C2×C4—6982no—
146C2×C4C22×D34912yes—
147C2×C4C22×D34911yescharacteristic
148C2×C4C22×D34912yes—
149C22—22no—
150C22—21no—
151C22—22no—
152C22—3494no—
153C22—6988no—
154C22—6981no—
155C22—6982no—
156C22—6982no—
157C22—6982no—
158C22D4×D34912yes—
159C22C23×D34911yescharacteristic
160C4—22no—
161C4—21no—
162C4—22no—
163C4—3492no—
164C4—3492no—
165C4—6981no—
166C4—6982no—
167C2—22no—
168C2—41no—
169C2—13961no—
170C2—6984no—
171C2C2×D4×D34911yescharacteristic
172C2#41×D34912yes—
subgroupG/H
center, FrattiniC22C23×D349
commutator, socle, cosocleC22×C349C24
Fitting#123×C349C2
hypercenter#123D349
hypocenterC349#525
LCSG ▷ C22×C349 ▷ C349
UCSC1 ◁ C22 ◁ #123

Complements

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

NH ≅ G/Ntype
1. C2×#19150232C2semidirect
2. C22×C4×D349C2semidirect
3. #26013622C2semidirect
4. #26013617C2semidirect
5. #26013612C2semidirect
6. #26013609C2semidirect
7. #26013607C2semidirect
8. #26013601C2semidirect
9. #26013599C2semidirect
10. #26013595C2semidirect
11. #123×C349C2semidirect
14. C2×Dic698C22semidirect
18. C2×C22⋊D349C22semidirect
20. C22×Dic349C22semidirect
24. C2×C4×D349C22semidirect
25. #19150212C22semidirect
26. #19150212C22semidirect
30. #19150210C22semidirect
31. C4×Dic349C22semidirect
34. C4⋊C4×C349C22semidirect
36. C22⋊C4×C349C22semidirect
47. C2×Dic349D4semidirect
72. C2×D349#41semidirect
86. C349#525semidirect
91. #123D349semidirect
116. C4⋊C4C2×D349semidirect
122. C22⋊C4C2×D349semidirect

Maximal subgroups

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

subgroupindex
C2×#191502322
C22×C4×D3492
#260136222
#260136172
#260136122
#260136092
#260136072
#260136012
#260135992
#260135952
#123×C3492
#525349