Finite Group Databasebeta

← #36374709

Subgroups — #36374709

Order 22336.

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

normal subgroups only

This group has 334 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×#19150232C211yesmaximal, characteristic, complemented
2C22×C22⋊D349C211yesmaximal, characteristic, complemented
3#26013635C212yesmaximal, complemented
4#26013625C212yesmaximal, complemented
5#26013612C212yesmaximal, complemented
6#26013611C212yesmaximal, complemented
7#26013609C211yesmaximal, characteristic, complemented
8#26013608C211yesmaximal, characteristic, complemented
9#26013607C211yesmaximal, characteristic, complemented
10#26013595C211yesmaximal, characteristic, complemented
11#120×C349C211yesmaximal, characteristic, complemented
12C23×D349C2211yescharacteristic
13#19150232—24no—
14C2×Dic698C2211yescharacteristic, complemented
15C2×D1396C2212yescomplemented
16C2×C22⋊D349—24no—
17C2×C22⋊D349C2211yescharacteristic
18C2×C22⋊D349C2212yes—
19C2×C22⋊D349C2212yes—
20C2×C22⋊D349C2211yescharacteristic, complemented
21C2×C22⋊D349C2212yescomplemented
22C2×C22⋊D349C2212yescomplemented
23C22×Dic349C2211yescharacteristic
24C22×Dic349C2211yescharacteristic, complemented
25C2×C4×D349C2211yescharacteristic, complemented
26#19150212C2212yescomplemented
27C22⋊Dic349C2211yescharacteristic
28C22⋊Dic349C2212yescomplemented
29#19150210C2212yescomplemented
30#19150210C2212yescomplemented
31C4×Dic349C2212yescomplemented
32C24×C349C2211yescharacteristic
33C2×D4×C349C2211yescharacteristic
34C2×D4×C349C2212yescomplemented
35C22⋊C4×C349C2211yescharacteristic
36C22⋊C4×C349C2212yescomplemented
37C22×D349—22no—
38C22×D349—42no—
39C22×D349C2312yes—
40C22×D349C2311yescharacteristic
41Dic698—22no—
42D1396—42no—
43C22⋊D349—22no—
44C22⋊D349—24no—
45C22⋊D349—24no—
46C22⋊D349—22no—
47C22⋊D349—42no—
48C22⋊D349—42no—
49C2×Dic349—22no—
50C2×Dic349D414yescomplemented
51C2×Dic349C2312yes—
52C2×Dic349C2311yescharacteristic
53C2×Dic349C2311yescharacteristic
54C2×Dic349C2311yescharacteristic
55C4×D349—22no—
56C23×C349—22no—
57C23×C349—42no—
58C23×C349C2312yes—
59C23×C349C2311yescharacteristic
60C23×C349C2311yescharacteristic
61D4×C349—22no—
62D4×C349—42no—
63C2×C4×C349C2312yes—
64C2×C4×C349C2311yescharacteristic
65C22×C349—22no—
66C22×C349—21no—
67C22×C349—22no—
68C22×C349—22no—
69C22×C349—22no—
70C22×C349—22no—
71C22×C349—44no—
72C22×C349C2×D412yes—
73C22×C349C2411yescharacteristic
74C2×D349—21no—
75C2×D349—22no—
76C2×D349—22no—
77C2×D349—22no—
78C2×D349—22no—
79C2×D349—44no—
80C4×C349—21no—
81C4×C349—22no—
82Dic349—21no—
83Dic349—22no—
84Dic349C2×D412yes—
85Dic349C2×D412yes—
86C2×C349—22no—
87C2×C349—41no—
88C2×C349—42no—
89C2×C349C22×D411yescharacteristic
90C2×C349H(5,𝔽2)12yes—
91D349—41no—
92D349—42no—
93C349#53511yescharacteristic, complemented, Sylow
94#535—3491nomaximal, Sylow
95C2×#41—3491no—
96C22×D4—3491no—
97#127—3492no—
98#124—3492no—
99#121—3492no—
100#121—3492no—
101#120—3491no—
102#120—3491no—
103#120—3491no—
104#120D34911yescharacteristic, complemented
105#117—3491no—
106C24—3491no—
107C24C2×D34911yescharacteristic
108#41—6984no—
109C2×Q8—3491no—
110C2×D4—3492no—
111C2×D4—3492no—
112C2×D4—3491no—
113C2×D4—3492no—
114C2×D4—3492no—
115C2×D4—3491no—
116C2×D4—3492no—
117C2×D4—6984no—
118C2×D4C2×D34911yescharacteristic
119C2×D4C2×D34912yescomplemented
120C22×C4—3491no—
121C22×C4—3491no—
122C22×C4—3491no—
123C4⋊C4—3492no—
124C22⋊C4—3491no—
125C22⋊C4—3492no—
126C22⋊C4—3492no—
127C22⋊C4—3492no—
128C22⋊C4C2×D34911yescharacteristic
129C22⋊C4C2×D34912yescomplemented
130C42—3492no—
131C23—22no—
132C23—42no—
133C23—3492no—
134C23—3491no—
135C23—13962no—
136C23—6982no—
137C23C22×D34912yes—
138C23C22×D34911yescharacteristic
139C23C22×D34911yescharacteristic
140Q8—6982no—
141D4—22no—
142D4—42no—
143D4—13962no—
144D4—13962no—
145D4—13962no—
146D4—6984no—
147D4—6984no—
148D4—6982no—
149D4—6982no—
150C2×C4—3494no—
151C2×C4—3491no—
152C2×C4—3491no—
153C2×C4—3492no—
154C2×C4—3491no—
155C2×C4—6982no—
156C2×C4—6982no—
157C2×C4C22×D34912yes—
158C2×C4C22×D34911yescharacteristic
159C22—22no—
160C22—21no—
161C22—22no—
162C22—22no—
163C22—22no—
164C22—22no—
165C22—44no—
166C22—13964no—
167C22—6981no—
168C22—6982no—
169C22—6982no—
170C22—6982no—
171C22—6982no—
172C22D4×D34912yes—
173C22C23×D34911yescharacteristic
174C4—21no—
175C4—22no—
176C4—3492no—
177C4—3492no—
178C4—6981no—
179C4—6982no—
180C2—22no—
181C2—42no—
182C2—41no—
183C2—13962no—
184C2—13961no—
185C2C2×D4×D34911yescharacteristic
186C2#2601368412yes—
subgroupG/H
center, FrattiniC22C23×D349
commutator, socle, cosocleC22×C349C24
Fitting#120×C349C2
hypercenter#120D349
hypocenterC349#535
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. C22×C22⋊D349C2semidirect
3. #26013635C2semidirect
4. #26013625C2semidirect
5. #26013612C2semidirect
6. #26013611C2semidirect
7. #26013609C2semidirect
8. #26013608C2semidirect
9. #26013607C2semidirect
10. #26013595C2semidirect
11. #120×C349C2semidirect
14. C2×Dic698C22semidirect
15. C2×D1396C22semidirect
20. C2×C22⋊D349C22semidirect
21. C2×C22⋊D349C22semidirect
22. C2×C22⋊D349C22semidirect
24. C22×Dic349C22semidirect
25. C2×C4×D349C22semidirect
26. #19150212C22semidirect
28. C22⋊Dic349C22semidirect
29. #19150210C22semidirect
30. #19150210C22semidirect
31. C4×Dic349C22semidirect
34. C2×D4×C349C22semidirect
36. C22⋊C4×C349C22semidirect
50. C2×Dic349D4semidirect
93. C349#535semidirect
104. #120D349semidirect
119. C2×D4C2×D349semidirect
129. 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×C22⋊D3492
#260136352
#260136252
#260136122
#260136112
#260136092
#260136082
#260136072
#260135952
#120×C3492
#535349