Finite Group Databasebeta

← #36374710

Subgroups — #36374710

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×#19150233C211yesmaximal, characteristic, complemented
2C2×D4×D349C211yesmaximal, characteristic, complemented
3#26013635C212yesmaximal, complemented
4#26013625C212yesmaximal, complemented
5#26013614C211yesmaximal, characteristic, complemented
6#26013613C211yesmaximal, characteristic, complemented
7#26013610C211yesmaximal, characteristic, complemented
8#26013609C212yesmaximal, complemented
9#26013607C212yesmaximal, complemented
10#26013597C211yesmaximal, characteristic, complemented
11#121×C349C211yesmaximal, characteristic, complemented
12C23×D349C2212yes—
13#19150233—24no—
14C2×Dic698C2211yescharacteristic, complemented
15C2×D1396C2211yescharacteristic
16C2×D1396C2211yescharacteristic
17C2×D1396C2212yescomplemented
18C2×C22⋊D349C2211yescharacteristic
19C2×C22⋊D349C2212yes—
20C2×C22⋊D349C2211yescharacteristic, complemented
21C2×C22⋊D349C2212yescomplemented
22C2×C22⋊D349C2212yescomplemented
23D4×D349—24no—
24C2×C4×D349C2211yescharacteristic
25C2×C4×D349C2211yescharacteristic, complemented
26C4⋊Dic349C2211yescharacteristic, complemented
27C22⋊Dic349C2212yescomplemented
28#19150210C2212yes—
29#19150210C2212yescomplemented
30#19150210C2212yescomplemented
31C4×Dic349C2212yescomplemented
32C2×D4×C349C2211yescharacteristic
33C2×D4×C349C2212yescomplemented
34C22×C4×C349C2211yescharacteristic, complemented
35C4⋊C4×C349C2211yescharacteristic, complemented
36C22⋊C4×C349C2212yescomplemented
37C22×D349—24no—
38C22×D349—42no—
39C22×D349—42no—
40C22×D349C2312yes—
41C22×D349C2311yescharacteristic
42C22×D349C2311yescharacteristic
43Dic698—22no—
44D1396—22no—
45D1396—22no—
46D1396—42no—
47C22⋊D349—22no—
48C22⋊D349—24no—
49C22⋊D349—22no—
50C22⋊D349—42no—
51C22⋊D349—42no—
52C2×Dic349C2312yes—
53C2×Dic349C2311yescharacteristic
54C2×Dic349C2311yescharacteristic
55C4×D349—22no—
56C4×D349D414yescomplemented
57C23×C349C2312yes—
58C23×C349C2311yescharacteristic
59D4×C349—22no—
60D4×C349—42no—
61C2×C4×C349—22no—
62C2×C4×C349C2311yescharacteristic
63C2×C4×C349C2312yes—
64C2×C4×C349C2311yescharacteristic
65C22×C349—21no—
66C22×C349—22no—
67C22×C349—21no—
68C22×C349—22no—
69C22×C349—21no—
70C22×C349—22no—
71C22×C349C2411yescharacteristic
72C2×D349—21no—
73C2×D349—21no—
74C2×D349—21no—
75C2×D349—22no—
76C2×D349—21no—
77C2×D349—22no—
78C2×D349—21no—
79C2×D349—22no—
80C2×D349—48no—
81C2×D349C2×D412yes—
82C4×C349—21no—
83C4×C349—22no—
84C4×C349C2×D412yes—
85Dic349—21no—
86Dic349—22no—
87Dic349C2×D412yes—
88C2×C349—42no—
89C2×C349—41no—
90C2×C349C22×D411yescharacteristic
91C2×C349H(5,𝔽2)11yescharacteristic
92C2×C349H(5,𝔽2)11yescharacteristic
93D349—22no—
94D349—41no—
95D349—42no—
96C349#53511yescharacteristic, complemented, Sylow
97#535—3491nomaximal, Sylow
98C2×#41—3491no—
99C22×D4—3491no—
100#127—3492no—
101#124—3492no—
102#121—3491no—
103#121—3491no—
104#121—3491no—
105#121D34911yescharacteristic, complemented
106#120—3492no—
107#120—3492no—
108#117—3491no—
109C24—3492no—
110#41—6984no—
111C2×Q8—3491no—
112C2×D4—3492no—
113C2×D4—3492no—
114C2×D4—3491no—
115C2×D4—3492no—
116C2×D4—3491no—
117C2×D4—3492no—
118C2×D4—3491no—
119C2×D4—3491no—
120C2×D4—6984no—
121C2×D4C2×D34911yescharacteristic
122C2×D4C2×D34912yescomplemented
123C22×C4—3491no—
124C22×C4—3491no—
125C22×C4C2×D34911yescharacteristic, complemented
126C4⋊C4—3491no—
127C4⋊C4C2×D34911yescharacteristic, complemented
128C22⋊C4—3492no—
129C22⋊C4—3492no—
130C22⋊C4—3492no—
131C22⋊C4—3492no—
132C22⋊C4C2×D34912yescomplemented
133C42—3492no—
134C23—3491no—
135C23—3492no—
136C23—3491no—
137C23—13962no—
138C23—13962no—
139C23—6984no—
140C23C22×D34912yes—
141C23C22×D34911yescharacteristic
142Q8—6982no—
143D4—22no—
144D4—42no—
145D4—13962no—
146D4—13962no—
147D4—13962no—
148D4—6982no—
149D4—6984no—
150D4—6982no—
151D4—6982no—
152D4—6982no—
153C2×C4—22no—
154C2×C4—3491no—
155C2×C4—3494no—
156C2×C4—3492no—
157C2×C4—3491no—
158C2×C4—6982no—
159C2×C4C22×D34912yes—
160C2×C4C22×D34911yescharacteristic
161C2×C4C22×D34911yescharacteristic
162C22—21no—
163C22—22no—
164C22—21no—
165C22—22no—
166C22—21no—
167C22—22no—
168C22—3492no—
169C22—13968no—
170C22—6981no—
171C22—6982no—
172C22—6981no—
173C22—6982no—
174C22—6981no—
175C22—6982no—
176C22—6981no—
177C22—6981no—
178C22C23×D34911yescharacteristic
179C4—21no—
180C4—22no—
181C4—3492no—
182C4—6981no—
183C4—6982no—
184C4D4×D34912yes—
185C2—42no—
186C2—41no—
187C2—13962no—
188C2—13961no—
189C2—6982no—
190C2C2×D4×D34911yescharacteristic
191C2#2601368411yescharacteristic
192C2#2601368511yescharacteristic
subgroupG/H
center, FrattiniC22C23×D349
commutator, socle, cosocleC22×C349C24
Fitting#121×C349C2
hypercenter#121D349
hypocenterC349#535
LCSG ▷ C22×C349 ▷ C349
UCSC1 ◁ C22 ◁ #121

Complements

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

NH ≅ G/Ntype
1. C2×#19150233C2semidirect
2. C2×D4×D349C2semidirect
3. #26013635C2semidirect
4. #26013625C2semidirect
5. #26013614C2semidirect
6. #26013613C2semidirect
7. #26013610C2semidirect
8. #26013609C2semidirect
9. #26013607C2semidirect
10. #26013597C2semidirect
11. #121×C349C2semidirect
14. C2×Dic698C22semidirect
17. C2×D1396C22semidirect
20. C2×C22⋊D349C22semidirect
21. C2×C22⋊D349C22semidirect
22. C2×C22⋊D349C22semidirect
25. C2×C4×D349C22semidirect
26. C4⋊Dic349C22semidirect
27. C22⋊Dic349C22semidirect
29. #19150210C22semidirect
30. #19150210C22semidirect
31. C4×Dic349C22semidirect
33. C2×D4×C349C22semidirect
34. C22×C4×C349C22semidirect
35. C4⋊C4×C349C22semidirect
36. C22⋊C4×C349C22semidirect
56. C4×D349D4semidirect
96. C349#535semidirect
105. #121D349semidirect
122. C2×D4C2×D349semidirect
125. C22×C4C2×D349semidirect
127. C4⋊C4C2×D349semidirect
132. C22⋊C4C2×D349semidirect

Maximal subgroups

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

subgroupindex
C2×#191502332
C2×D4×D3492
#260136352
#260136252
#260136142
#260136132
#260136102
#260136092
#260136072
#260135972
#121×C3492
#535349