Finite Group Databasebeta

← #36374704

Subgroups — #36374704

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
1C22×C22⋊D349C211yesmaximal, characteristic, complemented
2C2×D4×D349C211yesmaximal, characteristic, complemented
3#26013623C211yesmaximal, characteristic, complemented
4#26013621C212yesmaximal, complemented
5#26013620C211yesmaximal, characteristic, complemented
6#26013612C212yesmaximal, complemented
7#26013610C212yesmaximal, complemented
8#26013608C211yesmaximal, characteristic, complemented
9#26013607C212yesmaximal, complemented
10C22⋊C4×D349C211yesmaximal, characteristic, complemented
11#120×C349C211yesmaximal, characteristic, complemented
12C23×D349C2211yescharacteristic
13C23×D349C2211yescharacteristic, complemented
14C2×D1396C2211yescharacteristic, complemented
15C2×C22⋊D349—24no—
16C2×C22⋊D349C2212yes—
17C2×C22⋊D349C2211yescharacteristic
18C2×C22⋊D349C2211yescharacteristic, complemented
19C2×C22⋊D349C2212yescomplemented
20C2×C22⋊D349C2212yescomplemented
21D4×D349—24no—
22C22×Dic349C2211yescharacteristic, complemented
23C2×C4×D349C2211yescharacteristic, complemented
24C2×C4×D349C2212yescomplemented
25C4⋊Dic349C2212yescomplemented
26#19150212C2212yescomplemented
27C22⋊Dic349C2211yescharacteristic, complemented
28C22⋊Dic349C2212yescomplemented
29C22⋊Dic349C2212yescomplemented
30#19150210C2212yescomplemented
31#19150210C2212yescomplemented
32C24×C349C2211yescharacteristic, complemented
33C2×D4×C349C2211yescharacteristic
34C2×D4×C349C2212yescomplemented
35C22⋊C4×C349C2211yescharacteristic, complemented
36C22⋊C4×C349C2212yescomplemented
37C22×D349—22no—
38C22×D349—42no—
39C22×D349—42no—
40C22×D349D414yescomplemented
41C22×D349C2311yescharacteristic
42C22×D349C2311yescharacteristic
43C22×D349C2311yescharacteristic
44D1396—22no—
45C22⋊D349—22no—
46C22⋊D349—22no—
47C22⋊D349—24no—
48C22⋊D349—24no—
49C22⋊D349—42no—
50C2×Dic349—22no—
51C2×Dic349C2312yes—
52C2×Dic349C2312yes—
53C2×Dic349C2311yescharacteristic
54C4×D349—22no—
55C4×D349—42no—
56C23×C349—22no—
57C23×C349—42no—
58C23×C349C2312yes—
59C23×C349C2311yescharacteristic
60C23×C349C2311yescharacteristic
61D4×C349—22no—
62D4×C349—42no—
63C2×C4×C349C2311yescharacteristic
64C2×C4×C349C2312yes—
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—22no—
75C2×D349—22no—
76C2×D349—21no—
77C2×D349—22no—
78C2×D349—44no—
79C2×D349—44no—
80C2×D349C2×D412yes—
81C2×D349C2×D412yes—
82C4×C349—21no—
83C4×C349—22no—
84Dic349—21no—
85Dic349—22no—
86Dic349—22no—
87C2×C349—22no—
88C2×C349—41no—
89C2×C349—42no—
90C2×C349C22×D411yescharacteristic
91C2×C349H(5,𝔽2)12yes—
92D349—22no—
93D349—22no—
94D349—41no—
95C349#53411yescharacteristic, complemented, Sylow
96#534—3491nomaximal, Sylow
97C22×D4—3491no—
98C22×D4—3491no—
99#123—3491no—
100#123—3492no—
101#123—3491no—
102#121—3492no—
103#121—3492no—
104#120—3491no—
105#120—3492no—
106#120D34911yescharacteristic, complemented
107C2×C22⋊C4—3491no—
108C24—3491no—
109C24—3491no—
110C24C2×D34911yescharacteristic, complemented
111C2×D4—3492no—
112C2×D4—3492no—
113C2×D4—3491no—
114C2×D4—3491no—
115C2×D4—3492no—
116C2×D4—3491no—
117C2×D4—6984no—
118C2×D4—6984no—
119C2×D4C2×D34911yescharacteristic
120C2×D4C2×D34912yescomplemented
121C22×C4—3491no—
122C22×C4—3492no—
123C22×C4—3491no—
124C4⋊C4—3492no—
125C4⋊C4—3492no—
126C22⋊C4—3491no—
127C22⋊C4—3492no—
128C22⋊C4—3492no—
129C22⋊C4—3492no—
130C22⋊C4—3492no—
131C22⋊C4C2×D34911yescharacteristic, complemented
132C22⋊C4C2×D34912yescomplemented
133C23—22no—
134C23—42no—
135C23—3494no—
136C23—3491no—
137C23—3491no—
138C23—3491no—
139C23—13962no—
140C23—13962no—
141C23—6982no—
142C23C22×D34911yescharacteristic
143C23C22×D34912yes—
144C23C22×D34911yescharacteristic
145D4—22no—
146D4—42no—
147D4—13962no—
148D4—6982no—
149D4—6982no—
150D4—6984no—
151D4—6984no—
152D4—6982no—
153C2×C4—3492no—
154C2×C4—3491no—
155C2×C4—3492no—
156C2×C4—13962no—
157C2×C4—6982no—
158C2×C4—6982no—
159C2×C4C22×D34912yes—
160C2×C4C22×D34911yescharacteristic
161C22—22no—
162C22—21no—
163C22—22no—
164C22—22no—
165C22—22no—
166C22—22no—
167C22—44no—
168C22—3492no—
169C22—3492no—
170C22—13964no—
171C22—13964no—
172C22—6981no—
173C22—6982no—
174C22—6982no—
175C22—6982no—
176C22D4×D34912yes—
177C22C23×D34911yescharacteristic
178C4—21no—
179C4—22no—
180C4—6981no—
181C4—6982no—
182C4—6982no—
183C2—22no—
184C2—42no—
185C2—41no—
186C2—13961no—
187C2—6982no—
188C2—6982no—
189C2C2×D4×D34911yescharacteristic
190C2#2601368412yes—
subgroupG/H
center, FrattiniC22C23×D349
commutator, socle, cosocleC22×C349C24
Fitting#120×C349C2
hypercenter#120D349
hypocenterC349#534
LCSG ▷ C22×C349 ▷ C349
UCSC1 ◁ C22 ◁ #120

Complements

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

NH ≅ G/Ntype
1. C22×C22⋊D349C2semidirect
2. C2×D4×D349C2semidirect
3. #26013623C2semidirect
4. #26013621C2semidirect
5. #26013620C2semidirect
6. #26013612C2semidirect
7. #26013610C2semidirect
8. #26013608C2semidirect
9. #26013607C2semidirect
10. C22⋊C4×D349C2semidirect
11. #120×C349C2semidirect
13. C23×D349C22semidirect
14. C2×D1396C22semidirect
18. C2×C22⋊D349C22semidirect
19. C2×C22⋊D349C22semidirect
20. C2×C22⋊D349C22semidirect
22. C22×Dic349C22semidirect
23. C2×C4×D349C22semidirect
24. C2×C4×D349C22semidirect
25. C4⋊Dic349C22semidirect
26. #19150212C22semidirect
27. C22⋊Dic349C22semidirect
28. C22⋊Dic349C22semidirect
29. C22⋊Dic349C22semidirect
30. #19150210C22semidirect
31. #19150210C22semidirect
32. C24×C349C22semidirect
34. C2×D4×C349C22semidirect
35. C22⋊C4×C349C22semidirect
36. C22⋊C4×C349C22semidirect
40. C22×D349D4semidirect
95. C349#534semidirect
106. #120D349semidirect
110. C24C2×D349semidirect
120. C2×D4C2×D349semidirect
131. C22⋊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
C22×C22⋊D3492
C2×D4×D3492
#260136232
#260136212
#260136202
#260136122
#260136102
#260136082
#260136072
C22⋊C4×D3492
#120×C3492
#534349