Finite Group Databasebeta

← #14271728

Subgroups — #14271728

Order 1856.

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
1C22×C22⋊D29C211yesmaximal, characteristic, complemented
2C2×#90807C211yesmaximal, characteristic, complemented
3C29×#120C211yesmaximal, characteristic, complemented
4#11745315C211yesmaximal, characteristic, complemented
5#11745302C212yesmaximal, complemented
6#11745301C212yesmaximal, complemented
7#11745299C211yesmaximal, characteristic, complemented
8#11745247C212yesmaximal, complemented
9#11745246C212yesmaximal, complemented
10#11745244C211yesmaximal, characteristic, complemented
11#11745239C211yesmaximal, characteristic, complemented
12C24×C29C2211yescharacteristic
13C23×D29C2211yescharacteristic
14C2×D4×C29C2211yescharacteristic
15C2×D4×C29C2212yescomplemented
16C2×C22⋊D29—24no—
17C2×C22⋊D29C2212yes—
18C2×C22⋊D29C2212yes—
19C2×C22⋊D29C2211yescharacteristic
20C2×C22⋊D29C2211yescharacteristic, complemented
21C2×C22⋊D29C2212yescomplemented
22C2×C22⋊D29C2212yescomplemented
23C22×Dic29C2211yescharacteristic
24C22×Dic29C2211yescharacteristic, complemented
25#90807—24no—
26C2×D116C2212yescomplemented
27C2×C4×D29C2211yescharacteristic, complemented
28C2×Dic58C2211yescharacteristic, complemented
29C22⋊C4×C29C2211yescharacteristic
30C22⋊C4×C29C2212yescomplemented
31C22⋊Dic29C2211yescharacteristic
32C22⋊Dic29C2212yescomplemented
33#90781C2212yescomplemented
34#90781C2212yescomplemented
35#90779C2212yescomplemented
36C4×Dic29C2212yescomplemented
37C23×C29—22no—
38C23×C29—42no—
39C23×C29C2312yes—
40C23×C29C2311yescharacteristic
41C23×C29C2311yescharacteristic
42C22×D29—22no—
43C22×D29—42no—
44C22×D29C2312yes—
45C22×D29C2311yescharacteristic
46D4×C29—22no—
47D4×C29—42no—
48C2×C4×C29C2312yes—
49C2×C4×C29C2311yescharacteristic
50C22⋊D29—22no—
51C22⋊D29—22no—
52C22⋊D29—24no—
53C22⋊D29—24no—
54C22⋊D29—42no—
55C22⋊D29—42no—
56C2×Dic29—22no—
57C2×Dic29D414yescomplemented
58C2×Dic29C2312yes—
59C2×Dic29C2311yescharacteristic
60C2×Dic29C2311yescharacteristic
61C2×Dic29C2311yescharacteristic
62D116—42no—
63C4×D29—22no—
64Dic58—22no—
65C22×C29—22no—
66C22×C29—21no—
67C22×C29—22no—
68C22×C29—22no—
69C22×C29—22no—
70C22×C29—22no—
71C22×C29—44no—
72C22×C29C2×D412yes—
73C22×C29C2411yescharacteristic
74C2×D29—21no—
75C2×D29—22no—
76C2×D29—22no—
77C2×D29—22no—
78C2×D29—22no—
79C2×D29—44no—
80C4×C29—21no—
81C4×C29—22no—
82Dic29—21no—
83Dic29—22no—
84Dic29C2×D412yes—
85Dic29C2×D412yes—
86#535—291nomaximal, Sylow
87C2×C29—22no—
88C2×C29—41no—
89C2×C29—42no—
90C2×C29C22×D411yescharacteristic
91C2×C29H(5,𝔽2)12yes—
92D29—41no—
93D29—42no—
94C2×#41—291no—
95C22×D4—291no—
96#127—292no—
97#124—292no—
98#121—292no—
99#121—292no—
100#120—291no—
101#120—291no—
102#120—291no—
103#120D2911yescharacteristic, complemented
104#117—291no—
105C29#53511yescharacteristic, complemented, Sylow
106C24—291no—
107C24C2×D2911yescharacteristic
108#41—584no—
109C2×Q8—291no—
110C2×D4—291no—
111C2×D4—292no—
112C2×D4—292no—
113C2×D4—291no—
114C2×D4—292no—
115C2×D4—292no—
116C2×D4—292no—
117C2×D4—584no—
118C2×D4C2×D2911yescharacteristic
119C2×D4C2×D2912yescomplemented
120C22×C4—291no—
121C22×C4—291no—
122C22×C4—291no—
123C4⋊C4—292no—
124C22⋊C4—291no—
125C22⋊C4—292no—
126C22⋊C4—292no—
127C22⋊C4—292no—
128C22⋊C4C2×D2911yescharacteristic
129C22⋊C4C2×D2912yescomplemented
130C42—292no—
131C23—22no—
132C23—42no—
133C23—292no—
134C23—291no—
135C23—582no—
136C23—1162no—
137C23C22×D2912yes—
138C23C22×D2911yescharacteristic
139C23C22×D2911yescharacteristic
140Q8—582no—
141D4—22no—
142D4—42no—
143D4—584no—
144D4—584no—
145D4—582no—
146D4—582no—
147D4—1162no—
148D4—1162no—
149D4—1162no—
150C2×C4—294no—
151C2×C4—291no—
152C2×C4—291no—
153C2×C4—292no—
154C2×C4—291no—
155C2×C4—582no—
156C2×C4—582no—
157C2×C4C22×D2912yes—
158C2×C4C22×D2911yescharacteristic
159C22—22no—
160C22—21no—
161C22—22no—
162C22—22no—
163C22—22no—
164C22—22no—
165C22—44no—
166C22—581no—
167C22—582no—
168C22—582no—
169C22—582no—
170C22—582no—
171C22—1164no—
172C22D4×D2912yes—
173C22C23×D2911yescharacteristic
174C4—21no—
175C4—22no—
176C4—292no—
177C4—292no—
178C4—581no—
179C4—582no—
180C2—22no—
181C2—42no—
182C2—41no—
183C2—1162no—
184C2—1161no—
185C2C2×D4×D2911yescharacteristic
186C2#1174535912yes—
subgroupG/H
center, FrattiniC22C23×D29
commutator, socle, cosocleC22×C29C24
FittingC29×#120C2
hypercenter#120D29
hypocenterC29#535
LCSG ▷ C22×C29 ▷ C29
UCSC1 ◁ C22 ◁ #120

Complements

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

NH ≅ G/Ntype
1. C22×C22⋊D29C2semidirect
2. C2×#90807C2semidirect
3. C29×#120C2semidirect
4. #11745315C2semidirect
5. #11745302C2semidirect
6. #11745301C2semidirect
7. #11745299C2semidirect
8. #11745247C2semidirect
9. #11745246C2semidirect
10. #11745244C2semidirect
11. #11745239C2semidirect
15. C2×D4×C29C22semidirect
20. C2×C22⋊D29C22semidirect
21. C2×C22⋊D29C22semidirect
22. C2×C22⋊D29C22semidirect
24. C22×Dic29C22semidirect
26. C2×D116C22semidirect
27. C2×C4×D29C22semidirect
28. C2×Dic58C22semidirect
30. C22⋊C4×C29C22semidirect
32. C22⋊Dic29C22semidirect
33. #90781C22semidirect
34. #90781C22semidirect
35. #90779C22semidirect
36. C4×Dic29C22semidirect
57. C2×Dic29D4semidirect
103. #120D29semidirect
105. C29#535semidirect
119. C2×D4C2×D29semidirect
129. C22⋊C4C2×D29semidirect

Maximal subgroups

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

subgroupindex
C22×C22⋊D292
C2×#908072
C29×#1202
#117453152
#117453022
#117453012
#117452992
#117452472
#117452462
#117452442
#117452392
#53529