Finite Group Databasebeta

← D23×#490 #15683729

Subgroups — D23×#490 #15683729

Order 2944.

488 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 488 conjugacy classes of subgroups — too many to draw. Only its 125 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
1#127×D23C211yesmaximal, characteristic
2#125×D23C211yesmaximal, characteristic, complemented
3#13092783C211yesmaximal, characteristic, complemented
4#13092782C211yesmaximal, characteristic, complemented
5#13092781C211yesmaximal, characteristic, complemented
6#13092780C214yesmaximal, complemented
7#102×D23C214yesmaximal, complemented
8#97×D23C211yesmaximal, characteristic, complemented
9C23×#490C211yesmaximal, characteristic, complemented
10C2×D4×D23—22no—
11C2×D4×D23C2212yes—
12C23×#127C2211yescharacteristic
13C23×#125C2211yescharacteristic, complemented
14#10625111—22no—
15#10625111C2212yes—
16C2×C8×D23C2212yes—
17C4⋊C4×D23—22no—
18C4⋊C4×D23C2212yes—
19#10625061C2212yescomplemented
20#10625060—22no—
21#10625045C2211yescharacteristic
22C42×D23C2211yescharacteristic
23#10625041C2211yescharacteristic, complemented
24C23×#102C2214yescomplemented
25C23×#97C2211yescharacteristic, complemented
26#10625014C2214yescomplemented
27#10625003C2214yescomplemented
28#10624997C2212yescomplemented
29#10624991C2214yescomplemented
30#10624985C2211yescharacteristic, complemented
31C23×D23—22no—
32C2×D4×C23—22no—
33C2×D4×C23C2312yes—
34C2×C22⋊D23—22no—
35C2×C22⋊D23—24no—
36D4×D23—28no—
37D4×D23—48no—
38C2×D92—22no—
39C2×D92C2312yes—
40C2×C4×D23—22no—
41C2×C4×D23D412yes—
42C2×C4×D23C2311yescharacteristic
43C2×C8×C23C2312yes—
44C4⋊C4×C23—22no—
45C4⋊C4×C23C2312yes—
46C42×C23C2311yescharacteristic
47C4⋊Dic23—22no—
48C4⋊Dic23C2312yes—
49Dic23⋊C4—22no—
50Dic23⋊C4—22no—
51Dic23⋊C4—22no—
52C4×Dic23D412yes—
53C4×Dic23C2311yescharacteristic
54C2×C23⋊C8C2312yes—
55C8×D23—24no—
56C23×C23—22no—
57C22×D23—22no—
58C22×D23—44no—
59C22×D23—48no—
60C22×D23C2×D411yescharacteristic
61D4×C23—24no—
62D4×C23—44no—
63C2×C4×C23—22no—
64C2×C4×C23C2×D412yes—
65C2×C4×C23C2411yescharacteristic
66C22⋊D23—48no—
67C22⋊D23—44no—
68C2×Dic23—22no—
69C2×Dic23C2×D412yes—
70C2×Dic23C2×D411yescharacteristic
71D92—24no—
72D92—44no—
73C4×D23—24no—
74C4×D23—44no—
75C4×D23#4114yes—
76C8×C23—22no—
77C23⋊C8—22no—
78C2×#490—231nomaximal, Sylow
79C22×C23—42no—
80C22×C23—44no—
81C22×C23C22×D411yescharacteristic
82C2×D23—42no—
83C2×D23—44no—
84C2×D23—88no—
85C2×D23#12412yes—
86C2×D23AGL(1,ℤ/8)14yes—
87C4×C23—22no—
88C4×C23—42no—
89C4×C23C2×#4112yes—
90Dic23—22no—
91Dic23—42no—
92Dic23#12412yes—
93C2×#127—231no—
94C2×#125—231no—
95#490—231no—
96#490—231no—
97#490—234no—
98#490—231no—
99#490D2311yescharacteristic, direct product factor
100C2×#102—234no—
101C2×#97—231no—
102C2×C23—82no—
103C2×C23C2×#12411yescharacteristic
104C2×C23C2×AGL(1,ℤ/8)12yes—
105D23—82no—
106D23#49014yesdirect product factor
107C22×D4—232no—
108C22×D4—462no—
109C22×C8—232no—
110#127—232no—
111#127—231no—
112#127—462no—
113#127C2×D2311yescharacteristic
114#125—232no—
115#125—231no—
116#125—462no—
117#125C2×D2311yescharacteristic, complemented
118C2×C4⋊C4—232no—
119C2×C4⋊C4—462no—
120C2×C42—231no—
121#102—234no—
122#102—234no—
123#102—234no—
124#102C2×D2314yescomplemented
125#97—232no—
126#97—231no—
127#97C2×D2311yescharacteristic, complemented
128C23C2×#49011yescharacteristic, complemented, Sylow
129C24—462no—
130C2×D4—22no—
131C2×D4—232no—
132C2×D4—468no—
133C2×D4—464no—
134C2×D4—462no—
135C2×D4—462no—
136C2×D4—928no—
137C2×D4C22×D2312yes—
138C22×C4—231no—
139C22×C4—232no—
140C22×C4—462no—
141C2×C8—232no—
142C2×C8—464no—
143C2×C8C22×D2312yes—
144C4⋊C4—22no—
145C4⋊C4—232no—
146C4⋊C4—462no—
147C4⋊C4—462no—
148C4⋊C4—462no—
149C4⋊C4—462no—
150C4⋊C4C22×D2312yes—
151C42—232no—
152C42—231no—
153C42C22×D2311yescharacteristic
154C23—22no—
155C23—231no—
156C23—462no—
157C23—924no—
158C23—928no—
159D4—24no—
160D4—44no—
161D4—464no—
162D4—924no—
163D4—924no—
164D4—928no—
165C2×C4—22no—
166C2×C4—231no—
167C2×C4—234no—
168C2×C4—232no—
169C2×C4—464no—
170C2×C4—462no—
171C2×C4—924no—
172C2×C4D4×D2312yes—
173C2×C4C23×D2311yescharacteristic
174C8—22no—
175C8—462no—
176C22—42no—
177C22—44no—
178C22—232no—
179C22—234no—
180C22—922no—
181C22—924no—
182C22—1848no—
183C22C2×D4×D2311yescharacteristic
184C4—22no—
185C4—42no—
186C4—232no—
187C4—462no—
188C4—922no—
189C4#41×D2312yes—
190C2—82no—
191C2—234no—
192C2—1842no—
193C2#124×D2311yescharacteristic
194C2AGL(1,ℤ/8)×D2312yes—
subgroupG/H
centerC22C2×D4×D23
commutator, cosocleC2×C4×C23C24
FrattiniC2×C4C23×D23
FittingC23×#490C2
socleC22×C23C22×D4
prime radical#127×D23C2
hypercenter#490D23
hypocenterC23C2×#490
LCSG ▷ C2×C4×C23 ▷ C2×C23 ▷ C23
UCSC1 ◁ C22 ◁ C42 ◁ #490

Complements

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

NH ≅ G/Ntype
2. #125×D23C2semidirect
3. #13092783C2semidirect
4. #13092782C2semidirect
5. #13092781C2semidirect
6. #13092780C2semidirect
7. #102×D23C2semidirect
8. #97×D23C2semidirect
9. C23×#490C2semidirect
13. C23×#125C22semidirect
19. #10625061C22semidirect
23. #10625041C22semidirect
24. C23×#102C22semidirect
25. C23×#97C22semidirect
26. #10625014C22semidirect
27. #10625003C22semidirect
28. #10624997C22semidirect
29. #10624991C22semidirect
30. #10624985C22semidirect
99. #490D23direct (N × H)
106. D23#490direct (N × H)
117. #125C2×D23semidirect
124. #102C2×D23semidirect
127. #97C2×D23semidirect
128. C23C2×#490semidirect

Maximal subgroups

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

subgroupindex
#127×D232
#125×D232
#130927832
#130927822
#130927812
#130927802
#102×D232
#97×D232
C23×#4902
C2×#49023