Finite Group Databasebeta

← #15683769

Subgroups — #15683769

Order 2944.

484 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 484 conjugacy classes of subgroups — too many to draw. Only its 129 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#13092970C211yesmaximal, characteristic, complemented
2C2×#10625111C211yesmaximal, characteristic
3#13092780C214yesmaximal, complemented
4#13092762C214yesmaximal, complemented
5#13092423C211yesmaximal, characteristic, complemented
6#13092422C211yesmaximal, characteristic, complemented
7#13092421C211yesmaximal, characteristic, complemented
8M16×Dic23C211yesmaximal, characteristic, complemented
9C23×#418C211yesmaximal, characteristic, complemented
10C22×D4×C23C2211yescharacteristic
11C22×C22⋊D23—22no—
12C22×D92C2211yescharacteristic
13C2×M16×C23C2211yescharacteristic
14C23×#117C2211yescharacteristic
15#10625111—22no—
16#10625111C2212yes—
17#10625111C2212yes—
18#10625097C2211yescharacteristic
19C2×C4×Dic23C2211yescharacteristic
20C2×#67267C2211yescharacteristic
21#10625061C2212yescomplemented
22#10625059C2212yescomplemented
23#10625049—22no—
24C4×Dic46—22no—
25C23×#102C2214yescomplemented
26#10625014C2214yescomplemented
27#10625003C2214yescomplemented
28#10624997C2212yescomplemented
29C8×Dic23C2212yescomplemented
30#10624991C2214yescomplemented
31C24×C23—21no—
32C23×D23—21no—
33C2×D4×C23—21no—
34C2×D4×C23—22no—
35C2×D4×C23—22no—
36C2×D4×C23C2312yes—
37C22×C4×C23C2311yescharacteristic
38C2×C22⋊D23—24no—
39C2×C22⋊D23—24no—
40C2×C22⋊D23—48no—
41C22×Dic23D412yes—
42C2×D92—22no—
43C2×D92—22no—
44C2×D92—21no—
45C2×D92C2312yes—
46C2×Dic46—22no—
47M16×C23—22no—
48C2×C8×C23C2312yes—
49C4⋊C4×C23C2312yes—
50C22⋊C4×C23—21no—
51C42×C23—21no—
52C22⋊Dic23—21no—
53C4⋊Dic23—22no—
54C4⋊Dic23C2312yes—
55Dic23⋊C4—22no—
56Dic23⋊C4—22no—
57C4×Dic23—21no—
58C4×Dic23D412yes—
59C4×Dic23C2311yescharacteristic
60C4×Dic23C2311yescharacteristic
61#67267—22no—
62C2×C23⋊C8C2312yes—
63C23×C23—22no—
64C23×C23—44no—
65C23×C23—42no—
66C23×C23C2×D411yescharacteristic
67C22×D23—22no—
68C22×D23—44no—
69C22×D23—42no—
70D4×C23—22no—
71D4×C23—22no—
72D4×C23—42no—
73C2×C4×C23—22no—
74C2×C4×C23C2×D411yescharacteristic
75C2×C4×C23#4112yes—
76C2×C4×C23#4112yes—
77C2×C4×C23C2411yescharacteristic
78C22⋊D23—48no—
79C22⋊D23—48no—
80C2×Dic23—24no—
81C2×Dic23—22no—
82C2×Dic23C2×D412yes—
83C2×Dic23C2×D412yes—
84D92—22no—
85D92—22no—
86D92—42no—
87Dic46—44no—
88C8×C23—22no—
89C23⋊C8—22no—
90#3138—231nomaximal, Sylow
91C22×C23—22no—
92C22×C23—42no—
93C22×C23—44no—
94C22×C23—84no—
95C22×C23#12412yes—
96C22×C23C22×D411yescharacteristic
97C2×D23—42no—
98C2×D23—44no—
99C2×D23—84no—
100C4×C23—42no—
101C4×C23#12412yes—
102C4×C23C2×#4111yescharacteristic
103C4×C23C2×#4111yescharacteristic
104Dic23—22no—
105Dic23—42no—
106Dic23AGL(1,ℤ/8)14yes—
107#533—231no—
108C2×#127—231no—
109#490—234no—
110#486—234no—
111#418—231no—
112#418—231no—
113#418—231no—
114#418D2311yescharacteristic, complemented
115C4×M16—231no—
116C2×C23—22no—
117C2×C23—82no—
118C2×C23C2×#12411yescharacteristic
119C2×C23C2×AGL(1,ℤ/8)12yes—
120D23—82no—
121C22×D4—231no—
122C22×D4—462no—
123C22×D4C2×D2311yescharacteristic
124C2×M16—231no—
125C2×M16C2×D2311yescharacteristic
126C4⋊Q8—232no—
127#127—232no—
128#127—232no—
129#127—462no—
130#125—232no—
131C22⋊Q8—462no—
132C4×Q8—462no—
133#117—231no—
134#117C2×D2311yescharacteristic
135C2×C42—231no—
136#102—234no—
137#102—234no—
138#102—234no—
139#102C2×D2314yescomplemented
140#97—232no—
141C4×C8—232no—
142C23#313811yescharacteristic, complemented, Sylow
143C24—21no—
144C24—461no—
145C2×Q8—462no—
146C2×D4—22no—
147C2×D4—21no—
148C2×D4—22no—
149C2×D4—232no—
150C2×D4—464no—
151C2×D4—464no—
152C2×D4—462no—
153C2×D4—461no—
154C2×D4—462no—
155C2×D4—928no—
156C2×D4C22×D2312yes—
157C22×C4—232no—
158C22×C4C22×D2311yescharacteristic
159M16—22no—
160M16—462no—
161C2×C8—232no—
162C2×C8C22×D2312yes—
163C4⋊C4—232no—
164C4⋊C4—462no—
165C4⋊C4—462no—
166C4⋊C4—462no—
167C4⋊C4C22×D2312yes—
168C22⋊C4—21no—
169C22⋊C4—461no—
170C42—21no—
171C42—232no—
172C42—231no—
173C42—231no—
174C42—461no—
175C23—22no—
176C23—44no—
177C23—42no—
178C23—462no—
179C23—924no—
180C23—922no—
181C23D4×D2311yescharacteristic
182Q8—924no—
183D4—22no—
184D4—22no—
185D4—42no—
186D4—462no—
187D4—462no—
188D4—928no—
189D4—928no—
190D4—922no—
191C2×C4—22no—
192C2×C4—232no—
193C2×C4—232no—
194C2×C4—464no—
195C2×C4—462no—
196C2×C4#6728812yes—
197C2×C4D4×D2311yescharacteristic
198C2×C4#6729012yes—
199C2×C4C23×D2311yescharacteristic
200C8—22no—
201C8—462no—
202C22—22no—
203C22—42no—
204C22—44no—
205C22—84no—
206C22—922no—
207C22—924no—
208C22—1844no—
209C22#1062505612yes—
210C22C2×D4×D2311yescharacteristic
211C4—42no—
212C4—234no—
213C4—462no—
214C4—922no—
215C4#1062505612yes—
216C4C2×#6728811yescharacteristic
217C4C2×#6729011yescharacteristic
218C2—22no—
219C2—82no—
220C2—1842no—
221C2C2×#1062505611yescharacteristic
222C2AGL(1,ℤ/8)×D2312yes—
subgroupG/H
centerC22C2×D4×D23
commutator, cosocleC2×C4×C23C24
FrattiniC2×C4C23×D23
FittingC23×#418C2
socleC22×C23C22×D4
prime radicalC2×#10625111C2
hypercenter#418D23
hypocenterC23#3138
LCSG ▷ C2×C4×C23 ▷ C2×C23 ▷ C23
UCSC1 ◁ C22 ◁ C22×C4 ◁ #418

Complements

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

NH ≅ G/Ntype
1. #13092970C2semidirect
3. #13092780C2semidirect
4. #13092762C2semidirect
5. #13092423C2semidirect
6. #13092422C2semidirect
7. #13092421C2semidirect
8. M16×Dic23C2semidirect
9. C23×#418C2semidirect
21. #10625061C22semidirect
22. #10625059C22semidirect
25. C23×#102C22semidirect
26. #10625014C22semidirect
27. #10625003C22semidirect
28. #10624997C22semidirect
29. C8×Dic23C22semidirect
30. #10624991C22semidirect
114. #418D23semidirect
139. #102C2×D23semidirect
142. C23#3138semidirect

Maximal subgroups

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

subgroupindex
#130929702
C2×#106251112
#130927802
#130927622
#130924232
#130924222
#130924212
M16×Dic232
C23×#4182
#313823