Finite Group Databasebeta

← #15673360

Subgroups — #15673360

Order 2944.

548 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 548 conjugacy classes of subgroups — too many to draw. Only its 97 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#13093253C211yesmaximal, characteristic
2#13092433C212yesmaximal, complemented
3#13092414C212yesmaximal, complemented
4#13092391C211yesmaximal, characteristic, complemented
5C23×#408C211yesmaximal, characteristic, complemented
6C2×#41×C23C2211yescharacteristic
7#10625161—22no—
8#10625160—22no—
9#41×D23—24no—
10#10625158C412yescomplemented
11#10625155—22no—
12#10625155C412yescomplemented
13C2×#67288—22no—
14C2×#67288C2212yes—
15C2×M16×C23C2211yescharacteristic, complemented
16C22×C8×C23C2211yescharacteristic, complemented
17#10625097C2212yescomplemented
18C23×C22⋊C8—22no—
19#10625016—22no—
20#10625007—24no—
21#10625003—22no—
22#10624998—22no—
23#41×C23—22no—
24#41×C23—22no—
25C2×Q8×C23C2×C411yescharacteristic
26C2×D4×C23C2×C411yescharacteristic
27C2×D4×C23D412yes—
28C22×C4×C23D412yes—
29C22×C4×C23C2311yescharacteristic
30C2×C22⋊D23—24no—
31C2×C22⋊D23—22no—
32#67292—24no—
33Q8×D23—24no—
34#67290—28no—
35#67290—24no—
36D4×D23—28no—
37D4×D23—24no—
38#67288—24no—
39#67288—24no—
40#67288—24no—
41#67288—22no—
42#67288—22no—
43#67288C2×C414yes—
44#67288D414yes—
45C2×D92—22no—
46C2×D92D412yes—
47C2×C4×D23—24no—
48C2×C4×D23—22no—
49C2×Dic46—22no—
50C2×Dic46D412yes—
51M16×C23—22no—
52C2×C8×C23—22no—
53C2×C8×C23—21no—
54C2×C8×C23—21no—
55C22⋊Dic23—22no—
56C4⋊Dic23—22no—
57C4×Dic23—22no—
58C23×C23C22⋊C412yes—
59C23×C23C2×D411yescharacteristic
60C22×D23—24no—
61C22×D23—22no—
62Q8×C23—22no—
63D4×C23—24no—
64D4×C23—22no—
65C2×C4×C23—22no—
66C2×C4×C23—22no—
67C2×C4×C23C22⋊C412yes—
68C2×C4×C23C22×C411yescharacteristic
69C2×C4×C23C2×D412yes—
70C2×C4×C23C2×D412yes—
71C2×C4×C23C2×D411yescharacteristic
72C22⋊D23—28no—
73C22⋊D23—24no—
74C22⋊D23—24no—
75C2×Dic23—24no—
76C2×Dic23—22no—
77C2×Dic23—42no—
78D92—24no—
79D92—22no—
80D92C22⋊C414yes—
81C4×D23—28no—
82C4×D23—24no—
83C4×D23—24no—
84Dic46—24no—
85Dic46—22no—
86Dic46C22⋊C414yes—
87C8×C23—41no—
88C8×C23—41no—
89#1794—231nomaximal, Sylow
90C22×C23—22no—
91C22×C23—41no—
92C22×C23—42no—
93C22×C23C2×C22⋊C411yescharacteristic
94C22×C23#12012yes—
95C2×D23—24no—
96C2×D23—82no—
97C2×D23—84no—
98C4×C23—22no—
99C4×C23C2×C22⋊C412yes—
100C4×C23#12011yescharacteristic
101C4×C23#12011yescharacteristic
102Dic23—24no—
103Dic23—82no—
104#585—231no—
105#421—232no—
106#416—232no—
107#410—231no—
108#408D2311yescharacteristic, complemented
109C2×C23—22no—
110C2×C23—21no—
111C2×C23—42no—
112C2×C23#37911yescharacteristic
113D23—44no—
1142−1+4—232no—
1152−1+4—462no—
116H(5,𝔽2)—232no—
117H(5,𝔽2)—462no—
118H(5,𝔽2)—462no—
119C2×#41—232no—
120C2×#41—464no—
121C2×#41—462no—
122C2×#41C2×D2311yescharacteristic
123C2×M16C2×D2311yescharacteristic, complemented
124C22×C8C2×D2311yescharacteristic, complemented
125#117—232no—
126#104—464no—
127#103—462no—
128#102—462no—
129#99—462no—
130C22⋊C8—22no—
131C23#179411yescharacteristic, complemented, Sylow
132#41—22no—
133#41—22no—
134#41—234no—
135#41—234no—
136#41—464no—
137#41—464no—
138#41—464no—
139#41—468no—
140#41—464no—
141#41—464no—
142#41—462no—
143#41—462no—
144C2×Q8—232no—
145C2×Q8—462no—
146C2×Q8—464no—
147C2×Q8C4×D2311yescharacteristic
148C2×D4—232no—
149C2×D4—462no—
150C2×D4—464no—
151C2×D4—468no—
152C2×D4—464no—
153C2×D4—462no—
154C2×D4C4×D2311yescharacteristic
155C2×D4D9212yes—
156C22×C4—464no—
157C22×C4—462no—
158C22×C4C22⋊D2312yes—
159C22×C4C22×D2311yescharacteristic
160M16—22no—
161C2×C8—22no—
162C2×C8—21no—
163C2×C8—21no—
164C4⋊C4—462no—
165C22⋊C4—462no—
166C42—462no—
167C23—464no—
168C23—462no—
169C23(C2×D23)⋊C412yes—
170C23C2×D9211yescharacteristic
171Q8—22no—
172Q8—234no—
173Q8—464no—
174Q8—462no—
175D4—24no—
176D4—22no—
177D4—234no—
178D4—464no—
179D4—468no—
180D4—464no—
181D4—464no—
182D4—462no—
183C2×C4—22no—
184C2×C4—22no—
185C2×C4—468no—
186C2×C4—464no—
187C2×C4—462no—
188C2×C4—464no—
189C2×C4—464no—
190C2×C4—922no—
191C2×C4(C2×D23)⋊C412yes—
192C2×C4C2×C4×D2311yescharacteristic
193C2×C4D4×D2312yes—
194C2×C4D4×D2312yes—
195C2×C4C2×C22⋊D2311yescharacteristic
196C8—41no—
197C8—41no—
198C22—22no—
199C22—41no—
200C22—42no—
201C22—464no—
202C22—1842no—
203C22—1844no—
204C22#1062505312yes—
205C22C2×(C2×D23)⋊C411yescharacteristic
206C4—22no—
207C4—464no—
208C4—1842no—
209C4C22⋊C4×D2312yes—
210C4#1062510811yescharacteristic
211C4#1062510811yescharacteristic
212C2—22no—
213C2—21no—
214C2—42no—
215C2—924no—
216C2#1309227211yescharacteristic
subgroupG/H
centerC4#10625108
commutatorC2×C4×C23C22×C4
FrattiniC22×C4C22×D23
FittingC23×#408C2
socleC2×C23#379
prime radical#13093253C2
cosocleC22×C4×C23C23
hypercenter#408D23
hypocenterC23#1794
LCSG ▷ C2×C4×C23 ▷ C2×C23 ▷ C23
UCSC1 ◁ C4 ◁ C22×C4 ◁ #408

Complements

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

NH ≅ G/Ntype
2. #13092433C2semidirect
3. #13092414C2semidirect
4. #13092391C2semidirect
5. C23×#408C2semidirect
10. #10625158C4semidirect
12. #10625155C4semidirect
15. C2×M16×C23C22semidirect
16. C22×C8×C23C22semidirect
17. #10625097C22semidirect
108. #408D23semidirect
123. C2×M16C2×D23semidirect
124. C22×C8C2×D23semidirect
131. C23#1794semidirect

Maximal subgroups

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

subgroupindex
#130932532
#130924332
#130924142
#130923912
C23×#4082
#179423