Finite Group Databasebeta

← D23×#352 #15675031

Subgroups — D23×#352 #15675031

Order 2944.

278 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 278 conjugacy classes of subgroups — too many to draw. Only its 57 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#123×D23C211yesmaximal, characteristic
2#100×D23C211yesmaximal, characteristic
3#99×D23C211yesmaximal, characteristic
4#13092208C211yesmaximal, characteristic, complemented
5#13092207C211yesmaximal, characteristic, complemented
6#13092206C211yesmaximal, characteristic, complemented
7C23×#352C211yesmaximal, characteristic, complemented
8C2×D4×D23C2211yescharacteristic
9C22×C4×D23C411yescharacteristic, complemented
10C23×#123C2211yescharacteristic
11#10625106C2211yescharacteristic
12#10625100C411yescharacteristic, complemented
13M16×D23—21no—
14#10625065—21no—
15C4⋊C4×D23—21no—
16#10625057C411yescharacteristic, complemented
17#10625054—21no—
18C22⋊C4×D23—21no—
19C22⋊C4×D23—21no—
20C22⋊C4×D23C411yescharacteristic, complemented
21C23×#100C2211yescharacteristic
22C23×#99C2211yescharacteristic
23#10625016C2211yescharacteristic
24#10625015C2211yescharacteristic
25#10625005—21no—
26#10624988—21no—
27C23×D23C2×C411yescharacteristic
28C23×D23D411yescharacteristic
29C2×D4×C23C2311yescharacteristic
30C22×C4×C23C2×C411yescharacteristic, complemented
31C2×C22⋊D23C2×C411yescharacteristic
32C2×C22⋊D23D411yescharacteristic
33C22×Dic23C2×C411yescharacteristic, complemented
34D4×D23—41no—
35D4×D23—41no—
36C2×D92D411yescharacteristic
37C2×C4×D23—21no—
38C2×C4×D23—21no—
39C2×C4×D23—21no—
40C2×C4×D23—41no—
41C2×C4×D23—41no—
42C2×C4×D23—41no—
43C2×C4×D23D411yescharacteristic
44M16×C23—21no—
45C4⋊C4×C23—21no—
46C22⋊C4×C23—21no—
47C22⋊C4×C23—21no—
48C22⋊C4×C23C2×C411yescharacteristic, complemented
49C22⋊Dic23—21no—
50C22⋊Dic23—21no—
51C22⋊Dic23C2×C411yescharacteristic, complemented
52(C2×D23)⋊C4—21no—
53(C2×D23)⋊C4—21no—
54(C2×D23)⋊C4—21no—
55(C2×D23)⋊C4—41no—
56C4⋊Dic23—21no—
57Dic23⋊C4—21no—
58Dic23⋊C4—21no—
59#67267—21no—
60#67262—41no—
61C8×D23—41no—
62C23×C23C22×C411yescharacteristic
63C23×C23C2×D411yescharacteristic
64C22×D23—21no—
65C22×D23—21no—
66C22×D23—21no—
67C22×D23—21no—
68C22×D23—42no—
69C22×D23—42no—
70C22×D23C22⋊C411yescharacteristic
71C22×D23C22⋊C411yescharacteristic
72C22×D23C22⋊C411yescharacteristic
73D4×C23—41no—
74C2×C4×C23—21no—
75C2×C4×C23—21no—
76C2×C4×C23—41no—
77C2×C4×C23—41no—
78C2×C4×C23C2×D411yescharacteristic
79C22⋊D23—41no—
80C22⋊D23—41no—
81C2×Dic23—21no—
82C2×Dic23—21no—
83C2×Dic23—41no—
84C2×Dic23—41no—
85C2×Dic23C22⋊C411yescharacteristic
86D92—41no—
87C4×D23—21no—
88C4×D23—21no—
89C4×D23—42no—
90C4×D23—42no—
91C4×D23—41no—
92C4×D23—41no—
93C4×D23—82no—
94C8×C23—41no—
95C23⋊C8—41no—
96C2×#352—231nomaximal, Sylow
97C22×C23—21no—
98C22×C23—21no—
99C22×C23—41no—
100C22×C23—41no—
101C22×C23C2×C22⋊C411yescharacteristic
102C2×D23—22no—
103C2×D23—21no—
104C2×D23—21no—
105C2×D23—42no—
106C2×D23—42no—
107C2×D23—42no—
108C2×D23—41no—
109C2×D23—41no—
110C2×D23—81no—
111C2×D23#9911yescharacteristic
112C2×D23#9911yescharacteristic
113C4×C23—21no—
114C4×C23—41no—
115C4×C23—41no—
116C4×C23—81no—
117Dic23—21no—
118Dic23—41no—
119Dic23—41no—
120Dic23—81no—
121C2×#123—231no—
122C2×#100—231no—
123C2×#99—231no—
124#352—231no—
125#352—231no—
126#352—231no—
127#352D2311yescharacteristic, direct product factor
128C2×C23—21no—
129C2×C23—41no—
130C2×C23—41no—
131C2×C23C2×#9911yescharacteristic
132D23—21no—
133D23—41no—
134D23—41no—
135D23#35212yesdirect product factor
136C22×D4—231no—
137C23×C4—231no—
138C2×M16—461no—
139#123—231no—
140#123—231no—
141#123—231no—
142#123—461no—
143#123—461no—
144#123C2×D2311yescharacteristic
145C2×C4⋊C4—461no—
146C2×C22⋊C4—231no—
147C2×C22⋊C4—461no—
148C2×C22⋊C4—461no—
149#100—231no—
150#100—461no—
151#100C2×D2311yescharacteristic
152#99—231no—
153#99—461no—
154#99C2×D2311yescharacteristic
155C23C2×#35211yescharacteristic, complemented, Sylow
156C24—231no—
157C24—231no—
158C2×D4—231no—
159C2×D4—231no—
160C2×D4—231no—
161C2×D4—921no—
162C2×D4—921no—
163C2×D4C22×D2311yescharacteristic
164C22×C4—231no—
165C22×C4—231no—
166C22×C4—461no—
167C22×C4—461no—
168C22×C4—461no—
169C22×C4—921no—
170C22×C4—921no—
171C22×C4—921no—
172C22×C4C4×D2311yescharacteristic, complemented
173M16—21no—
174M16—461no—
175M16—921no—
176C2×C8—921no—
177C4⋊C4—21no—
178C4⋊C4—461no—
179C4⋊C4—461no—
180C4⋊C4—461no—
181C22⋊C4—21no—
182C22⋊C4—21no—
183C22⋊C4—231no—
184C22⋊C4—461no—
185C22⋊C4—461no—
186C22⋊C4—461no—
187C22⋊C4—461no—
188C22⋊C4—461no—
189C22⋊C4—921no—
190C22⋊C4C4×D2311yescharacteristic, complemented
191C23—231no—
192C23—231no—
193C23—231no—
194C23—461no—
195C23—461no—
196C23—461no—
197C23—461no—
198C23—922no—
199C23—922no—
200C23C2×C4×D2311yescharacteristic
201C23D4×D2311yescharacteristic
202D4—41no—
203D4—921no—
204D4—921no—
205D4—921no—
206C2×C4—21no—
207C2×C4—21no—
208C2×C4—41no—
209C2×C4—41no—
210C2×C4—231no—
211C2×C4—461no—
212C2×C4—461no—
213C2×C4—461no—
214C2×C4—461no—
215C2×C4—921no—
216C2×C4—921no—
217C2×C4—922no—
218C2×C4—922no—
219C2×C4—921no—
220C2×C4—921no—
221C2×C4—1842no—
222C2×C4D4×D2311yescharacteristic
223C8—41no—
224C8—921no—
225C22—21no—
226C22—21no—
227C22—41no—
228C22—41no—
229C22—231no—
230C22—231no—
231C22—462no—
232C22—461no—
233C22—461no—
234C22—921no—
235C22—921no—
236C22—922no—
237C22—922no—
238C22—922no—
239C22—1841no—
240C22C22⋊C4×D2311yescharacteristic
241C4—21no—
242C4—41no—
243C4—41no—
244C4—81no—
245C4—461no—
246C4—921no—
247C4—921no—
248C4—1841no—
249C2—21no—
250C2—41no—
251C2—41no—
252C2—232no—
253C2—461no—
254C2—921no—
255C2—921no—
256C2#99×D2311yescharacteristic
subgroupG/H
centerC2#99×D23
commutatorC23×C23C22×C4
FrattiniC2×D4C22×D23
FittingC23×#352C2
socleC2×C23C2×#99
prime radicalC2×D4×D23C22
cosocleC2×D4×C23C23
hypercenter#352D23
hypocenterC23C2×#352
LCSG ▷ C23×C23 ▷ C22×C23 ▷ C2×C23 ▷ C23
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #352

Complements

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

NH ≅ G/Ntype
4. #13092208C2semidirect
5. #13092207C2semidirect
6. #13092206C2semidirect
7. C23×#352C2semidirect
9. C22×C4×D23C4semidirect
12. #10625100C4semidirect
16. #10625057C4semidirect
20. C22⋊C4×D23C4semidirect
30. C22×C4×C23C2×C4semidirect
33. C22×Dic23C2×C4semidirect
48. C22⋊C4×C23C2×C4semidirect
51. C22⋊Dic23C2×C4semidirect
127. #352D23direct (N × H)
135. D23#352direct (N × H)
155. C23C2×#352semidirect
172. C22×C4C4×D23semidirect
190. C22⋊C4C4×D23semidirect

Maximal subgroups

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

subgroupindex
#123×D232
#100×D232
#99×D232
#130922082
#130922072
#130922062
C23×#3522
C2×#35223