Finite Group Databasebeta

← D23×#360 #15675127

Subgroups — D23×#360 #15675127

Order 2944.

232 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 232 conjugacy classes of subgroups — too many to draw. Only its 67 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#135×D23C211yesmaximal, characteristic
2M32×D23C211yesmaximal, characteristic, complemented
3#106×D23C211yesmaximal, characteristic, complemented
4#13092228C211yesmaximal, characteristic, complemented
5#13092227C211yesmaximal, characteristic, complemented
6#13092226C211yesmaximal, characteristic, complemented
7C23×#360C211yesmaximal, characteristic, complemented
8#41×D23—21no—
9C23×#135C2211yescharacteristic
10#10625121—21no—
11#10625090C411yescharacteristic, complemented
12Q16×D23C411yescharacteristic, complemented
13#10625087—21no—
14SD16×D23—21no—
15#10625083C411yescharacteristic, complemented
16D8×D23C411yescharacteristic, complemented
17#10625075C2211yescharacteristic
18C2×C8×D23C2211yescharacteristic
19C4⋊C4×D23—21no—
20C23×M32C2211yescharacteristic, complemented
21C23×#106C2211yescharacteristic, complemented
22#10624999C2211yescharacteristic, complemented
23#10624994C2211yescharacteristic, complemented
24#10624990—21no—
25#10624980—21no—
26C16×D23—21no—
27#41×C23—21no—
28#67292—21no—
29Q8×D23—21no—
30#67290—21no—
31#67290—41no—
32D4×D23—21no—
33D4×D23—41no—
34#67288—21no—
35#67288—41no—
36C2×C4×D23—41no—
37C2×C4×D23—41no—
38C2×C4×D23D411yescharacteristic
39Q16×C23C2×C411yescharacteristic, complemented
40SD16×C23—21no—
41D8×C23C2×C411yescharacteristic, complemented
42C2×C8×C23C2311yescharacteristic
43C4⋊C4×C23—21no—
44#67275—21no—
45#67274—21no—
46#67273—21no—
47#67272—21no—
48C4⋊Dic23—21no—
49Dic23⋊C4—41no—
50C2×C23⋊C8D411yescharacteristic
51Dic92C2×C411yescharacteristic, complemented
52D184C2×C411yescharacteristic, complemented
53#67263—21no—
54C8×D23C2×C411yescharacteristic
55C8×D23C2×C411yescharacteristic
56C8×D23D411yescharacteristic
57C8×D23D411yescharacteristic
58C16×C23—21no—
59C23⋊C16—21no—
60C22×D23—41no—
61C22×D23D811yescharacteristic
62Q8×C23—21no—
63D4×C23—21no—
64D4×C23—41no—
65C2×C4×C23—41no—
66C2×C4×C23—41no—
67C2×C4×C23C2×D411yescharacteristic
68C22⋊D23—41no—
69C22⋊D23—41no—
70C22⋊D23—41no—
71C2×Dic23—41no—
72C2×Dic23—41no—
73C2×Dic23D811yescharacteristic
74D92—21no—
75D92—41no—
76C4×D23—41no—
77C4×D23—41no—
78C4×D23—41no—
79C4×D23—82no—
80C4×D23C22⋊C411yescharacteristic
81C4×D23C22⋊C411yescharacteristic
82C4×D23SD1611yescharacteristic
83C4×D23SD1611yescharacteristic
84Dic46—21no—
85Dic46—41no—
86C8×C23C22×C411yescharacteristic
87C8×C23C2×D411yescharacteristic
88C23⋊C8C22⋊C411yescharacteristic
89C23⋊C8C22⋊C411yescharacteristic
90C2×#360—231nomaximal, Sylow
91C22×C23—41no—
92C22×C23C2×D811yescharacteristic
93C2×D23—22no—
94C2×D23—41no—
95C2×D23—82no—
96C2×D23#10211yescharacteristic
97C2×D23#10211yescharacteristic
98C4×C23—41no—
99C4×C23—81no—
100C4×C23C2×C22⋊C411yescharacteristic
101C4×C23C2×SD1611yescharacteristic
102Dic23—41no—
103Dic23—81no—
104Dic23#10211yescharacteristic
105Dic23#10211yescharacteristic
106C2×#135—231no—
107C2×M32—231no—
108C2×#106—231no—
109#360—231no—
110#360—231no—
111#360—231no—
112#360D2311yescharacteristic, direct product factor
113C2×C23—21no—
114C2×C23—81no—
115C2×C23C2×#10211yescharacteristic
116D23—21no—
117D23—81no—
118D23#36012yesdirect product factor
119C2×#41—461no—
120#135—231no—
121#135—231no—
122#135—231no—
123#135—461no—
124#135—461no—
125#135C2×D2311yescharacteristic
126C2×Q16—231no—
127C2×SD16—461no—
128C2×D8—231no—
129C22×C8—231no—
130C2×C4⋊C4—461no—
131M32—231no—
132M32—461no—
133M32C2×D2311yescharacteristic, complemented
134C2×C16—461no—
135#106—231no—
136#106—461no—
137#106C2×D2311yescharacteristic, complemented
138C23C2×#36011yescharacteristic, complemented, Sylow
139#41—21no—
140#41—461no—
141#41—461no—
142#41—461no—
143#41—921no—
144#41—921no—
145C2×Q8—461no—
146C2×D4—461no—
147C2×D4—921no—
148C22×C4—231no—
149C22×C4—921no—
150C22×C4—921no—
151Q16—231no—
152Q16—461no—
153Q16C4×D2311yescharacteristic, complemented
154SD16—21no—
155SD16—461no—
156SD16—461no—
157SD16—461no—
158D8—231no—
159D8—461no—
160D8C4×D2311yescharacteristic, complemented
161C2×C8—231no—
162C2×C8—231no—
163C2×C8—231no—
164C2×C8—231no—
165C2×C8—231no—
166C2×C8C22×D2311yescharacteristic
167C4⋊C4—21no—
168C4⋊C4—461no—
169C4⋊C4—921no—
170C16—21no—
171C16—461no—
172C23—231no—
173C23—921no—
174Q8—21no—
175Q8—461no—
176Q8—921no—
177D4—21no—
178D4—41no—
179D4—461no—
180D4—921no—
181D4—921no—
182D4—921no—
183D4—921no—
184C2×C4—41no—
185C2×C4—41no—
186C2×C4—231no—
187C2×C4—231no—
188C2×C4—231no—
189C2×C4—231no—
190C2×C4—231no—
191C2×C4—921no—
192C2×C4—921no—
193C2×C4—921no—
194C2×C4—921no—
195C2×C4—921no—
196C2×C4—1842no—
197C2×C4D4×D2311yescharacteristic
198C8—231no—
199C8—231no—
200C8C2×C4×D2311yescharacteristic
201C8D4×D2311yescharacteristic
202C22—41no—
203C22—231no—
204C22—231no—
205C22—462no—
206C22—921no—
207C22—1842no—
208C22D8×D2311yescharacteristic
209C4—41no—
210C4—81no—
211C4—231no—
212C4—231no—
213C4—921no—
214C4—1841no—
215C4C22⋊C4×D2311yescharacteristic
216C4SD16×D2311yescharacteristic
217C2—21no—
218C2—81no—
219C2—232no—
220C2—461no—
221C2—1841no—
222C2#102×D2311yescharacteristic
subgroupG/H
centerC2#102×D23
commutatorC8×C23C22×C4
FrattiniC2×C8C22×D23
FittingC23×#360C2
socleC2×C23C2×#102
prime radical#135×D23C2
cosocleC2×C8×C23C23
hypercenter#360D23
hypocenterC23C2×#360
LCSG ▷ C8×C23 ▷ C4×C23 ▷ C2×C23 ▷ C23
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ #360

Complements

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

NH ≅ G/Ntype
2. M32×D23C2semidirect
3. #106×D23C2semidirect
4. #13092228C2semidirect
5. #13092227C2semidirect
6. #13092226C2semidirect
7. C23×#360C2semidirect
11. #10625090C4semidirect
12. Q16×D23C4semidirect
15. #10625083C4semidirect
16. D8×D23C4semidirect
20. C23×M32C22semidirect
21. C23×#106C22semidirect
22. #10624999C22semidirect
23. #10624994C22semidirect
39. Q16×C23C2×C4semidirect
41. D8×C23C2×C4semidirect
51. Dic92C2×C4semidirect
52. D184C2×C4semidirect
112. #360D23direct (N × H)
118. D23#360direct (N × H)
133. M32C2×D23semidirect
137. #106C2×D23semidirect
138. C23C2×#360semidirect
153. Q16C4×D23semidirect
160. D8C4×D23semidirect

Maximal subgroups

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

subgroupindex
#135×D232
M32×D232
#106×D232
#130922282
#130922272
#130922262
C23×#3602
C2×#36023