Finite Group Databasebeta

← #15685175

Subgroups — #15685175

Order 2944.

448 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 448 conjugacy classes of subgroups — too many to draw. Only its 117 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#13093233C212yesmaximal, complemented
2#13093228C212yesmaximal, complemented
3#13092812C211yesmaximal, characteristic, complemented
4#13092690C212yesmaximal, complemented
5#13092679C212yesmaximal, complemented
6#13092601C212yesmaximal, complemented
7#13092589C211yesmaximal, characteristic, complemented
8#13092536C212yesmaximal, complemented
9C23×#455C211yesmaximal, characteristic, complemented
10C23×H(5,𝔽2)C2211yescharacteristic
11#10625161C2212yes—
12#10625158C2211yescharacteristic, complemented
13C23×#137C2212yes—
14C23×#124C2211yescharacteristic
15#10625122—22no—
16#10625122C2212yes—
17#10625121—22no—
18#10625121C2212yes—
19#10625119C2212yes—
20C2×#67275C2212yescomplemented
21C2×#67273—22no—
22C2×#67273C2212yes—
23#10625103—22no—
24#10625103C2212yes—
25#10625089—22no—
26Q16×D23—22no—
27#10625087—22no—
28#10625086—22no—
29#10625080C2212yescomplemented
30#10625078C2212yescomplemented
31#10625040C2211yescharacteristic, complemented
32C23×#104C2212yes—
33C23×#100C2211yescharacteristic
34#10625018C2211yescharacteristic, complemented
35#10625006C2212yescomplemented
36#10625004C2212yescomplemented
37#10624987C2212yescomplemented
38#10624985C2211yescharacteristic, complemented
39#41×C23—22no—
40#41×C23C2312yes—
41C2×Q8×C23C2311yescharacteristic
42C2×D4×C23—22no—
43C2×D4×C23—41no—
44C2×D4×C23C2311yescharacteristic
45#67292—22no—
46Q8×D23—22no—
47Q8×D23—22no—
48#67290—22no—
49#67290—42no—
50#67288—22no—
51#67288—41no—
52#67288C2312yes—
53C2×Dic46—22no—
54C2×Dic46C2312yes—
55Q16×C23—22no—
56SD16×C23—22no—
57M16×C23C2312yes—
58C22⋊C4×C23—41no—
59C42×C23C2311yescharacteristic
60#67275—22no—
61#67275—22no—
62#67275—22no—
63#67275D412yes—
64#67274D412yes—
65#67273—22no—
66#67273—22no—
67#67273—22no—
68#67273—22no—
69#67273D412yes—
70#67272—22no—
71#67272D412yes—
72C4⋊Dic23—41no—
73#67267—22no—
74#67267C2312yes—
75C2×C23⋊C8—22no—
76C2×C23⋊C8C2312yes—
77Dic92—22no—
78#67263—22no—
79#67262—22no—
80C8×D23—22no—
81C23×C23—21no—
82C23×C23—41no—
83Q8×C23—22no—
84Q8×C23C2×D412yes—
85D4×C23—22no—
86D4×C23—22no—
87D4×C23—42no—
88D4×C23C2×D412yes—
89C2×C4×C23—22no—
90C2×C4×C23—21no—
91C2×C4×C23—21no—
92C2×C4×C23—41no—
93C2×C4×C23C2411yescharacteristic
94C22⋊D23—22no—
95C22⋊D23—42no—
96C2×Dic23—22no—
97C2×Dic23—42no—
98D92—41no—
99D92C2×D412yes—
100C4×D23—22no—
101C4×D23—42no—
102C4×D23—42no—
103Dic46—22no—
104Dic46—22no—
105Dic46—41no—
106Dic46—42no—
107Dic46C2×D412yes—
108C8×C23—22no—
109C23⋊C8—22no—
110C23⋊C8C2×D412yes—
111C23⋊C8C2×D412yes—
112#3289—231nomaximal, Sylow
113C22×C23—22no—
114C22×C23—41no—
115C22×C23—81no—
116C22×C23—161no—
117C22×C23H(5,𝔽2)11yescharacteristic
118C2×D23—22no—
119C4×C23—22no—
120C4×C23—41no—
121C4×C23—41no—
122C4×C23C22×D412yes—
123Dic23—22no—
124Dic23—42no—
125#578—232no—
126#577—232no—
127#497—231no—
128#473—232no—
129#471—232no—
130#456—232no—
131#455D2311yescharacteristic, complemented
132#454—231no—
133#444—232no—
134C2×C23—21no—
135C2×C23—42no—
136C2×C23—81no—
137C2×C23D4211yescharacteristic
138D23—42no—
1392−1+4—232no—
1402−1+4—231no—
141H(5,𝔽2)C2×D2311yescharacteristic
142#137—232no—
143#137—232no—
144#137—462no—
145#137—462no—
146#137—462no—
147#137C2×D2312yes—
148AGL(1,ℤ/8)—232no—
149AGL(1,ℤ/8)—462no—
150#135—232no—
151#135—462no—
152#135—462no—
153C2×Q16—232no—
154C2×Q16—462no—
155C2×SD16—232no—
156C2×SD16—462no—
157#131—232no—
158#131—232no—
159C4⋊Q8—231no—
160#124C2×D2311yescharacteristic
161#108—232no—
162#104—232no—
163#104C2×D2312yes—
164#101—231no—
165#101—232no—
166#100C2×D2311yescharacteristic
167#97—231no—
168C23#328911yescharacteristic, complemented, Sylow
169#41—22no—
170#41—232no—
171#41—462no—
172#41—462no—
173#41—462no—
174#41—921no—
175#41—922no—
176#41C22×D2312yes—
177C2×Q8—232no—
178C2×Q8—462no—
179C2×Q8—462no—
180C2×Q8—462no—
181C2×Q8C22×D2311yescharacteristic
182C2×D4—22no—
183C2×D4—41no—
184C2×D4C22×D2311yescharacteristic
185Q16—22no—
186Q16—232no—
187Q16—462no—
188Q16—462no—
189Q16—462no—
190Q16—462no—
191SD16—22no—
192SD16—232no—
193SD16—232no—
194SD16—462no—
195SD16—462no—
196SD16—462no—
197SD16—462no—
198SD16—462no—
199D8—232no—
200D8—462no—
201M16—232no—
202M16—462no—
203M16—462no—
204M16C22×D2312yes—
205C2×C8—232no—
206C2×C8—462no—
207C2×C8—462no—
208C4⋊C4—921no—
209C22⋊C4—41no—
210C42C22×D2311yescharacteristic
211C23—21no—
212C23—41no—
213Q8—22no—
214Q8—232no—
215Q8—462no—
216Q8—462no—
217Q8—922no—
218Q8—921no—
219Q8D4×D2312yes—
220D4—22no—
221D4—22no—
222D4—42no—
223D4—232no—
224D4—462no—
225D4—921no—
226D4—922no—
227D4D4×D2312yes—
228C2×C4—22no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—41no—
232C2×C4—462no—
233C2×C4—462no—
234C2×C4—922no—
235C2×C4—922no—
236C2×C4—922no—
237C2×C4C23×D2311yescharacteristic
238C8—22no—
239C8—232no—
240C8—232no—
241C8—462no—
242C22—22no—
243C22—41no—
244C22—81no—
245C22—161no—
246C22—462no—
247C22#1062515511yescharacteristic
248C4—22no—
249C4—41no—
250C4—41no—
251C4—462no—
252C4—922no—
253C4C2×D4×D2312yes—
254C2—21no—
255C2—42no—
256C2—81no—
257C2—922no—
258C2#1309304911yescharacteristic
subgroupG/H
centerC2#13093049
commutator, cosocleC2×C4×C23C24
FrattiniC2×C4C23×D23
FittingC23×#455C2
socleC2×C23D42
hypercenter#455D23
hypocenterC23#3289
LCSG ▷ C2×C4×C23 ▷ C2×C23 ▷ C23
UCSC1 ◁ C2 ◁ C2×C4 ◁ #455

Complements

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

NH ≅ G/Ntype
1. #13093233C2semidirect
2. #13093228C2semidirect
3. #13092812C2semidirect
4. #13092690C2semidirect
5. #13092679C2semidirect
6. #13092601C2semidirect
7. #13092589C2semidirect
8. #13092536C2semidirect
9. C23×#455C2semidirect
12. #10625158C22semidirect
20. C2×#67275C22semidirect
29. #10625080C22semidirect
30. #10625078C22semidirect
31. #10625040C22semidirect
34. #10625018C22semidirect
35. #10625006C22semidirect
36. #10625004C22semidirect
37. #10624987C22semidirect
38. #10624985C22semidirect
131. #455D23semidirect
168. C23#3289semidirect

Maximal subgroups

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

subgroupindex
#130932332
#130932282
#130928122
#130926902
#130926792
#130926012
#130925892
#130925362
C23×#4552
#328923