Finite Group Databasebeta

← #15685174

Subgroups — #15685174

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#13093232C212yesmaximal, complemented
2#13093227C212yesmaximal, complemented
3#13092812C211yesmaximal, characteristic, complemented
4#13092691C212yesmaximal, complemented
5#13092678C212yesmaximal, complemented
6#13092601C212yesmaximal, complemented
7#13092594C211yesmaximal, characteristic, complemented
8#13092536C212yesmaximal, complemented
9C23×#454C211yesmaximal, characteristic, complemented
10C23×2−1+4C2211yescharacteristic, complemented
11#10625161C2212yes—
12#10625155C2211yescharacteristic
13C23×AGL(1,ℤ/8)C2212yes—
14C23×#124C2211yescharacteristic
15#10625122—22no—
16#10625122C2212yescomplemented
17#10625121—22no—
18#10625121C2212yes—
19#10625119C2212yescomplemented
20C2×#67275—22no—
21C2×#67275C2212yescomplemented
22#10625113—22no—
23#10625113C2212yes—
24C2×#67273C2212yes—
25#10625086—22no—
26SD16×D23—22no—
27#10625083—22no—
28#10625082—22no—
29#10625080C2212yes—
30#10625078C2212yes—
31#10625040C2211yescharacteristic, complemented
32C23×#104C2212yescomplemented
33C23×#101C2211yescharacteristic, complemented
34#10625015C2211yescharacteristic
35#10625006C2212yescomplemented
36#10625004C2212yescomplemented
37#10624987C2212yes—
38#10624985C2211yescharacteristic, complemented
39#41×C23—22no—
40#41×C23—41no—
41#41×C23C2312yes—
42C2×Q8×C23—22no—
43C2×Q8×C23C2311yescharacteristic
44C2×D4×C23C2311yescharacteristic
45C2×C22⋊D23—41no—
46Q8×D23—22no—
47#67290—22no—
48#67290—22no—
49#67290—42no—
50D4×D23—22no—
51#67288—22no—
52#67288C2312yes—
53C2×Dic46—22no—
54C2×Dic46C2312yes—
55SD16×C23—22no—
56D8×C23—22no—
57M16×C23C2312yes—
58C22⋊C4×C23—41no—
59C42×C23C2311yescharacteristic
60#67275—22no—
61#67275—22no—
62#67275—22no—
63#67275—22no—
64#67275D412yes—
65#67274—22no—
66#67274D412yes—
67#67273—22no—
68#67273—22no—
69#67273—22no—
70#67273D412yes—
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—
82C22×D23—41no—
83Q8×C23—22no—
84Q8×C23—41no—
85Q8×C23C2×D412yes—
86D4×C23—22no—
87D4×C23—22no—
88D4×C23—41no—
89D4×C23C2×D412yes—
90C2×C4×C23—22no—
91C2×C4×C23—21no—
92C2×C4×C23—21no—
93C2×C4×C23—42no—
94C2×C4×C23C2411yescharacteristic
95C22⋊D23—22no—
96C22⋊D23—42no—
97C22⋊D23—42no—
98C2×Dic23—22no—
99C2×Dic23—41no—
100C2×Dic23—42no—
101D92C2×D412yes—
102C4×D23—22no—
103C4×D23—42no—
104Dic46—22no—
105Dic46—22no—
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×C23H(5,𝔽2)11yescharacteristic
117C2×D23—22no—
118C2×D23—161no—
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#455—231no—
132#454D2311yescharacteristic, complemented
133#444—232no—
134C2×C23—21no—
135C2×C23—42no—
136C2×C23—81no—
137C2×C23D4211yescharacteristic
138D23—42no—
1392−1+4—232no—
1402−1+4C2×D2311yescharacteristic, complemented
141H(5,𝔽2)—231no—
142#137—232no—
143#137—232no—
144#137—232no—
145#137—462no—
146#137—462no—
147#137—462no—
148AGL(1,ℤ/8)—462no—
149AGL(1,ℤ/8)C2×D2312yes—
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×D2312yescomplemented
164#101—232no—
165#101C2×D2311yescharacteristic, complemented
166#100—231no—
167#97—231no—
168C23#328911yescharacteristic, complemented, Sylow
169#41—22no—
170#41—41no—
171#41—232no—
172#41—462no—
173#41—462no—
174#41—462no—
175#41—922no—
176#41C22×D2312yes—
177C2×Q8—22no—
178C2×Q8—232no—
179C2×Q8—462no—
180C2×Q8—462no—
181C2×Q8C22×D2311yescharacteristic
182C2×D4—462no—
183C2×D4—921no—
184C2×D4C22×D2311yescharacteristic
185Q16—232no—
186Q16—462no—
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—22no—
200D8—232no—
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—921no—
213Q8—22no—
214Q8—41no—
215Q8—232no—
216Q8—462no—
217Q8—462no—
218Q8—922no—
219Q8D4×D2312yes—
220D4—22no—
221D4—22no—
222D4—41no—
223D4—232no—
224D4—462no—
225D4—922no—
226D4—922no—
227D4D4×D2312yes—
228C2×C4—22no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—42no—
232C2×C4—462no—
233C2×C4—462no—
234C2×C4—922no—
235C2×C4—921no—
236C2×C4—922no—
237C2×C4C23×D2311yescharacteristic
238C8—22no—
239C8—232no—
240C8—232no—
241C8—462no—
242C22—22no—
243C22—41no—
244C22—81no—
245C22—462no—
246C22—3681no—
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×#454C2
socleC2×C23D42
hypercenter#454D23
hypocenterC23#3289
LCSG ▷ C2×C4×C23 ▷ C2×C23 ▷ C23
UCSC1 ◁ C2 ◁ C2×C4 ◁ #454

Complements

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

NH ≅ G/Ntype
1. #13093232C2semidirect
2. #13093227C2semidirect
3. #13092812C2semidirect
4. #13092691C2semidirect
5. #13092678C2semidirect
6. #13092601C2semidirect
7. #13092594C2semidirect
8. #13092536C2semidirect
9. C23×#454C2semidirect
10. C23×2−1+4C22semidirect
16. #10625122C22semidirect
19. #10625119C22semidirect
21. C2×#67275C22semidirect
31. #10625040C22semidirect
32. C23×#104C22semidirect
33. C23×#101C22semidirect
35. #10625006C22semidirect
36. #10625004C22semidirect
38. #10624985C22semidirect
132. #454D23semidirect
140. 2−1+4C2×D23semidirect
163. #104C2×D23semidirect
165. #101C2×D23semidirect
168. C23#3289semidirect

Maximal subgroups

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

subgroupindex
#130932322
#130932272
#130928122
#130926912
#130926782
#130926012
#130925942
#130925362
C23×#4542
#328923