Finite Group Databasebeta

← #37582983

Subgroups — #37582983

Order 24448.

572 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 572 conjugacy classes of subgroups — too many to draw. Only its 137 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
1C2×#20259354C212yesmaximal, complemented
2C2×#20259349C212yesmaximal, complemented
3#27133214C211yesmaximal, characteristic, complemented
4#124×D191C211yesmaximal, characteristic, complemented
5#27133055C211yesmaximal, characteristic, complemented
6#27133053C211yesmaximal, characteristic, complemented
7#27133052C211yesmaximal, characteristic, complemented
8#27133048C212yesmaximal, complemented
9#27133044C212yesmaximal, complemented
10#27132615C211yesmaximal, characteristic, complemented
11#497×C191C211yesmaximal, characteristic, complemented
12C2×#15690288C2212yes—
13C2×Q8×D191C2211yescharacteristic
14C2×D4×D191C2211yescharacteristic
15#20259354—28no—
16#20259349—28no—
17C2×#15690277C2212yescomplemented
18C2×#15690276C2212yescomplemented
19C2×#15690274C2212yes—
20C2×#15690274C2212yescomplemented
21C2×D1528C2212yes—
22C2×#15690272C2212yes—
23C2×#15690270C2212yes—
24#20259319C2211yescharacteristic, complemented
25#20259311C2211yescharacteristic
26#20259311C2212yescomplemented
27#20259310C2211yescharacteristic
28#20259309C2211yescharacteristic
29#20259308—22no—
30#20259297—22no—
31#20259284—22no—
32C22⋊C4×D191—22no—
33C42×D191C2211yescharacteristic
34#20259233C2211yescharacteristic, complemented
35C8×Dic191C2212yescomplemented
36C2×Q16×C191C2212yescomplemented
37C2×SD16×C191C2212yes—
38C4⋊Q8×C191C2211yescharacteristic, complemented
39#124×C191C2211yescharacteristic
40#97×C191C2211yescharacteristic, complemented
41C23×D191—21no—
42#15690288—24no—
43#15690288—24no—
44C2×Dic382—21no—
45C2×Dic382C2311yescharacteristic
46Q8×D191—24no—
47C2×D764—22no—
48C2×D764—22no—
49C2×D764C2312yes—
50C2×D764C2311yescharacteristic
51C2×C22⋊D191—21no—
52D4×D191—24no—
53C2×C4×D191—21no—
54C2×C4×D191—22no—
55C2×C4×D191D412yes—
56C2×C4×D191C2311yescharacteristic
57#15690277—24no—
58#15690276—24no—
59#15690274—24no—
60#15690274—24no—
61D1528—24no—
62#15690272—24no—
63#15690270D418yescomplemented
64C2×C191⋊C8C2312yes—
65C22⋊Dic191—22no—
66(C2×D191)⋊C4—22no—
67(C2×D191)⋊C4—24no—
68(C2×D191)⋊C4—22no—
69C4×Dic191D412yes—
70C4×Dic191C2311yescharacteristic
71C2×Q8×C191C2312yes—
72C2×Q8×C191C2311yescharacteristic
73C2×D4×C191C2311yescharacteristic
74Q16×C191—24no—
75SD16×C191—24no—
76C2×C8×C191C2312yes—
77C4⋊C4×C191—22no—
78C22⋊C4×C191—22no—
79C42×C191C2311yescharacteristic
80C22×D191—21no—
81C22×D191—22no—
82C22×D191—44no—
83C22×D191—42no—
84C22×D191C2×D411yescharacteristic
85Dic382—22no—
86Dic382—42no—
87D764—22no—
88D764—22no—
89D764—22no—
90D764—44no—
91D764—44no—
92C22⋊D191—42no—
93C2×Dic191—21no—
94C2×Dic191C2×D412yes—
95C2×Dic191C2×D411yescharacteristic
96C4×D191—42no—
97C4×D191—42no—
98C4×D191—44no—
99C4×D191C2×D414yes—
100C191⋊C8C2×D414yes—
101C23×C191—21no—
102Q8×C191—22no—
103Q8×C191—22no—
104Q8×C191—22no—
105D4×C191—22no—
106C2×C4×C191—21no—
107C2×C4×C191—22no—
108C2×C4×C191C2×D412yes—
109C2×C4×C191C2411yescharacteristic
110C8×C191C2×D414yes—
111C22×C191—41no—
112C22×C191—42no—
113C22×C191C22×D411yescharacteristic
114C2×D191—22no—
115C2×D191—42no—
116C2×D191—41no—
117C2×D191—44no—
118C2×D191—42no—
119C2×D191—84no—
120C2×D191#12712yes—
121C4×C191—22no—
122C4×C191—42no—
123C4×C191—41no—
124C4×C191C22×D412yes—
125Dic191—22no—
126Dic191—41no—
127Dic191#12712yes—
128C2×C191—81no—
129C2×C191C2×#12711yescharacteristic
130C2×C191#57512yes—
131D191—22no—
132D191—82no—
133D191—81no—
134C191#315311yescharacteristic, complemented, Sylow
135#3153—1911nomaximal, Sylow
136C2×#137—1912no—
137C2×AGL(1,ℤ/8)—1912no—
138#532—1911no—
139C2×#124—1911no—
140#497—1911no—
141#497D19111yescharacteristic, complemented
142#496—1911no—
143#496—1911no—
144#495—1912no—
145#495—1912no—
146C4×M16—1911no—
147C2×#41—1912no—
148C22×Q8—1911no—
149C22×D4—1911no—
150#137—3828no—
151AGL(1,ℤ/8)—3828no—
152C2×Q16—1912no—
153C2×Q16C2×D19112yescomplemented
154C2×SD16—1912no—
155C2×SD16—1912no—
156C2×SD16—1912no—
157C2×SD16C2×D19112yes—
158C2×D8—1912no—
159C2×D8—1912no—
160C2×M16—1912no—
161C4⋊Q8C2×D19111yescharacteristic, complemented
162#127—1911no—
163#124—3822no—
164#124—1911no—
165#124—1912no—
166#124—1911no—
167#124—1911no—
168#124C2×D19111yescharacteristic
169#121—3822no—
170C4×D4—3822no—
171C2×C22⋊C4—3822no—
172C2×C42—1911no—
173#97—1911no—
174#97C2×D19111yescharacteristic, complemented
175C4×C8—1912no—
176C24—3821no—
177#41—3824no—
178#41—3824no—
179C2×Q8—3821no—
180C2×Q8—3824no—
181C2×Q8—1911no—
182C2×Q8C22×D19111yescharacteristic
183C2×Q8C22×D19112yes—
184C2×D4—3822no—
185C2×D4—3824no—
186C2×D4—3821no—
187C2×D4—3822no—
188C2×D4—1911no—
189C2×D4—1912no—
190C2×D4C22×D19111yescharacteristic
191C22×C4—3821no—
192C22×C4—3822no—
193C22×C4—1911no—
194C22×C4—1912no—
195Q16—24no—
196Q16—3824no—
197SD16—24no—
198SD16—3824no—
199SD16—3824no—
200SD16—3824no—
201D8—3824no—
202D8—3824no—
203M16—1918no—
204C2×C8—1912no—
205C2×C8C22×D19112yes—
206C4⋊C4—22no—
207C22⋊C4—22no—
208C22⋊C4—3822no—
209C22⋊C4—3824no—
210C22⋊C4—3822no—
211C22⋊C4—3822no—
212C42—1912no—
213C42—1911no—
214C42C22×D19111yescharacteristic
215C23—21no—
216C23—3821no—
217C23—3822no—
218C23—1911no—
219C23—7644no—
220C23—7642no—
221Q8—22no—
222Q8—22no—
223Q8—22no—
224Q8—3822no—
225Q8—7642no—
226D4—22no—
227D4—3822no—
228D4—3822no—
229D4—3822no—
230D4—7644no—
231D4—7644no—
232D4—7642no—
233C2×C4—21no—
234C2×C4—22no—
235C2×C4—3821no—
236C2×C4—1912no—
237C2×C4—1911no—
238C2×C4—1914no—
239C2×C4—7642no—
240C2×C4—7644no—
241C2×C4—7642no—
242C2×C4D4×D19112yes—
243C2×C4C23×D19111yescharacteristic
244C8—1914no—
245C8D4×D19114yes—
246C22—41no—
247C22—42no—
248C22—3822no—
249C22—1912no—
250C22—15284no—
251C22—7642no—
252C22—7641no—
253C22—7644no—
254C22—7642no—
255C22C2×D4×D19111yescharacteristic
256C4—22no—
257C4—42no—
258C4—41no—
259C4—3822no—
260C4—1912no—
261C4—7641no—
262C4C2×D4×D19112yes—
263C2—81no—
264C2—3822no—
265C2—15281no—
266C2—15282no—
267C2#127×D19111yescharacteristic
268C2#2713346212yes—
subgroupG/H
centerC22C2×D4×D191
commutator, cosocleC2×C4×C191C24
FrattiniC2×C4C23×D191
Fitting#497×C191C2
socleC22×C191C22×D4
hypercenter#497D191
hypocenterC191#3153
LCSG ▷ C2×C4×C191 ▷ C2×C191 ▷ C191
UCSC1 ◁ C22 ◁ C42 ◁ #497

Complements

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

NH ≅ G/Ntype
1. C2×#20259354C2semidirect
2. C2×#20259349C2semidirect
3. #27133214C2semidirect
4. #124×D191C2semidirect
5. #27133055C2semidirect
6. #27133053C2semidirect
7. #27133052C2semidirect
8. #27133048C2semidirect
9. #27133044C2semidirect
10. #27132615C2semidirect
11. #497×C191C2semidirect
17. C2×#15690277C22semidirect
18. C2×#15690276C22semidirect
20. C2×#15690274C22semidirect
24. #20259319C22semidirect
26. #20259311C22semidirect
34. #20259233C22semidirect
35. C8×Dic191C22semidirect
36. C2×Q16×C191C22semidirect
38. C4⋊Q8×C191C22semidirect
40. #97×C191C22semidirect
63. #15690270D4semidirect
134. C191#3153semidirect
141. #497D191semidirect
153. C2×Q16C2×D191semidirect
161. C4⋊Q8C2×D191semidirect
174. #97C2×D191semidirect

Maximal subgroups

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

subgroupindex
C2×#202593542
C2×#202593492
#271332142
#124×D1912
#271330552
#271330532
#271330522
#271330482
#271330442
#271326152
#497×C1912
#3153191