Finite Group Databasebeta

← #37583002

Subgroups — #37583002

Order 24448.

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

normal subgroups only

This group has 596 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×#20259349C212yesmaximal, complemented
2C2×#20259345C211yesmaximal, characteristic, complemented
3C2×D8×D191C211yesmaximal, characteristic, complemented
4#27133231C212yesmaximal, complemented
5#27133053C212yesmaximal, complemented
6#27133049C212yesmaximal, complemented
7#27133048C211yesmaximal, characteristic, complemented
8#27133040C211yesmaximal, characteristic, complemented
9#27133038C211yesmaximal, characteristic, complemented
10#27132617C211yesmaximal, characteristic, complemented
11#495×C191C211yesmaximal, characteristic, complemented
12C2×#15690288C2212yes—
13C2×D4×D191C2212yes—
14#20259349—28no—
15#20259345—24no—
16C2×#15690274C2212yescomplemented
17C2×#15690274C2212yescomplemented
18C2×D1528C2211yescharacteristic
19C2×D1528C2211yescharacteristic, complemented
20C2×D1528C2212yescomplemented
21C2×#15690272C2212yes—
22C2×#15690272C2212yescomplemented
23D8×D191—24no—
24C2×#15690270C2211yescharacteristic
25C2×C8×D191C2211yescharacteristic
26#20259319C2212yescomplemented
27#20259318C2212yescomplemented
28#20259311C2212yescomplemented
29#20259295—24no—
30#20259292—24no—
31#20259279C2211yescharacteristic
32#20259233C2211yescharacteristic, complemented
33#20259232C2211yescharacteristic, complemented
34C8×Dic191C2211yescharacteristic, complemented
35C2×Q16×C191C2211yescharacteristic, complemented
36C2×SD16×C191C2212yescomplemented
37C2×D8×C191C2211yescharacteristic
38#124×C191C2212yescomplemented
39C4×C8×C191C2211yescharacteristic, complemented
40C23×D191—22no—
41#15690288—24no—
42#15690288—24no—
43C2×D764—22no—
44C2×D764—24no—
45C2×D764C2312yes—
46C2×D764C2312yes—
47C2×C22⋊D191—22no—
48C2×C22⋊D191—24no—
49D4×D191—24no—
50D4×D191—24no—
51C2×C4×D191—22no—
52C2×C4×D191C2311yescharacteristic
53#15690274—24no—
54#15690274—24no—
55D1528—22no—
56D1528—22no—
57D1528—24no—
58#15690272—24no—
59#15690272—24no—
60#15690270D414yescomplemented
61C2×C191⋊C8C2311yescharacteristic
62C2×C191⋊C8C2311yescharacteristic
63C8×D191D414yescomplemented
64Dic191⋊C4D412yes—
65(C2×D191)⋊C4—24no—
66(C2×D191)⋊C4D412yes—
67C4×Dic191C2311yescharacteristic
68C2×Q8×C191C2312yes—
69C2×D4×C191C2312yes—
70Q16×C191—22no—
71SD16×C191—24no—
72D8×C191—22no—
73C2×C8×C191C2311yescharacteristic
74C2×C8×C191C2311yescharacteristic
75C22⋊C4×C191—24no—
76C42×C191C2311yescharacteristic
77C22×D191—22no—
78C22×D191—22no—
79C22×D191—44no—
80C22×D191—84no—
81C22×D191C2×D411yescharacteristic
82D764—22no—
83D764—22no—
84D764—22no—
85D764—22no—
86D764—44no—
87D764—84no—
88C22⋊D191—44no—
89C22⋊D191—84no—
90C2×Dic191C2×D412yes—
91C2×Dic191C2×D411yescharacteristic
92C4×D191—44no—
93C4×D191C2×D412yes—
94C4×D191C2×D412yes—
95C191⋊C8C2×D412yes—
96C191⋊C8C2×D412yes—
97C23×C191—22no—
98Q8×C191—22no—
99Q8×C191—22no—
100D4×C191—22no—
101D4×C191—22no—
102C2×C4×C191—22no—
103C2×C4×C191C2×D412yes—
104C2×C4×C191C2411yescharacteristic
105C8×C191C2×D412yes—
106C8×C191C2×D412yes—
107C22×C191—42no—
108C22×C191—44no—
109C22×C191C22×D411yescharacteristic
110C2×D191—22no—
111C2×D191—42no—
112C2×D191—42no—
113C2×D191—44no—
114C2×D191—44no—
115C2×D191—88no—
116C2×D191#12712yes—
117C4×C191—22no—
118C4×C191—42no—
119C4×C191C22×D411yescharacteristic
120C4×C191C22×D411yescharacteristic
121Dic191—22no—
122Dic191#12712yes—
123C2×C191—82no—
124C2×C191C2×#12711yescharacteristic
125C2×C191#57612yes—
126D191—22no—
127D191—82no—
128D191—82no—
129C191#315511yescharacteristic, complemented, Sylow
130#3155—1911nomaximal, Sylow
131C2×AGL(1,ℤ/8)—1912no—
132C2×#135—1911no—
133C22×D8—1911no—
134#535—1912no—
135#496—1912no—
136#496—1912no—
137#495—1911no—
138#495D19111yescharacteristic, complemented
139#493—1911no—
140#493—1911no—
141#405—1911no—
142C2×#41—1912no—
143C22×D4—1912no—
144AGL(1,ℤ/8)—3828no—
145#135—3824no—
146C2×Q16C2×D19111yescharacteristic, complemented
147C2×SD16—1912no—
148C2×SD16—1912no—
149C2×SD16C2×D19112yescomplemented
150C2×D8—3824no—
151C2×D8—1912no—
152C2×D8—1912no—
153C2×D8—1911no—
154C2×D8—1911no—
155C2×D8—1912no—
156C2×D8C2×D19111yescharacteristic
157C2×M16—1911no—
158C22×C8—1911no—
159#127—1912no—
160#127—1912no—
161#124—1912no—
162#124C2×D19112yescomplemented
163#121—3824no—
164#120—3824no—
165#117—1911no—
166#97—1911no—
167#97—1911no—
168C4×C8—1911no—
169C4×C8C2×D19111yescharacteristic, complemented
170C24—3822no—
171#41—3824no—
172#41—3824no—
173C2×Q8C22×D19112yes—
174C2×D4—3822no—
175C2×D4—3824no—
176C2×D4—3824no—
177C2×D4—3824no—
178C2×D4—3822no—
179C2×D4—3824no—
180C2×D4—1912no—
181C2×D4—1912no—
182C2×D4C22×D19112yes—
183C22×C4—3822no—
184C22×C4—1911no—
185Q16—22no—
186SD16—24no—
187SD16—3824no—
188SD16—3824no—
189D8—22no—
190D8—3824no—
191D8—3824no—
192D8—3822no—
193D8—3822no—
194D8—3824no—
195M16—1914no—
196C2×C8—1911no—
197C2×C8—1911no—
198C2×C8—1914no—
199C2×C8C22×D19111yescharacteristic
200C2×C8C22×D19111yescharacteristic
201C4⋊C4—1912no—
202C22⋊C4—24no—
203C22⋊C4—3824no—
204C22⋊C4—1912no—
205C42—1911no—
206C42C22×D19111yescharacteristic
207C23—22no—
208C23—3822no—
209C23—3822no—
210C23—1911no—
211C23—15284no—
212C23—7644no—
213Q8—22no—
214Q8—22no—
215D4—22no—
216D4—22no—
217D4—3822no—
218D4—3822no—
219D4—3822no—
220D4—3822no—
221D4—15284no—
222D4—15284no—
223D4—7644no—
224D4—7644no—
225C2×C4—22no—
226C2×C4—1912no—
227C2×C4—1911no—
228C2×C4—1912no—
229C2×C4—1912no—
230C2×C4—7644no—
231C2×C4D4×D19112yes—
232C2×C4C23×D19111yescharacteristic
233C8—1912no—
234C8—1912no—
235C8D4×D19112yes—
236C8D4×D19112yes—
237C22—42no—
238C22—44no—
239C22—3822no—
240C22—1912no—
241C22—15288no—
242C22—7642no—
243C22—7642no—
244C22—7644no—
245C22—7644no—
246C22C2×D4×D19111yescharacteristic
247C4—22no—
248C4—42no—
249C4—3822no—
250C4—1912no—
251C4C2×D4×D19111yescharacteristic
252C4C2×D4×D19111yescharacteristic
253C2—82no—
254C2—3822no—
255C2—15282no—
256C2—15282no—
257C2#127×D19111yescharacteristic
258C2#2713346512yes—
subgroupG/H
centerC22C2×D4×D191
commutator, cosocleC2×C4×C191C24
FrattiniC2×C4C23×D191
Fitting#495×C191C2
socleC22×C191C22×D4
hypercenter#495D191
hypocenterC191#3155
LCSG ▷ C2×C4×C191 ▷ C2×C191 ▷ C191
UCSC1 ◁ C22 ◁ C42 ◁ #495

Complements

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

NH ≅ G/Ntype
1. C2×#20259349C2semidirect
2. C2×#20259345C2semidirect
3. C2×D8×D191C2semidirect
4. #27133231C2semidirect
5. #27133053C2semidirect
6. #27133049C2semidirect
7. #27133048C2semidirect
8. #27133040C2semidirect
9. #27133038C2semidirect
10. #27132617C2semidirect
11. #495×C191C2semidirect
16. C2×#15690274C22semidirect
17. C2×#15690274C22semidirect
19. C2×D1528C22semidirect
20. C2×D1528C22semidirect
22. C2×#15690272C22semidirect
26. #20259319C22semidirect
27. #20259318C22semidirect
28. #20259311C22semidirect
32. #20259233C22semidirect
33. #20259232C22semidirect
34. C8×Dic191C22semidirect
35. C2×Q16×C191C22semidirect
36. C2×SD16×C191C22semidirect
38. #124×C191C22semidirect
39. C4×C8×C191C22semidirect
60. #15690270D4semidirect
63. C8×D191D4semidirect
129. C191#3155semidirect
138. #495D191semidirect
146. C2×Q16C2×D191semidirect
149. C2×SD16C2×D191semidirect
162. #124C2×D191semidirect
169. C4×C8C2×D191semidirect

Maximal subgroups

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

subgroupindex
C2×#202593492
C2×#202593452
C2×D8×D1912
#271332312
#271330532
#271330492
#271330482
#271330402
#271330382
#271326172
#495×C1912
#3155191