Finite Group Databasebeta

← #37582058

Subgroups — #37582058

Order 24448.

504 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 504 conjugacy classes of subgroups — too many to draw. Only its 133 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×#20259350C211yesmaximal, characteristic, complemented
2C2×#20259342C211yesmaximal, characteristic, complemented
3#27133227C211yesmaximal, characteristic, complemented
4#27133144C211yesmaximal, characteristic, complemented
5#27132888C212yesmaximal, complemented
6#27132881C212yesmaximal, complemented
7#27132862C212yesmaximal, complemented
8#27132857C212yesmaximal, complemented
9#27132691C211yesmaximal, characteristic, complemented
10C2×#20259242C211yesmaximal, characteristic, complemented
11#417×C191C211yesmaximal, characteristic, complemented
12C2×#15690287C2211yescharacteristic
13C22×D764C2211yescharacteristic
14#20259350—24no—
15#20259342—24no—
16C2×#15690277C2211yescharacteristic, complemented
17C2×#15690276C2212yes—
18C2×#15690275C2211yescharacteristic, complemented
19C2×#15690274C2211yescharacteristic, complemented
20C2×D1528C2212yes—
21C2×#15690272C2211yescharacteristic, complemented
22C22×C191⋊C8C2211yescharacteristic
23#20259318C2212yes—
24#20259311C2212yes—
25#20259296—21no—
26#20259296—21no—
27#20259290—22no—
28#20259287C2212yescomplemented
29#20259284C2212yescomplemented
30C4×C22⋊D191—22no—
31#20259281C2211yescharacteristic
32#20259280—21no—
33C4⋊C4×D191—21no—
34Dic191⋊C8C2212yescomplemented
35Dic191⋊C8C2212yescomplemented
36#20259242C2212yes—
37#20259242C2212yes—
38C2×#41×C191C2211yescharacteristic
39C2×M16×C191C2211yescharacteristic
40C2×C4⋊C4×C191C2211yescharacteristic
41#103×C191C2212yescomplemented
42#102×C191C2212yescomplemented
43C23×D191—21no—
44#15690287—22no—
45#15690287D414yescomplemented
46C2×Dic382C2311yescharacteristic
47C2×D764—22no—
48C2×D764—21no—
49C2×D764—22no—
50C2×D764C2312yes—
51C2×D764C2311yescharacteristic
52C2×C22⋊D191—22no—
53C2×C22⋊D191—22no—
54C2×C22⋊D191—21no—
55C2×C4×D191—22no—
56C2×C4×D191—21no—
57#15690277—22no—
58#15690276—24no—
59#15690275—22no—
60#15690274—22no—
61D1528—24no—
62#15690272—22no—
63C2×C191⋊C8—22no—
64C2×C191⋊C8C2311yescharacteristic
65C2×C191⋊C8C2311yescharacteristic
66C4⋊Dic191D411yescharacteristic
67C4⋊Dic191D411yescharacteristic
68Dic191⋊C4—22no—
69C22⋊Dic191D412yes—
70(C2×D191)⋊C4—22no—
71(C2×D191)⋊C4—22no—
72(C2×D191)⋊C4—22no—
73C4×Dic191—22no—
74C4×Dic191C2312yes—
75#41×C191—22no—
76#41×C191—22no—
77C2×Q8×C191C2311yescharacteristic
78C2×D4×C191—21no—
79C2×D4×C191C2311yescharacteristic
80C22×C4×C191—21no—
81C22×C4×C191—21no—
82C22×C4×C191C2311yescharacteristic
83M16×C191—22no—
84C2×C8×C191C2312yes—
85C4⋊C4×C191—21no—
86C4⋊C4×C191C2312yes—
87C22×D191—21no—
88C22×D191—22no—
89C22×D191—42no—
90C22×D191—81no—
91C22×D191—81no—
92Dic382—21no—
93Dic382C2×D412yes—
94D764—22no—
95D764—21no—
96D764—22no—
97D764—42no—
98D764C2×D412yes—
99C22⋊D191—24no—
100C22⋊D191—82no—
101C22⋊D191—82no—
102C2×Dic191—21no—
103C2×Dic191C2×D412yes—
104C2×Dic191C2×D412yes—
105C4×D191—24no—
106C4×D191—82no—
107C191⋊C8—21no—
108C191⋊C8—21no—
109C23×C191—21no—
110C23×C191C2×D411yescharacteristic
111Q8×C191—21no—
112Q8×C191—21no—
113D4×C191—21no—
114D4×C191—21no—
115D4×C191—42no—
116C2×C4×C191—22no—
117C2×C4×C191—21no—
118C2×C4×C191—42no—
119C2×C4×C191—81no—
120C2×C4×C191C2×D411yescharacteristic
121C2×C4×C191C2×D412yes—
122C2×C4×C191#4112yes—
123C2×C4×C191C2411yescharacteristic
124C8×C191—22no—
125C22×C191—21no—
126C22×C191—21no—
127C22×C191—41no—
128C22×C191—41no—
129C22×C191—41no—
130C22×C191#12112yes—
131C22×C191C22×D411yescharacteristic
132C2×D191—22no—
133C2×D191—41no—
134C2×D191—41no—
135C2×D191—42no—
136C2×D191—42no—
137C2×D191—42no—
138C2×D191—84no—
139C4×C191—41no—
140C4×C191—42no—
141C4×C191#12112yes—
142C4×C191C22×D411yescharacteristic
143C4×C191C2×#4111yescharacteristic
144Dic191—22no—
145Dic191—22no—
146Dic191—22no—
147C2×C191—22no—
148C2×C191—81no—
149C2×C191C2×#12111yescharacteristic
150C2×C191#57611yescharacteristic
151C2×C191#57711yescharacteristic
152D191—42no—
153D191—82no—
154C191#304511yescharacteristic, complemented, Sylow
155#3045—1911nomaximal, Sylow
156C2×AGL(1,ℤ/8)—1911no—
157C2×#135—1911no—
158#535—1911no—
159#520—1911no—
160#464—1912no—
161#463—1912no—
162#460—1912no—
163#459—1912no—
164#424—1911no—
165#417D19111yescharacteristic, complemented
166C2×#102—1911no—
167C2×#41—1911no—
168C2×#41C2×D19111yescharacteristic
169C22×D4—1911no—
170AGL(1,ℤ/8)—3824no—
171#135—3824no—
172C2×Q16—1911no—
173C2×SD16—1911no—
174C2×SD16—1911no—
175C2×SD16—1912no—
176C2×D8—1911no—
177C2×D8—1912no—
178C2×M16C2×D19111yescharacteristic
179C22×C8—1911no—
180#127—1912no—
181#124—1912no—
182#121—3821no—
183#121—3821no—
184#120—3822no—
185C4×Q8—1912no—
186C4×D4—3822no—
187C4×D4—1912no—
188#117—3821no—
189#117—1911no—
190C2×C4⋊C4—3821no—
191C2×C4⋊C4C2×D19111yescharacteristic
192C4⋊C8—1912no—
193C4⋊C8—1912no—
194#103C2×D19112yescomplemented
195#102—1912no—
196#102—1912no—
197#102C2×D19112yescomplemented
198C24—3821no—
199#41—22no—
200#41—22no—
201#41—3822no—
202#41—1914no—
203C2×Q8—1911no—
204C2×Q8C22×D19111yescharacteristic
205C2×D4—21no—
206C2×D4—3821no—
207C2×D4—3822no—
208C2×D4—3822no—
209C2×D4—3822no—
210C2×D4—3821no—
211C2×D4—3822no—
212C2×D4—1912no—
213C2×D4—1911no—
214C2×D4C22×D19111yescharacteristic
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—3822no—
218C22×C4—3821no—
219C22×C4C22×D19111yescharacteristic
220Q16—3822no—
221SD16—3824no—
222SD16—3822no—
223SD16—3822no—
224D8—3824no—
225D8—3822no—
226M16—22no—
227C2×C8—3822no—
228C2×C8—1911no—
229C2×C8—1911no—
230C2×C8C22×D19112yes—
231C4⋊C4—21no—
232C4⋊C4—3822no—
233C4⋊C4—1911no—
234C4⋊C4—1911no—
235C4⋊C4C22×D19112yes—
236C22⋊C4—3822no—
237C22⋊C4—3822no—
238C22⋊C4—3822no—
239C22⋊C4—1912no—
240C42—3822no—
241C42—1912no—
242C23—21no—
243C23—3821no—
244C23—3822no—
245C23—15281no—
246C23—15281no—
247C23—7642no—
248C23D4×D19111yescharacteristic
249Q8—21no—
250Q8—21no—
251Q8—3821no—
252Q8—1912no—
253D4—21no—
254D4—21no—
255D4—42no—
256D4—3824no—
257D4—3822no—
258D4—3821no—
259D4—3822no—
260D4—1912no—
261D4—15282no—
262D4—15282no—
263D4—7642no—
264C2×C4—22no—
265C2×C4—21no—
266C2×C4—42no—
267C2×C4—81no—
268C2×C4—3821no—
269C2×C4—3824no—
270C2×C4—1912no—
271C2×C4—1912no—
272C2×C4—15282no—
273C2×C4D4×D19112yes—
274C2×C4D4×D19111yescharacteristic
275C2×C4#1569028712yes—
276C2×C4C23×D19111yescharacteristic
277C8—22no—
278C8—3821no—
279C8—3821no—
280C22—21no—
281C22—21no—
282C22—41no—
283C22—41no—
284C22—41no—
285C22—3822no—
286C22—15284no—
287C22—7642no—
288C22—7642no—
289C22—7642no—
290C22—7641no—
291C22—7641no—
292C22#2025929512yes—
293C22C2×D4×D19111yescharacteristic
294C4—42no—
295C4—41no—
296C4—3822no—
297C4—3822no—
298C4—3822no—
299C4#2025929512yes—
300C4C2×D4×D19111yescharacteristic
301C4C2×#1569028711yescharacteristic
302C2—22no—
303C2—81no—
304C2—15282no—
305C2—7642no—
306C2C2×#2025929511yescharacteristic
307C2#2713346711yescharacteristic
308C2#2713347211yescharacteristic
subgroupG/H
centerC22C2×D4×D191
commutator, cosocleC2×C4×C191C24
FrattiniC2×C4C23×D191
Fitting#417×C191C2
socleC22×C191C22×D4
hypercenter#417D191
hypocenterC191#3045
LCSG ▷ C2×C4×C191 ▷ C2×C191 ▷ C191
UCSC1 ◁ C22 ◁ C22×C4 ◁ #417

Complements

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

NH ≅ G/Ntype
1. C2×#20259350C2semidirect
2. C2×#20259342C2semidirect
3. #27133227C2semidirect
4. #27133144C2semidirect
5. #27132888C2semidirect
6. #27132881C2semidirect
7. #27132862C2semidirect
8. #27132857C2semidirect
9. #27132691C2semidirect
10. C2×#20259242C2semidirect
11. #417×C191C2semidirect
16. C2×#15690277C22semidirect
18. C2×#15690275C22semidirect
19. C2×#15690274C22semidirect
21. C2×#15690272C22semidirect
28. #20259287C22semidirect
29. #20259284C22semidirect
34. Dic191⋊C8C22semidirect
35. Dic191⋊C8C22semidirect
41. #103×C191C22semidirect
42. #102×C191C22semidirect
45. #15690287D4semidirect
154. C191#3045semidirect
165. #417D191semidirect
194. #103C2×D191semidirect
197. #102C2×D191semidirect

Maximal subgroups

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

subgroupindex
C2×#202593502
C2×#202593422
#271332272
#271331442
#271328882
#271328812
#271328622
#271328572
#271326912
C2×#202592422
#417×C1912
#3045191