Finite Group Databasebeta

← #37583003

Subgroups — #37583003

Order 24448.

596 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 596 conjugacy classes of subgroups — too many to draw. Only its 149 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×#20259348C211yesmaximal, characteristic, complemented
2C2×#20259347C211yesmaximal, characteristic, complemented
3C2×#20259342C211yesmaximal, characteristic, complemented
4C22×#15690272C211yesmaximal, characteristic, complemented
5#27133232C211yesmaximal, characteristic, complemented
6#27133053C212yesmaximal, complemented
7#27133050C212yesmaximal, complemented
8#27133045C212yesmaximal, complemented
9#27133039C212yesmaximal, complemented
10#27132619C211yesmaximal, characteristic, complemented
11#535×C191C211yesmaximal, characteristic, complemented
12C2×#15690287C2211yescharacteristic
13C22×D764C2211yescharacteristic
14#20259348—24no—
15#20259347—24no—
16#20259342—24no—
17C2×#15690277C2211yescharacteristic, complemented
18C2×#15690275C2211yescharacteristic, complemented
19C2×#15690275C2212yescomplemented
20C2×#15690274C2211yescharacteristic, complemented
21C2×#15690274C2212yescomplemented
22C2×#15690272—24no—
23C2×#15690272C2212yes—
24C2×#15690272C2212yes—
25C2×#15690272C2211yescharacteristic, complemented
26C2×#15690272C2212yescomplemented
27C2×#15690272C2212yescomplemented
28C2×#15690271C2211yescharacteristic
29C22×C191⋊C8C2211yescharacteristic
30#20259319C2212yescomplemented
31#20259310C2212yescomplemented
32#20259297—22no—
33#20259292—22no—
34#20259233C2212yescomplemented
35C4×C191⋊C8C2212yescomplemented
36C2×#41×C191C2211yescharacteristic
37C22×D4×C191C2211yescharacteristic
38#127×C191C2212yescomplemented
39#124×C191C2212yescomplemented
40#121×C191—22no—
41#120×C191—22no—
42#117×C191C2211yescharacteristic
43C23×D191—21no—
44#15690287—22no—
45#15690287—22no—
46C2×Dic382C2311yescharacteristic
47C2×D764—22no—
48C2×D764—22no—
49C2×D764—21no—
50C2×D764—24no—
51C2×D764C2312yes—
52C2×D764C2311yescharacteristic
53C2×C22⋊D191—21no—
54C2×C4×D191—21no—
55#15690277—22no—
56#15690275—22no—
57#15690275—24no—
58#15690274—22no—
59#15690274—24no—
60#15690272—24no—
61#15690272—24no—
62#15690272—22no—
63#15690272—24no—
64#15690272—24no—
65#15690271D414yescomplemented
66C2×C191⋊C8D414yescomplemented
67C2×C191⋊C8C2311yescharacteristic
68C2×C191⋊C8C2311yescharacteristic
69C2×C191⋊C8C2312yes—
70(C2×D191)⋊C4—24no—
71C24×C191—21no—
72#41×C191—22no—
73#41×C191—22no—
74C2×Q8×C191C2311yescharacteristic
75C2×D4×C191—21no—
76C2×D4×C191—22no—
77C2×D4×C191—22no—
78C2×D4×C191—21no—
79C2×D4×C191—24no—
80C2×D4×C191C2312yes—
81C2×D4×C191C2311yescharacteristic
82C22×C4×C191—21no—
83C22×C4×C191C2311yescharacteristic
84C4⋊C4×C191D412yes—
85C22⋊C4×C191—24no—
86C22⋊C4×C191D412yes—
87C42×C191C2312yes—
88C22×D191—22no—
89C22×D191—21no—
90C22×D191—42no—
91C22×D191—82no—
92Dic382—21no—
93Dic382—21no—
94D764—22no—
95D764—21no—
96D764—22no—
97D764—21no—
98D764—42no—
99D764—84no—
100C22⋊D191—42no—
101C2×Dic191—21no—
102C4×D191—42no—
103C191⋊C8C2×D412yes—
104C191⋊C8C2×D412yes—
105C191⋊C8C2×D414yes—
106C23×C191—22no—
107C23×C191—21no—
108C23×C191—42no—
109C23×C191—82no—
110C23×C191C2×D411yescharacteristic
111Q8×C191—21no—
112Q8×C191—21no—
113D4×C191—22no—
114D4×C191—22no—
115D4×C191—21no—
116D4×C191—21no—
117D4×C191—42no—
118D4×C191—42no—
119D4×C191—84no—
120C2×C4×C191—21no—
121C2×C4×C191—42no—
122C2×C4×C191C2×D412yes—
123C2×C4×C191C2×D412yes—
124C2×C4×C191C2×D414yes—
125C2×C4×C191C2×D411yescharacteristic
126C2×C4×C191C2411yescharacteristic
127C22×C191—22no—
128C22×C191—41no—
129C22×C191—42no—
130C22×C191—42no—
131C22×C191—44no—
132C22×C191—84no—
133C22×C191#12712yes—
134C22×C191C22×D411yescharacteristic
135C2×D191—41no—
136C2×D191—42no—
137C2×D191—42no—
138C2×D191—44no—
139C2×D191—84no—
140C4×C191—24no—
141C4×C191—41no—
142C4×C191#12712yes—
143C4×C191C22×D411yescharacteristic
144C4×C191C22×D411yescharacteristic
145Dic191—41no—
146C2×C191—22no—
147C2×C191—81no—
148C2×C191—82no—
149C2×C191C2×#12711yescharacteristic
150C2×C191#57612yes—
151D191—81no—
152D191—82no—
153C191#315511yescharacteristic, complemented, Sylow
154#3155—1911nomaximal, Sylow
155C2×AGL(1,ℤ/8)—1911no—
156C2×AGL(1,ℤ/8)—1911no—
157C2×#135—1911no—
158C22×D8—1911no—
159#535—1911no—
160#535D19111yescharacteristic, complemented
161#496—1912no—
162#496—1912no—
163#495—1912no—
164#493—1912no—
165#405—1911no—
166C2×#41—1911no—
167C2×#41C2×D19111yescharacteristic
168C22×D4—1911no—
169C22×D4C2×D19111yescharacteristic
170AGL(1,ℤ/8)—3824no—
171AGL(1,ℤ/8)—3824no—
172#135—3824no—
173C2×Q16—1911no—
174C2×SD16—1911no—
175C2×SD16—1912no—
176C2×SD16—1911no—
177C2×SD16—1912no—
178C2×D8—3824no—
179C2×D8—1912no—
180C2×D8—1912no—
181C2×D8—1911no—
182C2×D8—1912no—
183C2×D8—1912no—
184C2×M16—1911no—
185C22×C8—1911no—
186#127—1912no—
187#127C2×D19112yescomplemented
188#124—1912no—
189#124C2×D19112yescomplemented
190#121—22no—
191#121—3822no—
192#120—22no—
193#120—3822no—
194#117C2×D19111yescharacteristic
195#97—1912no—
196C4×C8—1912no—
197C24—21no—
198C24—3821no—
199#41—22no—
200#41—22no—
201#41—3822no—
202#41—3822no—
203C2×Q8—1911no—
204C2×Q8C22×D19111yescharacteristic
205C2×D4—21no—
206C2×D4—22no—
207C2×D4—22no—
208C2×D4—21no—
209C2×D4—24no—
210C2×D4—3821no—
211C2×D4—3822no—
212C2×D4—3822no—
213C2×D4—3824no—
214C2×D4—3821no—
215C2×D4—1912no—
216C2×D4—1911no—
217C2×D4C22×D19112yes—
218C2×D4C22×D19111yescharacteristic
219C22×C4—21no—
220C22×C4—3821no—
221C22×C4C22×D19111yescharacteristic
222Q16—3822no—
223SD16—3822no—
224SD16—3824no—
225SD16—3822no—
226SD16—3824no—
227D8—3824no—
228D8—3824no—
229D8—3822no—
230D8—3824no—
231D8—3824no—
232M16—1914no—
233C2×C8—1914no—
234C2×C8—1911no—
235C2×C8—1912no—
236C2×C8—1911no—
237C4⋊C4C22⋊D19112yes—
238C22⋊C4—24no—
239C22⋊C4—3824no—
240C22⋊C4C22⋊D19112yes—
241C42C22×D19112yes—
242C23—22no—
243C23—21no—
244C23—42no—
245C23—82no—
246C23—3822no—
247C23—3821no—
248C23—15282no—
249C23—7642no—
250C23C2×C22⋊D19111yescharacteristic
251Q8—21no—
252Q8—21no—
253Q8—3821no—
254Q8—3821no—
255D4—22no—
256D4—21no—
257D4—22no—
258D4—21no—
259D4—42no—
260D4—42no—
261D4—84no—
262D4—3822no—
263D4—3821no—
264D4—3822no—
265D4—3821no—
266D4—15284no—
267D4—7642no—
268D4—7642no—
269C2×C4—21no—
270C2×C4—42no—
271C2×C4—3821no—
272C2×C4—7642no—
273C2×C4D4×D19112yes—
274C2×C4D4×D19112yes—
275C2×C4C2×C22⋊D19114yes—
276C2×C4C2×C22⋊D19111yescharacteristic
277C2×C4C23×D19111yescharacteristic
278C8—1912no—
279C8—1912no—
280C8—1914no—
281C22—22no—
282C22—41no—
283C22—42no—
284C22—42no—
285C22—44no—
286C22—84no—
287C22—15284no—
288C22—7641no—
289C22—7642no—
290C22—7642no—
291C22—7644no—
292C22#2025931812yes—
293C22C22×C22⋊D19111yescharacteristic
294C4—24no—
295C4—41no—
296C4—7641no—
297C4#2025931812yes—
298C4C2×D4×D19111yescharacteristic
299C4C2×D4×D19111yescharacteristic
300C2—22no—
301C2—82no—
302C2—81no—
303C2—15282no—
304C2—15281no—
305C2C2×#2025931811yescharacteristic
306C2#2713346612yes—
subgroupG/H
centerC22C22×C22⋊D191
commutator, cosocleC2×C4×C191C24
FrattiniC2×C4C23×D191
Fitting#535×C191C2
socleC22×C191C22×D4
hypercenter#535D191
hypocenterC191#3155
LCSG ▷ C2×C4×C191 ▷ C2×C191 ▷ C191
UCSC1 ◁ C22 ◁ #117 ◁ #535

Complements

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

NH ≅ G/Ntype
1. C2×#20259348C2semidirect
2. C2×#20259347C2semidirect
3. C2×#20259342C2semidirect
4. C22×#15690272C2semidirect
5. #27133232C2semidirect
6. #27133053C2semidirect
7. #27133050C2semidirect
8. #27133045C2semidirect
9. #27133039C2semidirect
10. #27132619C2semidirect
11. #535×C191C2semidirect
17. C2×#15690277C22semidirect
18. C2×#15690275C22semidirect
19. C2×#15690275C22semidirect
20. C2×#15690274C22semidirect
21. C2×#15690274C22semidirect
25. C2×#15690272C22semidirect
26. C2×#15690272C22semidirect
27. C2×#15690272C22semidirect
30. #20259319C22semidirect
31. #20259310C22semidirect
34. #20259233C22semidirect
35. C4×C191⋊C8C22semidirect
38. #127×C191C22semidirect
39. #124×C191C22semidirect
65. #15690271D4semidirect
66. C2×C191⋊C8D4semidirect
153. C191#3155semidirect
160. #535D191semidirect
187. #127C2×D191semidirect
189. #124C2×D191semidirect

Maximal subgroups

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

subgroupindex
C2×#202593482
C2×#202593472
C2×#202593422
C22×#156902722
#271332322
#271330532
#271330502
#271330452
#271330392
#271326192
#535×C1912
#3155191