Finite Group Databasebeta

← #37581869

Subgroups — #37581869

Order 24448.

724 conjugacy classes of subgroups; only the 161 normal subgroups are shown.

normal subgroups only

This group has 724 conjugacy classes of subgroups — too many to draw. Only its 161 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#27133461C212yesmaximal, complemented
2#27133209C211yesmaximal, characteristic, complemented
3#27132846C212yesmaximal, complemented
4#27132841C212yesmaximal, complemented
5#27132836C212yesmaximal, complemented
6#27132831C212yesmaximal, complemented
7C2×#20259249C212yesmaximal, complemented
8#27132647C211yesmaximal, characteristic, complemented
9#585×C191C211yesmaximal, characteristic, complemented
10C2×#15690287C2211yescharacteristic, complemented
11#20259353—22no—
12#20259352C2214yescomplemented
13#20259348—22no—
14#20259347C2214yescomplemented
15#20259342—24no—
16C2×#15690271C2212yescomplemented
17#20259320C2211yescharacteristic, complemented
18#20259318C2211yescharacteristic, complemented
19#20259311C2211yescharacteristic, complemented
20#20259309C2211yescharacteristic, complemented
21#20259294—22no—
22C4×C22⋊D191—22no—
23C2×C4×Dic191C2211yescharacteristic, complemented
24#20259249C2214yescomplemented
25#20259249C2214yescomplemented
26#20259241C2212yescomplemented
27#20259239C2212yescomplemented
282−1+4×C191—22no—
292−1+4×C191C2212yes—
30H(5,𝔽2)×C191—22no—
31H(5,𝔽2)×C191—22no—
32H(5,𝔽2)×C191C2212yes—
33C2×#41×C191—24no—
34C2×#41×C191—22no—
35C2×#41×C191C2212yes—
36C2×#41×C191C2211yescharacteristic
37#15690287—22no—
38#15690287—22no—
39C2×Dic382C2311yescharacteristic
40C2×D764C2311yescharacteristic
41C2×C22⋊D191—21no—
42C2×C22⋊D191—22no—
43C22×Dic191—21no—
44C2×C4×D191—21no—
45#15690277—24no—
46#15690275—24no—
47#15690274—24no—
48#15690272—24no—
49#15690271—22no—
50#15690271C2314yes—
51C2×C191⋊C8—22no—
52Dic191⋊C4—22no—
53C22⋊Dic191—22no—
54(C2×D191)⋊C4—22no—
55C4×Dic191—21no—
56C4×Dic191C2311yescharacteristic
57C4×Dic191C2311yescharacteristic
58#41×C191—24no—
59#41×C191—28no—
60#41×C191—24no—
61#41×C191—24no—
62#41×C191—24no—
63#41×C191—22no—
64#41×C191—22no—
65#41×C191—24no—
66#41×C191—22no—
67#41×C191—22no—
68#41×C191D414yescomplemented
69#41×C191C2314yes—
70C2×Q8×C191—22no—
71C2×Q8×C191—24no—
72C2×Q8×C191D412yescomplemented
73C2×Q8×C191C2311yescharacteristic
74C2×D4×C191—28no—
75C2×D4×C191—22no—
76C2×D4×C191—24no—
77C2×D4×C191—22no—
78C2×D4×C191—24no—
79C2×D4×C191D412yes—
80C2×D4×C191D412yescomplemented
81C2×D4×C191C2311yescharacteristic
82C22×C4×C191—24no—
83C22×C4×C191—22no—
84C22×C4×C191D412yes—
85C22×C4×C191C2311yescharacteristic
86C22×D191—21no—
87Dic382—22no—
88D764—22no—
89C22⋊D191—44no—
90C22⋊D191—42no—
91C2×Dic191—21no—
92C2×Dic191—21no—
93C2×Dic191—21no—
94C2×Dic191—42no—
95C4×D191—41no—
96C4×D191—41no—
97C191⋊C8—24no—
98C23×C191—24no—
99C23×C191—22no—
100C23×C191C2×D412yes—
101C23×C191C2×D411yescharacteristic
102Q8×C191—24no—
103Q8×C191—22no—
104Q8×C191—22no—
105Q8×C191C2×D414yes—
106D4×C191—28no—
107D4×C191—24no—
108D4×C191—24no—
109D4×C191—24no—
110D4×C191—24no—
111D4×C191—22no—
112D4×C191—22no—
113D4×C191C2×D414yes—
114C2×C4×C191—24no—
115C2×C4×C191—28no—
116C2×C4×C191—22no—
117C2×C4×C191—24no—
118C2×C4×C191—22no—
119C2×C4×C191—22no—
120C2×C4×C191—24no—
121C2×C4×C191C2×D412yes—
122C2×C4×C191C2×D411yescharacteristic
123C2×C4×C191C2×D412yes—
124C2×C4×C191C2×D412yes—
125C2×C4×C191C2411yescharacteristic
126C22×C191—22no—
127C22×C191—24no—
128C22×C191—41no—
129C22×C191—42no—
130C22×C191—82no—
131C22×C191—84no—
132C22×C191#12012yes—
133C22×C191C22×D411yescharacteristic
134C2×D191—42no—
135C2×D191—41no—
136C4×C191—24no—
137C4×C191—22no—
138C4×C191#12011yescharacteristic
139C4×C191#12011yescharacteristic
140C4×C191C22×D412yes—
141Dic191—22no—
142Dic191—41no—
143Dic191—41no—
144C2×C191—21no—
145C2×C191—22no—
146C2×C191—42no—
147C2×C191—44no—
148C2×C191C2×#12011yescharacteristic
149D191—81no—
150C191#301811yescharacteristic, complemented, Sylow
151#3018—1911nomaximal, Sylow
152#585D19111yescharacteristic, complemented
153#575—1912no—
154#532—1911no—
155#456—1912no—
156#455—1912no—
157#454—1912no—
158#453—1912no—
159C2×#104—1912no—
160#413—1911no—
1612−1+4—22no—
1622−1+4C2×D19112yes—
163H(5,𝔽2)—22no—
164H(5,𝔽2)—22no—
165H(5,𝔽2)C2×D19112yes—
166C2×#41—24no—
167C2×#41—22no—
168C2×#41—1911no—
169C2×#41C2×D19112yes—
170C2×#41C2×D19111yescharacteristic
171#137—3822no—
172#137—1914no—
173AGL(1,ℤ/8)—3822no—
174AGL(1,ℤ/8)—1914no—
175#135—3824no—
176C2×M16—1912no—
177C4⋊Q8—1911no—
178#127—1911no—
179#124—1911no—
180#124—1911no—
181#121—3822no—
182C4×D4—3822no—
183C2×C42—1911no—
184#104—1914no—
185#104—1914no—
186#101—1912no—
187#100—1912no—
188#41—24no—
189#41—28no—
190#41—24no—
191#41—24no—
192#41—24no—
193#41—22no—
194#41—22no—
195#41—24no—
196#41—22no—
197#41—22no—
198#41—3822no—
199#41—3822no—
200#41C22⋊D19114yescomplemented
201#41C22×D19114yes—
202C2×Q8—22no—
203C2×Q8—24no—
204C2×Q8—1911no—
205C2×Q8C22⋊D19112yescomplemented
206C2×Q8C22×D19111yescharacteristic
207C2×D4—28no—
208C2×D4—22no—
209C2×D4—24no—
210C2×D4—22no—
211C2×D4—24no—
212C2×D4—3822no—
213C2×D4—3821no—
214C2×D4—1911no—
215C2×D4C22⋊D19112yes—
216C2×D4C22⋊D19112yescomplemented
217C2×D4C22×D19111yescharacteristic
218C22×C4—24no—
219C22×C4—22no—
220C22×C4—3821no—
221C22×C4—3821no—
222C22×C4C22⋊D19112yes—
223C22×C4C22×D19111yescharacteristic
224Q16—3824no—
225SD16—3824no—
226SD16—3824no—
227D8—3824no—
228M16—3822no—
229M16—1914no—
230C2×C8—3822no—
231C4⋊C4—3822no—
232C22⋊C4—3822no—
233C22⋊C4—3822no—
234C42—3821no—
235C42—1911no—
236C42—1911no—
237C23—24no—
238C23—22no—
239C23—3821no—
240C23C2×C22⋊D19112yes—
241C23C2×C22⋊D19111yescharacteristic
242Q8—24no—
243Q8—22no—
244Q8—22no—
245Q8—3822no—
246Q8C2×C22⋊D19114yes—
247D4—28no—
248D4—24no—
249D4—24no—
250D4—24no—
251D4—24no—
252D4—22no—
253D4—22no—
254D4—3822no—
255D4—7644no—
256D4—7642no—
257D4C2×C22⋊D19114yes—
258C2×C4—24no—
259C2×C4—28no—
260C2×C4—22no—
261C2×C4—24no—
262C2×C4—22no—
263C2×C4—24no—
264C2×C4—22no—
265C2×C4—3821no—
266C2×C4—3821no—
267C2×C4—3821no—
268C2×C4—7642no—
269C2×C4—7641no—
270C2×C4—7641no—
271C2×C4C2×C22⋊D19112yes—
272C2×C4C2×C22⋊D19112yes—
273C2×C4C2×C22⋊D19112yes—
274C2×C4C2×C22⋊D19111yescharacteristic
275C2×C4C23×D19111yescharacteristic
276C8—3824no—
277C22—22no—
278C22—24no—
279C22—41no—
280C22—42no—
281C22—82no—
282C22—84no—
283C22—7642no—
284C22—7641no—
285C22#2025929112yes—
286C22C22×C22⋊D19111yescharacteristic
287C4—24no—
288C4—22no—
289C4—3822no—
290C4—7641no—
291C4—7641no—
292C4#2025929111yescharacteristic
293C4#2025929111yescharacteristic
294C4C22×C22⋊D19112yes—
295C2—21no—
296C2—22no—
297C2—42no—
298C2—44no—
299C2—15281no—
300C2C2×#2025929111yescharacteristic
subgroupG/H
centerC4#20259291
commutator, cosocleC2×C4×C191C24
FrattiniC2×C4C23×D191
Fitting#585×C191C2
socleC2×C191C2×#120
hypercenter#585D191
hypocenterC191#3018
LCSG ▷ C2×C4×C191 ▷ C2×C191 ▷ C191
UCSC1 ◁ C4 ◁ C22×C4 ◁ #585

Complements

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

NH ≅ G/Ntype
1. #27133461C2semidirect
2. #27133209C2semidirect
3. #27132846C2semidirect
4. #27132841C2semidirect
5. #27132836C2semidirect
6. #27132831C2semidirect
7. C2×#20259249C2semidirect
8. #27132647C2semidirect
9. #585×C191C2semidirect
10. C2×#15690287C22semidirect
12. #20259352C22semidirect
14. #20259347C22semidirect
16. C2×#15690271C22semidirect
17. #20259320C22semidirect
18. #20259318C22semidirect
19. #20259311C22semidirect
20. #20259309C22semidirect
23. C2×C4×Dic191C22semidirect
24. #20259249C22semidirect
25. #20259249C22semidirect
26. #20259241C22semidirect
27. #20259239C22semidirect
68. #41×C191D4semidirect
72. C2×Q8×C191D4semidirect
80. C2×D4×C191D4semidirect
150. C191#3018semidirect
152. #585D191semidirect
200. #41C22⋊D191semidirect
205. C2×Q8C22⋊D191semidirect
216. C2×D4C22⋊D191semidirect

Maximal subgroups

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

subgroupindex
#271334612
#271332092
#271328462
#271328412
#271328362
#271328312
C2×#202592492
#271326472
#585×C1912
#3018191