Finite Group Databasebeta

← #489×D191 #37582846

Subgroups — #489×D191 #37582846

Order 24448.

436 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 436 conjugacy classes of subgroups — too many to draw. Only its 125 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
1C4⋊Q8×D191C211yesmaximal, characteristic, complemented
2#124×D191C211yesmaximal, characteristic
3#27133025C211yesmaximal, characteristic, complemented
4#27133024C212yesmaximal, complemented
5#27133023C211yesmaximal, characteristic, complemented
6#27133022C211yesmaximal, characteristic, complemented
7#27133021C212yesmaximal, complemented
8#103×D191C212yesmaximal, complemented
9#102×D191C212yesmaximal
10#97×D191C211yesmaximal, characteristic, complemented
11#489×C191C211yesmaximal, characteristic, complemented
12C2×Q8×D191—21no—
13C2×Q8×D191C2211yescharacteristic
14C2×D4×D191C2211yescharacteristic
15C2×C8×D191C2212yes—
16#20259322C2211yescharacteristic, complemented
17#20259321—21no—
18#20259321C2212yescomplemented
19#20259320—21no—
20#20259311C2211yescharacteristic
21#20259310C2211yescharacteristic
22#20259309C2211yescharacteristic
23#20259308—21no—
24#20259308—21no—
25C4⋊C4×D191—21no—
26C4⋊C4×D191C2212yes—
27C22⋊C4×D191—21no—
28C22⋊C4×D191—21no—
29C42×D191C2211yescharacteristic
30#20259247C2212yescomplemented
31#20259246C2212yescomplemented
32#20259245C2212yescomplemented
33#20259244C2212yes—
34#20259243C2212yes—
35#20259242C2212yes—
36#20259233C2211yescharacteristic, complemented
37#20259232C2212yescomplemented
38C4⋊Q8×C191C2211yescharacteristic, complemented
39#124×C191C2211yescharacteristic
40#103×C191C2212yescomplemented
41#102×C191C2212yes—
42#97×C191C2211yescharacteristic, complemented
43C23×D191—21no—
44C2×Dic382—22no—
45C2×Dic382—21no—
46C2×Dic382—21no—
47C2×Dic382C2311yescharacteristic
48Q8×D191—24no—
49Q8×D191—44no—
50C2×D764C2311yescharacteristic
51C2×C22⋊D191—21no—
52D4×D191—24no—
53C2×C4×D191—22no—
54C2×C4×D191—21no—
55C2×C4×D191D411yescharacteristic
56C2×C4×D191D411yescharacteristic
57C2×C4×D191C2311yescharacteristic
58C2×C191⋊C8C2312yes—
59C8×D191—24no—
60C4⋊Dic191—21no—
61C4⋊Dic191C2312yes—
62Dic191⋊C4—22no—
63Dic191⋊C4—22no—
64C22⋊Dic191—21no—
65C22⋊Dic191—21no—
66(C2×D191)⋊C4—21no—
67(C2×D191)⋊C4—21no—
68(C2×D191)⋊C4—21no—
69(C2×D191)⋊C4—21no—
70C4×Dic191D411yescharacteristic
71C4×Dic191D411yescharacteristic
72C4×Dic191C2311yescharacteristic
73C2×Q8×C191—21no—
74C2×Q8×C191C2311yescharacteristic
75C2×D4×C191C2311yescharacteristic
76C2×C8×C191C2312yes—
77C4⋊C4×C191—21no—
78C4⋊C4×C191C2312yes—
79C22⋊C4×C191—21no—
80C22⋊C4×C191—21no—
81C42×C191C2311yescharacteristic
82C22×D191—21no—
83C22×D191—42no—
84C22×D191—42no—
85C22×D191—42no—
86C22×D191C2×D411yescharacteristic
87Dic382—22no—
88Dic382—44no—
89Dic382—42no—
90Dic382—42no—
91D764—22no—
92C22⋊D191—42no—
93C2×Dic191—21no—
94C2×Dic191—22no—
95C2×Dic191C2×D411yescharacteristic
96C2×Dic191C2×D411yescharacteristic
97C2×Dic191C2×D411yescharacteristic
98C4×D191—22no—
99C4×D191—22no—
100C4×D191—42no—
101C4×D191—44no—
102C4×D191#4114yes—
103C191⋊C8—22no—
104C23×C191—21no—
105Q8×C191—22no—
106Q8×C191—42no—
107D4×C191—22no—
108C2×C4×C191—21no—
109C2×C4×C191—22no—
110C2×C4×C191C2×D411yescharacteristic
111C2×C4×C191C2×D411yescharacteristic
112C2×C4×C191C2411yescharacteristic
113C8×C191—22no—
114C22×C191—41no—
115C22×C191—41no—
116C22×C191—41no—
117C22×C191C22×D411yescharacteristic
118C2×D191—41no—
119C2×D191—41no—
120C2×D191—41no—
121C2×D191—84no—
122C2×D191#12412yes—
123C2×D191AGL(1,ℤ/8)12yes—
124C2×D191#13712yes—
125C4×C191—21no—
126C4×C191—21no—
127C4×C191—41no—
128C4×C191—42no—
129C4×C191C2×#4112yes—
130Dic191—21no—
131Dic191—21no—
132Dic191—41no—
133Dic191—42no—
134Dic191#12412yes—
135C2×C191—81no—
136C2×C191C2×#12411yescharacteristic
137C2×C191C2×AGL(1,ℤ/8)11yescharacteristic
138C2×C191C2×#13711yescharacteristic
139D191—81no—
140D191#48914yesdirect product factor
141C191C2×#48911yescharacteristic, complemented, Sylow
142C2×#489—1911nomaximal, Sylow
143C2×C4⋊Q8—1911no—
144C2×#124—1911no—
145#489—1911no—
146#489—1911no—
147#489—1912no—
148#489—1912no—
149#489—1911no—
150#489D19111yescharacteristic, direct product factor
151C2×#103—1912no—
152C2×#102—1912no—
153C2×#97—1911no—
154C22×Q8—3821no—
155C22×Q8—1911no—
156C22×D4—1911no—
157C22×C8—1912no—
158C4⋊Q8—3821no—
159C4⋊Q8—3821no—
160C4⋊Q8—1912no—
161C4⋊Q8—1911no—
162C4⋊Q8C2×D19111yescharacteristic, complemented
163#124—3821no—
164#124—3821no—
165#124—1911no—
166#124—1911no—
167#124—1911no—
168#124C2×D19111yescharacteristic
169C2×C4⋊C4—3821no—
170C2×C4⋊C4—1912no—
171C2×C22⋊C4—3821no—
172C2×C22⋊C4—3821no—
173C2×C42—1911no—
174#103—1912no—
175#103—1912no—
176#103—1912no—
177#103C2×D19112yescomplemented
178#102—1912no—
179#102—1912no—
180#102—1912no—
181#102C2×D19112yes—
182#97—1912no—
183#97—1911no—
184#97C2×D19111yescharacteristic, complemented
185C24—3821no—
186C2×Q8—21no—
187C2×Q8—3821no—
188C2×Q8—3822no—
189C2×Q8—3821no—
190C2×Q8—3824no—
191C2×Q8—1911no—
192C2×Q8—7644no—
193C2×Q8C22×D19111yescharacteristic
194C2×D4—3824no—
195C2×D4—3821no—
196C2×D4—1911no—
197C2×D4C22×D19111yescharacteristic
198C22×C4—3821no—
199C22×C4—3822no—
200C22×C4—1911no—
201C22×C4—1911no—
202C22×C4—1911no—
203C2×C8—3824no—
204C2×C8—1912no—
205C2×C8C22×D19112yes—
206C4⋊C4—21no—
207C4⋊C4—3822no—
208C4⋊C4—3822no—
209C4⋊C4—3821no—
210C4⋊C4—1912no—
211C4⋊C4C22×D19112yes—
212C22⋊C4—21no—
213C22⋊C4—21no—
214C22⋊C4—3821no—
215C22⋊C4—3821no—
216C22⋊C4—3821no—
217C22⋊C4—3821no—
218C22⋊C4—3821no—
219C22⋊C4—3821no—
220C42—1911no—
221C42—1911no—
222C42—1911no—
223C42C22×D19111yescharacteristic
224C23—21no—
225C23—3821no—
226C23—1911no—
227C23—7642no—
228C23—7642no—
229C23—7642no—
230Q8—22no—
231Q8—42no—
232Q8—3822no—
233Q8—7642no—
234Q8—7644no—
235Q8—7642no—
236D4—22no—
237D4—3822no—
238D4—7642no—
239C2×C4—21no—
240C2×C4—22no—
241C2×C4—3822no—
242C2×C4—3822no—
243C2×C4—3821no—
244C2×C4—3822no—
245C2×C4—1911no—
246C2×C4—1911no—
247C2×C4—1911no—
248C2×C4—1914no—
249C2×C4—7642no—
250C2×C4—7644no—
251C2×C4D4×D19111yescharacteristic
252C2×C4D4×D19111yescharacteristic
253C2×C4C23×D19111yescharacteristic
254C8—22no—
255C8—3822no—
256C22—41no—
257C22—41no—
258C22—41no—
259C22—1912no—
260C22—1912no—
261C22—1912no—
262C22—15284no—
263C22—7641no—
264C22—7641no—
265C22—7641no—
266C22C2×D4×D19111yescharacteristic
267C4—21no—
268C4—21no—
269C4—42no—
270C4—41no—
271C4—3821no—
272C4—3821no—
273C4—1912no—
274C4—7642no—
275C4—7641no—
276C4#41×D19112yes—
277C2—81no—
278C2—1914no—
279C2—15281no—
280C2#124×D19111yescharacteristic
281C2AGL(1,ℤ/8)×D19111yescharacteristic
282C2#137×D19111yescharacteristic
subgroupG/H
centerC22C2×D4×D191
commutator, cosocleC2×C4×C191C24
FrattiniC2×C4C23×D191
Fitting#489×C191C2
socleC22×C191C22×D4
prime radicalC2×D4×D191C22
hypercenter#489D191
hypocenterC191C2×#489
LCSG ▷ C2×C4×C191 ▷ C2×C191 ▷ C191
UCSC1 ◁ C22 ◁ C42 ◁ #489

Complements

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

NH ≅ G/Ntype
1. C4⋊Q8×D191C2semidirect
3. #27133025C2semidirect
4. #27133024C2semidirect
5. #27133023C2semidirect
6. #27133022C2semidirect
7. #27133021C2semidirect
8. #103×D191C2semidirect
10. #97×D191C2semidirect
11. #489×C191C2semidirect
16. #20259322C22semidirect
18. #20259321C22semidirect
30. #20259247C22semidirect
31. #20259246C22semidirect
32. #20259245C22semidirect
36. #20259233C22semidirect
37. #20259232C22semidirect
38. C4⋊Q8×C191C22semidirect
40. #103×C191C22semidirect
42. #97×C191C22semidirect
140. D191#489direct (N × H)
141. C191C2×#489semidirect
150. #489D191direct (N × H)
162. C4⋊Q8C2×D191semidirect
177. #103C2×D191semidirect
184. #97C2×D191semidirect

Maximal subgroups

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

subgroupindex
C4⋊Q8×D1912
#124×D1912
#271330252
#271330242
#271330232
#271330222
#271330212
#103×D1912
#102×D1912
#97×D1912
#489×C1912
C2×#489191