Finite Group Databasebeta

← #37583482

Subgroups — #37583482

Order 24448.

410 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 410 conjugacy classes of subgroups — too many to draw. Only its 113 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#27133194C211yesmaximal, characteristic, complemented
2C2×#20259309C211yesmaximal, characteristic, complemented
3#27133046C212yesmaximal, complemented
4#27133016C212yesmaximal, complemented
5#27132884C212yesmaximal, complemented
6#27132876C212yesmaximal, complemented
7#27132821C211yesmaximal, characteristic, complemented
8#27132814C211yesmaximal, characteristic, complemented
9#27132801C211yesmaximal, characteristic, complemented
10#27132713C211yesmaximal, characteristic, complemented
11#447×C191C211yesmaximal, characteristic
12C22×Dic382C2211yescharacteristic
13C2×Dic764C2212yescomplemented
14C2×#15690275C2212yes—
15C2×#15690275C2212yescomplemented
16#20259315—21no—
17#20259313C2211yescharacteristic, complemented
18#20259309—21no—
19#20259309—21no—
20#20259309C2212yes—
21#20259309C2212yes—
22#20259309C2211yescharacteristic, complemented
23#20259303—21no—
24#20259301C2211yescharacteristic
25#20259287C2211yescharacteristic, complemented
26D4×Dic191—21no—
27D4×Dic191C2211yescharacteristic, complemented
28#20259278—21no—
29C2×C22⋊Dic191—21no—
30C2×C22⋊Dic191—21no—
31C2×C4×Dic191C2211yescharacteristic
32Dic191⋊C8C2212yescomplemented
33#20259247C2212yescomplemented
34#20259245C2212yescomplemented
35#20259243C2212yes—
36C22⋊(C191⋊C8)C2211yescharacteristic
37C8×Dic191C2212yescomplemented
38C22×D4×C191C2211yescharacteristic
39C2×D8×C191C2212yes—
40#121×C191C2211yescharacteristic
41#102×C191C2212yes—
42C22⋊C8×C191C2211yescharacteristic
43C2×Dic382—21no—
44C2×Dic382—42no—
45C2×Dic382C2311yescharacteristic
46C2×Dic382C2312yes—
47C22×Dic191—21no—
48C22×Dic191—21no—
49C22×Dic191D411yescharacteristic
50C22×Dic191D411yescharacteristic
51Dic764—42no—
52#15690275—42no—
53#15690275—42no—
54C2×C191⋊C8C2312yes—
55C4⋊Dic191—21no—
56C4⋊Dic191C2311yescharacteristic
57Dic191⋊C4—21no—
58Dic191⋊C4—21no—
59Dic191⋊C4—21no—
60C22⋊Dic191—22no—
61C22⋊Dic191—22no—
62C22⋊Dic191—22no—
63C22⋊Dic191—22no—
64C22⋊Dic191—21no—
65C22⋊Dic191—21no—
66C22⋊Dic191—21no—
67C22⋊Dic191—21no—
68C4×Dic191—21no—
69C4×Dic191D411yescharacteristic
70C4×Dic191D411yescharacteristic
71C4×Dic191C2311yescharacteristic
72C4×Dic191C2311yescharacteristic
73C24×C191—21no—
74C2×D4×C191—21no—
75C2×D4×C191—21no—
76C2×D4×C191—42no—
77C2×D4×C191C2312yes—
78C2×D4×C191C2311yescharacteristic
79C22×C4×C191C2311yescharacteristic
80D8×C191—42no—
81C2×C8×C191C2312yes—
82C4⋊C4×C191C2311yescharacteristic
83C22⋊C4×C191—21no—
84Dic382—22no—
85Dic382—22no—
86Dic382—22no—
87Dic382—42no—
88C2×Dic191—22no—
89C2×Dic191—21no—
90C2×Dic191—22no—
91C2×Dic191—21no—
92C2×Dic191—41no—
93C2×Dic191—42no—
94C2×Dic191—42no—
95C2×Dic191C2×D411yescharacteristic
96C2×Dic191C2×D411yescharacteristic
97C2×Dic191C2×D411yescharacteristic
98C2×Dic191C2×D411yescharacteristic
99C191⋊C8—22no—
100C23×C191—21no—
101C23×C191—22no—
102C23×C191—42no—
103C23×C191—42no—
104C23×C191—42no—
105C23×C191C2×D411yescharacteristic
106D4×C191—22no—
107D4×C191—22no—
108D4×C191—22no—
109D4×C191—42no—
110D4×C191—42no—
111C2×C4×C191—22no—
112C2×C4×C191—21no—
113C2×C4×C191C2×D411yescharacteristic
114C2×C4×C191C2411yescharacteristic
115C8×C191—22no—
116C22×C191—21no—
117C22×C191—21no—
118C22×C191—42no—
119C22×C191—41no—
120C22×C191—41no—
121C22×C191—42no—
122C22×C191—42no—
123C22×C191—41no—
124C22×C191—84no—
125C22×C191#13512yes—
126C22×C191C22×D411yescharacteristic
127C4×C191—21no—
128C4×C191—41no—
129C4×C191H(5,𝔽2)12yes—
130Dic191—21no—
131Dic191—21no—
132Dic191—21no—
133Dic191—21no—
134Dic191—42no—
135Dic191—41no—
136Dic191—41no—
137C2×C191—22no—
138C2×C191—81no—
139C2×C191—82no—
140C2×C191#53511yescharacteristic
141C2×C191C2×#13511yescharacteristic
142C2×C191#57511yescharacteristic
143C191#321111yescharacteristic, complemented, Sylow
144#3211—1911nomaximal, Sylow
145#529—1911no—
146C2×#124—1911no—
147#495—1912no—
148#488—1912no—
149#464—1912no—
150#463—1912no—
151#451—1911no—
152#450—1911no—
153#448—1911no—
154#447D19111yescharacteristic
155#431—1911no—
156C22×Q8—1911no—
157C22×D4C2×D19111yescharacteristic
158C2×Q16—1912no—
159C2×SD16—1912no—
160C2×SD16—1912no—
161C2×D8C2×D19112yes—
162#126—3821no—
163#125—1911no—
164#124—3821no—
165#124—3821no—
166#124—1912no—
167#124—1912no—
168#124—1911no—
169#123—3821no—
170C22⋊Q8—1911no—
171#121C2×D19111yescharacteristic
172C4×Q8—1911no—
173C4×D4—3821no—
174C4×D4—1911no—
175#117—3821no—
176C2×C22⋊C4—3821no—
177C2×C22⋊C4—3821no—
178C2×C42—1911no—
179C4⋊C8—1912no—
180#103—1912no—
181#103—1912no—
182#102—1912no—
183#102C2×D19112yes—
184C22⋊C8—1911no—
185C22⋊C8C2×D19111yescharacteristic
186C4×C8—1912no—
187C24—21no—
188C2×Q8—3821no—
189C2×Q8—1912no—
190C2×Q8—1911no—
191C2×Q8—7642no—
192C2×D4—21no—
193C2×D4—21no—
194C2×D4—42no—
195C2×D4C22×D19112yes—
196C2×D4C22×D19111yescharacteristic
197C22×C4—3821no—
198C22×C4—3821no—
199C22×C4—1911no—
200C22×C4—1911no—
201C22×C4C22×D19111yescharacteristic
202Q16—7642no—
203SD16—7642no—
204SD16—7642no—
205D8—42no—
206C2×C8—1912no—
207C2×C8C22×D19112yes—
208C4⋊C4—3821no—
209C4⋊C4—3821no—
210C4⋊C4—3821no—
211C4⋊C4—3821no—
212C4⋊C4—1911no—
213C4⋊C4C22×D19111yescharacteristic
214C22⋊C4—21no—
215C22⋊C4—3821no—
216C22⋊C4—3822no—
217C22⋊C4—3822no—
218C22⋊C4—3822no—
219C22⋊C4—3822no—
220C22⋊C4—3821no—
221C22⋊C4—3821no—
222C22⋊C4—3821no—
223C42—3821no—
224C42—1911no—
225C42—1911no—
226C42—1911no—
227C42—1911no—
228C23—21no—
229C23—22no—
230C23—42no—
231C23—42no—
232C23—42no—
233C23D4×D19111yescharacteristic
234Q8—3822no—
235Q8—3822no—
236Q8—3822no—
237Q8—7642no—
238D4—22no—
239D4—22no—
240D4—22no—
241D4—42no—
242D4—42no—
243C2×C4—22no—
244C2×C4—21no—
245C2×C4—3822no—
246C2×C4—3821no—
247C2×C4—3822no—
248C2×C4—3821no—
249C2×C4—1911no—
250C2×C4—1911no—
251C2×C4—1911no—
252C2×C4—1911no—
253C2×C4—7641no—
254C2×C4—7642no—
255C2×C4—7642no—
256C2×C4D4×D19111yescharacteristic
257C2×C4C23×D19111yescharacteristic
258C8—22no—
259C8—3822no—
260C22—21no—
261C22—21no—
262C22—41no—
263C22—42no—
264C22—41no—
265C22—42no—
266C22—42no—
267C22—41no—
268C22—84no—
269C22#2025934112yes—
270C22C2×D4×D19111yescharacteristic
271C4—21no—
272C4—41no—
273C4—3821no—
274C4—3821no—
275C4—3821no—
276C4—3821no—
277C4—7641no—
278C4—7642no—
279C4—7641no—
280C4#2025936712yes—
281C2—22no—
282C2—82no—
283C2—81no—
284C2#2713322811yescharacteristic
285C2C2×#2025934111yescharacteristic
286C2#2713345911yescharacteristic
subgroupG/H
centerC22C2×D4×D191
commutator, cosocleC2×C4×C191C24
FrattiniC2×C4C23×D191
Fitting, prime radical#447×C191C2
socleC22×C191C22×D4
hypercenter#447D191
hypocenterC191#3211
LCSG ▷ C2×C4×C191 ▷ C2×C191 ▷ C191
UCSC1 ◁ C22 ◁ C22×C4 ◁ #447

Complements

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

NH ≅ G/Ntype
1. #27133194C2semidirect
2. C2×#20259309C2semidirect
3. #27133046C2semidirect
4. #27133016C2semidirect
5. #27132884C2semidirect
6. #27132876C2semidirect
7. #27132821C2semidirect
8. #27132814C2semidirect
9. #27132801C2semidirect
10. #27132713C2semidirect
13. C2×Dic764C22semidirect
15. C2×#15690275C22semidirect
17. #20259313C22semidirect
22. #20259309C22semidirect
25. #20259287C22semidirect
27. D4×Dic191C22semidirect
32. Dic191⋊C8C22semidirect
33. #20259247C22semidirect
34. #20259245C22semidirect
37. C8×Dic191C22semidirect
143. C191#3211semidirect

Maximal subgroups

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

subgroupindex
#271331942
C2×#202593092
#271330462
#271330162
#271328842
#271328762
#271328212
#271328142
#271328012
#271327132
#447×C1912
#3211191