Finite Group Databasebeta

← #37583460

Subgroups — #37583460

Order 24448.

488 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 488 conjugacy classes of subgroups — too many to draw. Only its 123 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#27133217C211yesmaximal, characteristic, complemented
2#127×D191C211yesmaximal, characteristic
3#27133034C212yesmaximal, complemented
4#27133008C212yesmaximal, complemented
5#27132864C211yesmaximal, characteristic, complemented
6#27132863C211yesmaximal, characteristic, complemented
7#27132860C211yesmaximal, characteristic, complemented
8#27132815C212yesmaximal, complemented
9#27132812C212yesmaximal, complemented
10#27132715C211yesmaximal, characteristic, complemented
11#460×C191C211yesmaximal, characteristic, complemented
12C2×D4×D191—21no—
13C2×D4×D191—21no—
14C2×D4×D191C2212yes—
15C2×#15690276C2212yescomplemented
16C2×#15690275C2212yescomplemented
17C2×#15690274C2212yescomplemented
18#20259321C2211yescharacteristic, complemented
19#20259320C2211yescharacteristic, complemented
20#20259319C2211yescharacteristic
21#20259318—21no—
22#20259318—21no—
23#20259318C2212yes—
24#20259312—21no—
25#20259302C2211yescharacteristic
26#20259300—21no—
27#20259298C2211yescharacteristic
28C4×Dic382—21no—
29C4×Dic382C2211yescharacteristic, complemented
30#20259279—21no—
31C42×D191C2211yescharacteristic
32C4⋊(C191⋊C8)C2211yescharacteristic, complemented
33#20259244C2212yescomplemented
34#20259243C2212yescomplemented
35#20259242C2212yescomplemented
36(C2×D191)⋊C8C2212yes—
37C8×Dic191C2212yescomplemented
38C2×SD16×C191C2212yescomplemented
39#127×C191C2211yescharacteristic
40C4×Q8×C191C2211yescharacteristic, complemented
41C4⋊C8×C191C2211yescharacteristic, complemented
42#102×C191C2212yescomplemented
43C23×D191—22no—
44C2×Dic382—21no—
45C2×Dic382C2311yescharacteristic
46C2×D764—21no—
47C2×D764—21no—
48C2×D764C2312yes—
49C2×C22⋊D191—22no—
50C2×C22⋊D191—22no—
51C2×C22⋊D191—22no—
52D4×D191—42no—
53D4×D191—42no—
54D4×D191—44no—
55D4×D191—44no—
56C2×C4×D191D411yescharacteristic
57C2×C4×D191D411yescharacteristic
58C2×C4×D191C2311yescharacteristic
59#15690276—42no—
60#15690275—42no—
61#15690274—42no—
62C2×C191⋊C8C2312yes—
63C4⋊Dic191—21no—
64C4⋊Dic191C2311yescharacteristic
65Dic191⋊C4—21no—
66Dic191⋊C4—21no—
67Dic191⋊C4—21no—
68Dic191⋊C4—21no—
69Dic191⋊C4—21no—
70(C2×D191)⋊C4—21no—
71(C2×D191)⋊C4—21no—
72C4×Dic191—21no—
73C4×Dic191D411yescharacteristic
74C4×Dic191D411yescharacteristic
75C4×Dic191C2311yescharacteristic
76C2×Q8×C191C2311yescharacteristic
77C2×D4×C191—21no—
78C2×D4×C191—21no—
79C2×D4×C191C2312yes—
80SD16×C191—42no—
81C2×C8×C191C2312yes—
82C4⋊C4×C191—21no—
83C4⋊C4×C191C2311yescharacteristic
84C42×C191—21no—
85C42×C191C2311yescharacteristic
86C22×D191—22no—
87C22×D191—44no—
88C22×D191—44no—
89C22×D191—44no—
90C22×D191C2×D411yescharacteristic
91Dic382—21no—
92Dic382—21no—
93Dic382—42no—
94D764—22no—
95D764—22no—
96D764—42no—
97D764—42no—
98C22⋊D191—44no—
99C22⋊D191—44no—
100C22⋊D191—44no—
101C2×Dic191—21no—
102C2×Dic191—21no—
103C2×Dic191C2×D411yescharacteristic
104C2×Dic191C2×D411yescharacteristic
105C2×Dic191C2×D411yescharacteristic
106C4×D191—21no—
107C4×D191—21no—
108C4×D191—22no—
109C4×D191SD1614yescomplemented
110C191⋊C8—22no—
111C23×C191—22no—
112Q8×C191—21no—
113Q8×C191—21no—
114D4×C191—22no—
115D4×C191—22no—
116D4×C191—42no—
117D4×C191—42no—
118C2×C4×C191—21no—
119C2×C4×C191—21no—
120C2×C4×C191C2×D411yescharacteristic
121C2×C4×C191C2×D411yescharacteristic
122C2×C4×C191C2411yescharacteristic
123C8×C191—22no—
124C22×C191—42no—
125C22×C191—42no—
126C22×C191—42no—
127C22×C191C22×D411yescharacteristic
128C2×D191—21no—
129C2×D191—21no—
130C2×D191—42no—
131C2×D191—42no—
132C2×D191—42no—
133C2×D191—88no—
134C2×D191C2×SD1612yes—
135C4×C191—21no—
136C4×C191—41no—
137C4×C191—41no—
138C4×C191C2×SD1612yes—
139C4×C191H(5,𝔽2)11yescharacteristic
140C4×C191H(5,𝔽2)11yescharacteristic
141Dic191—21no—
142Dic191—41no—
143Dic191—41no—
144Dic191C2×SD1612yes—
145Dic191AGL(1,ℤ/8)12yes—
146C2×C191—82no—
147C2×C191#53511yescharacteristic
148C2×C191C22×SD1611yescharacteristic
149C2×C191C2×AGL(1,ℤ/8)11yescharacteristic
150D191—22no—
151D191—82no—
152C191#320811yescharacteristic, complemented, Sylow
153#3208—1911nomaximal, Sylow
154#533—1911no—
155C2×#127—1911no—
156#492—1912no—
157#486—1912no—
158#460—1911no—
159#460—1911no—
160#460—1911no—
161#460D19111yescharacteristic, complemented
162#450—1912no—
163#450—1912no—
164#431—1911no—
165C22×D4—3821no—
166C22×D4—3821no—
167C22×D4—1912no—
168C2×SD16—1912no—
169C2×SD16—1912no—
170C2×SD16—1912no—
171C2×SD16C2×D19112yescomplemented
172C4⋊Q8—1911no—
173C4⋊Q8—1911no—
174#127—3821no—
175#127—3821no—
176#127—1912no—
177#127—1911no—
178#127C2×D19111yescharacteristic
179#125—3821no—
180C22⋊Q8—3821no—
181C22⋊Q8—1911no—
182C22⋊Q8—1911no—
183C4×Q8—3821no—
184C4×Q8—1911no—
185C4×Q8C2×D19111yescharacteristic, complemented
186#117—3821no—
187C2×C42—1911no—
188C4⋊C8—1911no—
189C4⋊C8C2×D19111yescharacteristic, complemented
190#102—1912no—
191#102—1912no—
192#102—1912no—
193#102C2×D19112yescomplemented
194C22⋊C8—1912no—
195C4×C8—1912no—
196C24—3822no—
197C2×Q8—3821no—
198C2×Q8—1911no—
199C2×Q8C22×D19111yescharacteristic
200C2×D4—21no—
201C2×D4—21no—
202C2×D4—3822no—
203C2×D4—3822no—
204C2×D4—3822no—
205C2×D4—3821no—
206C2×D4—3821no—
207C2×D4—1912no—
208C2×D4—7642no—
209C2×D4—7642no—
210C2×D4—7644no—
211C2×D4—7644no—
212C2×D4C22×D19112yes—
213C22×C4—1911no—
214C22×C4—1911no—
215C22×C4—1911no—
216SD16—42no—
217SD16—7642no—
218SD16—7642no—
219SD16—7642no—
220C2×C8—1912no—
221C2×C8C22×D19112yes—
222C4⋊C4—21no—
223C4⋊C4—3821no—
224C4⋊C4—3821no—
225C4⋊C4—3821no—
226C4⋊C4—3821no—
227C4⋊C4—3821no—
228C4⋊C4—3821no—
229C4⋊C4—1911no—
230C4⋊C4C22×D19111yescharacteristic
231C22⋊C4—3821no—
232C22⋊C4—3821no—
233C42—21no—
234C42—3821no—
235C42—1911no—
236C42—1911no—
237C42—1911no—
238C42C22×D19111yescharacteristic
239C23—22no—
240C23—3822no—
241C23—1911no—
242C23—7644no—
243C23—7644no—
244C23—7644no—
245Q8—21no—
246Q8—21no—
247Q8—3821no—
248Q8—3821no—
249Q8—7642no—
250D4—22no—
251D4—22no—
252D4—42no—
253D4—42no—
254D4—3822no—
255D4—3822no—
256D4—7642no—
257D4—7644no—
258D4—7644no—
259D4—7644no—
260D4—7642no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—3822no—
264C2×C4—3821no—
265C2×C4—3821no—
266C2×C4—3821no—
267C2×C4—3821no—
268C2×C4—1911no—
269C2×C4—1911no—
270C2×C4—1911no—
271C2×C4—1914no—
272C2×C4D4×D19111yescharacteristic
273C2×C4D4×D19111yescharacteristic
274C2×C4C23×D19111yescharacteristic
275C8—22no—
276C8—3822no—
277C22—42no—
278C22—42no—
279C22—42no—
280C22—3821no—
281C22—3821no—
282C22—1912no—
283C22—15288no—
284C22—7642no—
285C22—7642no—
286C22—7642no—
287C22C2×D4×D19111yescharacteristic
288C4—21no—
289C4—41no—
290C4—41no—
291C4—3821no—
292C4—1912no—
293C4—1912no—
294C4—7641no—
295C4—7641no—
296C4SD16×D19112yes—
297C4#2025936711yescharacteristic
298C4#2025936811yescharacteristic
299C2—82no—
300C2—3822no—
301C2—15282no—
302C2#2713322911yescharacteristic
303C2C2×SD16×D19111yescharacteristic
304C2AGL(1,ℤ/8)×D19111yescharacteristic
subgroupG/H
centerC22C2×D4×D191
commutator, cosocleC2×C4×C191C24
FrattiniC2×C4C23×D191
Fitting#460×C191C2
socleC22×C191C22×D4
prime radical#127×D191C2
hypercenter#460D191
hypocenterC191#3208
LCSG ▷ C2×C4×C191 ▷ C2×C191 ▷ C191
UCSC1 ◁ C22 ◁ C42 ◁ #460

Complements

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

NH ≅ G/Ntype
1. #27133217C2semidirect
3. #27133034C2semidirect
4. #27133008C2semidirect
5. #27132864C2semidirect
6. #27132863C2semidirect
7. #27132860C2semidirect
8. #27132815C2semidirect
9. #27132812C2semidirect
10. #27132715C2semidirect
11. #460×C191C2semidirect
15. C2×#15690276C22semidirect
16. C2×#15690275C22semidirect
17. C2×#15690274C22semidirect
18. #20259321C22semidirect
19. #20259320C22semidirect
29. C4×Dic382C22semidirect
32. C4⋊(C191⋊C8)C22semidirect
33. #20259244C22semidirect
34. #20259243C22semidirect
35. #20259242C22semidirect
37. C8×Dic191C22semidirect
38. C2×SD16×C191C22semidirect
40. C4×Q8×C191C22semidirect
41. C4⋊C8×C191C22semidirect
42. #102×C191C22semidirect
109. C4×D191SD16semidirect
152. C191#3208semidirect
161. #460D191semidirect
171. C2×SD16C2×D191semidirect
185. C4×Q8C2×D191semidirect
189. C4⋊C8C2×D191semidirect
193. #102C2×D191semidirect

Maximal subgroups

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

subgroupindex
#271332172
#127×D1912
#271330342
#271330082
#271328642
#271328632
#271328602
#271328152
#271328122
#271327152
#460×C1912
#3208191