Finite Group Databasebeta

← #37581997

Subgroups — #37581997

Order 24448.

584 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 584 conjugacy classes of subgroups — too many to draw. Only its 133 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×#20259349C212yesmaximal, complemented
2#127×D191C211yesmaximal, characteristic, complemented
3C4×#15690288C211yesmaximal, characteristic, complemented
4#27132863C211yesmaximal, characteristic, complemented
5#27132861C212yesmaximal, complemented
6#27132859C211yesmaximal, characteristic, complemented
7#27132857C212yesmaximal, complemented
8#27132855C211yesmaximal, characteristic, complemented
9#27132685C211yesmaximal, characteristic, complemented
10#27132666C212yesmaximal, complemented
11#460×C191C211yesmaximal, characteristic, complemented
12C2×#15690288C2211yescharacteristic
13C2×D4×D191—21no—
14C2×D4×D191—21no—
15C2×D4×D191C2212yes—
16#20259349—28no—
17C2×#15690274C2212yescomplemented
18C2×D1528C2212yescomplemented
19C2×#15690272C2212yescomplemented
20C2×#15690270C2212yes—
21#20259319C2211yescharacteristic
22#20259318—21no—
23#20259318—21no—
24#20259318C2212yes—
25Q8×Dic191C2211yescharacteristic, complemented
26C4×D764—21no—
27C4×D764C2211yescharacteristic, complemented
28#20259284—21no—
29#20259284C2211yescharacteristic, complemented
30#20259280—21no—
31#20259280C2211yescharacteristic
32C42×D191—21no—
33C42×D191C2211yescharacteristic
34C4⋊(C191⋊C8)C2211yescharacteristic, complemented
35Dic191⋊C8C2212yescomplemented
36#20259244C2212yescomplemented
37#20259243C2212yescomplemented
38#20259242C2212yescomplemented
39C2×SD16×C191C2212yescomplemented
40#127×C191C2211yescharacteristic
41C4×Q8×C191C2211yescharacteristic, complemented
42C4⋊C8×C191C2211yescharacteristic, complemented
43#102×C191C2212yescomplemented
44C23×D191—22no—
45#15690288—22no—
46#15690288D414yescomplemented
47C2×D764—21no—
48C2×D764—21no—
49C2×D764—21no—
50C2×D764C2312yes—
51C2×D764C2311yescharacteristic
52C2×C22⋊D191—22no—
53C2×C22⋊D191—22no—
54C2×C22⋊D191—22no—
55D4×D191—24no—
56D4×D191—24no—
57D4×D191—44no—
58D4×D191—44no—
59C2×C4×D191—21no—
60C2×C4×D191—21no—
61C2×C4×D191—21no—
62C2×C4×D191D411yescharacteristic
63C2×C4×D191D411yescharacteristic
64C2×C4×D191C2311yescharacteristic
65#15690274—24no—
66D1528—24no—
67#15690272—24no—
68#15690270—24no—
69C2×C191⋊C8C2312yes—
70C4⋊Dic191—21no—
71C4⋊Dic191C2311yescharacteristic
72(C2×D191)⋊C4—21no—
73(C2×D191)⋊C4—21no—
74(C2×D191)⋊C4—21no—
75C4×Dic191—21no—
76C4×Dic191—21no—
77C4×Dic191D411yescharacteristic
78C4×Dic191D411yescharacteristic
79C4×Dic191C2311yescharacteristic
80C2×Q8×C191C2311yescharacteristic
81C2×D4×C191—21no—
82C2×D4×C191—21no—
83C2×D4×C191C2312yes—
84SD16×C191—24no—
85C2×C8×C191C2312yes—
86C4⋊C4×C191—21no—
87C4⋊C4×C191C2311yescharacteristic
88C42×C191—21no—
89C42×C191C2311yescharacteristic
90C22×D191—21no—
91C22×D191—22no—
92C22×D191—44no—
93C22×D191—44no—
94C22×D191—44no—
95C22×D191C2×D411yescharacteristic
96D764—24no—
97D764—22no—
98D764—21no—
99D764—22no—
100D764—42no—
101D764—42no—
102D764C2×D412yescomplemented
103C22⋊D191—44no—
104C22⋊D191—44no—
105C22⋊D191—44no—
106C2×Dic191—21no—
107C2×Dic191C2×D411yescharacteristic
108C2×Dic191C2×D411yescharacteristic
109C2×Dic191C2×D411yescharacteristic
110C4×D191—22no—
111C4×D191—24no—
112C4×D191—22no—
113C4×D191—42no—
114C4×D191—42no—
115C4×D191C2×D412yes—
116C4×D191#4112yes—
117C191⋊C8—22no—
118C23×C191—22no—
119Q8×C191—21no—
120Q8×C191C2×D412yescomplemented
121D4×C191—22no—
122D4×C191—22no—
123D4×C191—42no—
124D4×C191—42no—
125C2×C4×C191—21no—
126C2×C4×C191—21no—
127C2×C4×C191C2×D411yescharacteristic
128C2×C4×C191C2×D411yescharacteristic
129C2×C4×C191C2411yescharacteristic
130C8×C191—22no—
131C22×C191—42no—
132C22×C191—42no—
133C22×C191—42no—
134C22×C191C22×D411yescharacteristic
135C2×D191—21no—
136C2×D191—21no—
137C2×D191—22no—
138C2×D191—41no—
139C2×D191—41no—
140C2×D191—42no—
141C2×D191—42no—
142C2×D191—42no—
143C2×D191—88no—
144C2×D191#12112yes—
145C4×C191—22no—
146C4×C191—21no—
147C4×C191—41no—
148C4×C191AGL(1,ℤ/8)12yescomplemented
149C4×C191C22×D411yescharacteristic
150C4×C191C2×#4111yescharacteristic
151Dic191—21no—
152Dic191—41no—
153Dic191#12112yes—
154Dic191AGL(1,ℤ/8)12yescomplemented
155C2×C191—82no—
156C2×C191C2×#12111yescharacteristic
157C2×C191C2×AGL(1,ℤ/8)11yescharacteristic
158C2×C191C2×AGL(1,ℤ/8)11yescharacteristic
159D191—22no—
160D191—42no—
161D191—82no—
162C191#303811yescharacteristic, complemented, Sylow
163#3038—1911nomaximal, Sylow
164C2×AGL(1,ℤ/8)—1912no—
165C2×#127—1911no—
166C4×#41—1911no—
167#460—1912no—
168#460—1911no—
169#460D19111yescharacteristic, complemented
170#459—1911no—
171#459—1911no—
172#459—1912no—
173#423—1911no—
174#418—1912no—
175C2×#41—1911no—
176C22×D4—3821no—
177C22×D4—3821no—
178C22×D4—1912no—
179AGL(1,ℤ/8)—3828no—
180C2×SD16—1912no—
181C2×SD16C2×D19112yescomplemented
182C2×D8—1912no—
183C2×D8—1912no—
184C2×M16—1912no—
185#127—3821no—
186#127—3821no—
187#127—1912no—
188#127—1911no—
189#127C2×D19111yescharacteristic
190C4×Q8—1911no—
191C4×Q8C2×D19111yescharacteristic, complemented
192C4×D4—3821no—
193C4×D4—3821no—
194C4×D4—1911no—
195C4×D4—1911no—
196#117—3821no—
197#117—1911no—
198C2×C42—3821no—
199C2×C42—1911no—
200C4⋊C8—1912no—
201C4⋊C8—1911no—
202C4⋊C8C2×D19111yescharacteristic, complemented
203#102—1912no—
204#102—1912no—
205#102—1912no—
206#102C2×D19112yescomplemented
207C24—3822no—
208#41—3822no—
209#41—1914no—
210C2×Q8C22×D19111yescharacteristic
211C2×D4—21no—
212C2×D4—21no—
213C2×D4—3821no—
214C2×D4—3824no—
215C2×D4—3824no—
216C2×D4—3822no—
217C2×D4—3822no—
218C2×D4—3822no—
219C2×D4—3821no—
220C2×D4—3821no—
221C2×D4—1912no—
222C2×D4—1911no—
223C2×D4—7644no—
224C2×D4—7644no—
225C2×D4C22×D19112yes—
226C22×C4—3821no—
227C22×C4—3821no—
228C22×C4—3821no—
229C22×C4—1911no—
230C22×C4—1911no—
231C22×C4—1911no—
232SD16—24no—
233SD16—3824no—
234D8—3824no—
235D8—3824no—
236M16—3824no—
237C2×C8—1912no—
238C2×C8C22×D19112yes—
239C4⋊C4—21no—
240C4⋊C4—3821no—
241C4⋊C4—1911no—
242C4⋊C4C22×D19111yescharacteristic
243C22⋊C4—3821no—
244C22⋊C4—3821no—
245C22⋊C4—3821no—
246C42—21no—
247C42—3821no—
248C42—3821no—
249C42—1911no—
250C42—1911no—
251C42—1911no—
252C42C22×D19111yescharacteristic
253C23—22no—
254C23—3821no—
255C23—3822no—
256C23—1911no—
257C23—7644no—
258C23—7644no—
259C23—7644no—
260Q8—21no—
261Q8D4×D19112yescomplemented
262D4—22no—
263D4—22no—
264D4—42no—
265D4—42no—
266D4—3824no—
267D4—3822no—
268D4—3821no—
269D4—3822no—
270D4—1912no—
271D4—7642no—
272D4—7644no—
273D4—7644no—
274D4—7644no—
275D4—7642no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—3824no—
279C2×C4—3822no—
280C2×C4—3822no—
281C2×C4—3821no—
282C2×C4—1911no—
283C2×C4—1911no—
284C2×C4—1911no—
285C2×C4—1912no—
286C2×C4—1912no—
287C2×C4—7642no—
288C2×C4—7642no—
289C2×C4D4×D19111yescharacteristic
290C2×C4D4×D19111yescharacteristic
291C2×C4C23×D19111yescharacteristic
292C8—22no—
293C8—3822no—
294C22—42no—
295C22—42no—
296C22—42no—
297C22—3821no—
298C22—3821no—
299C22—3822no—
300C22—1912no—
301C22—15288no—
302C22—7642no—
303C22—7642no—
304C22—7642no—
305C22—7641no—
306C22—7641no—
307C22C2×D4×D19111yescharacteristic
308C4—21no—
309C4—22no—
310C4—41no—
311C4—3821no—
312C4—1912no—
313C4—1912no—
314C4—7641no—
315C4#2025934912yescomplemented
316C4C2×D4×D19111yescharacteristic
317C4#41×D19111yescharacteristic
318C2—82no—
319C2—3822no—
320C2—15282no—
321C2—7642no—
322C2#121×D19111yescharacteristic
323C2AGL(1,ℤ/8)×D19111yescharacteristic
324C2C2×#2025934911yescharacteristic
subgroupG/H
centerC22C2×D4×D191
commutator, cosocleC2×C4×C191C24
FrattiniC2×C4C23×D191
Fitting#460×C191C2
socleC22×C191C22×D4
hypercenter#460D191
hypocenterC191#3038
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. C2×#20259349C2semidirect
2. #127×D191C2semidirect
3. C4×#15690288C2semidirect
4. #27132863C2semidirect
5. #27132861C2semidirect
6. #27132859C2semidirect
7. #27132857C2semidirect
8. #27132855C2semidirect
9. #27132685C2semidirect
10. #27132666C2semidirect
11. #460×C191C2semidirect
17. C2×#15690274C22semidirect
18. C2×D1528C22semidirect
19. C2×#15690272C22semidirect
25. Q8×Dic191C22semidirect
27. C4×D764C22semidirect
29. #20259284C22semidirect
34. C4⋊(C191⋊C8)C22semidirect
35. Dic191⋊C8C22semidirect
36. #20259244C22semidirect
37. #20259243C22semidirect
38. #20259242C22semidirect
39. C2×SD16×C191C22semidirect
41. C4×Q8×C191C22semidirect
42. C4⋊C8×C191C22semidirect
43. #102×C191C22semidirect
46. #15690288D4semidirect
102. D764C2×D4semidirect
120. Q8×C191C2×D4semidirect
148. C4×C191AGL(1,ℤ/8)semidirect
154. Dic191AGL(1,ℤ/8)semidirect
162. C191#3038semidirect
169. #460D191semidirect
181. C2×SD16C2×D191semidirect
191. C4×Q8C2×D191semidirect
202. C4⋊C8C2×D191semidirect
206. #102C2×D191semidirect
261. Q8D4×D191semidirect
315. C4#20259349semidirect

Maximal subgroups

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

subgroupindex
C2×#202593492
#127×D1912
C4×#156902882
#271328632
#271328612
#271328592
#271328572
#271328552
#271326852
#271326662
#460×C1912
#3038191