Finite Group Databasebeta

← #37582137

Subgroups — #37582137

Order 24448.

548 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 548 conjugacy classes of subgroups — too many to draw. Only its 135 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×#20259348C211yesmaximal, characteristic, complemented
2C2×M16×D191C211yesmaximal, characteristic, complemented
3#27133175C211yesmaximal, characteristic, complemented
4C2×#20259293C211yesmaximal, characteristic, complemented
5#27132915C212yesmaximal, complemented
6#27132908C212yesmaximal, complemented
7#27132898C212yesmaximal, complemented
8#27132891C212yesmaximal, complemented
9#27132706C211yesmaximal, characteristic, complemented
10#27132665C211yesmaximal, characteristic, complemented
11#418×C191C211yesmaximal, characteristic, complemented
12C2×#15690287C2211yescharacteristic
13C22×C22⋊D191—21no—
14C22×C4×D191C2211yescharacteristic
15#20259348—24no—
16C2×#15690275C2212yescomplemented
17C2×#15690272C2212yescomplemented
18C2×#15690271C2211yescharacteristic
19C2×#15690270C2212yescomplemented
20M16×D191—24no—
21C2×C8×D191C2212yescomplemented
22#20259306—21no—
23#20259302C2211yescharacteristic, complemented
24#20259302C2211yescharacteristic, complemented
25#20259297C2211yescharacteristic, complemented
26#20259297C2211yescharacteristic, complemented
27#20259293—21no—
28#20259293—21no—
29#20259293C2212yes—
30#20259293C2212yes—
31#20259292—21no—
32C4×D764—21no—
33C4×D764—21no—
34C2×C4⋊Dic191C2211yescharacteristic
35C2×C22⋊Dic191—21no—
36#20259255C2212yescomplemented
37#20259253C2212yescomplemented
38#20259247C2212yescomplemented
39#20259244C2212yescomplemented
40C22×D4×C191C2211yescharacteristic
41C2×M16×C191C2211yescharacteristic
42#117×C191C2211yescharacteristic
43#102×C191C2212yescomplemented
44#102×C191C2212yescomplemented
45C23×D191D411yescharacteristic
46#15690287—22no—
47#15690287—22no—
48C2×Dic382C2311yescharacteristic
49C2×D764—21no—
50C2×D764—21no—
51C2×D764C2311yescharacteristic
52C2×C22⋊D191—22no—
53C2×C22⋊D191—22no—
54C2×C22⋊D191—21no—
55C2×C22⋊D191—44no—
56C22×Dic191—21no—
57C22×Dic191D411yescharacteristic
58C2×C4×D191—22no—
59C2×C4×D191—21no—
60C2×C4×D191—22no—
61C2×C4×D191D411yescharacteristic
62C2×C4×D191D411yescharacteristic
63C2×C4×D191C2311yescharacteristic
64C2×C4×D191C2311yescharacteristic
65#15690275—24no—
66#15690272—24no—
67#15690271D414yescomplemented
68#15690270—24no—
69C2×C191⋊C8C2312yes—
70C8×D191—24no—
71C4⋊Dic191—21no—
72C4⋊Dic191—21no—
73C4⋊Dic191—21no—
74C4⋊Dic191C2312yes—
75C22⋊Dic191—22no—
76C22⋊Dic191—22no—
77(C2×D191)⋊C4—21no—
78(C2×D191)⋊C4—21no—
79(C2×D191)⋊C4—21no—
80(C2×D191)⋊C4—21no—
81C24×C191—21no—
82C2×D4×C191—22no—
83C2×D4×C191—21no—
84C2×D4×C191—22no—
85C2×D4×C191C2312yes—
86C22×C4×C191C2311yescharacteristic
87M16×C191—22no—
88C2×C8×C191C2312yes—
89C4⋊C4×C191C2311yescharacteristic
90C4⋊C4×C191C2311yescharacteristic
91C22⋊C4×C191—21no—
92C42×C191—21no—
93C22×D191—22no—
94C22×D191—22no—
95C22×D191—21no—
96C22×D191—41no—
97C22×D191C2×D411yescharacteristic
98C22×D191C2×D411yescharacteristic
99Dic382—21no—
100Dic382—21no—
101D764—21no—
102D764—21no—
103D764—42no—
104D764—42no—
105C22⋊D191—44no—
106C22⋊D191—42no—
107C22⋊D191—44no—
108C2×Dic191—22no—
109C2×Dic191—21no—
110C2×Dic191—22no—
111C2×Dic191—42no—
112C2×Dic191C2×D411yescharacteristic
113C2×Dic191C2×D411yescharacteristic
114C4×D191—22no—
115C4×D191—22no—
116C4×D191—21no—
117C4×D191—21no—
118C4×D191—21no—
119C4×D191—21no—
120C4×D191—42no—
121C191⋊C8C2×D414yescomplemented
122C23×C191—22no—
123C23×C191—42no—
124C23×C191—42no—
125C23×C191—42no—
126C23×C191C2×D411yescharacteristic
127D4×C191—22no—
128D4×C191—22no—
129D4×C191—42no—
130C2×C4×C191—21no—
131C2×C4×C191—21no—
132C2×C4×C191C2×D411yescharacteristic
133C2×C4×C191C2×D412yes—
134C2×C4×C191#4112yes—
135C2×C4×C191C2411yescharacteristic
136C8×C191—22no—
137C22×C191—21no—
138C22×C191—21no—
139C22×C191—42no—
140C22×C191—42no—
141C22×C191—42no—
142C22×C191—84no—
143C22×C191#12112yes—
144C22×C191C22×D411yescharacteristic
145C2×D191—21no—
146C2×D191—21no—
147C2×D191—21no—
148C2×D191—21no—
149C2×D191—21no—
150C2×D191—44no—
151C2×D191—41no—
152C2×D191—41no—
153C2×D191—41no—
154C2×D191AGL(1,ℤ/8)12yescomplemented
155C4×C191—41no—
156C4×C191—41no—
157C4×C191#12112yes—
158C4×C191C22×D411yescharacteristic
159C4×C191C2×#4111yescharacteristic
160Dic191—21no—
161Dic191—21no—
162Dic191—41no—
163Dic191—42no—
164C2×C191—22no—
165C2×C191—82no—
166C2×C191C2×#12111yescharacteristic
167C2×C191C2×AGL(1,ℤ/8)11yescharacteristic
168C2×C191#57511yescharacteristic
169D191—22no—
170D191—41no—
171D191—81no—
172C191#305511yescharacteristic, complemented, Sylow
173#3055—1911nomaximal, Sylow
174C2×AGL(1,ℤ/8)—1911no—
175C22×M16—1911no—
176#525—1911no—
177C2×#121—1911no—
178#469—1912no—
179#468—1912no—
180#466—1912no—
181#465—1912no—
182#428—1911no—
183#418D19111yescharacteristic, complemented
184#417—1911no—
185C2×#41—1911no—
186C22×D4—3821no—
187C22×D4C2×D19111yescharacteristic
188C23×C4—1911no—
189AGL(1,ℤ/8)—3824no—
190C2×SD16—1912no—
191C2×D8—1912no—
192C2×M16—3824no—
193C2×M16—1911no—
194C2×M16—1912no—
195C2×M16C2×D19111yescharacteristic
196C22×C8—1912no—
197#123—3821no—
198C22⋊Q8—1911no—
199C22⋊Q8—1911no—
200#121—3821no—
201#121—3821no—
202#121—1912no—
203#121—1912no—
204#121—1911no—
205#121—1911no—
206#120—3821no—
207C4×D4—3821no—
208C4×D4—3821no—
209#117C2×D19111yescharacteristic
210C2×C4⋊C4—1911no—
211C2×C22⋊C4—3821no—
212#107—1912no—
213#106—1912no—
214#103—1912no—
215#102—1912no—
216#102C2×D19112yescomplemented
217#102C2×D19112yescomplemented
218C24—21no—
219C24—1911no—
220#41—3822no—
221#41—3822no—
222C2×Q8—1911no—
223C2×D4—22no—
224C2×D4—21no—
225C2×D4—22no—
226C2×D4—3821no—
227C2×D4—3821no—
228C2×D4—3822no—
229C2×D4—3822no—
230C2×D4—3821no—
231C2×D4—1911no—
232C2×D4—7644no—
233C2×D4C22×D19112yes—
234C22×C4—3822no—
235C22×C4—3822no—
236C22×C4—3821no—
237C22×C4—3821no—
238C22×C4—1911no—
239C22×C4—1911no—
240C22×C4—1911no—
241C22×C4—1911no—
242C22×C4—1911no—
243C22×C4C22×D19111yescharacteristic
244SD16—3824no—
245D8—3824no—
246M16—22no—
247M16—3824no—
248M16—1914no—
249C2×C8—3824no—
250C2×C8—1912no—
251C2×C8C22×D19112yes—
252C4⋊C4—3821no—
253C4⋊C4—3821no—
254C4⋊C4—3821no—
255C4⋊C4—1912no—
256C4⋊C4C22×D19111yescharacteristic
257C4⋊C4C22×D19111yescharacteristic
258C22⋊C4—21no—
259C22⋊C4—3821no—
260C22⋊C4—3821no—
261C22⋊C4—3821no—
262C22⋊C4—3821no—
263C22⋊C4—3822no—
264C22⋊C4—3822no—
265C42—21no—
266C23—22no—
267C23—42no—
268C23—42no—
269C23—42no—
270C23—3822no—
271C23—3822no—
272C23—3821no—
273C23—1911no—
274C23—1911no—
275C23—7641no—
276C23D4×D19111yescharacteristic
277Q8—3821no—
278Q8—3821no—
279D4—22no—
280D4—22no—
281D4—42no—
282D4—3821no—
283D4—3821no—
284D4—7642no—
285D4—7642no—
286D4—7644no—
287D4—7644no—
288D4—7642no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—3822no—
292C2×C4—3821no—
293C2×C4—3821no—
294C2×C4—3821no—
295C2×C4—3821no—
296C2×C4—3822no—
297C2×C4—3822no—
298C2×C4—3821no—
299C2×C4—3822no—
300C2×C4—1911no—
301C2×C4—1911no—
302C2×C4—7642no—
303C2×C4—7642no—
304C2×C4D4×D19112yes—
305C2×C4D4×D19111yescharacteristic
306C2×C4#1569028712yes—
307C2×C4C23×D19111yescharacteristic
308C8—22no—
309C8—1914no—
310C22—21no—
311C22—21no—
312C22—42no—
313C22—42no—
314C22—42no—
315C22—84no—
316C22—3821no—
317C22—3821no—
318C22—3821no—
319C22—3821no—
320C22—3821no—
321C22—1912no—
322C22—7644no—
323C22—7641no—
324C22—7641no—
325C22—7641no—
326C22#2025929512yes—
327C22C2×D4×D19111yescharacteristic
328C4—41no—
329C4—41no—
330C4—3821no—
331C4—3821no—
332C4—7642no—
333C4—7641no—
334C4#2025929512yes—
335C4C2×D4×D19111yescharacteristic
336C4C2×#1569028711yescharacteristic
337C2—22no—
338C2—82no—
339C2—3822no—
340C2—15281no—
341C2—7641no—
342C2C2×#2025929511yescharacteristic
343C2AGL(1,ℤ/8)×D19111yescharacteristic
344C2#2713345911yescharacteristic
subgroupG/H
centerC22C2×D4×D191
commutator, cosocleC2×C4×C191C24
FrattiniC2×C4C23×D191
Fitting#418×C191C2
socleC22×C191C22×D4
hypercenter#418D191
hypocenterC191#3055
LCSG ▷ C2×C4×C191 ▷ C2×C191 ▷ C191
UCSC1 ◁ C22 ◁ C22×C4 ◁ #418

Complements

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

NH ≅ G/Ntype
1. C2×#20259348C2semidirect
2. C2×M16×D191C2semidirect
3. #27133175C2semidirect
4. C2×#20259293C2semidirect
5. #27132915C2semidirect
6. #27132908C2semidirect
7. #27132898C2semidirect
8. #27132891C2semidirect
9. #27132706C2semidirect
10. #27132665C2semidirect
11. #418×C191C2semidirect
16. C2×#15690275C22semidirect
17. C2×#15690272C22semidirect
19. C2×#15690270C22semidirect
21. C2×C8×D191C22semidirect
23. #20259302C22semidirect
24. #20259302C22semidirect
25. #20259297C22semidirect
26. #20259297C22semidirect
36. #20259255C22semidirect
37. #20259253C22semidirect
38. #20259247C22semidirect
39. #20259244C22semidirect
43. #102×C191C22semidirect
44. #102×C191C22semidirect
67. #15690271D4semidirect
121. C191⋊C8C2×D4semidirect
154. C2×D191AGL(1,ℤ/8)semidirect
172. C191#3055semidirect
183. #418D191semidirect
216. #102C2×D191semidirect
217. #102C2×D191semidirect

Maximal subgroups

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

subgroupindex
C2×#202593482
C2×M16×D1912
#271331752
C2×#202592932
#271329152
#271329082
#271328982
#271328912
#271327062
#271326652
#418×C1912
#3055191