Finite Group Databasebeta

← #37582142

Subgroups — #37582142

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×#20259347C211yesmaximal, characteristic, complemented
2C2×M16×D191C211yesmaximal, characteristic, complemented
3#27133170C211yesmaximal, characteristic, complemented
4C2×#20259297C211yesmaximal, characteristic, complemented
5#27132915C212yesmaximal, complemented
6#27132912C212yesmaximal, complemented
7#27132898C212yesmaximal, complemented
8#27132895C212yesmaximal, complemented
9#27132703C211yesmaximal, characteristic, complemented
10#27132669C211yesmaximal, characteristic, complemented
11#417×C191C211yesmaximal, characteristic, complemented
12C22×D764—21no—
13C22×D764C2211yescharacteristic
14C22×C4×D191C2211yescharacteristic
15#20259347—24no—
16C2×#15690274C2212yescomplemented
17C2×#15690272C2212yescomplemented
18C2×#15690271C2211yescharacteristic
19C2×#15690270C2212yescomplemented
20M16×D191—24no—
21C2×C8×D191C2212yescomplemented
22#20259303—21no—
23#20259298C2211yescharacteristic, complemented
24#20259298C2211yescharacteristic, complemented
25#20259297—21no—
26#20259297—21no—
27#20259297C2212yes—
28#20259297C2212yes—
29#20259293C2211yescharacteristic, complemented
30#20259293C2211yescharacteristic, complemented
31#20259290—21no—
32C4×C22⋊D191—21no—
33C4×C22⋊D191—21no—
34#20259281C2211yescharacteristic
35C2×(C2×D191)⋊C4—21no—
36#20259255C2212yescomplemented
37#20259253C2212yescomplemented
38#20259244C2212yescomplemented
39#20259244C2212yescomplemented
40C2×#41×C191C2211yescharacteristic
41C2×M16×C191C2211yescharacteristic
42C2×C4⋊C4×C191C2211yescharacteristic
43#103×C191C2212yescomplemented
44#102×C191C2212yescomplemented
45C23×D191—21no—
46C23×D191D411yescharacteristic
47C2×D764—22no—
48C2×D764—22no—
49C2×D764—22no—
50C2×D764—21no—
51C2×D764—22no—
52C2×D764—44no—
53C2×D764C2312yes—
54C2×C22⋊D191—21no—
55C2×C22⋊D191—21no—
56C22×Dic191D411yescharacteristic
57C2×C4×D191—22no—
58C2×C4×D191—22no—
59C2×C4×D191D411yescharacteristic
60C2×C4×D191D411yescharacteristic
61C2×C4×D191C2311yescharacteristic
62C2×C4×D191C2311yescharacteristic
63#15690274—24no—
64#15690272—24no—
65#15690271D414yescomplemented
66#15690270—24no—
67C2×C191⋊C8C2312yes—
68C8×D191—24no—
69C4⋊Dic191C2311yescharacteristic
70C4⋊Dic191C2311yescharacteristic
71Dic191⋊C4—21no—
72Dic191⋊C4—21no—
73C22⋊Dic191—21no—
74C22⋊Dic191—21no—
75C22⋊Dic191—21no—
76(C2×D191)⋊C4—21no—
77(C2×D191)⋊C4—21no—
78(C2×D191)⋊C4—22no—
79(C2×D191)⋊C4—22no—
80C4×Dic191—21no—
81#41×C191—22no—
82#41×C191—22no—
83C2×Q8×C191C2311yescharacteristic
84C2×D4×C191—21no—
85C2×D4×C191C2311yescharacteristic
86C22×C4×C191—21no—
87C22×C4×C191—21no—
88C22×C4×C191C2311yescharacteristic
89M16×C191—22no—
90C2×C8×C191C2312yes—
91C4⋊C4×C191—21no—
92C4⋊C4×C191C2312yes—
93C22×D191—22no—
94C22×D191—22no—
95C22×D191—22no—
96C22×D191—41no—
97C22×D191—42no—
98C22×D191—42no—
99C22×D191—42no—
100C22×D191C2×D411yescharacteristic
101C22×D191C2×D411yescharacteristic
102D764—22no—
103D764—22no—
104D764—44no—
105D764—44no—
106D764—42no—
107C22⋊D191—42no—
108C22⋊D191—42no—
109C2×Dic191—22no—
110C2×Dic191—21no—
111C2×Dic191—21no—
112C2×Dic191C2×D411yescharacteristic
113C2×Dic191C2×D411yescharacteristic
114C4×D191—22no—
115C4×D191—22no—
116C4×D191—21no—
117C4×D191—21no—
118C4×D191—21no—
119C4×D191—21no—
120C191⋊C8C2×D414yescomplemented
121C23×C191—21no—
122C23×C191C2×D411yescharacteristic
123Q8×C191—21no—
124Q8×C191—21no—
125D4×C191—21no—
126D4×C191—21no—
127D4×C191—42no—
128C2×C4×C191—22no—
129C2×C4×C191—21no—
130C2×C4×C191—42no—
131C2×C4×C191—42no—
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—41no—
140C22×C191—41no—
141C22×C191—41no—
142C22×C191#12112yes—
143C22×C191C22×D411yescharacteristic
144C2×D191—21no—
145C2×D191—21no—
146C2×D191—21no—
147C2×D191—21no—
148C2×D191—21no—
149C2×D191—44no—
150C2×D191—42no—
151C2×D191—42no—
152C2×D191—42no—
153C2×D191—84no—
154C2×D191AGL(1,ℤ/8)12yescomplemented
155C4×C191—41no—
156C4×C191—42no—
157C4×C191#12112yes—
158C4×C191C22×D411yescharacteristic
159C4×C191C2×#4111yescharacteristic
160Dic191—21no—
161Dic191—21no—
162Dic191—41no—
163Dic191—41no—
164C2×C191—22no—
165C2×C191—81no—
166C2×C191C2×#12111yescharacteristic
167C2×C191C2×AGL(1,ℤ/8)11yescharacteristic
168C2×C191#57511yescharacteristic
169D191—22no—
170D191—41no—
171D191—82no—
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#418—1911no—
184#417D19111yescharacteristic, complemented
185C2×#41C2×D19111yescharacteristic
186C22×D4—3821no—
187C22×D4—1911no—
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—1911no—
203#121—1911no—
204#121—1912no—
205#121—1912no—
206#120—3821no—
207C4×D4—3821no—
208C4×D4—3821no—
209#117—1911no—
210C2×C4⋊C4C2×D19111yescharacteristic
211C2×C22⋊C4—3821no—
212#107—1912no—
213#106—1912no—
214#103C2×D19112yescomplemented
215#102—1912no—
216#102—1912no—
217#102C2×D19112yescomplemented
218C24—3821no—
219C24—1911no—
220#41—22no—
221#41—22no—
222C2×Q8C22×D19111yescharacteristic
223C2×D4—21no—
224C2×D4—3822no—
225C2×D4—3822no—
226C2×D4—3821no—
227C2×D4—3821no—
228C2×D4—3822no—
229C2×D4—3821no—
230C2×D4—3822no—
231C2×D4—1912no—
232C2×D4—7644no—
233C2×D4C22×D19111yescharacteristic
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—3822no—
237C22×C4—3822no—
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—21no—
253C4⋊C4—3821no—
254C4⋊C4—3821no—
255C4⋊C4—1911no—
256C4⋊C4—1911no—
257C4⋊C4C22×D19112yes—
258C22⋊C4—3821no—
259C22⋊C4—3821no—
260C22⋊C4—3822no—
261C22⋊C4—3822no—
262C22⋊C4—3821no—
263C22⋊C4—3821no—
264C22⋊C4—3821no—
265C42—3821no—
266C23—21no—
267C23—3822no—
268C23—3822no—
269C23—3822no—
270C23—1911no—
271C23—1911no—
272C23—7641no—
273C23—7642no—
274C23—7642no—
275C23—7642no—
276C23D4×D19111yescharacteristic
277Q8—21no—
278Q8—21no—
279D4—21no—
280D4—21no—
281D4—42no—
282D4—3822no—
283D4—3822no—
284D4—7644no—
285D4—7644no—
286D4—7642no—
287D4—7642no—
288D4—7642no—
289C2×C4—22no—
290C2×C4—21no—
291C2×C4—42no—
292C2×C4—42no—
293C2×C4—3822no—
294C2×C4—3821no—
295C2×C4—3821no—
296C2×C4—3821no—
297C2×C4—3821no—
298C2×C4—3822no—
299C2×C4—3822no—
300C2×C4—3821no—
301C2×C4—3821no—
302C2×C4—1911no—
303C2×C4—1911no—
304C2×C4D4×D19112yes—
305C2×C4D4×D19111yescharacteristic
306C2×C4#1569028712yes—
307C2×C4C23×D19111yescharacteristic
308C8—22no—
309C8—1914no—
310C22—21no—
311C22—21no—
312C22—41no—
313C22—41no—
314C22—41no—
315C22—3821no—
316C22—3821no—
317C22—3821no—
318C22—3821no—
319C22—3821no—
320C22—1912no—
321C22—15284no—
322C22—7644no—
323C22—7642no—
324C22—7642no—
325C22—7642no—
326C22#2025929512yes—
327C22C2×D4×D19111yescharacteristic
328C4—42no—
329C4—41no—
330C4—3821no—
331C4—3821no—
332C4—7641no—
333C4—7641no—
334C4#2025929512yes—
335C4C2×D4×D19111yescharacteristic
336C4C2×#1569028711yescharacteristic
337C2—22no—
338C2—81no—
339C2—3822no—
340C2—15282no—
341C2—7641no—
342C2C2×#2025929511yescharacteristic
343C2AGL(1,ℤ/8)×D19111yescharacteristic
344C2#2713346211yescharacteristic
subgroupG/H
centerC22C2×D4×D191
commutator, cosocleC2×C4×C191C24
FrattiniC2×C4C23×D191
Fitting#417×C191C2
socleC22×C191C22×D4
hypercenter#417D191
hypocenterC191#3055
LCSG ▷ C2×C4×C191 ▷ C2×C191 ▷ C191
UCSC1 ◁ C22 ◁ C22×C4 ◁ #417

Complements

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

NH ≅ G/Ntype
1. C2×#20259347C2semidirect
2. C2×M16×D191C2semidirect
3. #27133170C2semidirect
4. C2×#20259297C2semidirect
5. #27132915C2semidirect
6. #27132912C2semidirect
7. #27132898C2semidirect
8. #27132895C2semidirect
9. #27132703C2semidirect
10. #27132669C2semidirect
11. #417×C191C2semidirect
16. C2×#15690274C22semidirect
17. C2×#15690272C22semidirect
19. C2×#15690270C22semidirect
21. C2×C8×D191C22semidirect
23. #20259298C22semidirect
24. #20259298C22semidirect
29. #20259293C22semidirect
30. #20259293C22semidirect
36. #20259255C22semidirect
37. #20259253C22semidirect
38. #20259244C22semidirect
39. #20259244C22semidirect
43. #103×C191C22semidirect
44. #102×C191C22semidirect
65. #15690271D4semidirect
120. C191⋊C8C2×D4semidirect
154. C2×D191AGL(1,ℤ/8)semidirect
172. C191#3055semidirect
184. #417D191semidirect
214. #103C2×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×#202593472
C2×M16×D1912
#271331702
C2×#202592972
#271329152
#271329122
#271328982
#271328952
#271327032
#271326692
#417×C1912
#3055191