Finite Group Databasebeta

← #37575869

Subgroups — #37575869

Order 24448.

1232 conjugacy classes of subgroups; only the 171 normal subgroups are shown.

normal subgroups only

This group has 1232 conjugacy classes of subgroups — too many to draw. Only its 171 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
1C22×D4×D191C211yesmaximal, characteristic, complemented
2C2×#20259294C212yesmaximal, complemented
3C2×#20259290C212yesmaximal, complemented
4C2×(C22⋊D191)⋊C4C211yesmaximal, characteristic, complemented
5#27132581C211yesmaximal, characteristic, complemented
6#27132577C211yesmaximal, characteristic, complemented
7#27132572C211yesmaximal, characteristic, complemented
8#27132563C212yesmaximal, complemented
9#27132522C211yesmaximal, characteristic, complemented
10#27132518C212yesmaximal, complemented
11#392×C191C211yesmaximal, characteristic, complemented
12C24×D191C2212yes—
13C22×D764C2211yescharacteristic
14C22×C22⋊D191C2212yes—
15C22×C22⋊D191C2211yescharacteristic
16C22×C22⋊D191C2212yescomplemented
17C2×D4×D191—28no—
18C2×D4×D191—24no—
19C23×Dic191C2211yescharacteristic, complemented
20C22×C4×D191C2211yescharacteristic
21#20259294—28no—
22#20259290—28no—
23(C22⋊D191)⋊C4—24no—
24C2×Dic191⋊C4C2212yescomplemented
25C2×Dic191⋊C4C2211yescharacteristic, complemented
26C2×C22⋊Dic191C2212yescomplemented
27C2×C22⋊Dic191C2212yescomplemented
28C2×(C2×D191)⋊C4C2211yescharacteristic
29C2×(C2×D191)⋊C4C2212yes—
30C2×(C2×D191)⋊C4C2212yescomplemented
31C2×(C2×D191)⋊C4C2212yescomplemented
32C2×(C2×D191)⋊C4C2212yescomplemented
33C2×C4×Dic191C2211yescharacteristic, complemented
34C191⋊#95C2211yescharacteristic, complemented
35C22×D4×C191C2211yescharacteristic
36C22×D4×C191C2212yescomplemented
37C2×C22⋊C4×C191C2211yescharacteristic, complemented
38C2×C22⋊C4×C191C2212yescomplemented
39#95×C191C2211yescharacteristic, complemented
40C23×D191—24no—
41C23×D191—24no—
42C23×D191—24no—
43C23×D191—24no—
44C23×D191—24no—
45C23×D191—42no—
46C23×D191—42no—
47C23×D191C2312yes—
48C23×D191C2311yescharacteristic
49C2×D764—24no—
50C2×D764—22no—
51C2×C22⋊D191—22no—
52C2×C22⋊D191—22no—
53C2×C22⋊D191—24no—
54C2×C22⋊D191—24no—
55C2×C22⋊D191—24no—
56C2×C22⋊D191—24no—
57C2×C22⋊D191—42no—
58C2×C22⋊D191D418yescomplemented
59C2×C22⋊D191D414yescomplemented
60D4×D191—232no—
61C22×Dic191—22no—
62C22×Dic191—24no—
63C22×Dic191C2311yescharacteristic
64C22×Dic191C2312yes—
65C22×Dic191C2311yescharacteristic
66C22×Dic191C2311yescharacteristic
67C2×C4×D191—24no—
68C2×C4×D191D414yescomplemented
69Dic191⋊C4—24no—
70Dic191⋊C4—22no—
71C22⋊Dic191—24no—
72C22⋊Dic191—24no—
73(C2×D191)⋊C4—22no—
74(C2×D191)⋊C4—24no—
75(C2×D191)⋊C4—24no—
76(C2×D191)⋊C4—24no—
77(C2×D191)⋊C4—42no—
78C4×Dic191—22no—
79C24×C191C2312yes—
80C24×C191C2311yescharacteristic
81C2×D4×C191—24no—
82C2×D4×C191—24no—
83C2×D4×C191—24no—
84C2×D4×C191—22no—
85C2×D4×C191—42no—
86C22×C4×C191C2312yes—
87C22×C4×C191C2311yescharacteristic
88C22×C4×C191C2311yescharacteristic
89C22⋊C4×C191—22no—
90C22⋊C4×C191—42no—
91C22×D191—22no—
92C22×D191—22no—
93C22×D191—216no—
94C22×D191—22no—
95C22×D191—22no—
96C22×D191—22no—
97C22×D191—22no—
98C22×D191—22no—
99C22×D191—22no—
100C22×D191—48no—
101C22×D191—48no—
102C22×D191—48no—
103C22×D191—48no—
104C22×D191—48no—
105C22×D191—48no—
106C22×D191C2×D414yes—
107C22×D191C2×D412yes—
108C22×D191C2×D412yes—
109C22×D191#4112yes—
110D764—28no—
111C22⋊D191—28no—
112C22⋊D191—216no—
113C22⋊D191—48no—
114C2×Dic191—21no—
115C2×Dic191—21no—
116C2×Dic191—22no—
117C2×Dic191—22no—
118C2×Dic191—21no—
119C2×Dic191—22no—
120C2×Dic191—22no—
121C2×Dic191—44no—
122C2×Dic191C2×D412yes—
123C2×Dic191C2×D414yes—
124C2×Dic191C2×D412yescomplemented
125C4×D191—28no—
126C23×C191—21no—
127C23×C191—22no—
128C23×C191—22no—
129C23×C191—22no—
130C23×C191—22no—
131C23×C191—21no—
132C23×C191—22no—
133C23×C191—22no—
134C23×C191—22no—
135C23×C191—22no—
136C23×C191C2×D414yes—
137C23×C191C2×D412yescomplemented
138C23×C191C2411yescharacteristic
139D4×C191—28no—
140D4×C191—48no—
141C2×C4×C191—22no—
142C2×C4×C191—21no—
143C2×C4×C191—22no—
144C2×C4×C191—22no—
145C2×C4×C191—22no—
146C2×C4×C191—22no—
147C2×C4×C191C2×D412yes—
148C22×C191—24no—
149C22×C191—22no—
150C22×C191—24no—
151C22×C191—24no—
152C22×C191—24no—
153C22×C191—41no—
154C22×C191—42no—
155C22×C191—41no—
156C22×C191—42no—
157C22×C191—42no—
158C22×C191—42no—
159C22×C191—42no—
160C22×C191C22×D411yescharacteristic
161C22×C191C22×D412yes—
162C22×C191C22×D411yescharacteristic
163C22×C191C2×#4111yescharacteristic
164C22×C191H(5,𝔽2)12yes—
165C2×D191—22no—
166C2×D191—22no—
167C2×D191—24no—
168C2×D191—24no—
169C2×D191—24no—
170C2×D191—24no—
171C2×D191—24no—
172C2×D191—432no—
173C2×D191—42no—
174C2×D191—42no—
175C2×D191—42no—
176C2×D191—42no—
177C2×D191#12114yescomplemented
178C4×C191—22no—
179C4×C191—41no—
180C4×C191—42no—
181Dic191—22no—
182Dic191—41no—
183Dic191—42no—
184Dic191#12014yescomplemented
185C2×C191—44no—
186C2×C191—42no—
187C2×C191C2×#12011yescharacteristic
188C2×C191C2×#12111yescharacteristic
189C2×C191#53511yescharacteristic
190C2×C191D4212yes—
191C2×C191#54612yes—
192D191—24no—
193D191—44no—
194C191#240811yescharacteristic, complemented, Sylow
195#2408—1911nomaximal, Sylow
196C23×D4—1911no—
197C2×#121—1912no—
198C2×#120—1912no—
199C2×C4×D4—1911no—
200#394—1911no—
201#394—1911no—
202#392—1911no—
203#392D19111yescharacteristic, complemented
204#390—1912no—
205#380—1911no—
206#379—1912no—
207C25—1912no—
208C22×D4—3828no—
209C22×D4—3824no—
210C22×D4—1911no—
211C22×D4—1912no—
212C22×D4—1912no—
213C22×D4—1911no—
214C22×D4C2×D19111yescharacteristic
215C22×D4C2×D19112yescomplemented
216C23×C4—1911no—
217C23×C4—1911no—
218#121—3828no—
219#120—3828no—
220C4×D4—3824no—
221C2×C4⋊C4—1911no—
222C2×C4⋊C4—1912no—
223C2×C22⋊C4—1912no—
224C2×C22⋊C4—1912no—
225C2×C22⋊C4—1911no—
226C2×C22⋊C4—1912no—
227C2×C22⋊C4—1912no—
228C2×C22⋊C4—1912no—
229C2×C22⋊C4—1912no—
230C2×C22⋊C4C2×D19111yescharacteristic, complemented
231C2×C22⋊C4C2×D19112yescomplemented
232C2×C42—1911no—
233#95—1911no—
234#95C2×D19111yescharacteristic, complemented
235C24—3824no—
236C24—3824no—
237C24—3824no—
238C24—3824no—
239C24—3824no—
240C24—1912no—
241C24—1911no—
242C24—7642no—
243C24—7642no—
244C24C22×D19112yes—
245C24C22×D19111yescharacteristic
246C2×D4—24no—
247C2×D4—24no—
248C2×D4—24no—
249C2×D4—22no—
250C2×D4—42no—
251C2×D4—3824no—
252C2×D4—38232no—
253C2×D4—3822no—
254C2×D4—3822no—
255C2×D4—3824no—
256C2×D4—3824no—
257C2×D4—3824no—
258C2×D4—3824no—
259C2×D4—3822no—
260C2×D4—1918no—
261C2×D4—1914no—
262C2×D4—7642no—
263C22×C4—3824no—
264C22×C4—3822no—
265C22×C4—3824no—
266C22×C4—1912no—
267C22×C4—1911no—
268C22×C4—1911no—
269C22×C4—1914no—
270C22×C4—1911no—
271C22×C4C22×D19112yes—
272C22×C4C22×D19111yescharacteristic
273C22×C4C22×D19111yescharacteristic
274C4⋊C4—3824no—
275C4⋊C4—3822no—
276C22⋊C4—22no—
277C22⋊C4—42no—
278C22⋊C4—3822no—
279C22⋊C4—3824no—
280C22⋊C4—3824no—
281C22⋊C4—3824no—
282C22⋊C4—3824no—
283C22⋊C4—3824no—
284C22⋊C4—7642no—
285C42—3822no—
286C23—21no—
287C23—22no—
288C23—22no—
289C23—22no—
290C23—22no—
291C23—21no—
292C23—22no—
293C23—22no—
294C23—22no—
295C23—22no—
296C23—38216no—
297C23—3822no—
298C23—3822no—
299C23—3822no—
300C23—3822no—
301C23—3822no—
302C23—3822no—
303C23—3822no—
304C23—3822no—
305C23—1914no—
306C23—1912no—
307C23—1912no—
308C23—1912no—
309C23—7648no—
310C23—7648no—
311C23—7648no—
312C23—7648no—
313C23—7648no—
314C23—7648no—
315C23D4×D19114yes—
316C23D4×D19112yescomplemented
317C23C23×D19111yescharacteristic
318D4—28no—
319D4—48no—
320D4—3828no—
321D4—3828no—
322D4—38216no—
323D4—7648no—
324C2×C4—22no—
325C2×C4—21no—
326C2×C4—22no—
327C2×C4—22no—
328C2×C4—22no—
329C2×C4—22no—
330C2×C4—3828no—
331C2×C4—3821no—
332C2×C4—3821no—
333C2×C4—3822no—
334C2×C4—3822no—
335C2×C4—3821no—
336C2×C4—3822no—
337C2×C4—3822no—
338C2×C4—1912no—
339C2×C4—1912no—
340C2×C4—1914no—
341C2×C4—7644no—
342C2×C4D4×D19112yes—
343C22—24no—
344C22—24no—
345C22—24no—
346C22—24no—
347C22—22no—
348C22—42no—
349C22—41no—
350C22—42no—
351C22—42no—
352C22—42no—
353C22—41no—
354C22—42no—
355C22—3822no—
356C22—3824no—
357C22—3824no—
358C22—3822no—
359C22—3824no—
360C22—3824no—
361C22—3824no—
362C22—1914no—
363C22—76432no—
364C22—7642no—
365C22—7642no—
366C22—7642no—
367C22—7642no—
368C22C2×D4×D19111yescharacteristic
369C22C2×D4×D19111yescharacteristic
370C22C2×D4×D19112yes—
371C22#41×D19111yescharacteristic
372C22#2025936712yes—
373C4—22no—
374C4—41no—
375C4—42no—
376C4—3822no—
377C4—1914no—
378C4—7642no—
379C4—7641no—
380C2—44no—
381C2—42no—
382C2—3824no—
383C2—7644no—
384C2#120×D19111yescharacteristic
385C2#121×D19111yescharacteristic
386C2#2713322611yescharacteristic
387C2#2713329512yes—
388C2#2713330212yes—
subgroupG/H
center, FrattiniC23C23×D191
commutator, socle, cosocleC23×C191C24
Fitting#392×C191C2
hypercenter#392D191
hypocenterC191#2408
LCSG ▷ C23×C191 ▷ C191
UCSC1 ◁ C23 ◁ #392

Complements

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

NH ≅ G/Ntype
1. C22×D4×D191C2semidirect
2. C2×#20259294C2semidirect
3. C2×#20259290C2semidirect
4. C2×(C22⋊D191)⋊C4C2semidirect
5. #27132581C2semidirect
6. #27132577C2semidirect
7. #27132572C2semidirect
8. #27132563C2semidirect
9. #27132522C2semidirect
10. #27132518C2semidirect
11. #392×C191C2semidirect
16. C22×C22⋊D191C22semidirect
19. C23×Dic191C22semidirect
24. C2×Dic191⋊C4C22semidirect
25. C2×Dic191⋊C4C22semidirect
26. C2×C22⋊Dic191C22semidirect
27. C2×C22⋊Dic191C22semidirect
30. C2×(C2×D191)⋊C4C22semidirect
31. C2×(C2×D191)⋊C4C22semidirect
32. C2×(C2×D191)⋊C4C22semidirect
33. C2×C4×Dic191C22semidirect
34. C191⋊#95C22semidirect
36. C22×D4×C191C22semidirect
37. C2×C22⋊C4×C191C22semidirect
38. C2×C22⋊C4×C191C22semidirect
39. #95×C191C22semidirect
58. C2×C22⋊D191D4semidirect
59. C2×C22⋊D191D4semidirect
68. C2×C4×D191D4semidirect
124. C2×Dic191C2×D4semidirect
137. C23×C191C2×D4semidirect
177. C2×D191#121semidirect
184. Dic191#120semidirect
194. C191#2408semidirect
203. #392D191semidirect
215. C22×D4C2×D191semidirect
230. C2×C22⋊C4C2×D191semidirect
231. C2×C22⋊C4C2×D191semidirect
234. #95C2×D191semidirect
316. C23D4×D191semidirect

Maximal subgroups

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

subgroupindex
C22×D4×D1912
C2×#202592942
C2×#202592902
C2×(C22⋊D191)⋊C42
#271325812
#271325772
#271325722
#271325632
#271325222
#271325182
#392×C1912
#2408191