Finite Group Databasebeta

← #37576166

Subgroups — #37576166

Order 24448.

1088 conjugacy classes of subgroups; only the 161 normal subgroups are shown.

normal subgroups only

This group has 1088 conjugacy classes of subgroups — too many to draw. Only its 161 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
2C2×#20259308C212yesmaximal, complemented
3C2×#20259300C211yesmaximal, characteristic, complemented
4#27132581C211yesmaximal, characteristic, complemented
5#27132580C211yesmaximal, characteristic, complemented
6#27132577C211yesmaximal, characteristic, complemented
7#27132563C212yesmaximal, complemented
8#27132518C212yesmaximal, complemented
9#27132518C212yesmaximal, complemented
10#27132510C211yesmaximal, characteristic, complemented
11#394×C191C211yesmaximal, characteristic, complemented
12C24×D191C2212yes—
13C22×Dic382C2211yescharacteristic, complemented
14C22×D764C2211yescharacteristic
15C22×C22⋊D191C2212yes—
16C2×D4×D191—28no—
17C2×D4×D191—24no—
18C22×C4×D191C2211yescharacteristic
19#20259308—28no—
20#20259300—24no—
21C2×Dic191⋊C4C2211yescharacteristic, complemented
22C2×Dic191⋊C4C2211yescharacteristic, complemented
23C2×C22⋊Dic191C2212yescomplemented
24C2×C22⋊Dic191C2212yescomplemented
25C2×(C2×D191)⋊C4C2212yes—
26C2×(C2×D191)⋊C4C2211yescharacteristic
27C2×(C2×D191)⋊C4C2211yescharacteristic
28C2×(C2×D191)⋊C4C2212yescomplemented
29C2×(C2×D191)⋊C4C2212yescomplemented
30C2×(C2×D191)⋊C4C2212yescomplemented
31C2×(C2×D191)⋊C4C2212yescomplemented
32C2×C4×Dic191C2212yescomplemented
33C191⋊#95C2211yescharacteristic, complemented
34C22×D4×C191C2211yescharacteristic
35C2×C4⋊C4×C191C2211yescharacteristic, complemented
36C2×C22⋊C4×C191C2212yescomplemented
37C2×C22⋊C4×C191C2212yescomplemented
38#95×C191C2211yescharacteristic, complemented
39C23×D191—24no—
40C23×D191—24no—
41C23×D191—24no—
42C23×D191—24no—
43C23×D191—24no—
44C23×D191—42no—
45C23×D191—42no—
46C23×D191C2312yes—
47C23×D191C2311yescharacteristic
48C2×Dic382—22no—
49C2×Dic382—24no—
50C2×D764—24no—
51C2×D764—22no—
52C2×C22⋊D191—24no—
53C2×C22⋊D191—24no—
54C2×C22⋊D191D418yes—
55D4×D191—232no—
56C22×Dic191C2311yescharacteristic
57C22×Dic191C2312yes—
58C22×Dic191C2311yescharacteristic
59C22×Dic191C2311yescharacteristic
60C2×C4×D191—24no—
61C2×C4×D191D414yes—
62Dic191⋊C4—22no—
63Dic191⋊C4—22no—
64C22⋊Dic191—24no—
65C22⋊Dic191—42no—
66(C2×D191)⋊C4—22no—
67(C2×D191)⋊C4—22no—
68(C2×D191)⋊C4—24no—
69(C2×D191)⋊C4—24no—
70(C2×D191)⋊C4—24no—
71(C2×D191)⋊C4—42no—
72(C2×D191)⋊C4—42no—
73C4×Dic191—24no—
74C24×C191C2312yes—
75C2×D4×C191—24no—
76C2×D4×C191—22no—
77C22×C4×C191C2311yescharacteristic
78C22×C4×C191C2312yes—
79C22×C4×C191C2311yescharacteristic
80C22×C4×C191C2311yescharacteristic
81C4⋊C4×C191—22no—
82C22⋊C4×C191—24no—
83C22⋊C4×C191—42no—
84C22×D191—22no—
85C22×D191—21no—
86C22×D191—21no—
87C22×D191—216no—
88C22×D191—22no—
89C22×D191—22no—
90C22×D191—22no—
91C22×D191—22no—
92C22×D191—22no—
93C22×D191—22no—
94C22×D191—48no—
95C22×D191—48no—
96C22×D191—48no—
97C22×D191—48no—
98C22×D191—48no—
99C22×D191—48no—
100C22×D191C2×D414yes—
101C22×D191C2×D412yes—
102C22×D191#4114yes—
103Dic382—44no—
104D764—28no—
105C22⋊D191—216no—
106C2×Dic191—22no—
107C2×Dic191—22no—
108C2×Dic191—21no—
109C2×Dic191—21no—
110C2×Dic191—22no—
111C2×Dic191—21no—
112C2×Dic191—22no—
113C2×Dic191—21no—
114C2×Dic191C2×D414yes—
115C2×Dic191C2×D412yes—
116C4×D191—28no—
117C23×C191—22no—
118C23×C191—22no—
119C23×C191—22no—
120C23×C191—22no—
121C23×C191—22no—
122C23×C191—22no—
123C23×C191C2×D414yes—
124C23×C191C2411yescharacteristic
125D4×C191—28no—
126C2×C4×C191—22no—
127C2×C4×C191—22no—
128C2×C4×C191—22no—
129C2×C4×C191—21no—
130C2×C4×C191—21no—
131C2×C4×C191—22no—
132C2×C4×C191—21no—
133C2×C4×C191—22no—
134C2×C4×C191—21no—
135C2×C4×C191C2×D412yes—
136C22×C191—24no—
137C22×C191—24no—
138C22×C191—24no—
139C22×C191—42no—
140C22×C191—42no—
141C22×C191—42no—
142C22×C191—42no—
143C22×C191C22×D411yescharacteristic
144C22×C191C22×D412yes—
145C22×C191C2×#4112yes—
146C22×C191H(5,𝔽2)11yescharacteristic
147C22×C191H(5,𝔽2)11yescharacteristic
148C2×D191—22no—
149C2×D191—22no—
150C2×D191—24no—
151C2×D191—24no—
152C2×D191—24no—
153C2×D191—24no—
154C2×D191—24no—
155C2×D191—432no—
156C2×D191—42no—
157C2×D191—42no—
158C2×D191—42no—
159C2×D191—42no—
160C2×D191#12414yescomplemented
161C4×C191—22no—
162C4×C191—41no—
163C4×C191—41no—
164C4×C191—42no—
165Dic191—41no—
166Dic191—41no—
167Dic191—42no—
168Dic191#12014yes—
169C2×C191—44no—
170C2×C191C2×#12011yescharacteristic
171C2×C191C2×#12411yescharacteristic
172C2×C191#53511yescharacteristic
173C2×C191#54612yes—
174C2×C191#54612yes—
175D191—24no—
176D191—44no—
177C191#243511yescharacteristic, complemented, Sylow
178#2435—1911nomaximal, Sylow
179C23×D4—1911no—
180C2×#124—1912no—
181C2×C22⋊Q8—1911no—
182#394—1911no—
183#394—1911no—
184#394—1911no—
185#394D19111yescharacteristic, complemented
186#390—1912no—
187#379—1912no—
188#379—1912no—
189C4×C22⋊C4—1911no—
190C25—1912no—
191C22×Q8—1911no—
192C22×D4—3828no—
193C22×D4—3824no—
194C22×D4—1912no—
195C22×D4—1911no—
196C22×D4C2×D19111yescharacteristic
197C23×C4—1911no—
198#124—3828no—
199C22⋊Q8—3824no—
200C2×C4⋊C4—1911no—
201C2×C4⋊C4—1911no—
202C2×C4⋊C4C2×D19111yescharacteristic, complemented
203C2×C22⋊C4—1912no—
204C2×C22⋊C4—1911no—
205C2×C22⋊C4—1911no—
206C2×C22⋊C4—1912no—
207C2×C22⋊C4—1912no—
208C2×C22⋊C4—1912no—
209C2×C22⋊C4—1912no—
210C2×C22⋊C4—1912no—
211C2×C22⋊C4—1912no—
212C2×C22⋊C4C2×D19112yescomplemented
213C2×C22⋊C4C2×D19112yescomplemented
214C2×C42—1912no—
215#95—1911no—
216#95C2×D19111yescharacteristic, complemented
217C24—3824no—
218C24—3824no—
219C24—3824no—
220C24—3824no—
221C24—3824no—
222C24—1912no—
223C24—1911no—
224C24—7642no—
225C24—7642no—
226C24C22×D19112yes—
227C2×Q8—3822no—
228C2×Q8—3824no—
229C2×D4—24no—
230C2×D4—22no—
231C2×D4—3824no—
232C2×D4—38232no—
233C2×D4—3824no—
234C2×D4—3824no—
235C2×D4—3822no—
236C2×D4—1918no—
237C22×C4—3824no—
238C22×C4—1912no—
239C22×C4—1911no—
240C22×C4—1911no—
241C22×C4—1914no—
242C22×C4—1911no—
243C22×C4C22×D19112yes—
244C22×C4C22×D19111yescharacteristic
245C22×C4C22×D19111yescharacteristic
246C22×C4C22×D19111yescharacteristic
247C4⋊C4—22no—
248C4⋊C4—3822no—
249C4⋊C4—3822no—
250C22⋊C4—24no—
251C22⋊C4—42no—
252C22⋊C4—3824no—
253C22⋊C4—3822no—
254C22⋊C4—3822no—
255C22⋊C4—3824no—
256C22⋊C4—3824no—
257C22⋊C4—3824no—
258C22⋊C4—7642no—
259C22⋊C4—7642no—
260C22⋊C4—7642no—
261C42—3824no—
262C23—22no—
263C23—22no—
264C23—22no—
265C23—22no—
266C23—22no—
267C23—22no—
268C23—38216no—
269C23—3822no—
270C23—3821no—
271C23—3821no—
272C23—3822no—
273C23—3822no—
274C23—3822no—
275C23—3822no—
276C23—3822no—
277C23—3822no—
278C23—1914no—
279C23—1912no—
280C23—1914no—
281C23—7648no—
282C23—7648no—
283C23—7648no—
284C23—7648no—
285C23—7648no—
286C23—7648no—
287C23D4×D19114yes—
288C23C23×D19111yescharacteristic
289Q8—7644no—
290D4—28no—
291D4—3828no—
292D4—38216no—
293C2×C4—22no—
294C2×C4—22no—
295C2×C4—22no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—22no—
299C2×C4—21no—
300C2×C4—22no—
301C2×C4—21no—
302C2×C4—3828no—
303C2×C4—3822no—
304C2×C4—3822no—
305C2×C4—3821no—
306C2×C4—3821no—
307C2×C4—3822no—
308C2×C4—3821no—
309C2×C4—3821no—
310C2×C4—3822no—
311C2×C4—1914no—
312C2×C4—1912no—
313C2×C4D4×D19112yes—
314C22—24no—
315C22—24no—
316C22—24no—
317C22—42no—
318C22—42no—
319C22—42no—
320C22—42no—
321C22—3822no—
322C22—3824no—
323C22—3824no—
324C22—3822no—
325C22—3824no—
326C22—3824no—
327C22—3824no—
328C22—1914no—
329C22—76432no—
330C22—7642no—
331C22—7642no—
332C22—7642no—
333C22—7642no—
334C22C2×D4×D19111yescharacteristic
335C22C2×D4×D19112yes—
336C22#41×D19112yes—
337C22#2025936711yescharacteristic
338C22#2025936811yescharacteristic
339C4—22no—
340C4—41no—
341C4—42no—
342C4—41no—
343C4—1914no—
344C4—7641no—
345C4—7642no—
346C4—7641no—
347C2—44no—
348C2—3824no—
349C2—7644no—
350C2#120×D19111yescharacteristic
351C2#124×D19111yescharacteristic
352C2#2713322911yescharacteristic
353C2#2713330212yes—
354C2#2713330512yes—
subgroupG/H
center, FrattiniC23C23×D191
commutator, socle, cosocleC23×C191C24
Fitting#394×C191C2
prime radicalC22×D4×D191C2
hypercenter#394D191
hypocenterC191#2435
LCSG ▷ C23×C191 ▷ C191
UCSC1 ◁ C23 ◁ #394

Complements

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

NH ≅ G/Ntype
2. C2×#20259308C2semidirect
3. C2×#20259300C2semidirect
4. #27132581C2semidirect
5. #27132580C2semidirect
6. #27132577C2semidirect
7. #27132563C2semidirect
8. #27132518C2semidirect
9. #27132518C2semidirect
10. #27132510C2semidirect
11. #394×C191C2semidirect
13. C22×Dic382C22semidirect
21. C2×Dic191⋊C4C22semidirect
22. C2×Dic191⋊C4C22semidirect
23. C2×C22⋊Dic191C22semidirect
24. C2×C22⋊Dic191C22semidirect
28. C2×(C2×D191)⋊C4C22semidirect
29. C2×(C2×D191)⋊C4C22semidirect
30. C2×(C2×D191)⋊C4C22semidirect
31. C2×(C2×D191)⋊C4C22semidirect
32. C2×C4×Dic191C22semidirect
33. C191⋊#95C22semidirect
35. C2×C4⋊C4×C191C22semidirect
36. C2×C22⋊C4×C191C22semidirect
37. C2×C22⋊C4×C191C22semidirect
38. #95×C191C22semidirect
160. C2×D191#124semidirect
177. C191#2435semidirect
185. #394D191semidirect
202. C2×C4⋊C4C2×D191semidirect
212. C2×C22⋊C4C2×D191semidirect
213. C2×C22⋊C4C2×D191semidirect
216. #95C2×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×#202593082
C2×#202593002
#271325812
#271325802
#271325772
#271325632
#271325182
#271325102
#394×C1912
#2435191