Finite Group Databasebeta

← #37583269

Subgroups — #37583269

Order 24448.

520 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 520 conjugacy classes of subgroups — too many to draw. Only its 121 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×#20259293C211yesmaximal, characteristic, complemented
2#121×D191C211yesmaximal, characteristic, complemented
3#27132977C211yesmaximal, characteristic, complemented
4#27132975C212yesmaximal, complemented
5#27132974C211yesmaximal, characteristic, complemented
6#27132898C212yesmaximal, complemented
7#27132896C212yesmaximal, complemented
8#27132797C211yesmaximal, characteristic, complemented
9#27132795C212yesmaximal, complemented
10C22⋊C8×D191C211yesmaximal, characteristic, complemented
11#447×C191C211yesmaximal, characteristic, complemented
12C22×C22⋊D191—21no—
13C2×D4×D191—21no—
14C2×D4×D191C2211yescharacteristic, complemented
15C22×C4×D191C2211yescharacteristic
16C2×#15690272C2212yescomplemented
17C2×C8×D191C2212yescomplemented
18#20259297C2211yescharacteristic, complemented
19#20259296C2211yescharacteristic
20#20259295—21no—
21#20259294—21no—
22#20259293—21no—
23#20259293—21no—
24#20259293C2212yes—
25#20259293C2211yescharacteristic, complemented
26#20259293C2212yescomplemented
27C2×C4⋊Dic191C2211yescharacteristic, complemented
28C4⋊C4×D191C2211yescharacteristic, complemented
29C2×C22⋊Dic191—21no—
30C22⋊C4×D191—21no—
31#20259256C2212yescomplemented
32#20259255C2212yescomplemented
33#20259244C2212yescomplemented
34#20259243C2212yescomplemented
35#20259243C2212yescomplemented
36C22⋊(C191⋊C8)C2211yescharacteristic, complemented
37(C2×D191)⋊C8C2212yescomplemented
38C22×D4×C191C2211yescharacteristic, complemented
39C2×D8×C191C2212yescomplemented
40#121×C191C2211yescharacteristic
41#102×C191C2212yescomplemented
42C22⋊C8×C191C2211yescharacteristic, complemented
43C23×D191—21no—
44C23×D191D411yescharacteristic
45C2×D764—21no—
46C2×D764C2311yescharacteristic
47C2×C22⋊D191—21no—
48C2×C22⋊D191—22no—
49C2×C22⋊D191—22no—
50C2×C22⋊D191—21no—
51C2×C22⋊D191—21no—
52C2×C22⋊D191—44no—
53D4×D191—42no—
54D4×D191—44no—
55C22×Dic191—21no—
56C22×Dic191D411yescharacteristic
57C2×C4×D191—21no—
58C2×C4×D191—42no—
59C2×C4×D191D411yescharacteristic
60C2×C4×D191D411yescharacteristic
61C2×C4×D191C2311yescharacteristic
62C2×C4×D191C2311yescharacteristic
63#15690272—42no—
64C2×C191⋊C8C2312yes—
65C8×D191—42no—
66C4⋊Dic191—21no—
67C4⋊Dic191C2311yescharacteristic
68C4⋊Dic191C2312yes—
69Dic191⋊C4—21no—
70C22⋊Dic191—21no—
71C22⋊Dic191—22no—
72C22⋊Dic191—22no—
73(C2×D191)⋊C4—21no—
74(C2×D191)⋊C4—21no—
75C24×C191—21no—
76C2×D4×C191—21no—
77C2×D4×C191—21no—
78C2×D4×C191—42no—
79C2×D4×C191C2312yes—
80C2×D4×C191C2311yescharacteristic
81C22×C4×C191C2311yescharacteristic
82D8×C191—42no—
83C2×C8×C191C2312yes—
84C4⋊C4×C191C2311yescharacteristic
85C22⋊C4×C191—21no—
86C22×D191—21no—
87C22×D191—41no—
88C22×D191—41no—
89C22×D191—42no—
90C22×D191—42no—
91C22×D191—42no—
92C22×D191D814yescomplemented
93C22×D191C2×D411yescharacteristic
94C22×D191C2×D411yescharacteristic
95D764—22no—
96D764—42no—
97C22⋊D191—42no—
98C22⋊D191—42no—
99C22⋊D191—42no—
100C22⋊D191—44no—
101C22⋊D191—44no—
102C2×Dic191—21no—
103C2×Dic191—22no—
104C2×Dic191—41no—
105C2×Dic191—42no—
106C2×Dic191C2×D411yescharacteristic
107C2×Dic191C2×D411yescharacteristic
108C4×D191—22no—
109C4×D191—22no—
110C4×D191—41no—
111C4×D191—41no—
112C4×D191—42no—
113C191⋊C8—22no—
114C23×C191—21no—
115C23×C191—22no—
116C23×C191—42no—
117C23×C191—42no—
118C23×C191—42no—
119C23×C191C2×D411yescharacteristic
120D4×C191—22no—
121D4×C191—22no—
122D4×C191—22no—
123D4×C191—42no—
124D4×C191—42no—
125C2×C4×C191—22no—
126C2×C4×C191—21no—
127C2×C4×C191C2×D411yescharacteristic
128C2×C4×C191C2411yescharacteristic
129C8×C191—22no—
130C22×C191—21no—
131C22×C191—21no—
132C22×C191—42no—
133C22×C191—41no—
134C22×C191—41no—
135C22×C191—42no—
136C22×C191—42no—
137C22×C191—41no—
138C22×C191—84no—
139C22×C191C2×D812yes—
140C22×C191C22×D411yescharacteristic
141C2×D191—21no—
142C2×D191—21no—
143C2×D191—21no—
144C2×D191—21no—
145C2×D191—44no—
146C2×D191—41no—
147C2×D191—41no—
148C2×D191—41no—
149C2×D191—84no—
150C2×D191C2×D812yes—
151C2×D191C2×D812yes—
152C4×C191—21no—
153C4×C191—41no—
154C4×C191H(5,𝔽2)12yes—
155Dic191—21no—
156Dic191—21no—
157Dic191—41no—
158Dic191—42no—
159C2×C191—22no—
160C2×C191—81no—
161C2×C191—82no—
162C2×C191#53411yescharacteristic
163C2×C191C22×D811yescharacteristic
164C2×C191#57511yescharacteristic
165D191—22no—
166D191—22no—
167D191—81no—
168C191#318611yescharacteristic, complemented, Sylow
169#3186—1911nomaximal, Sylow
170C2×#121—1911no—
171C2×#121—1911no—
172#480—1911no—
173#480—1912no—
174#480—1911no—
175#466—1912no—
176#466—1912no—
177#447—1911no—
178#447—1912no—
179#447D19111yescharacteristic, complemented
180C2×C22⋊C8—1911no—
181C22×D4—3821no—
182C22×D4—3821no—
183C22×D4—1911no—
184C22×D4C2×D19111yescharacteristic, complemented
185C23×C4—1911no—
186C2×D8—1912no—
187C2×D8C2×D19112yescomplemented
188C22×C8—1912no—
189#121—3821no—
190#121—3821no—
191#121—3821no—
192#121—3821no—
193#121—1912no—
194#121—1911no—
195#121—1912no—
196#121—1911no—
197#121—1911no—
198#121C2×D19111yescharacteristic
199C2×C4⋊C4—1911no—
200C2×C4⋊C4—1911no—
201C2×C22⋊C4—3821no—
202C2×C22⋊C4—3821no—
203#107—1912no—
204#107—1912no—
205#102—1912no—
206#102—1912no—
207#102—1912no—
208#102C2×D19112yescomplemented
209C22⋊C8—1911no—
210C22⋊C8—1912no—
211C22⋊C8C2×D19111yescharacteristic, complemented
212C24—21no—
213C24—3821no—
214C24—1911no—
215C2×D4—21no—
216C2×D4—21no—
217C2×D4—42no—
218C2×D4—3821no—
219C2×D4—3821no—
220C2×D4—3822no—
221C2×D4—3822no—
222C2×D4—3821no—
223C2×D4—3821no—
224C2×D4—1911no—
225C2×D4—7642no—
226C2×D4—7644no—
227C2×D4—7644no—
228C2×D4C22×D19111yescharacteristic
229C2×D4C22×D19112yes—
230C22×C4—3821no—
231C22×C4—3821no—
232C22×C4—1911no—
233C22×C4—1911no—
234C22×C4—1911no—
235C22×C4—1911no—
236C22×C4—1911no—
237C22×C4—7642no—
238C22×C4C22×D19111yescharacteristic
239D8—42no—
240D8—7642no—
241C2×C8—1912no—
242C2×C8—7642no—
243C2×C8C22×D19112yes—
244C4⋊C4—3821no—
245C4⋊C4—3821no—
246C4⋊C4—1911no—
247C4⋊C4—1912no—
248C4⋊C4C22×D19111yescharacteristic
249C22⋊C4—21no—
250C22⋊C4—3821no—
251C22⋊C4—3821no—
252C22⋊C4—3822no—
253C22⋊C4—3822no—
254C22⋊C4—3821no—
255C23—21no—
256C23—22no—
257C23—42no—
258C23—42no—
259C23—42no—
260C23—3821no—
261C23—1911no—
262C23—1911no—
263C23—1914no—
264C23—7641no—
265C23—7641no—
266C23—7642no—
267C23—7642no—
268C23—7642no—
269C23D4×D19111yescharacteristic
270D4—22no—
271D4—22no—
272D4—22no—
273D4—42no—
274D4—42no—
275D4—3822no—
276D4—7642no—
277D4—7642no—
278D4—7642no—
279D4—7644no—
280D4—7644no—
281D4—7642no—
282C2×C4—22no—
283C2×C4—21no—
284C2×C4—3822no—
285C2×C4—3822no—
286C2×C4—3822no—
287C2×C4—3821no—
288C2×C4—1911no—
289C2×C4—1911no—
290C2×C4—7641no—
291C2×C4—7641no—
292C2×C4—7642no—
293C2×C4—7641no—
294C2×C4—7642no—
295C2×C4D4×D19111yescharacteristic
296C2×C4C23×D19111yescharacteristic
297C8—22no—
298C8—3822no—
299C22—21no—
300C22—21no—
301C22—42no—
302C22—41no—
303C22—42no—
304C22—41no—
305C22—41no—
306C22—42no—
307C22—84no—
308C22—3821no—
309C22—3821no—
310C22—3821no—
311C22—3821no—
312C22—1912no—
313C22—1912no—
314C22—15284no—
315C22—7644no—
316C22—7641no—
317C22—7641no—
318C22—7641no—
319C22D8×D19112yes—
320C22C2×D4×D19111yescharacteristic
321C4—21no—
322C4—41no—
323C4—3821no—
324C4—3821no—
325C4—7641no—
326C4—7642no—
327C4#2025936712yes—
328C2—22no—
329C2—82no—
330C2—81no—
331C2—3822no—
332C2—3822no—
333C2—15281no—
334C2#2713322311yescharacteristic
335C2C2×D8×D19111yescharacteristic
336C2#2713345911yescharacteristic
subgroupG/H
centerC22C2×D4×D191
commutator, cosocleC2×C4×C191C24
FrattiniC2×C4C23×D191
Fitting#447×C191C2
socleC22×C191C22×D4
hypercenter#447D191
hypocenterC191#3186
LCSG ▷ C2×C4×C191 ▷ C2×C191 ▷ C191
UCSC1 ◁ C22 ◁ C22×C4 ◁ #447

Complements

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

NH ≅ G/Ntype
1. C2×#20259293C2semidirect
2. #121×D191C2semidirect
3. #27132977C2semidirect
4. #27132975C2semidirect
5. #27132974C2semidirect
6. #27132898C2semidirect
7. #27132896C2semidirect
8. #27132797C2semidirect
9. #27132795C2semidirect
10. C22⋊C8×D191C2semidirect
11. #447×C191C2semidirect
14. C2×D4×D191C22semidirect
16. C2×#15690272C22semidirect
17. C2×C8×D191C22semidirect
18. #20259297C22semidirect
25. #20259293C22semidirect
26. #20259293C22semidirect
27. C2×C4⋊Dic191C22semidirect
28. C4⋊C4×D191C22semidirect
31. #20259256C22semidirect
32. #20259255C22semidirect
33. #20259244C22semidirect
34. #20259243C22semidirect
35. #20259243C22semidirect
36. C22⋊(C191⋊C8)C22semidirect
37. (C2×D191)⋊C8C22semidirect
38. C22×D4×C191C22semidirect
39. C2×D8×C191C22semidirect
41. #102×C191C22semidirect
42. C22⋊C8×C191C22semidirect
92. C22×D191D8semidirect
168. C191#3186semidirect
179. #447D191semidirect
184. C22×D4C2×D191semidirect
187. C2×D8C2×D191semidirect
208. #102C2×D191semidirect
211. C22⋊C8C2×D191semidirect

Maximal subgroups

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

subgroupindex
C2×#202592932
#121×D1912
#271329772
#271329752
#271329742
#271328982
#271328962
#271327972
#271327952
C22⋊C8×D1912
#447×C1912
#3186191