Finite Group Databasebeta

← #37582441

Subgroups — #37582441

Order 24448.

458 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 458 conjugacy classes of subgroups — too many to draw. Only its 129 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
1#27133170C211yesmaximal, characteristic
2C2×C4⋊C4×D191C211yesmaximal, characteristic, complemented
3#27132997C212yesmaximal, complemented
4#27132992C212yesmaximal, complemented
5#27132980C212yesmaximal, complemented
6#27132975C212yesmaximal, complemented
7#27132705C211yesmaximal, characteristic, complemented
8#27132703C211yesmaximal, characteristic, complemented
9#27132664C211yesmaximal, characteristic, complemented
10#27132626C211yesmaximal, characteristic, complemented
11#417×C191C211yesmaximal, characteristic, complemented
12C22×C4×D191—21no—
13C22×C4×D191C2211yescharacteristic
14C2×#15690271C2211yescharacteristic
15#20259303—21no—
16#20259298C2211yescharacteristic
17#20259298C2211yescharacteristic
18#20259293C2211yescharacteristic
19#20259293C2211yescharacteristic
20#20259290—21no—
21C4×C22⋊D191—21no—
22C4×C22⋊D191—21no—
23#20259281C2211yescharacteristic
24C2×C4⋊Dic191C2211yescharacteristic
25C2×Dic191⋊C4—21no—
26C4⋊C4×D191—21no—
27C4⋊C4×D191—21no—
28C4⋊C4×D191C2212yes—
29C4⋊C4×D191C2212yes—
30#20259256C2212yescomplemented
31#20259255C2212yescomplemented
32#20259254C2212yescomplemented
33#20259253C2212yescomplemented
34#20259246C2212yescomplemented
35#20259243C2212yescomplemented
36(C2×D191)⋊C8C2212yes—
37(C2×D191)⋊C8C2212yes—
38C2×#41×C191C2211yescharacteristic
39C2×M16×C191C2211yescharacteristic
40C2×C4⋊C4×C191C2211yescharacteristic
41#103×C191C2212yescomplemented
42#102×C191C2212yescomplemented
43C23×D191D411yescharacteristic
44C2×C22⋊D191—21no—
45C2×C22⋊D191—21no—
46C22×Dic191—21no—
47C22×Dic191D411yescharacteristic
48C2×C4×D191—22no—
49C2×C4×D191—22no—
50C2×C4×D191—22no—
51C2×C4×D191—22no—
52C2×C4×D191—44no—
53C2×C4×D191D411yescharacteristic
54C2×C4×D191D411yescharacteristic
55C2×C4×D191C2311yescharacteristic
56C2×C4×D191C2311yescharacteristic
57#15690271—22no—
58C2×C191⋊C8C2312yes—
59C4⋊Dic191—21no—
60C4⋊Dic191C2312yes—
61C4⋊Dic191C2311yescharacteristic
62C4⋊Dic191C2311yescharacteristic
63Dic191⋊C4—22no—
64Dic191⋊C4—22no—
65Dic191⋊C4—21no—
66Dic191⋊C4—21no—
67C22⋊Dic191—21no—
68C22⋊Dic191—21no—
69C22⋊Dic191—21no—
70(C2×D191)⋊C4—21no—
71(C2×D191)⋊C4—21no—
72C4×Dic191—21no—
73#41×C191—22no—
74#41×C191—22no—
75C2×Q8×C191C2311yescharacteristic
76C2×D4×C191—21no—
77C2×D4×C191C2311yescharacteristic
78C22×C4×C191—21no—
79C22×C4×C191—21no—
80C22×C4×C191C2311yescharacteristic
81M16×C191—22no—
82C2×C8×C191C2312yes—
83C4⋊C4×C191—21no—
84C4⋊C4×C191C2312yes—
85C22×D191—22no—
86C22×D191—22no—
87C22×D191—22no—
88C22×D191C2×D411yescharacteristic
89C22×D191C2×D411yescharacteristic
90C22⋊D191—42no—
91C22⋊D191—42no—
92C2×Dic191—22no—
93C2×Dic191—21no—
94C2×Dic191—21no—
95C2×Dic191—22no—
96C2×Dic191—42no—
97C2×Dic191C2×D411yescharacteristic
98C2×Dic191C2×D411yescharacteristic
99C4×D191—22no—
100C4×D191—22no—
101C4×D191—21no—
102C4×D191—21no—
103C4×D191—21no—
104C4×D191—21no—
105C4×D191—44no—
106C4×D191—44no—
107C191⋊C8—22no—
108C23×C191—21no—
109C23×C191C2×D411yescharacteristic
110Q8×C191—21no—
111Q8×C191—21no—
112D4×C191—21no—
113D4×C191—21no—
114D4×C191—42no—
115C2×C4×C191—22no—
116C2×C4×C191—21no—
117C2×C4×C191—42no—
118C2×C4×C191—42no—
119C2×C4×C191C2×D411yescharacteristic
120C2×C4×C191#4112yes—
121C2×C4×C191#4112yes—
122C2×C4×C191C2411yescharacteristic
123C8×C191—22no—
124C22×C191—21no—
125C22×C191—21no—
126C22×C191—41no—
127C22×C191—41no—
128C22×C191—41no—
129C22×C191#12312yes—
130C22×C191C22×D411yescharacteristic
131C2×D191—28no—
132C2×D191—21no—
133C2×D191—21no—
134C2×D191—21no—
135C2×D191—21no—
136C2×D191AGL(1,ℤ/8)12yes—
137C2×D191#13712yes—
138C4×C191—41no—
139C4×C191—42no—
140C4×C191#12312yes—
141C4×C191C2×#4111yescharacteristic
142C4×C191C2×#4111yescharacteristic
143Dic191—21no—
144Dic191—21no—
145Dic191—42no—
146Dic191—41no—
147Dic191—41no—
148C2×C191—22no—
149C2×C191—81no—
150C2×C191C2×#12311yescharacteristic
151C2×C191C2×AGL(1,ℤ/8)11yescharacteristic
152C2×C191C2×#13711yescharacteristic
153D191—22no—
154D191—22no—
155C191#309211yescharacteristic, complemented, Sylow
156#3092—1911nomaximal, Sylow
157#525—1911no—
158C22×C4⋊C4—1911no—
159#484—1912no—
160#483—1912no—
161#481—1912no—
162#480—1912no—
163#428—1911no—
164#428—1911no—
165#417—1911no—
166#417D19111yescharacteristic, complemented
167#407—1911no—
168C2×#41C2×D19111yescharacteristic
169C23×C4—3821no—
170C23×C4—1911no—
171C2×M16—1911no—
172C2×M16C2×D19111yescharacteristic
173#123—3821no—
174C22⋊Q8—1911no—
175C22⋊Q8—1911no—
176#121—1911no—
177#121—1911no—
178#120—3821no—
179C4×D4—3821no—
180C4×D4—3821no—
181#117—1911no—
182C2×C4⋊C4—3821no—
183C2×C4⋊C4—3821no—
184C2×C4⋊C4—3821no—
185C2×C4⋊C4—1911no—
186C2×C4⋊C4—1912no—
187C2×C4⋊C4—1912no—
188C2×C4⋊C4C2×D19111yescharacteristic
189#107—1912no—
190#107—1912no—
191#106—1912no—
192#106—1912no—
193#103—1912no—
194#103C2×D19112yescomplemented
195#102—1912no—
196#102C2×D19112yescomplemented
197C22⋊C8—1912no—
198C22⋊C8—1912no—
199C24—1911no—
200#41—22no—
201#41—22no—
202C2×Q8C22×D19111yescharacteristic
203C2×D4—21no—
204C2×D4—3821no—
205C2×D4—3821no—
206C2×D4C22×D19111yescharacteristic
207C22×C4—21no—
208C22×C4—21no—
209C22×C4—3822no—
210C22×C4—3822no—
211C22×C4—3822no—
212C22×C4—3822no—
213C22×C4—3821no—
214C22×C4—1911no—
215C22×C4—1911no—
216C22×C4—1911no—
217C22×C4—1911no—
218C22×C4—1911no—
219C22×C4—7644no—
220C22×C4C22×D19111yescharacteristic
221M16—22no—
222M16—3822no—
223C2×C8—1912no—
224C2×C8C22×D19112yes—
225C4⋊C4—21no—
226C4⋊C4—3822no—
227C4⋊C4—3822no—
228C4⋊C4—3821no—
229C4⋊C4—3821no—
230C4⋊C4—3821no—
231C4⋊C4—1912no—
232C4⋊C4—1911no—
233C4⋊C4—1911no—
234C4⋊C4C22×D19112yes—
235C22⋊C4—3821no—
236C22⋊C4—3821no—
237C22⋊C4—3821no—
238C22⋊C4—3821no—
239C22⋊C4—3821no—
240C42—3821no—
241C23—21no—
242C23—3822no—
243C23—3822no—
244C23—3822no—
245C23—1911no—
246C23—1911no—
247C23D4×D19111yescharacteristic
248Q8—21no—
249Q8—21no—
250D4—21no—
251D4—21no—
252D4—42no—
253D4—7642no—
254D4—7642no—
255C2×C4—21no—
256C2×C4—22no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—3822no—
260C2×C4—3821no—
261C2×C4—3821no—
262C2×C4—3821no—
263C2×C4—3821no—
264C2×C4—3822no—
265C2×C4—3822no—
266C2×C4—3821no—
267C2×C4—3821no—
268C2×C4—3822no—
269C2×C4—1911no—
270C2×C4—1911no—
271C2×C4—7644no—
272C2×C4—7644no—
273C2×C4—7642no—
274C2×C4D4×D19111yescharacteristic
275C2×C4#1569028612yes—
276C2×C4#1569028712yes—
277C2×C4C23×D19111yescharacteristic
278C8—22no—
279C8—3822no—
280C22—21no—
281C22—21no—
282C22—41no—
283C22—41no—
284C22—41no—
285C22—3828no—
286C22—3821no—
287C22—3821no—
288C22—3821no—
289C22—3821no—
290C22—1912no—
291C22—1912no—
292C22#2025930412yes—
293C22C2×D4×D19111yescharacteristic
294C4—42no—
295C4—41no—
296C4—3821no—
297C4—3821no—
298C4—7642no—
299C4—7641no—
300C4—7641no—
301C4#2025930412yes—
302C4C2×#1569028611yescharacteristic
303C4C2×#1569028711yescharacteristic
304C2—22no—
305C2—81no—
306C2—3822no—
307C2—3822no—
308C2C2×#2025930411yescharacteristic
309C2AGL(1,ℤ/8)×D19111yescharacteristic
310C2#137×D19111yescharacteristic
subgroupG/H
centerC22C2×D4×D191
commutator, cosocleC2×C4×C191C24
FrattiniC2×C4C23×D191
Fitting#417×C191C2
socleC22×C191C22×D4
prime radical#27133170C2
hypercenter#417D191
hypocenterC191#3092
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
2. C2×C4⋊C4×D191C2semidirect
3. #27132997C2semidirect
4. #27132992C2semidirect
5. #27132980C2semidirect
6. #27132975C2semidirect
7. #27132705C2semidirect
8. #27132703C2semidirect
9. #27132664C2semidirect
10. #27132626C2semidirect
11. #417×C191C2semidirect
30. #20259256C22semidirect
31. #20259255C22semidirect
32. #20259254C22semidirect
33. #20259253C22semidirect
34. #20259246C22semidirect
35. #20259243C22semidirect
41. #103×C191C22semidirect
42. #102×C191C22semidirect
155. C191#3092semidirect
166. #417D191semidirect
194. #103C2×D191semidirect
196. #102C2×D191semidirect

Maximal subgroups

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

subgroupindex
#271331702
C2×C4⋊C4×D1912
#271329972
#271329922
#271329802
#271329752
#271327052
#271327032
#271326642
#271326262
#417×C1912
#3092191