Finite Group Databasebeta

← #470×D191 #37582118

Subgroups — #470×D191 #37582118

Order 24448.

492 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 492 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×Q16×D191C211yesmaximal, characteristic, complemented
2C2×M16×D191C211yesmaximal, characteristic
3C22⋊Q8×D191C212yesmaximal
4#27132922C211yesmaximal, characteristic, complemented
5#27132921C211yesmaximal, characteristic, complemented
6#27132920C212yesmaximal, complemented
7#27132919C212yesmaximal, complemented
8#27132918C211yesmaximal, characteristic, complemented
9#106×D191C211yesmaximal, characteristic, complemented
10#103×D191C212yesmaximal, complemented
11#470×C191C211yesmaximal, characteristic, complemented
12C2×Q8×D191C2212yes—
13C22×C4×D191C2211yescharacteristic
14C2×Dic764C2211yescharacteristic, complemented
15C2×#15690277C2212yescomplemented
16Q16×D191—24no—
17C2×#15690271C2211yescharacteristic
18C2×#15690270C2211yescharacteristic
19C2×#15690270C2211yescharacteristic
20M16×D191—24no—
21C2×C8×D191C2211yescharacteristic
22C2×C8×D191C2211yescharacteristic
23#20259302C2212yes—
24#20259301C2212yes—
25#20259300—22no—
26#20259299—22no—
27#20259298C2212yes—
28C4⋊C4×D191—22no—
29C4⋊C4×D191C2212yes—
30C22⋊C4×D191—22no—
31#20259254C2211yescharacteristic, complemented
32#20259253C2212yescomplemented
33#20259247C2212yescomplemented
34#20259246C2212yescomplemented
35#20259245C2212yescomplemented
36C2×Q16×C191C2211yescharacteristic, complemented
37C2×M16×C191C2211yescharacteristic
38C22⋊Q8×C191C2212yes—
39#106×C191C2211yescharacteristic, complemented
40#103×C191C2212yescomplemented
41C23×D191D411yescharacteristic
42C2×Dic382—22no—
43C2×Dic382C2312yes—
44Q8×D191—24no—
45Q8×D191—24no—
46C22×Dic191D411yescharacteristic
47C2×C4×D191—22no—
48C2×C4×D191—22no—
49C2×C4×D191—22no—
50C2×C4×D191—22no—
51C2×C4×D191D411yescharacteristic
52C2×C4×D191D411yescharacteristic
53C2×C4×D191C2311yescharacteristic
54C2×C4×D191C2311yescharacteristic
55Dic764—22no—
56#15690277—24no—
57#15690271—22no—
58#15690270—22no—
59#15690270—22no—
60C2×C191⋊C8C2311yescharacteristic
61C2×C191⋊C8C2311yescharacteristic
62C8×D191—22no—
63C8×D191D414yes—
64C4⋊Dic191—22no—
65C4⋊Dic191C2312yes—
66Dic191⋊C4—22no—
67Dic191⋊C4—22no—
68Dic191⋊C4—22no—
69C22⋊Dic191—22no—
70(C2×D191)⋊C4—22no—
71(C2×D191)⋊C4—22no—
72C2×Q8×C191C2312yes—
73C22×C4×C191C2311yescharacteristic
74Q16×C191—22no—
75M16×C191—22no—
76C2×C8×C191C2311yescharacteristic
77C2×C8×C191C2311yescharacteristic
78C4⋊C4×C191—22no—
79C4⋊C4×C191C2312yes—
80C22⋊C4×C191—22no—
81C22×D191—22no—
82C22×D191—24no—
83C22×D191C2×D411yescharacteristic
84C22×D191C2×D411yescharacteristic
85Dic382—22no—
86Dic382—22no—
87Dic382—44no—
88C2×Dic191—22no—
89C2×Dic191—22no—
90C2×Dic191—22no—
91C2×Dic191C2×D411yescharacteristic
92C2×Dic191C2×D411yescharacteristic
93C4×D191—22no—
94C4×D191—21no—
95C4×D191—21no—
96C4×D191—22no—
97C4×D191—44no—
98C4×D191—44no—
99C4×D191C2×D412yes—
100C4×D191#4112yes—
101C191⋊C8—21no—
102C191⋊C8C2×D412yes—
103C23×C191C2×D411yescharacteristic
104Q8×C191—22no—
105Q8×C191—22no—
106C2×C4×C191—21no—
107C2×C4×C191—21no—
108C2×C4×C191—22no—
109C2×C4×C191—22no—
110C2×C4×C191C2×D411yescharacteristic
111C2×C4×C191C2411yescharacteristic
112C8×C191—21no—
113C8×C191C2×D412yes—
114C22×C191—21no—
115C22×C191—22no—
116C22×C191C22×D411yescharacteristic
117C2×D191—21no—
118C2×D191—22no—
119C2×D191—44no—
120C2×D191#12112yes—
121C2×D191#13714yes—
122C4×C191—21no—
123C4×C191—42no—
124C4×C191—42no—
125C4×C191C22×D411yescharacteristic
126C4×C191C2×#4111yescharacteristic
127Dic191—21no—
128Dic191—42no—
129Dic191—42no—
130Dic191#12112yes—
131C2×C191—41no—
132C2×C191C2×#12111yescharacteristic
133C2×C191C2×#13712yes—
134D191—41no—
135D191#47014yesdirect product factor
136C191C2×#47011yescharacteristic, complemented, Sylow
137C2×#470—1911nomaximal, Sylow
138C22×Q16—1911no—
139C22×M16—1911no—
140C2×C22⋊Q8—1912no—
141#470—1911no—
142#470—1911no—
143#470—1912no—
144#470—1911no—
145#470—1912no—
146#470D19111yescharacteristic, direct product factor
147C2×#106—1911no—
148C2×#103—1912no—
149C22×Q8—1912no—
150C23×C4—1911no—
151C2×Q16—3824no—
152C2×Q16—1912no—
153C2×Q16—1911no—
154C2×Q16C2×D19111yescharacteristic, complemented
155C2×M16—3824no—
156C2×M16—1911no—
157C2×M16—1911no—
158C2×M16—1911no—
159C2×M16C2×D19111yescharacteristic
160C22×C8—1911no—
161C22×C8—1911no—
162C22⋊Q8—3822no—
163C22⋊Q8—3822no—
164C22⋊Q8—1912no—
165C22⋊Q8—1912no—
166C22⋊Q8—1912no—
167C22⋊Q8C2×D19112yes—
168C2×C4⋊C4—3822no—
169C2×C4⋊C4—1912no—
170C2×C22⋊C4—3822no—
171#106—1911no—
172#106—1912no—
173#106C2×D19111yescharacteristic, complemented
174#103—1912no—
175#103—1912no—
176#103—1912no—
177#103C2×D19112yescomplemented
178C24—1911no—
179C2×Q8—3822no—
180C2×Q8—3824no—
181C2×Q8—3824no—
182C2×Q8—1912no—
183C2×Q8C22×D19112yes—
184C22×C4—3822no—
185C22×C4—3822no—
186C22×C4—3822no—
187C22×C4—3822no—
188C22×C4—1911no—
189C22×C4—1911no—
190C22×C4—1911no—
191C22×C4—1911no—
192C22×C4—1911no—
193C22×C4C22×D19111yescharacteristic
194Q16—22no—
195Q16—3824no—
196Q16—3822no—
197M16—22no—
198M16—3822no—
199M16—3822no—
200M16—3822no—
201C2×C8—3822no—
202C2×C8—1911no—
203C2×C8—1911no—
204C2×C8—1914no—
205C2×C8C22×D19111yescharacteristic
206C2×C8C22×D19111yescharacteristic
207C4⋊C4—22no—
208C4⋊C4—3822no—
209C4⋊C4—3822no—
210C4⋊C4—3822no—
211C4⋊C4—3822no—
212C4⋊C4—1912no—
213C4⋊C4C22×D19112yes—
214C22⋊C4—22no—
215C22⋊C4—3822no—
216C22⋊C4—3822no—
217C22⋊C4—3822no—
218C23—3822no—
219C23—3824no—
220C23—1911no—
221C23—1911no—
222C23D4×D19111yescharacteristic
223Q8—22no—
224Q8—22no—
225Q8—3822no—
226Q8—3822no—
227Q8—7644no—
228C2×C4—21no—
229C2×C4—21no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—3822no—
233C2×C4—3822no—
234C2×C4—3822no—
235C2×C4—3821no—
236C2×C4—3821no—
237C2×C4—3822no—
238C2×C4—3822no—
239C2×C4—1911no—
240C2×C4—1911no—
241C2×C4—1912no—
242C2×C4—1912no—
243C2×C4—7644no—
244C2×C4—7644no—
245C2×C4D4×D19111yescharacteristic
246C2×C4C23×D19111yescharacteristic
247C8—21no—
248C8—3821no—
249C8—1912no—
250C8D4×D19112yes—
251C22—22no—
252C22—21no—
253C22—3822no—
254C22—3821no—
255C22—1914no—
256C22—1912no—
257C22—7644no—
258C22C2×D4×D19111yescharacteristic
259C4—21no—
260C4—42no—
261C4—42no—
262C4—3821no—
263C4—1912no—
264C4—7642no—
265C4—7642no—
266C4C2×D4×D19111yescharacteristic
267C4#41×D19111yescharacteristic
268C2—41no—
269C2—1914no—
270C2—7641no—
271C2#121×D19111yescharacteristic
272C2#137×D19112yes—
subgroupG/H
centerC22C2×D4×D191
commutator, cosocleC2×C4×C191C24
FrattiniC2×C4C23×D191
Fitting#470×C191C2
socleC22×C191C22×D4
prime radicalC23×D191D4
hypercenter#470D191
hypocenterC191C2×#470
LCSG ▷ C2×C4×C191 ▷ C2×C191 ▷ C191
UCSC1 ◁ C22 ◁ C22×C4 ◁ #470

Complements

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

NH ≅ G/Ntype
1. C2×Q16×D191C2semidirect
4. #27132922C2semidirect
5. #27132921C2semidirect
6. #27132920C2semidirect
7. #27132919C2semidirect
8. #27132918C2semidirect
9. #106×D191C2semidirect
10. #103×D191C2semidirect
11. #470×C191C2semidirect
14. C2×Dic764C22semidirect
15. C2×#15690277C22semidirect
31. #20259254C22semidirect
32. #20259253C22semidirect
33. #20259247C22semidirect
34. #20259246C22semidirect
35. #20259245C22semidirect
36. C2×Q16×C191C22semidirect
39. #106×C191C22semidirect
40. #103×C191C22semidirect
135. D191#470direct (N × H)
136. C191C2×#470semidirect
146. #470D191direct (N × H)
154. C2×Q16C2×D191semidirect
173. #106C2×D191semidirect
177. #103C2×D191semidirect

Maximal subgroups

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

subgroupindex
C2×Q16×D1912
C2×M16×D1912
C22⋊Q8×D1912
#271329222
#271329212
#271329202
#271329192
#271329182
#106×D1912
#103×D1912
#470×C1912
C2×#470191