Finite Group Databasebeta

← #37575657

Subgroups — #37575657

Order 24448.

688 conjugacy classes of subgroups; only the 173 normal subgroups are shown.

normal subgroups only

This group has 688 conjugacy classes of subgroups — too many to draw. Only its 173 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×#20259290C211yesmaximal, characteristic
2#27132563C212yesmaximal, complemented
3#27132554C212yesmaximal, complemented
4#27132553C212yesmaximal, complemented
5#27132551C212yesmaximal, complemented
6#27132543C211yesmaximal, characteristic, complemented
7#27132540C212yesmaximal, complemented
8#27132518C212yesmaximal, complemented
9#390×C191C211yesmaximal, characteristic, complemented
10C24×D191C2211yescharacteristic
11C22×C22⋊D191C2212yes—
12C23×Dic191C2211yescharacteristic, complemented
13#20259290C418yescomplemented
14C2×Dic191⋊C4C2212yescomplemented
15C2×C22⋊Dic191C2212yes—
16C2×C22⋊Dic191C2211yescharacteristic
17C2×C22⋊Dic191C2211yescharacteristic, complemented
18C2×(C2×D191)⋊C4C2212yes—
19C2×(C2×D191)⋊C4C2212yes—
20C2×(C2×D191)⋊C4C2212yes—
21C2×(C2×D191)⋊C4C2212yes—
22C2×(C2×D191)⋊C4C2212yes—
23C2×C4×Dic191C2212yescomplemented
24C191⋊#95C2212yescomplemented
25C191⋊#95C2212yescomplemented
26C191⋊#95C2212yescomplemented
27C22×D4×C191C2211yescharacteristic
28C2×C4⋊C4×C191C2211yescharacteristic, complemented
29C2×C22⋊C4×C191C2212yes—
30C2×C22⋊C4×C191C2212yescomplemented
31C2×C42×C191C2211yescharacteristic, complemented
32C23×D191—42no—
33C23×D191—41no—
34C23×D191—41no—
35C23×D191—42no—
36C23×D191C2×C414yescomplemented
37C23×D191C2312yes—
38C2×C22⋊D191—42no—
39C2×C22⋊D191—42no—
40C2×C22⋊D191C2×C418yes—
41C22×Dic191—41no—
42C22×Dic191—42no—
43C22×Dic191C2312yes—
44C22×Dic191C2312yes—
45C22×Dic191C2312yes—
46Dic191⋊C4—42no—
47C22⋊Dic191—42no—
48C22⋊Dic191—41no—
49C22⋊Dic191C2×C414yescomplemented
50(C2×D191)⋊C4—42no—
51(C2×D191)⋊C4—42no—
52(C2×D191)⋊C4—42no—
53(C2×D191)⋊C4—42no—
54(C2×D191)⋊C4C2×C418yescomplemented
55C4×Dic191—42no—
56C24×C191C2311yescharacteristic
57C24×C191C2311yescharacteristic
58C2×D4×C191—41no—
59C2×D4×C191—41no—
60C2×D4×C191C2×C414yes—
61C22×C4×C191C2312yes—
62C22×C4×C191C2312yes—
63C22×C4×C191C2311yescharacteristic
64C4⋊C4×C191—41no—
65C22⋊C4×C191—42no—
66C22⋊C4×C191—42no—
67C42×C191—41no—
68C22×D191—22no—
69C22×D191—22no—
70C22×D191—22no—
71C22×D191—22no—
72C22×D191—22no—
73C22×D191—22no—
74C22×D191—48no—
75C22×D191—44no—
76C22×D191—44no—
77C22×D191—48no—
78C22×D191—44no—
79C22×D191C22×C414yes—
80C22⋊D191—48no—
81C2×Dic191—22no—
82C2×Dic191—22no—
83C2×Dic191—22no—
84C2×Dic191—22no—
85C2×Dic191—22no—
86C2×Dic191—22no—
87C2×Dic191—22no—
88C2×Dic191—22no—
89C2×Dic191—44no—
90C2×Dic191C22×C414yes—
91C23×C191—22no—
92C23×C191—22no—
93C23×C191—21no—
94C23×C191—21no—
95C23×C191—21no—
96C23×C191—21no—
97C23×C191—22no—
98C23×C191—22no—
99C23×C191C22×C412yes—
100C23×C191C22×C412yes—
101C23×C191C2411yescharacteristic
102D4×C191—44no—
103C2×C4×C191—21no—
104C2×C4×C191—21no—
105C2×C4×C191—22no—
106C2×C4×C191—22no—
107C2×C4×C191—22no—
108C2×C4×C191—22no—
109C2×C4×C191—22no—
110C2×C4×C191—22no—
111C2×C4×C191C22×C412yes—
112C22×C191—22no—
113C22×C191—22no—
114C22×C191—24no—
115C22×C191—24no—
116C22×C191—41no—
117C22×C191—42no—
118C22×C191—41no—
119C22×C191—42no—
120C22×C191—41no—
121C22×C191—41no—
122C22×C191C23×C411yescharacteristic
123C22×C191H(5,𝔽2)12yes—
124C22×C191H(5,𝔽2)11yescharacteristic
125C22×C191H(5,𝔽2)11yescharacteristic
126C22×C191H(5,𝔽2)12yes—
127C2×D191—24no—
128C2×D191—24no—
129C2×D191—24no—
130C2×D191—416no—
131C2×D191—42no—
132C2×D191—42no—
133C2×D191—42no—
134C2×D191—42no—
135C4×C191—22no—
136C4×C191—42no—
137C4×C191—42no—
138Dic191—24no—
139Dic191—42no—
140Dic191—42no—
141C2×C191—42no—
142C2×C191—42no—
143C2×C191#51812yes—
144C2×C191#51811yescharacteristic
145C2×C191#56011yescharacteristic
146C2×C191#56012yes—
147C2×C191H(3,𝔽4)11yescharacteristic
148D191—44no—
149C191#237511yescharacteristic, complemented, Sylow
150#2375—1911nomaximal, Sylow
151C2×#120—1911no—
152#390—1912no—
153#390D19111yescharacteristic, complemented
154#388—1912no—
155#388—1912no—
156#388—1912no—
157#386—1911no—
158#386—1912no—
159#379—1912no—
160C25—1911no—
161C22×D4—1912no—
162C22×D4C2×D19111yescharacteristic
163C23×C4—1911no—
164#120—1918no—
165C2×C4⋊C4—1912no—
166C2×C4⋊C4C2×D19111yescharacteristic, complemented
167C2×C22⋊C4—1911no—
168C2×C22⋊C4—1912no—
169C2×C22⋊C4—1912no—
170C2×C22⋊C4—1911no—
171C2×C22⋊C4—1912no—
172C2×C22⋊C4—1912no—
173C2×C22⋊C4—1912no—
174C2×C22⋊C4—1912no—
175C2×C22⋊C4C2×D19112yes—
176C2×C22⋊C4C2×D19112yescomplemented
177C2×C42—1912no—
178C2×C42C2×D19111yescharacteristic, complemented
179#95—1912no—
180#95—1912no—
181#95—1912no—
182C24—1912no—
183C24—1914no—
184C24—7642no—
185C24—7641no—
186C24—7641no—
187C24—7642no—
188C24C22×D19111yescharacteristic
189C24C22×D19111yescharacteristic
190C2×D4—41no—
191C2×D4—41no—
192C2×D4—1918no—
193C2×D4—7642no—
194C2×D4—7642no—
195C2×D4C4×D19114yes—
196C22×C4—1912no—
197C22×C4—1912no—
198C22×C4—1912no—
199C22×C4—7641no—
200C22×C4—7642no—
201C22×C4C22×D19112yes—
202C22×C4C22×D19112yes—
203C22×C4C22×D19111yescharacteristic
204C4⋊C4—41no—
205C4⋊C4—7642no—
206C22⋊C4—42no—
207C22⋊C4—42no—
208C22⋊C4—1914no—
209C22⋊C4—1918no—
210C22⋊C4—7642no—
211C22⋊C4—7642no—
212C22⋊C4—7642no—
213C22⋊C4—7642no—
214C22⋊C4—7641no—
215C22⋊C4—7642no—
216C42—41no—
217C42—7642no—
218C23—22no—
219C23—22no—
220C23—21no—
221C23—21no—
222C23—21no—
223C23—21no—
224C23—22no—
225C23—22no—
226C23—3822no—
227C23—3822no—
228C23—3822no—
229C23—3822no—
230C23—3822no—
231C23—3822no—
232C23—1914no—
233C23—7648no—
234C23—7644no—
235C23—7644no—
236C23—7648no—
237C23—7644no—
238C23C2×C4×D19112yes—
239C23C2×C4×D19112yes—
240C23C23×D19111yescharacteristic
241D4—44no—
242D4—7648no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—3822no—
252C2×C4—3822no—
253C2×C4—3822no—
254C2×C4—3822no—
255C2×C4—3822no—
256C2×C4—3822no—
257C2×C4—3822no—
258C2×C4—3822no—
259C2×C4—1914no—
260C2×C4—7644no—
261C2×C4C2×C4×D19112yes—
262C22—24no—
263C22—24no—
264C22—22no—
265C22—22no—
266C22—41no—
267C22—41no—
268C22—42no—
269C22—42no—
270C22—41no—
271C22—41no—
272C22—3824no—
273C22—3824no—
274C22—3824no—
275C22—76416no—
276C22—7642no—
277C22—7642no—
278C22—7642no—
279C22—7642no—
280C22C22×C4×D19111yescharacteristic
281C22#2025936712yes—
282C22#2025936711yescharacteristic
283C22#2025936711yescharacteristic
284C22#2025936812yes—
285C4—22no—
286C4—42no—
287C4—42no—
288C4—3824no—
289C4—7642no—
290C4—7642no—
291C2—42no—
292C2—42no—
293C2—7644no—
294C2#2713313512yes—
295C2#2713313611yescharacteristic
296C2#2713340011yescharacteristic
297C2#2713340112yes—
298C2#2713340511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D191
commutatorC22×C191C23×C4
Fitting#390×C191C2
socle, cosocleC23×C191C24
prime radicalC2×#20259290C2
hypercenter#390D191
hypocenterC191#2375
LCSG ▷ C22×C191 ▷ C191
UCSC1 ◁ C23 ◁ #390

Complements

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

NH ≅ G/Ntype
2. #27132563C2semidirect
3. #27132554C2semidirect
4. #27132553C2semidirect
5. #27132551C2semidirect
6. #27132543C2semidirect
7. #27132540C2semidirect
8. #27132518C2semidirect
9. #390×C191C2semidirect
12. C23×Dic191C22semidirect
13. #20259290C4semidirect
14. C2×Dic191⋊C4C22semidirect
17. C2×C22⋊Dic191C22semidirect
23. C2×C4×Dic191C22semidirect
24. C191⋊#95C22semidirect
25. C191⋊#95C22semidirect
26. C191⋊#95C22semidirect
28. C2×C4⋊C4×C191C22semidirect
30. C2×C22⋊C4×C191C22semidirect
31. C2×C42×C191C22semidirect
36. C23×D191C2×C4semidirect
49. C22⋊Dic191C2×C4semidirect
54. (C2×D191)⋊C4C2×C4semidirect
149. C191#2375semidirect
153. #390D191semidirect
166. C2×C4⋊C4C2×D191semidirect
176. C2×C22⋊C4C2×D191semidirect
178. C2×C42C2×D191semidirect

Maximal subgroups

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

subgroupindex
C2×#202592902
#271325632
#271325542
#271325532
#271325512
#271325432
#271325402
#271325182
#390×C1912
#2375191