Finite Group Databasebeta

← #37576156

Subgroups — #37576156

Order 24448.

1348 conjugacy classes of subgroups; only the 193 normal subgroups are shown.

normal subgroups only

This group has 1348 conjugacy classes of subgroups — too many to draw. Only its 193 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, complemented
2C2×#20259319C211yesmaximal, characteristic, complemented
3C2×#20259318C211yesmaximal, characteristic, complemented
4C2×#20259297C211yesmaximal, characteristic, complemented
5C2×#20259292C214yesmaximal, complemented
6#27132572C214yesmaximal, complemented
7#27132564C211yesmaximal, characteristic, complemented
8C4×(C2×D191)⋊C4C211yesmaximal, characteristic, complemented
9#390×C191C211yesmaximal, characteristic, complemented
10C24×D191C2212yes—
11C22×D764C2212yes—
12C22×D764C2211yescharacteristic
13C22×D764C2211yescharacteristic, complemented
14C22×D764C2214yescomplemented
15C22×C22⋊D191C2212yes—
16C22×C22⋊D191C2214yescomplemented
17C2×D4×D191—24no—
18C2×D4×D191—24no—
19C2×D4×D191—24no—
20C22×C4×D191C2211yescharacteristic
21#20259319—24no—
22#20259318—24no—
23#20259297—24no—
24#20259292—216no—
25C2×(C2×D191)⋊C4C2212yes—
26C2×(C2×D191)⋊C4C2214yescomplemented
27C2×(C2×D191)⋊C4C2212yescomplemented
28C2×C4×Dic191C2211yescharacteristic, complemented
29C191⋊#95C2212yescomplemented
30C22×D4×C191C2211yescharacteristic
31C2×C4⋊C4×C191C2211yescharacteristic, complemented
32C2×C22⋊C4×C191C2214yescomplemented
33C2×C42×C191C2211yescharacteristic, complemented
34C23×D191—24no—
35C23×D191—24no—
36C23×D191—28no—
37C23×D191—24no—
38C23×D191—44no—
39C23×D191C2312yes—
40C23×D191C2312yes—
41C23×D191C2311yescharacteristic
42C2×D764—28no—
43C2×D764—24no—
44C2×D764—28no—
45C2×D764—22no—
46C2×D764—22no—
47C2×D764—22no—
48C2×D764—22no—
49C2×D764—22no—
50C2×D764—28no—
51C2×D764—44no—
52C2×D764D414yescomplemented
53C2×C22⋊D191—24no—
54C2×C22⋊D191—24no—
55C2×C22⋊D191—28no—
56C2×C22⋊D191—24no—
57C2×C22⋊D191—28no—
58C2×C22⋊D191—44no—
59D4×D191—232no—
60C22×Dic191C2311yescharacteristic
61C22×Dic191C2312yes—
62C2×C4×D191—22no—
63C2×C4×D191—22no—
64C2×C4×D191D414yescomplemented
65(C2×D191)⋊C4—24no—
66(C2×D191)⋊C4—28no—
67(C2×D191)⋊C4D418yescomplemented
68C4×Dic191—22no—
69C24×C191C2312yes—
70C2×D4×C191—22no—
71C2×D4×C191—22no—
72C2×D4×C191D414yescomplemented
73C22×C4×C191C2312yes—
74C22×C4×C191C2312yes—
75C22×C4×C191C2311yescharacteristic
76C4⋊C4×C191—22no—
77C22⋊C4×C191—28no—
78C42×C191—22no—
79C22×D191—21no—
80C22×D191—21no—
81C22×D191—22no—
82C22×D191—216no—
83C22×D191—24no—
84C22×D191—22no—
85C22×D191—22no—
86C22×D191—24no—
87C22×D191—22no—
88C22×D191—22no—
89C22×D191—24no—
90C22×D191—22no—
91C22×D191—24no—
92C22×D191—416no—
93C22×D191—416no—
94C22×D191—48no—
95C22×D191—48no—
96C22×D191C2×D414yes—
97C22×D191C2×D414yes—
98C22×D191C2×D412yes—
99D764—28no—
100D764—416no—
101D764—48no—
102D764—44no—
103C22⋊D191—216no—
104C22⋊D191—416no—
105C2×Dic191—22no—
106C2×Dic191—21no—
107C2×Dic191—21no—
108C2×Dic191—22no—
109C2×Dic191C2×D412yes—
110C2×Dic191C2×D414yescomplemented
111C4×D191—28no—
112C23×C191—24no—
113C23×C191—22no—
114C23×C191—22no—
115C23×C191—24no—
116C23×C191C2×D414yes—
117C23×C191C2411yescharacteristic
118D4×C191—28no—
119C2×C4×C191—22no—
120C2×C4×C191—22no—
121C2×C4×C191—24no—
122C2×C4×C191—22no—
123C2×C4×C191C2×D412yes—
124C2×C4×C191C2×D412yes—
125C2×C4×C191C2×D412yes—
126C2×C4×C191C2×D414yescomplemented
127C22×C191—24no—
128C22×C191—28no—
129C22×C191—42no—
130C22×C191—42no—
131C22×C191—44no—
132C22×C191C22×D411yescharacteristic
133C22×C191C22×D411yescharacteristic
134C22×C191C22×D412yes—
135C22×C191C22×D411yescharacteristic
136C22×C191H(5,𝔽2)12yes—
137C2×D191—22no—
138C2×D191—22no—
139C2×D191—24no—
140C2×D191—24no—
141C2×D191—24no—
142C2×D191—28no—
143C2×D191—432no—
144C2×D191—42no—
145C2×D191—42no—
146C2×D191—44no—
147C2×D191—42no—
148C2×D191—42no—
149C2×D191—42no—
150C2×D191—44no—
151C2×D191—44no—
152C2×D191#12714yescomplemented
153C4×C191—24no—
154C4×C191—42no—
155C4×C191#12014yescomplemented
156Dic191—22no—
157Dic191—24no—
158C2×C191—44no—
159C2×C191C2×#12011yescharacteristic
160C2×C191C2×#12711yescharacteristic
161C2×C191#53511yescharacteristic
162C2×C191D4212yes—
163C2×C191D4212yes—
164D191—24no—
165D191—44no—
166D191—82no—
167C191#243311yescharacteristic, complemented, Sylow
168#2433—1911nomaximal, Sylow
169C23×D4—1911no—
170C2×#127—1911no—
171C2×#127—1911no—
172C2×#121—1911no—
173C2×#120—1914no—
174#392—1914no—
175#390—1911no—
176#390D19111yescharacteristic, complemented
177C4×C22⋊C4—1911no—
178C25—1912no—
179C22×D4—3824no—
180C22×D4—3824no—
181C22×D4—3824no—
182C22×D4—1912no—
183C22×D4—1914no—
184C22×D4—1912no—
185C22×D4—1911no—
186C22×D4—1911no—
187C22×D4—1914no—
188C22×D4C2×D19111yescharacteristic
189C23×C4—1911no—
190#127—3824no—
191#127—3824no—
192#121—3824no—
193#120—38216no—
194C2×C4⋊C4C2×D19111yescharacteristic, complemented
195C2×C22⋊C4—1914no—
196C2×C22⋊C4—1912no—
197C2×C22⋊C4—1912no—
198C2×C22⋊C4C2×D19114yescomplemented
199C2×C42—1911no—
200C2×C42C2×D19111yescharacteristic, complemented
201#95—1912no—
202C24—3824no—
203C24—3824no—
204C24—3828no—
205C24—3824no—
206C24—1912no—
207C24—1912no—
208C24—1911no—
209C24—7644no—
210C24C22×D19112yes—
211C2×D4—22no—
212C2×D4—22no—
213C2×D4—3828no—
214C2×D4—3824no—
215C2×D4—3828no—
216C2×D4—38232no—
217C2×D4—3824no—
218C2×D4—3824no—
219C2×D4—3828no—
220C2×D4—3824no—
221C2×D4—3828no—
222C2×D4—3822no—
223C2×D4—3822no—
224C2×D4—3822no—
225C2×D4—3828no—
226C2×D4—3822no—
227C2×D4—3822no—
228C2×D4—1914no—
229C2×D4—7644no—
230C2×D4—7644no—
231C2×D4D76414yescomplemented
232C22×C4—3822no—
233C22×C4—3822no—
234C22×C4—1911no—
235C22×C4—1912no—
236C22×C4—1914no—
237C22×C4C22×D19112yes—
238C22×C4C22×D19112yes—
239C22×C4C22×D19111yescharacteristic
240C4⋊C4—22no—
241C22⋊C4—28no—
242C22⋊C4—3828no—
243C22⋊C4—3824no—
244C22⋊C4—1918no—
245C42—22no—
246C42—3822no—
247C23—24no—
248C23—22no—
249C23—22no—
250C23—24no—
251C23—38216no—
252C23—3821no—
253C23—3821no—
254C23—3822no—
255C23—3824no—
256C23—3822no—
257C23—3822no—
258C23—3824no—
259C23—3822no—
260C23—3824no—
261C23—3822no—
262C23—3822no—
263C23—3824no—
264C23—1914no—
265C23—1914no—
266C23—1912no—
267C23—76416no—
268C23—76416no—
269C23—7648no—
270C23—7648no—
271C23C2×D76414yes—
272C23C23×D19111yescharacteristic
273D4—28no—
274D4—38216no—
275D4—3828no—
276D4—76416no—
277D4—7648no—
278D4—76416no—
279D4—7644no—
280C2×C4—22no—
281C2×C4—22no—
282C2×C4—24no—
283C2×C4—22no—
284C2×C4—3821no—
285C2×C4—3822no—
286C2×C4—3821no—
287C2×C4—3822no—
288C2×C4—3828no—
289C2×C4—1914no—
290C2×C4—1912no—
291C2×C4D4×D19112yes—
292C2×C4D4×D19112yes—
293C2×C4D4×D19114yescomplemented
294C2×C4C2×D76412yes—
295C22—28no—
296C22—24no—
297C22—42no—
298C22—42no—
299C22—44no—
300C22—3822no—
301C22—3824no—
302C22—3824no—
303C22—3822no—
304C22—3828no—
305C22—3824no—
306C22—1914no—
307C22—76432no—
308C22—7642no—
309C22—7642no—
310C22—7644no—
311C22—7644no—
312C22—7642no—
313C22—7642no—
314C22—7642no—
315C22—7644no—
316C22C2×D4×D19111yescharacteristic
317C22C2×D4×D19112yes—
318C22C2×D4×D19111yescharacteristic
319C22C22×D76411yescharacteristic
320C22#2025936812yes—
321C4—24no—
322C4—42no—
323C4—3824no—
324C4—3822no—
325C4#2025929214yescomplemented
326C2—44no—
327C2—3824no—
328C2—15282no—
329C2—7644no—
330C2C2×#2025929211yescharacteristic
331C2#127×D19111yescharacteristic
332C2#2713323111yescharacteristic
333C2#2713329712yes—
334C2D4×D76412yes—
subgroupG/H
center, FrattiniC23C23×D191
commutator, socle, cosocleC23×C191C24
Fitting#390×C191C2
hypercenter#390D191
hypocenterC191#2433
LCSG ▷ C23×C191 ▷ C191
UCSC1 ◁ C23 ◁ #390

Complements

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

NH ≅ G/Ntype
1. C22×D4×D191C2semidirect
2. C2×#20259319C2semidirect
3. C2×#20259318C2semidirect
4. C2×#20259297C2semidirect
5. C2×#20259292C2semidirect
6. #27132572C2semidirect
7. #27132564C2semidirect
8. C4×(C2×D191)⋊C4C2semidirect
9. #390×C191C2semidirect
13. C22×D764C22semidirect
14. C22×D764C22semidirect
16. C22×C22⋊D191C22semidirect
26. C2×(C2×D191)⋊C4C22semidirect
27. C2×(C2×D191)⋊C4C22semidirect
28. C2×C4×Dic191C22semidirect
29. C191⋊#95C22semidirect
31. C2×C4⋊C4×C191C22semidirect
32. C2×C22⋊C4×C191C22semidirect
33. C2×C42×C191C22semidirect
52. C2×D764D4semidirect
64. C2×C4×D191D4semidirect
67. (C2×D191)⋊C4D4semidirect
72. C2×D4×C191D4semidirect
110. C2×Dic191C2×D4semidirect
126. C2×C4×C191C2×D4semidirect
152. C2×D191#127semidirect
155. C4×C191#120semidirect
167. C191#2433semidirect
176. #390D191semidirect
194. C2×C4⋊C4C2×D191semidirect
198. C2×C22⋊C4C2×D191semidirect
200. C2×C42C2×D191semidirect
231. C2×D4D764semidirect
293. C2×C4D4×D191semidirect
325. C4#20259292semidirect

Maximal subgroups

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

subgroupindex
C22×D4×D1912
C2×#202593192
C2×#202593182
C2×#202592972
C2×#202592922
#271325722
#271325642
C4×(C2×D191)⋊C42
#390×C1912
#2433191