Finite Group Databasebeta

← #37582153

Subgroups — #37582153

Order 24448.

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

normal subgroups only

This group has 496 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×#20259352C211yesmaximal, characteristic, complemented
2C2×M16×D191C211yesmaximal, characteristic, complemented
3#27133170C211yesmaximal, characteristic
4C2×#20259302C211yesmaximal, characteristic, complemented
5#27132920C212yesmaximal, complemented
6#27132908C212yesmaximal, complemented
7#27132903C212yesmaximal, complemented
8#27132891C212yesmaximal, complemented
9#27132703C211yesmaximal, characteristic, complemented
10#27132673C211yesmaximal, characteristic, complemented
11#417×C191C211yesmaximal, characteristic, complemented
12C22×Dic382C2211yescharacteristic
13C22×C4×D191C2211yescharacteristic
14#20259352—24no—
15C2×#15690277C2212yescomplemented
16C2×#15690275C2212yescomplemented
17C2×#15690271C2211yescharacteristic
18C2×#15690270C2212yes—
19M16×D191—24no—
20C2×C8×D191C2212yes—
21#20259303—21no—
22#20259302—21no—
23#20259302—21no—
24#20259302C2212yes—
25#20259302C2212yes—
26#20259298C2211yescharacteristic
27#20259298C2211yescharacteristic
28#20259293C2211yescharacteristic
29#20259293C2211yescharacteristic
30#20259290—21no—
31C4×C22⋊D191—21no—
32C4×C22⋊D191—21no—
33#20259281C2211yescharacteristic
34C2×C4⋊Dic191—21no—
35C2×(C2×D191)⋊C4—21no—
36#20259255C2212yescomplemented
37#20259253C2212yescomplemented
38#20259247C2212yescomplemented
39#20259247C2212yescomplemented
40C2×#41×C191C2211yescharacteristic
41C2×M16×C191C2211yescharacteristic
42C2×C4⋊C4×C191C2211yescharacteristic
43#103×C191C2212yescomplemented
44#102×C191C2212yescomplemented
45C23×D191D411yescharacteristic
46C2×Dic382—21no—
47C2×Dic382—22no—
48C2×Dic382—22no—
49C2×Dic382C2312yes—
50C2×C22⋊D191—21no—
51C2×C22⋊D191—21no—
52C22×Dic191—21no—
53C22×Dic191D411yescharacteristic
54C2×C4×D191—22no—
55C2×C4×D191—22no—
56C2×C4×D191D411yescharacteristic
57C2×C4×D191D411yescharacteristic
58C2×C4×D191C2311yescharacteristic
59C2×C4×D191C2311yescharacteristic
60#15690277—24no—
61#15690275—24no—
62#15690271D414yescomplemented
63#15690270—24no—
64C2×C191⋊C8C2312yes—
65C8×D191—24no—
66C4⋊Dic191—22no—
67C4⋊Dic191—22no—
68C4⋊Dic191C2311yescharacteristic
69C4⋊Dic191C2311yescharacteristic
70Dic191⋊C4—21no—
71Dic191⋊C4—21no—
72C22⋊Dic191—21no—
73C22⋊Dic191—21no—
74C22⋊Dic191—21no—
75(C2×D191)⋊C4—22no—
76(C2×D191)⋊C4—22no—
77(C2×D191)⋊C4—21no—
78(C2×D191)⋊C4—21no—
79C4×Dic191—21no—
80#41×C191—22no—
81#41×C191—22no—
82C2×Q8×C191C2311yescharacteristic
83C2×D4×C191—21no—
84C2×D4×C191C2311yescharacteristic
85C22×C4×C191—21no—
86C22×C4×C191—21no—
87C22×C4×C191C2311yescharacteristic
88M16×C191—22no—
89C2×C8×C191C2312yes—
90C4⋊C4×C191—21no—
91C4⋊C4×C191C2312yes—
92C22×D191—22no—
93C22×D191—22no—
94C22×D191—41no—
95C22×D191C2×D411yescharacteristic
96C22×D191C2×D411yescharacteristic
97Dic382—22no—
98Dic382—22no—
99Dic382—42no—
100C22⋊D191—42no—
101C22⋊D191—42no—
102C2×Dic191—22no—
103C2×Dic191—21no—
104C2×Dic191—21no—
105C2×Dic191—22no—
106C2×Dic191—42no—
107C2×Dic191C2×D411yescharacteristic
108C2×Dic191C2×D411yescharacteristic
109C4×D191—22no—
110C4×D191—22no—
111C4×D191—21no—
112C4×D191—21no—
113C4×D191—21no—
114C4×D191—21no—
115C191⋊C8C2×D414yes—
116C23×C191—21no—
117C23×C191C2×D411yescharacteristic
118Q8×C191—21no—
119Q8×C191—21no—
120D4×C191—21no—
121D4×C191—21no—
122D4×C191—42no—
123C2×C4×C191—22no—
124C2×C4×C191—21no—
125C2×C4×C191—42no—
126C2×C4×C191—42no—
127C2×C4×C191C2×D411yescharacteristic
128C2×C4×C191C2×D412yes—
129C2×C4×C191#4112yes—
130C2×C4×C191C2411yescharacteristic
131C8×C191—22no—
132C22×C191—21no—
133C22×C191—21no—
134C22×C191—41no—
135C22×C191—41no—
136C22×C191—41no—
137C22×C191#12112yes—
138C22×C191C22×D411yescharacteristic
139C2×D191—21no—
140C2×D191—21no—
141C2×D191—21no—
142C2×D191—21no—
143C2×D191—21no—
144C2×D191—44no—
145C2×D191#13712yescomplemented
146C4×C191—41no—
147C4×C191—42no—
148C4×C191#12112yes—
149C4×C191C22×D411yescharacteristic
150C4×C191C2×#4111yescharacteristic
151Dic191—21no—
152Dic191—21no—
153Dic191—42no—
154Dic191—41no—
155Dic191—41no—
156C2×C191—22no—
157C2×C191—81no—
158C2×C191C2×#12111yescharacteristic
159C2×C191C2×#13711yescharacteristic
160C2×C191#57511yescharacteristic
161D191—22no—
162D191—41no—
163C191#305611yescharacteristic, complemented, Sylow
164#3056—1911nomaximal, Sylow
165C2×#137—1911no—
166C22×M16—1911no—
167#525—1911no—
168C2×C22⋊Q8—1911no—
169#470—1912no—
170#468—1912no—
171#467—1912no—
172#465—1912no—
173#428—1911no—
174#419—1911no—
175#417D19111yescharacteristic, complemented
176C2×#41C2×D19111yescharacteristic
177C22×Q8—1911no—
178C23×C4—1911no—
179#137—3824no—
180C2×Q16—1912no—
181C2×SD16—1912no—
182C2×M16—3824no—
183C2×M16—1911no—
184C2×M16—1912no—
185C2×M16C2×D19111yescharacteristic
186C22×C8—1912no—
187#123—3821no—
188C22⋊Q8—3821no—
189C22⋊Q8—3821no—
190C22⋊Q8—1911no—
191C22⋊Q8—1911no—
192C22⋊Q8—1912no—
193C22⋊Q8—1912no—
194#121—1911no—
195#121—1911no—
196#120—3821no—
197C4×D4—3821no—
198C4×D4—3821no—
199#117—1911no—
200C2×C4⋊C4—3821no—
201C2×C4⋊C4C2×D19111yescharacteristic
202C2×C22⋊C4—3821no—
203#107—1912no—
204#106—1912no—
205#103—1912no—
206#103—1912no—
207#103C2×D19112yescomplemented
208#102C2×D19112yescomplemented
209C24—1911no—
210#41—22no—
211#41—22no—
212C2×Q8—3821no—
213C2×Q8—3822no—
214C2×Q8—3822no—
215C2×Q8—1912no—
216C2×Q8C22×D19111yescharacteristic
217C2×D4—21no—
218C2×D4—3821no—
219C2×D4—3821no—
220C2×D4C22×D19111yescharacteristic
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—3822no—
224C22×C4—3822no—
225C22×C4—3821no—
226C22×C4—1911no—
227C22×C4—1911no—
228C22×C4—1911no—
229C22×C4—1911no—
230C22×C4—1911no—
231C22×C4C22×D19111yescharacteristic
232Q16—3824no—
233SD16—3824no—
234M16—22no—
235M16—3824no—
236M16—1914no—
237C2×C8—3824no—
238C2×C8—1912no—
239C2×C8C22×D19112yes—
240C4⋊C4—21no—
241C4⋊C4—3821no—
242C4⋊C4—3821no—
243C4⋊C4—3822no—
244C4⋊C4—3822no—
245C4⋊C4—1911no—
246C4⋊C4—1911no—
247C4⋊C4C22×D19112yes—
248C22⋊C4—3822no—
249C22⋊C4—3822no—
250C22⋊C4—3821no—
251C22⋊C4—3821no—
252C22⋊C4—3821no—
253C22⋊C4—3821no—
254C22⋊C4—3821no—
255C42—3821no—
256C23—21no—
257C23—3822no—
258C23—3822no—
259C23—1911no—
260C23—1911no—
261C23—7641no—
262C23D4×D19111yescharacteristic
263Q8—21no—
264Q8—21no—
265Q8—3822no—
266Q8—3822no—
267Q8—7642no—
268D4—21no—
269D4—21no—
270D4—42no—
271D4—7642no—
272D4—7642no—
273C2×C4—22no—
274C2×C4—21no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—3822no—
278C2×C4—3821no—
279C2×C4—3821no—
280C2×C4—3821no—
281C2×C4—3821no—
282C2×C4—3822no—
283C2×C4—3822no—
284C2×C4—3821no—
285C2×C4—3821no—
286C2×C4—3822no—
287C2×C4—1911no—
288C2×C4—1911no—
289C2×C4—7642no—
290C2×C4D4×D19112yes—
291C2×C4D4×D19111yescharacteristic
292C2×C4#1569028712yes—
293C2×C4C23×D19111yescharacteristic
294C8—22no—
295C8—1914no—
296C22—21no—
297C22—21no—
298C22—41no—
299C22—41no—
300C22—41no—
301C22—3821no—
302C22—3821no—
303C22—3821no—
304C22—3821no—
305C22—3821no—
306C22—1912no—
307C22—7644no—
308C22#2025929512yes—
309C22C2×D4×D19111yescharacteristic
310C4—42no—
311C4—41no—
312C4—3821no—
313C4—3821no—
314C4—7641no—
315C4—7641no—
316C4—7642no—
317C4#2025929512yes—
318C4C2×D4×D19111yescharacteristic
319C4C2×#1569028711yescharacteristic
320C2—22no—
321C2—81no—
322C2—3822no—
323C2—7641no—
324C2C2×#2025929511yescharacteristic
325C2#137×D19111yescharacteristic
326C2#2713345911yescharacteristic
subgroupG/H
centerC22C2×D4×D191
commutator, cosocleC2×C4×C191C24
FrattiniC2×C4C23×D191
Fitting#417×C191C2
socleC22×C191C22×D4
prime radical#27133170C2
hypercenter#417D191
hypocenterC191#3056
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
1. C2×#20259352C2semidirect
2. C2×M16×D191C2semidirect
4. C2×#20259302C2semidirect
5. #27132920C2semidirect
6. #27132908C2semidirect
7. #27132903C2semidirect
8. #27132891C2semidirect
9. #27132703C2semidirect
10. #27132673C2semidirect
11. #417×C191C2semidirect
15. C2×#15690277C22semidirect
16. C2×#15690275C22semidirect
36. #20259255C22semidirect
37. #20259253C22semidirect
38. #20259247C22semidirect
39. #20259247C22semidirect
43. #103×C191C22semidirect
44. #102×C191C22semidirect
62. #15690271D4semidirect
145. C2×D191#137semidirect
163. C191#3056semidirect
175. #417D191semidirect
207. #103C2×D191semidirect
208. #102C2×D191semidirect

Maximal subgroups

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

subgroupindex
C2×#202593522
C2×M16×D1912
#271331702
C2×#202593022
#271329202
#271329082
#271329032
#271328912
#271327032
#271326732
#417×C1912
#3056191