Finite Group Databasebeta

← #37582209

Subgroups — #37582209

Order 24448.

472 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 472 conjugacy classes of subgroups — too many to draw. Only its 133 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×#20259353C211yesmaximal, characteristic, complemented
2C2×#20259328C211yesmaximal, characteristic, complemented
3#27133244C211yesmaximal, characteristic
4#27133237C211yesmaximal, characteristic, complemented
5#27132920C212yesmaximal, complemented
6#27132912C212yesmaximal, complemented
7#27132903C212yesmaximal, complemented
8#27132895C212yesmaximal, complemented
9#27132703C211yesmaximal, characteristic, complemented
10#27132660C211yesmaximal, characteristic, complemented
11C2×#103×C191C211yesmaximal, characteristic, complemented
12C2×#15690287C2211yescharacteristic
13C2×#15690287C2211yescharacteristic
14#20259353—24no—
15C2×#15690277C2212yes—
16C2×#15690274C2212yes—
17#20259328—24no—
18C2×#15690271C2211yescharacteristic
19C2×#15690270C2212yes—
20C2×C8×D191C2212yes—
21#20259320—21no—
22#20259311—21no—
23#20259305—22no—
24#20259302—21no—
25#20259302C2212yes—
26#20259298C2212yes—
27#20259298C2212yes—
28#20259297—21no—
29#20259297C2212yes—
30#20259296—22no—
31#20259281C2211yescharacteristic
32#20259255C2212yes—
33#20259253C2212yes—
34#20259247C2211yescharacteristic
35#20259247C2211yescharacteristic
36#20259244C2211yescharacteristic
37#20259244C2211yescharacteristic
38C22×Q8×C191C2211yescharacteristic
39C22×C8×C191C2211yescharacteristic
40C2×C4⋊C4×C191C2211yescharacteristic
41#103×C191C2212yes—
42#103×C191C2212yes—
43#15690287—22no—
44#15690287—22no—
45#15690287—22no—
46#15690287—22no—
47C2×Dic382D411yescharacteristic
48C2×Dic382C2311yescharacteristic
49C2×D764—22no—
50C2×D764D411yescharacteristic
51C2×D764C2311yescharacteristic
52C2×C22⋊D191—21no—
53C2×C22⋊D191D412yes—
54C2×C4×D191—21no—
55C2×C4×D191C2312yes—
56#15690277—24no—
57#15690274—24no—
58#15690271D414yescomplemented
59#15690270—24no—
60C2×C191⋊C8C2312yes—
61C8×D191—24no—
62C4⋊Dic191—22no—
63C4⋊Dic191C2311yescharacteristic
64C4⋊Dic191C2311yescharacteristic
65Dic191⋊C4—22no—
66Dic191⋊C4—22no—
67C22⋊Dic191—21no—
68(C2×D191)⋊C4—22no—
69(C2×D191)⋊C4—22no—
70(C2×D191)⋊C4—22no—
71(C2×D191)⋊C4—22no—
72C4×Dic191—21no—
73C2×Q8×C191—21no—
74C2×Q8×C191—22no—
75C2×Q8×C191—22no—
76C2×Q8×C191C2312yes—
77C22×C4×C191—21no—
78C22×C4×C191—21no—
79C22×C4×C191C2311yescharacteristic
80C2×C8×C191—22no—
81C2×C8×C191C2311yescharacteristic
82C2×C8×C191C2311yescharacteristic
83C4⋊C4×C191—21no—
84C4⋊C4×C191C2312yes—
85C22×D191—21no—
86C22×D191C2×D412yes—
87Dic382—22no—
88Dic382—21no—
89Dic382—21no—
90D764—22no—
91D764—21no—
92D764—21no—
93D764—82no—
94C22⋊D191—24no—
95C22⋊D191—42no—
96C2×Dic191—21no—
97C2×Dic191—21no—
98C2×Dic191—21no—
99C2×Dic191C2×D412yes—
100C4×D191—22no—
101C4×D191—22no—
102C4×D191—42no—
103C191⋊C8C2×D414yes—
104C23×C191C2×D411yescharacteristic
105Q8×C191—22no—
106Q8×C191—22no—
107Q8×C191—42no—
108C2×C4×C191—22no—
109C2×C4×C191—22no—
110C2×C4×C191—42no—
111C2×C4×C191—81no—
112C2×C4×C191C2×D411yescharacteristic
113C2×C4×C191C2×D412yes—
114C2×C4×C191#4112yes—
115C2×C4×C191C2411yescharacteristic
116C8×C191—21no—
117C8×C191—21no—
118C22×C191—21no—
119C22×C191—21no—
120C22×C191#12112yes—
121C22×C191C22×D411yescharacteristic
122C2×D191—22no—
123C2×D191—22no—
124C2×D191—22no—
125C2×D191—41no—
126C2×D191—41no—
127C2×D191—41no—
128C4×C191—42no—
129C4×C191—42no—
130C4×C191#12112yes—
131C4×C191C22×D411yescharacteristic
132C4×C191C2×#4111yescharacteristic
133Dic191—22no—
134Dic191—41no—
135Dic191—41no—
136Dic191—41no—
137C2×C191—22no—
138C2×C191C2×#12111yescharacteristic
139C2×C191#57711yescharacteristic
140C2×C191#57811yescharacteristic
141D191—42no—
142D191—81no—
143C191#306311yescharacteristic, complemented, Sylow
144#3063—1911nomaximal, Sylow
145C2×#137—1911no—
146C2×#131—1911no—
147#537—1911no—
148#536—1911no—
149#470—1912no—
150#468—1912no—
151#467—1912no—
152#465—1912no—
153#428—1911no—
154#416—1911no—
155C2×#103D19111yescharacteristic, complemented
156C2×#41—1911no—
157C2×#41—1911no—
158C22×Q8C2×D19111yescharacteristic
159#137—3824no—
160C2×Q16—1912no—
161C2×SD16—1912no—
162#131—3824no—
163C2×M16—1911no—
164C2×M16—1912no—
165C22×C8—1912no—
166C22×C8C2×D19111yescharacteristic
167C4⋊Q8—3821no—
168#124—3821no—
169#123—3822no—
170C22⋊Q8—3821no—
171C22⋊Q8—1912no—
172C22⋊Q8—1912no—
173C22⋊Q8—1912no—
174#121—3821no—
175#121—3822no—
176#121—1912no—
177#117—1911no—
178C2×C4⋊C4C2×D19111yescharacteristic
179#107—1912no—
180#106—1912no—
181#103—1911no—
182#103—1911no—
183#103C2×D19112yes—
184#103C2×D19112yes—
185#102—1911no—
186#102—1911no—
187#41—3822no—
188#41—3822no—
189#41—3822no—
190#41—3822no—
191C2×Q8—21no—
192C2×Q8—22no—
193C2×Q8—22no—
194C2×Q8—1911no—
195C2×Q8—1911no—
196C2×Q8C22×D19112yes—
197C2×D4—3822no—
198C2×D4—3821no—
199C2×D4—1912no—
200C2×D4—1911no—
201C2×D4—1911no—
202C22×C4—21no—
203C22×C4—21no—
204C22×C4—3821no—
205C22×C4—1912no—
206C22×C4C22×D19111yescharacteristic
207Q16—3824no—
208SD16—3824no—
209M16—3824no—
210M16—1914no—
211C2×C8—22no—
212C2×C8—3824no—
213C2×C8—1912no—
214C2×C8C22×D19111yescharacteristic
215C2×C8C22×D19111yescharacteristic
216C4⋊C4—21no—
217C4⋊C4—3822no—
218C4⋊C4—3822no—
219C4⋊C4—3822no—
220C4⋊C4—1911no—
221C4⋊C4—1911no—
222C4⋊C4C22×D19112yes—
223C22⋊C4—3822no—
224C22⋊C4—3822no—
225C22⋊C4—3822no—
226C22⋊C4—3822no—
227C22⋊C4—3821no—
228C42—3821no—
229C23—3821no—
230C23—1912no—
231C23D4×D19111yescharacteristic
232Q8—22no—
233Q8—22no—
234Q8—42no—
235Q8—3822no—
236Q8—3821no—
237Q8—3821no—
238D4—3824no—
239D4—3821no—
240D4—3821no—
241D4—3822no—
242D4—15282no—
243D4—7642no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—42no—
247C2×C4—81no—
248C2×C4—3822no—
249C2×C4—3822no—
250C2×C4—3821no—
251C2×C4—3821no—
252C2×C4—3821no—
253C2×C4—1912no—
254C2×C4—7642no—
255C2×C4D4×D19112yes—
256C2×C4D4×D19111yescharacteristic
257C2×C4#1569028712yes—
258C2×C4C23×D19111yescharacteristic
259C8—21no—
260C8—21no—
261C8—1914no—
262C22—21no—
263C22—21no—
264C22—3822no—
265C22—3822no—
266C22—3822no—
267C22—7641no—
268C22—7641no—
269C22—7641no—
270C22#2025929512yes—
271C22C2×D4×D19111yescharacteristic
272C4—42no—
273C4—42no—
274C4—3822no—
275C4—7641no—
276C4—7641no—
277C4—7641no—
278C4#2025929512yes—
279C4C2×D4×D19111yescharacteristic
280C4C2×#1569028711yescharacteristic
281C2—22no—
282C2—15281no—
283C2—7642no—
284C2C2×#2025929511yescharacteristic
285C2#2713346911yescharacteristic
286C2#2713347611yescharacteristic
subgroupG/H
centerC22C2×D4×D191
commutator, cosocleC2×C4×C191C24
FrattiniC2×C4C23×D191
FittingC2×#103×C191C2
socleC22×C191C22×D4
prime radical#27133244C2
hypercenterC2×#103D191
hypocenterC191#3063
LCSG ▷ C2×C4×C191 ▷ C2×C191 ▷ C191
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#103

Complements

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

NH ≅ G/Ntype
1. C2×#20259353C2semidirect
2. C2×#20259328C2semidirect
4. #27133237C2semidirect
5. #27132920C2semidirect
6. #27132912C2semidirect
7. #27132903C2semidirect
8. #27132895C2semidirect
9. #27132703C2semidirect
10. #27132660C2semidirect
11. C2×#103×C191C2semidirect
58. #15690271D4semidirect
143. C191#3063semidirect
155. C2×#103D191semidirect

Maximal subgroups

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

subgroupindex
C2×#202593532
C2×#202593282
#271332442
#271332372
#271329202
#271329122
#271329032
#271328952
#271327032
#271326602
C2×#103×C1912
#3063191