Finite Group Databasebeta

← #18006403

Subgroups — #18006403

Order 4736.

494 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 494 conjugacy classes of subgroups — too many to draw. Only its 139 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
1D4×D148C211yesmaximal, characteristic, complemented
2#14557881C211yesmaximal, characteristic, complemented
3#14557846C212yesmaximal, complemented
4#14557810C211yesmaximal, characteristic, complemented
5#14557767C212yesmaximal, complemented
6#14557728C211yesmaximal, characteristic, complemented
7#14557666C212yesmaximal, complemented
8C4×#10596939C211yesmaximal, characteristic, complemented
9D4×#64436C211yesmaximal, characteristic, complemented
10C2×#11924456C212yesmaximal, complemented
11C37×#547C211yesmaximal, characteristic, complemented
12C22×D148C2212yes—
13C2×D4×D37—21no—
14C2×#10596939C2211yescharacteristic
15C2×#10596939C2212yescomplemented
16C22×#64436C2212yescomplemented
17#11924533C2211yescharacteristic
18#11924511—21no—
19#11924510C2212yes—
20#11924506—22no—
21C4×D148C2211yescharacteristic
22#11924469C2212yescomplemented
23#11924469C2211yescharacteristic, complemented
24C4⋊#64436C2211yescharacteristic, complemented
25#11924457C2211yescharacteristic, complemented
26#11924456C2211yescharacteristic
27#11924456C2212yes—
28#11924456C2212yescomplemented
29#11924456C2212yescomplemented
30#11924456C2212yescomplemented
31C22⋊#64436C2212yescomplemented
32C4×#64436C2211yescharacteristic, complemented
33C2×#41×C37—21no—
34C4⋊Q8×C37C2211yescharacteristic, complemented
35#123×C37—22no—
36C22⋊Q8×C37—21no—
37#121×C37C2212yescomplemented
38C4×D4×C37C2211yescharacteristic
39C4×D4×C37C2211yescharacteristic, complemented
40C2×C4⋊C4×C37C2212yescomplemented
41C23×D37—22no—
42#41×C37—44no—
43C2×Q8×C37—21no—
44C2×D4×C37—22no—
45C2×D4×C37D411yescharacteristic
46C2×D4×C37C2311yescharacteristic
47C22×C4×C37—22no—
48C22×C4×C37—21no—
49C22×C4×C37C2312yes—
50C2×C22⋊D37—22no—
51D4×D37—44no—
52C2×D148—24no—
53C2×D148—22no—
54C2×D148—21no—
55C2×D148—21no—
56C2×D148—42no—
57C2×D148C2312yes—
58C2×D148C2311yescharacteristic
59C2×C4×D37—21no—
60C4⋊C4×C37—22no—
61C4⋊C4×C37—21no—
62C4⋊C4×C37D411yescharacteristic
63C4⋊C4×C37C2311yescharacteristic
64C4⋊C4×C37C2312yes—
65C22⋊C4×C37—21no—
66C22⋊C4×C37—22no—
67C22⋊C4×C37D412yes—
68C42×C37C2311yescharacteristic
69#10596939—22no—
70#10596939—42no—
71#10596938—21no—
72#10596938—22no—
73C4⋊Dic37C2311yescharacteristic
74C2×#64436—42no—
75C2×#64436C2311yescharacteristic
76C2×#64436C2312yes—
77C2×#64436C2311yescharacteristic
78C23×C37—21no—
79C23×C37C2×D412yes—
80C22×D37—21no—
81C22×D37—22no—
82C22×D37—44no—
83C22×D37—44no—
84C22×D37—82no—
85Q8×C37—42no—
86D4×C37—21no—
87D4×C37—21no—
88D4×C37—44no—
89D4×C37D814yescomplemented
90C2×C4×C37—22no—
91C2×C4×C37—22no—
92C2×C4×C37—21no—
93C2×C4×C37—22no—
94C2×C4×C37—44no—
95C2×C4×C37—42no—
96C2×C4×C37C2×D411yescharacteristic
97C2×C4×C37C2×D412yes—
98C2×C4×C37C2×D411yescharacteristic
99C2×C4×C37C2411yescharacteristic
100C22⋊D37—44no—
101C2×Dic37—21no—
102D148—22no—
103D148—21no—
104D148—22no—
105D148—44no—
106D148—42no—
107D148—81no—
108D148#4112yescomplemented
109C4×D37—42no—
110#64436—21no—
111#64436—21no—
112#64436—22no—
113C22×C37—22no—
114C22×C37—22no—
115C22×C37—41no—
116C22×C37—41no—
117C22×C37—41no—
118C22×C37C2×D814yes—
119C22×C37C22×D411yescharacteristic
120C2×D37—22no—
121C2×D37—41no—
122C2×D37—41no—
123C2×D37—42no—
124C2×D37—42no—
125C2×D37—42no—
126C2×D37—88no—
127C4×C37—22no—
128C4×C37—21no—
129C4×C37—41no—
130C4×C37—42no—
131C4×C37C2×D812yes—
132C4×C37C2×#4111yescharacteristic
133C4×C37H(5,𝔽2)11yescharacteristic
134Dic37—41no—
135#3294—371nomaximal, Sylow
136C2×C37—24no—
137C2×C37—81no—
138C2×C37#54611yescharacteristic
139C2×C37C22×D811yescharacteristic
140C2×C37#57711yescharacteristic
141D37—42no—
142D37—82no—
143#547D3711yescharacteristic, complemented
144D42—371no—
145#486—371no—
146#480—372no—
147#474—371no—
148#466—372no—
149#459—371no—
150#447—372no—
151C4×D8—371no—
152C8×D4—371no—
153C2×#102—372no—
154C37#329411yescharacteristic, complemented, Sylow
155C2×#41—21no—
156C22×D4—372no—
157C22×D4—741no—
158C2×D8—371no—
159C2×D8—372no—
160C22×C8—372no—
161C4⋊Q8C2×D3711yescharacteristic, complemented
162#127—371no—
163#123—22no—
164C22⋊Q8—21no—
165#121—372no—
166#121—741no—
167#121C2×D3712yescomplemented
168#120—742no—
169C4×D4—371no—
170C4×D4C2×D3711yescharacteristic
171C4×D4C2×D3711yescharacteristic, complemented
172C2×C4⋊C4C2×D3712yescomplemented
173#107—371no—
174#107—372no—
175C4⋊C8—371no—
176#102—371no—
177#102—371no—
178#102—372no—
179#102—372no—
180#102—372no—
181#102—372no—
182C22⋊C8—372no—
183C4×C8—371no—
184C24—742no—
185#41—44no—
186C2×Q8—21no—
187C2×D4—22no—
188C2×D4—371no—
189C2×D4—372no—
190C2×D4—744no—
191C2×D4—742no—
192C2×D4—741no—
193C2×D4—742no—
194C2×D4—741no—
195C2×D4—1482no—
196C2×D4—1484no—
197C2×D4C22⋊D3711yescharacteristic
198C2×D4C22×D3711yescharacteristic
199C22×C4—22no—
200C22×C4—21no—
201C22×C4—741no—
202C22×C4C22×D3712yes—
203D8—742no—
204D8—1482no—
205C2×C8—371no—
206C2×C8—371no—
207C2×C8—372no—
208C2×C8—1482no—
209C4⋊C4—22no—
210C4⋊C4—21no—
211C4⋊C4—371no—
212C4⋊C4C22⋊D3711yescharacteristic
213C4⋊C4C22×D3711yescharacteristic
214C4⋊C4C22×D3712yes—
215C22⋊C4—21no—
216C22⋊C4—22no—
217C22⋊C4—741no—
218C22⋊C4—742no—
219C22⋊C4C22⋊D3712yes—
220C42C22×D3711yescharacteristic
221C23—21no—
222C23—2962no—
223C23—741no—
224C23—742no—
225C23—1484no—
226C23—1484no—
227C23C2×C22⋊D3712yes—
228Q8—42no—
229D4—21no—
230D4—21no—
231D4—44no—
232D4—372no—
233D4—2961no—
234D4—742no—
235D4—741no—
236D4—742no—
237D4—1484no—
238D4—1484no—
239D4—1482no—
240D4#1059693914yescomplemented
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—21no—
244C2×C4—22no—
245C2×C4—44no—
246C2×C4—42no—
247C2×C4—741no—
248C2×C4—1482no—
249C2×C4C2×C22⋊D3711yescharacteristic
250C2×C4C2×C22⋊D3712yes—
251C2×C4C2×C22⋊D3711yescharacteristic
252C2×C4C23×D3711yescharacteristic
253C8—741no—
254C8—741no—
255C8—742no—
256C22—22no—
257C22—22no—
258C22—41no—
259C22—41no—
260C22—41no—
261C22—2968no—
262C22—742no—
263C22—1482no—
264C22—1482no—
265C22—1482no—
266C22—1481no—
267C22—1481no—
268C22C2×#1059693914yes—
269C22C22×C22⋊D3711yescharacteristic
270C4—21no—
271C4—22no—
272C4—41no—
273C4—42no—
274C4—1481no—
275C4C2×#1059693912yes—
276C4#41×D3711yescharacteristic
277C4#1192458211yescharacteristic
278C2—24no—
279C2—81no—
280C2—2962no—
281C2—1482no—
282C2#1455817111yescharacteristic
283C2C22×#1059693911yescharacteristic
284C2#1455834111yescharacteristic
subgroupG/H
centerC22C22×C22⋊D37
commutator, cosocleC2×C4×C37C24
FrattiniC2×C4C23×D37
FittingC37×#547C2
socleC22×C37C22×D4
hypercenter#547D37
hypocenterC37#3294
LCSG ▷ C2×C4×C37 ▷ C2×C37 ▷ C37
UCSC1 ◁ C22 ◁ C4×D4 ◁ #547

Complements

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

NH ≅ G/Ntype
1. D4×D148C2semidirect
2. #14557881C2semidirect
3. #14557846C2semidirect
4. #14557810C2semidirect
5. #14557767C2semidirect
6. #14557728C2semidirect
7. #14557666C2semidirect
8. C4×#10596939C2semidirect
9. D4×#64436C2semidirect
10. C2×#11924456C2semidirect
11. C37×#547C2semidirect
15. C2×#10596939C22semidirect
16. C22×#64436C22semidirect
22. #11924469C22semidirect
23. #11924469C22semidirect
24. C4⋊#64436C22semidirect
25. #11924457C22semidirect
28. #11924456C22semidirect
29. #11924456C22semidirect
30. #11924456C22semidirect
31. C22⋊#64436C22semidirect
32. C4×#64436C22semidirect
34. C4⋊Q8×C37C22semidirect
37. #121×C37C22semidirect
39. C4×D4×C37C22semidirect
40. C2×C4⋊C4×C37C22semidirect
89. D4×C37D8semidirect
108. D148#41semidirect
143. #547D37semidirect
154. C37#3294semidirect
161. C4⋊Q8C2×D37semidirect
167. #121C2×D37semidirect
171. C4×D4C2×D37semidirect
172. C2×C4⋊C4C2×D37semidirect
240. D4#10596939semidirect

Maximal subgroups

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

subgroupindex
D4×D1482
#145578812
#145578462
#145578102
#145577672
#145577282
#145576662
C4×#105969392
D4×#644362
C2×#119244562
C37×#5472
#329437