Finite Group Databasebeta

← #18006325

Subgroups — #18006325

Order 4736.

494 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 494 conjugacy classes of subgroups — too many to draw. Only its 125 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×#11924557C212yesmaximal, complemented
2#14558203C211yesmaximal, characteristic, complemented
3#14558186C211yesmaximal, characteristic, complemented
4#14557759C212yesmaximal, complemented
5#14557740C211yesmaximal, characteristic, complemented
6#14557725C211yesmaximal, characteristic, complemented
7#14557692C212yesmaximal, complemented
8#14557677C212yesmaximal, complemented
9C4×#10596930C211yesmaximal, characteristic, complemented
10C8×D148C211yesmaximal, characteristic, complemented
11C37×#492C211yesmaximal, characteristic, complemented
12C2×#10596966C2212yes—
13C2×#10596964C2212yes—
14#11924557—28no—
15C2×#10596942C2212yes—
16C2×#10596930C2211yescharacteristic
17C2×#10596939C2212yes—
18C2×C8×D37C2212yes—
19#11924532—21no—
20#11924521—22no—
21#11924512C2212yes—
22#11924511—21no—
23#11924508—22no—
24#11924507C2212yes—
25C4×Dic74C2211yescharacteristic
26C4×D148C2211yescharacteristic
27C4×D148C2211yescharacteristic
28D4×Dic37—21no—
29C4⋊C4×D37—21no—
30#11924468C2211yescharacteristic
31C4⋊#64436C2211yescharacteristic
32#11924461C2211yescharacteristic
33#11924460C2212yes—
34#11924458C2211yescharacteristic
35#11924457C2212yes—
36#11924448C2212yes—
37C2×SD16×C37C2212yes—
38C2×SD16×C37C2212yes—
39C4⋊Q8×C37C2211yescharacteristic
40#127×C37C2211yescharacteristic
41C4×C8×C37C2211yescharacteristic
42C2×Q8×C37C2312yes—
43C2×D4×C37—21no—
44C2×D4×C37—21no—
45C2×D4×C37C2312yes—
46C2×C22⋊D37—22no—
47C2×C22⋊D37—22no—
48C22×Dic37—22no—
49#10596966—24no—
50#10596966—42no—
51#10596964—24no—
52#10596964—42no—
53C2×D148—22no—
54C2×D148D411yescharacteristic
55C2×D148C2311yescharacteristic
56C2×C4×D37—21no—
57C2×C4×D37—22no—
58C2×C4×D37C2312yes—
59C2×Dic74C2311yescharacteristic
60SD16×C37—24no—
61SD16×C37—42no—
62C2×C8×C37C2311yescharacteristic
63C2×C8×C37C2311yescharacteristic
64C4⋊C4×C37—21no—
65C4⋊C4×C37—21no—
66C42×C37C2311yescharacteristic
67C22⋊Dic37—22no—
68#10596942—24no—
69#10596939—24no—
70#10596938—22no—
71#10596938—21no—
72#10596938D412yes—
73C4⋊Dic37D411yescharacteristic
74C4⋊Dic37C2311yescharacteristic
75C4⋊Dic37C2311yescharacteristic
76#10596936—21no—
77#10596936—22no—
78C4×Dic37—21no—
79C2×#64436C2312yes—
80#10596930D414yescomplemented
81C8×D37—24no—
82C23×C37—22no—
83C22×D37—21no—
84C22×D37C2×D412yes—
85Q8×C37—22no—
86Q8×C37—22no—
87D4×C37—22no—
88D4×C37—22no—
89D4×C37—42no—
90D4×C37—81no—
91C2×C4×C37—22no—
92C2×C4×C37C2×D411yescharacteristic
93C2×C4×C37C2×D411yescharacteristic
94C2×C4×C37C2411yescharacteristic
95C22⋊D37—44no—
96C22⋊D37—82no—
97C2×Dic37—21no—
98C2×Dic37—21no—
99C2×Dic37—21no—
100C2×Dic37—44no—
101C2×Dic37C2×D412yes—
102D148—22no—
103D148—21no—
104D148—44no—
105D148C2×D412yes—
106C4×D37—22no—
107C4×D37—22no—
108C4×D37—44no—
109C4×D37—42no—
110Dic74—21no—
111Dic74C2×D412yes—
112C8×C37—21no—
113C8×C37C2×D412yes—
114#64436—22no—
115C22×C37—42no—
116C22×C37—42no—
117C22×C37—42no—
118C22×C37C22×D411yescharacteristic
119C2×D37—22no—
120C2×D37—22no—
121C2×D37—22no—
122C2×D37—22no—
123C2×D37—41no—
124C2×D37—41no—
125C4×C37—21no—
126C4×C37—42no—
127C4×C37#13512yescomplemented
128C4×C37C22×D411yescharacteristic
129C4×C37H(5,𝔽2)11yescharacteristic
130Dic37—22no—
131Dic37—22no—
132Dic37—41no—
133Dic37—41no—
134#3284—371nomaximal, Sylow
135C2×C37—82no—
136C2×C37D4211yescharacteristic
137C2×C37C2×#13511yescharacteristic
138C2×C37#57711yescharacteristic
139D37—42no—
140D37—42no—
141C2×#135—372no—
142#550—371no—
143#547—371no—
144#492D3711yescharacteristic, complemented
145#465—372no—
146#462—371no—
147#459—371no—
148#452—372no—
149#449—372no—
150C4×SD16—371no—
151C8×D4—371no—
152C37#328411yescharacteristic, complemented, Sylow
153C2×#41—372no—
154C2×#41—372no—
155#135—748no—
156C2×Q16—372no—
157C2×SD16—371no—
158C2×SD16C2×D3712yes—
159C2×SD16C2×D3712yes—
160C2×D8—372no—
161C22×C8—372no—
162C4⋊Q8C2×D3711yescharacteristic
163#127—741no—
164#127C2×D3711yescharacteristic
165#123—742no—
166C22⋊Q8—372no—
167#121—372no—
168#121—742no—
169#121—741no—
170C4×Q8—371no—
171C4×D4—371no—
172C4×D4—371no—
173C4×D4—741no—
174C2×C4⋊C4—741no—
175#106—371no—
176C4⋊C8—371no—
177#103—372no—
178#103—371no—
179#102—372no—
180#102—371no—
181C22⋊C8—372no—
182C4×C8C2×D3711yescharacteristic
183#41—744no—
184#41—744no—
185#41—1482no—
186#41—1482no—
187C2×Q8—371no—
188C2×Q8C22×D3712yes—
189C2×D4—21no—
190C2×D4—21no—
191C2×D4—371no—
192C2×D4—371no—
193C2×D4—742no—
194C2×D4—742no—
195C2×D4—742no—
196C2×D4C22×D3712yes—
197C22×C4—372no—
198C22×C4—742no—
199C22×C4—742no—
200C22×C4—741no—
201Q16—744no—
202SD16—24no—
203SD16—42no—
204SD16—374no—
205D8—744no—
206C2×C8—372no—
207C2×C8—744no—
208C2×C8C22×D3711yescharacteristic
209C2×C8C22×D3711yescharacteristic
210C4⋊C4—21no—
211C4⋊C4—21no—
212C4⋊C4—371no—
213C4⋊C4—371no—
214C4⋊C4—371no—
215C4⋊C4—742no—
216C4⋊C4—741no—
217C22⋊C4—372no—
218C22⋊C4—742no—
219C22⋊C4—742no—
220C22⋊C4—741no—
221C42—741no—
222C42C22×D3711yescharacteristic
223C23—22no—
224C23—372no—
225C23—741no—
226Q8—22no—
227Q8—22no—
228Q8—372no—
229Q8—741no—
230D4—22no—
231D4—22no—
232D4—42no—
233D4—81no—
234D4—372no—
235D4—2962no—
236D4—742no—
237D4—741no—
238D4—1484no—
239D4—1484no—
240C2×C4—22no—
241C2×C4—372no—
242C2×C4—741no—
243C2×C4—742no—
244C2×C4—742no—
245C2×C4—741no—
246C2×C4—741no—
247C2×C4—1484no—
248C2×C4—1484no—
249C2×C4—1482no—
250C2×C4D4×D3711yescharacteristic
251C2×C4D4×D3711yescharacteristic
252C2×C4C23×D3711yescharacteristic
253C8—21no—
254C8—742no—
255C8D4×D3712yes—
256C22—42no—
257C22—42no—
258C22—42no—
259C22—742no—
260C22—742no—
261C22—742no—
262C22—742no—
263C22—1481no—
264C22—1481no—
265C22C2×D4×D3711yescharacteristic
266C4—21no—
267C4—42no—
268C4—742no—
269C4—742no—
270C4—1481no—
271C4—1481no—
272C4#1192455712yescomplemented
273C4C2×D4×D3711yescharacteristic
274C4#1192458111yescharacteristic
275C2—82no—
276C2—1482no—
277C2—1482no—
278C2#1455816811yescharacteristic
279C2C2×#1192455711yescharacteristic
280C2#1455833911yescharacteristic
subgroupG/H
centerC22C2×D4×D37
commutator, cosocleC2×C4×C37C24
FrattiniC2×C4C23×D37
FittingC37×#492C2
socleC22×C37C22×D4
hypercenter#492D37
hypocenterC37#3284
LCSG ▷ C2×C4×C37 ▷ C2×C37 ▷ C37
UCSC1 ◁ C22 ◁ C42 ◁ #492

Complements

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

NH ≅ G/Ntype
1. C2×#11924557C2semidirect
2. #14558203C2semidirect
3. #14558186C2semidirect
4. #14557759C2semidirect
5. #14557740C2semidirect
6. #14557725C2semidirect
7. #14557692C2semidirect
8. #14557677C2semidirect
9. C4×#10596930C2semidirect
10. C8×D148C2semidirect
11. C37×#492C2semidirect
80. #10596930D4semidirect
127. C4×C37#135semidirect
144. #492D37semidirect
152. C37#3284semidirect
272. C4#11924557semidirect

Maximal subgroups

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

subgroupindex
C2×#119245572
#145582032
#145581862
#145577592
#145577402
#145577252
#145576922
#145576772
C4×#105969302
C8×D1482
C37×#4922
#328437