Finite Group Databasebeta

← #18006338

Subgroups — #18006338

Order 4736.

482 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 482 conjugacy classes of subgroups — too many to draw. Only its 131 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×#11924559C211yesmaximal, characteristic, complemented
2C2×Q16×D37C211yesmaximal, characteristic, complemented
3#14558195C212yesmaximal, complemented
4#14557917C211yesmaximal, characteristic, complemented
5#14557775C211yesmaximal, characteristic, complemented
6#14557770C211yesmaximal, characteristic, complemented
7#14557755C212yesmaximal, complemented
8#14557698C212yesmaximal, complemented
9#14557673C212yesmaximal, complemented
10C8×D148C211yesmaximal, characteristic, complemented
11C4×Q16×C37C211yesmaximal, characteristic, complemented
12C2×#10596966C2212yes—
13C2×Q8×D37C2212yes—
14#11924559—24no—
15C2×Dic148C2211yescharacteristic
16C2×#10596942C2212yes—
17Q16×D37—24no—
18C2×#10596930C2212yes—
19C2×#10596941C2212yes—
20C2×D296C2211yescharacteristic
21C2×C8×D37C2211yescharacteristic
22C2×C8×D37C2211yescharacteristic
23#11924524—22no—
24#11924524C2212yes—
25#11924516—22no—
26#11924516C2212yes—
27#11924511—22no—
28#11924511C2212yes—
29C4×D148—22no—
30C4×D148C2211yescharacteristic
31C4⋊#64436C2211yescharacteristic
32#11924459C2212yes—
33#11924456C2212yes—
34#11924448C2211yescharacteristic
35#11924448C2211yescharacteristic
36C2×Q16×C37C2211yescharacteristic
37C4×Q8×C37C2212yes—
38#107×C37C2211yescharacteristic
39#103×C37C2212yes—
40C4×C8×C37C2211yescharacteristic
41C2×Q8×C37C2312yes—
42#10596966—24no—
43#10596966—42no—
44Q8×D37—24no—
45Q8×D37—42no—
46C2×D148—22no—
47C2×D148—22no—
48C2×D148D411yescharacteristic
49C2×D148C2312yes—
50C2×C4×D37—22no—
51C2×C4×D37—22no—
52C2×C4×D37C2311yescharacteristic
53C2×C4×D37C2311yescharacteristic
54C2×Dic74—22no—
55C2×Dic74C2312yes—
56Q16×C37D414yescomplemented
57C2×C8×C37C2311yescharacteristic
58C2×C8×C37C2311yescharacteristic
59C4⋊C4×C37—22no—
60C4⋊C4×C37C2312yes—
61C42×C37—22no—
62C42×C37C2311yescharacteristic
63#10596942—24no—
64#10596941—24no—
65#10596938—22no—
66#10596938—22no—
67#10596938—22no—
68#10596938—22no—
69#10596938D411yescharacteristic
70#10596938D411yescharacteristic
71C4⋊Dic37—22no—
72C4⋊Dic37D411yescharacteristic
73C2×#64436C2311yescharacteristic
74C2×#64436C2311yescharacteristic
75Dic148—22no—
76D296—22no—
77#10596930—42no—
78C8×D37—22no—
79C8×D37—22no—
80C22×D37—22no—
81C22×D37C2×D411yescharacteristic
82C22×D37C2×D411yescharacteristic
83Q8×C37—22no—
84Q8×C37C2×D414yes—
85C2×C4×C37—22no—
86C2×C4×C37—22no—
87C2×C4×C37C2×D411yescharacteristic
88C2×C4×C37C2×D411yescharacteristic
89C2×C4×C37C2411yescharacteristic
90C2×Dic37—22no—
91C2×Dic37C2×D411yescharacteristic
92C2×Dic37C2×D411yescharacteristic
93D148—22no—
94D148—22no—
95D148—22no—
96D148—44no—
97D148—44no—
98C4×D37—21no—
99C4×D37—21no—
100C4×D37—21no—
101C4×D37—21no—
102C4×D37—44no—
103C4×D37—44no—
104Dic74—22no—
105Dic74—22no—
106Dic74—44no—
107C8×C37—21no—
108C8×C37C2×D412yes—
109#64436—21no—
110#64436—21no—
111C22×C37C22×D411yescharacteristic
112C2×D37—21no—
113C2×D37—21no—
114C2×D37—21no—
115C2×D37—21no—
116C2×D37—21no—
117C2×D37—42no—
118C2×D37—42no—
119C2×D37—42no—
120C2×D37#13512yescomplemented
121C4×C37—24no—
122C4×C37—21no—
123C4×C37—21no—
124C4×C37—42no—
125C4×C37C22×D411yescharacteristic
126C4×C37H(5,𝔽2)11yescharacteristic
127Dic37—21no—
128Dic37—21no—
129Dic37—42no—
130#3285—371nomaximal, Sylow
131C2×C37D4211yescharacteristic
132C2×C37C2×#13511yescharacteristic
133C2×C37#57811yescharacteristic
134D37—22no—
135D37—41no—
136D37—82no—
137C2×#135—371no—
138C22×Q16—371no—
139#548—372no—
140#495—371no—
141#467—371no—
142#466—371no—
143#464—372no—
144#452—372no—
145#448—372no—
146C4×Q16D3711yescharacteristic, complemented
147C8×D4—371no—
148C37#328511yescharacteristic, complemented, Sylow
149C2×#41—372no—
150C22×Q8—372no—
151#135—744no—
152C2×Q16—372no—
153C2×Q16—371no—
154C2×Q16—744no—
155C2×Q16C2×D3711yescharacteristic
156C2×SD16—372no—
157C2×SD16—372no—
158C2×D8—371no—
159C22×C8—371no—
160C22×C8—371no—
161#124—372no—
162#124—742no—
163C22⋊Q8—372no—
164C22⋊Q8—742no—
165#121—372no—
166#121—742no—
167C4×Q8C2×D3712yes—
168C4×D4—371no—
169C4×D4—742no—
170#107C2×D3711yescharacteristic
171C4⋊C8—371no—
172#103—372no—
173#103C2×D3712yes—
174#102—372no—
175C22⋊C8—371no—
176C22⋊C8—371no—
177C4×C8C2×D3711yescharacteristic
178#41—744no—
179#41—1482no—
180C2×Q8—372no—
181C2×Q8—742no—
182C2×Q8—744no—
183C2×Q8—1482no—
184C2×Q8C22×D3712yes—
185C2×D4—371no—
186C2×D4—372no—
187C2×D4—742no—
188C2×D4—742no—
189C22×C4—371no—
190C22×C4—371no—
191C22×C4—742no—
192C22×C4—742no—
193Q16—744no—
194Q16—742no—
195Q16D14814yescomplemented
196SD16—744no—
197SD16—1482no—
198D8—742no—
199C2×C8—371no—
200C2×C8—371no—
201C2×C8—742no—
202C2×C8—742no—
203C2×C8C22×D3711yescharacteristic
204C2×C8C22×D3711yescharacteristic
205C4⋊C4—22no—
206C4⋊C4—371no—
207C4⋊C4—742no—
208C4⋊C4C22×D3712yes—
209C22⋊C4—371no—
210C22⋊C4—371no—
211C22⋊C4—742no—
212C22⋊C4—742no—
213C22⋊C4—742no—
214C22⋊C4—742no—
215C42—22no—
216C42C22×D3711yescharacteristic
217C23—371no—
218C23—371no—
219C23—742no—
220Q8—22no—
221Q8—742no—
222Q8—742no—
223Q8—1484no—
224Q8C2×D14814yes—
225D4—742no—
226D4—742no—
227D4—742no—
228D4—1484no—
229D4—1484no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—371no—
233C2×C4—371no—
234C2×C4—741no—
235C2×C4—741no—
236C2×C4—741no—
237C2×C4—741no—
238C2×C4—742no—
239C2×C4—1484no—
240C2×C4—1484no—
241C2×C4D4×D3711yescharacteristic
242C2×C4D4×D3711yescharacteristic
243C2×C4C23×D3711yescharacteristic
244C8—21no—
245C8—741no—
246C8—741no—
247C8C2×D14812yes—
248C22—372no—
249C22—741no—
250C22—741no—
251C22—741no—
252C22—741no—
253C22—741no—
254C22—1482no—
255C22—1482no—
256C22—1482no—
257C22C2×D4×D3711yescharacteristic
258C4—21no—
259C4—21no—
260C4—24no—
261C4—42no—
262C4—741no—
263C4—741no—
264C4—1482no—
265C4C22×D14811yescharacteristic
266C4#1192458211yescharacteristic
267C2—2962no—
268C2—742no—
269C2—1481no—
270C2D4×D14811yescharacteristic
271C2#135×D3711yescharacteristic
272C2#1455834611yescharacteristic
subgroupG/H
centerC22C2×D4×D37
commutator, cosocleC2×C4×C37C24
FrattiniC2×C4C23×D37
FittingC4×Q16×C37C2
socleC22×C37C22×D4
hypercenterC4×Q16D37
hypocenterC37#3285
LCSG ▷ C2×C4×C37 ▷ C2×C37 ▷ C37
UCSC1 ◁ C22 ◁ C42 ◁ C4×Q16

Complements

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

NH ≅ G/Ntype
1. C2×#11924559C2semidirect
2. C2×Q16×D37C2semidirect
3. #14558195C2semidirect
4. #14557917C2semidirect
5. #14557775C2semidirect
6. #14557770C2semidirect
7. #14557755C2semidirect
8. #14557698C2semidirect
9. #14557673C2semidirect
10. C8×D148C2semidirect
11. C4×Q16×C37C2semidirect
56. Q16×C37D4semidirect
120. C2×D37#135semidirect
146. C4×Q16D37semidirect
148. C37#3285semidirect
195. Q16D148semidirect

Maximal subgroups

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

subgroupindex
C2×#119245592
C2×Q16×D372
#145581952
#145579172
#145577752
#145577702
#145577552
#145576982
#145576732
C8×D1482
C4×Q16×C372
#328537