Finite Group Databasebeta

← #18007310

Subgroups — #18007310

Order 4736.

372 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 372 conjugacy classes of subgroups — too many to draw. Only its 123 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
1Q8×D148C211yesmaximal, characteristic, complemented
2#14557892C212yesmaximal, complemented
3#14557883C211yesmaximal, characteristic, complemented
4#14557828C212yesmaximal, complemented
5#14557813C211yesmaximal, characteristic, complemented
6#14557745C212yesmaximal
7#14557735C211yesmaximal, characteristic, complemented
8#14557664C211yesmaximal, characteristic, complemented
9#14557631C212yesmaximal, complemented
10C4×#10596939C211yesmaximal, characteristic
11C37×#556C211yesmaximal, characteristic, complemented
12C2×Q8×D37—21no—
13C2×#10596941C2212yes—
14C2×#10596939C2211yescharacteristic
15#11924536C2211yescharacteristic
16#11924536C2212yes—
17#11924516—21no—
18#11924516—22no—
19C4×D148C2211yescharacteristic
20C4×D148C2212yes—
21#11924469C2212yes—
22#11924469C2211yescharacteristic
23C4⋊#64436C2211yescharacteristic
24C4⋊#64436C2212yes—
25#11924460C2212yes—
26#11924460C2212yes—
27#11924460C2212yes—
28#11924457C2212yes—
29#11924457C2211yescharacteristic
30#11924456C2211yescharacteristic
31#11924447C2212yes—
32C4×#64436C2211yescharacteristic
33C4⋊Q8×C37C2211yescharacteristic
34#126×C37—22no—
35#124×C37C2212yes—
36C22⋊Q8×C37—21no—
37C4×Q8×C37C2212yes—
38C4×Q8×C37C2211yescharacteristic
39C4×D4×C37C2211yescharacteristic
40#117×C37—21no—
41C2×Q8×C37D411yescharacteristic
42C2×Q8×C37C2312yes—
43C2×D4×C37C2311yescharacteristic
44C22×C4×C37—21no—
45Q8×D37—44no—
46C2×D148C2311yescharacteristic
47C2×C4×D37—22no—
48C2×C4×D37—21no—
49C2×Dic74—22no—
50C2×Dic74—21no—
51C4⋊C4×C37—21no—
52C4⋊C4×C37—22no—
53C4⋊C4×C37—21no—
54C4⋊C4×C37D411yescharacteristic
55C4⋊C4×C37D412yes—
56C4⋊C4×C37C2311yescharacteristic
57C22⋊C4×C37—22no—
58C22⋊C4×C37—21no—
59C22⋊C4×C37—22no—
60C42×C37—22no—
61C42×C37C2311yescharacteristic
62C42×C37C2312yes—
63#10596941—24no—
64#10596939—22no—
65#10596938—22no—
66#10596938—21no—
67C4⋊Dic37—22no—
68C4⋊Dic37—21no—
69C4⋊Dic37C2311yescharacteristic
70C4⋊Dic37C2312yes—
71C2×#64436C2311yescharacteristic
72C2×#64436C2312yes—
73C2×#64436C2311yescharacteristic
74C23×C37—21no—
75C22×D37—21no—
76Q8×C37—22no—
77Q8×C37—22no—
78Q8×C37—22no—
79D4×C37—21no—
80D4×C37—21no—
81C2×C4×C37—21no—
82C2×C4×C37—22no—
83C2×C4×C37—42no—
84C2×C4×C37C2×D412yes—
85C2×C4×C37C2×D411yescharacteristic
86C2×C4×C37C2×D412yes—
87C2×C4×C37C2×D411yescharacteristic
88C2×C4×C37C2411yescharacteristic
89C2×Dic37—21no—
90C2×Dic37—22no—
91D148—21no—
92D148#4112yes—
93C4×D37—44no—
94C4×D37—42no—
95Dic74—44no—
96Dic74—42no—
97#64436—21no—
98#64436—22no—
99#64436—21no—
100C22×C37—41no—
101C22×C37—41no—
102C22×C37—41no—
103C22×C37C22×D411yescharacteristic
104C2×D37—22no—
105C2×D37—41no—
106C2×D37—41no—
107C4×C37—22no—
108C4×C37—22no—
109C4×C37—21no—
110C4×C37—42no—
111C4×C37—41no—
112C4×C37AGL(1,ℤ/8)12yes—
113C4×C37C2×#4111yescharacteristic
114C4×C372−1+411yescharacteristic
115Dic37—41no—
116Dic37—42no—
117#3367—371nomaximal, Sylow
118C2×C37—81no—
119C2×C37#54811yescharacteristic
120C2×C37C2×AGL(1,ℤ/8)11yescharacteristic
121C2×C37#57811yescharacteristic
122D37—42no—
123#556D3711yescharacteristic, complemented
124D4×Q8—371no—
125#489—372no—
126#487—371no—
127#477—372no—
128#474—371no—
129#463—372no—
130#461—371no—
131#446—371no—
132#440—372no—
133C4×D8—371no—
134C37#336711yescharacteristic, complemented, Sylow
135C22×Q8—741no—
136C2×SD16—372no—
137C2×D8—371no—
138C4⋊Q8—371no—
139C4⋊Q8—372no—
140C4⋊Q8C2×D3711yescharacteristic
141#126—22no—
142#124C2×D3712yes—
143C22⋊Q8—21no—
144C22⋊Q8—741no—
145C22⋊Q8—742no—
146C4×Q8C2×D3712yes—
147C4×Q8C2×D3711yescharacteristic
148C4×D4—371no—
149C4×D4—372no—
150C4×D4C2×D3711yescharacteristic
151#117—21no—
152#107—371no—
153#107—372no—
154C4⋊C8—371no—
155C4⋊C8—372no—
156#103—372no—
157#103—372no—
158#103—372no—
159#102—371no—
160#102—372no—
161#102—371no—
162#97—372no—
163C4×C8—371no—
164C2×Q8—742no—
165C2×Q8—741no—
166C2×Q8—1484no—
167C2×Q8C22⋊D3711yescharacteristic
168C2×Q8C22×D3712yes—
169C2×D4—371no—
170C2×D4C22×D3711yescharacteristic
171C22×C4—21no—
172C22×C4—742no—
173C22×C4—741no—
174SD16—744no—
175D8—742no—
176C2×C8—371no—
177C2×C8—371no—
178C2×C8—372no—
179C4⋊C4—21no—
180C4⋊C4—22no—
181C4⋊C4—21no—
182C4⋊C4—371no—
183C4⋊C4—372no—
184C4⋊C4—742no—
185C4⋊C4—741no—
186C4⋊C4C22⋊D3711yescharacteristic
187C4⋊C4C22⋊D3712yes—
188C4⋊C4C22×D3711yescharacteristic
189C22⋊C4—22no—
190C22⋊C4—21no—
191C22⋊C4—22no—
192C22⋊C4—742no—
193C22⋊C4—741no—
194C42—22no—
195C42C22×D3711yescharacteristic
196C42C22×D3712yes—
197C23—21no—
198C23—741no—
199Q8—22no—
200Q8—22no—
201Q8—22no—
202Q8—1484no—
203Q8—1482no—
204D4—21no—
205D4—21no—
206D4—372no—
207D4—741no—
208C2×C4—21no—
209C2×C4—22no—
210C2×C4—42no—
211C2×C4—742no—
212C2×C4—741no—
213C2×C4—1484no—
214C2×C4—1482no—
215C2×C4C2×C22⋊D3712yes—
216C2×C4C2×C22⋊D3711yescharacteristic
217C2×C4C2×C22⋊D3712yes—
218C2×C4C2×C22⋊D3711yescharacteristic
219C2×C4C23×D3711yescharacteristic
220C8—741no—
221C8—742no—
222C8—741no—
223C22—41no—
224C22—41no—
225C22—41no—
226C22—742no—
227C22—1481no—
228C22—1481no—
229C22C22×C22⋊D3711yescharacteristic
230C4—22no—
231C4—22no—
232C4—21no—
233C4—41no—
234C4—42no—
235C4—1482no—
236C4—1481no—
237C4#1192456112yes—
238C4#41×D3711yescharacteristic
239C4#1192458311yescharacteristic
240C2—81no—
241C2—1482no—
242C2#1455819011yescharacteristic
243C2C2×#1192456111yescharacteristic
244C2#1455834811yescharacteristic
subgroupG/H
centerC22C22×C22⋊D37
commutator, cosocleC2×C4×C37C24
FrattiniC2×C4C23×D37
FittingC37×#556C2
socleC22×C37C22×D4
prime radicalC2×#10596939C22
hypercenter#556D37
hypocenterC37#3367
LCSG ▷ C2×C4×C37 ▷ C2×C37 ▷ C37
UCSC1 ◁ C22 ◁ C4×Q8 ◁ #556

Complements

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

NH ≅ G/Ntype
1. Q8×D148C2semidirect
2. #14557892C2semidirect
3. #14557883C2semidirect
4. #14557828C2semidirect
5. #14557813C2semidirect
7. #14557735C2semidirect
8. #14557664C2semidirect
9. #14557631C2semidirect
11. C37×#556C2semidirect
123. #556D37semidirect
134. C37#3367semidirect

Maximal subgroups

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

subgroupindex
Q8×D1482
#145578922
#145578832
#145578282
#145578132
#145577452
#145577352
#145576642
#145576312
C4×#105969392
C37×#5562
#336737