Finite Group Databasebeta

← #17997307

Subgroups — #17997307

Order 4736.

612 conjugacy classes of subgroups; only the 205 normal subgroups are shown.

normal subgroups only

This group has 612 conjugacy classes of subgroups — too many to draw. Only its 205 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×#11924524C211yesmaximal, characteristic
2C2×C4×Dic74C211yesmaximal, characteristic, complemented
3C2×C4×D148C211yesmaximal, characteristic
4#14557439C211yesmaximal, characteristic, complemented
5#14557435C211yesmaximal, characteristic
6#14557425C214yesmaximal, complemented
7#14557407C214yesmaximal, complemented
8#14557403C211yesmaximal, characteristic, complemented
9C37×#376C211yesmaximal, characteristic, complemented
10C22×Dic74C2211yescharacteristic
11C22×D148C2211yescharacteristic
12C22×C4×D37C2212yes—
13#11924524C418yescomplemented
14C4×Dic74—24no—
15C4×D148—24no—
16C2×C4⋊Dic37C2211yescharacteristic
17C2×C4⋊Dic37C2211yescharacteristic
18C2×C4⋊Dic37C2214yes—
19C2×#10596936C2212yes—
20C2×#10596938C2212yes—
21C2×#10596938C2214yes—
22C2×#10596938C2214yes—
23C2×C4×Dic37C2212yes—
24#11924443C2214yes—
25C2×C42×C37C2211yescharacteristic
26C2×C42×C37C2211yescharacteristic
27C2×C42×C37C2211yescharacteristic
28#95×C37C2212yes—
29#95×C37C2212yes—
30C23×D37C2312yes—
31C22×C4×C37C2312yes—
32C22×C4×C37C2312yes—
33C22×C4×C37C2311yescharacteristic
34C22×C4×C37C2311yescharacteristic
35C22×C4×C37C2311yescharacteristic
36C22×Dic37C2312yes—
37C22×Dic37C2312yes—
38C22×Dic37C2312yes—
39C2×D148—22no—
40C2×D148—22no—
41C2×D148C2×C414yes—
42C2×C4×D37—24no—
43C2×C4×D37—44no—
44C2×Dic74—22no—
45C2×Dic74—22no—
46C2×Dic74C2×C414yescomplemented
47C42×C37—22no—
48C42×C37C2×C414yescomplemented
49C42×C37D414yes—
50#10596938—24no—
51#10596938—44no—
52#10596938C2×C4116yescomplemented
53C4⋊Dic37—22no—
54C4⋊Dic37—22no—
55C4⋊Dic37—44no—
56#10596936—24no—
57C4×Dic37—24no—
58C23×C37C2411yescharacteristic
59C22×D37—22no—
60C22×D37—22no—
61C22×D37—24no—
62C22×D37—24no—
63C22×D37C22×C414yes—
64C2×C4×C37—22no—
65C2×C4×C37—22no—
66C2×C4×C37—22no—
67C2×C4×C37—22no—
68C2×C4×C37—22no—
69C2×C4×C37—22no—
70C2×C4×C37—22no—
71C2×C4×C37C22×C414yes—
72C2×C4×C37C22×C412yes—
73C2×C4×C37C2×D412yes—
74C2×C4×C37C2×D412yes—
75C2×C4×C37C2×D412yes—
76C2×C4×C37#4112yes—
77C2×Dic37—22no—
78C2×Dic37—22no—
79C2×Dic37—22no—
80C2×Dic37—22no—
81C2×Dic37—24no—
82C2×Dic37—24no—
83C2×Dic37C22×C414yes—
84D148—28no—
85C4×D37—48no—
86Dic74—28no—
87C22×C37C23×C411yescharacteristic
88C22×C37C22×D411yescharacteristic
89C22×C37C2×#4111yescharacteristic
90C22×C37H(5,𝔽2)12yes—
91C22×C372−1+412yes—
92C2×D37—24no—
93C2×D37—28no—
94C2×D37—42no—
95C2×D37—42no—
96C2×D37—44no—
97C4×C37—24no—
98C4×C37—22no—
99C4×C37—22no—
100C4×C37—42no—
101C4×C37C4×D414yes—
102Dic37—24no—
103Dic37—42no—
104Dic37—42no—
105#2326—371nomaximal, Sylow
106C2×C37C2×C4×D411yescharacteristic
107C2×C37#52012yes—
108C2×C37#53511yescharacteristic
109C2×C37#53611yescharacteristic
110C2×C37#54212yes—
111D37—44no—
112C2×#124—371no—
113C2×C4×Q8—371no—
114C2×C4×D4—371no—
115#391—371no—
116#390—371no—
117#388—374no—
118#385—374no—
119#384—371no—
120#376D3711yescharacteristic, complemented
121C37#232611yescharacteristic, complemented, Sylow
122C22×Q8—371no—
123C22×D4—371no—
124C23×C4—372no—
125#124—378no—
126C4×Q8—744no—
127C4×D4—744no—
128C2×C4⋊C4—371no—
129C2×C4⋊C4—372no—
130C2×C4⋊C4—371no—
131C2×C4⋊C4—374no—
132C2×C22⋊C4—374no—
133C2×C22⋊C4—372no—
134C2×C22⋊C4—374no—
135C2×C42—372no—
136C2×C42C2×D3711yescharacteristic
137C2×C42C2×D3711yescharacteristic
138C2×C42C2×D3711yescharacteristic
139#95—374no—
140#95C2×D3712yes—
141#95C2×D3712yes—
142C24—372no—
143C2×Q8—374no—
144C2×Q8—742no—
145C2×Q8—742no—
146C2×D4—374no—
147C2×D4—742no—
148C2×D4—742no—
149C22×C4—372no—
150C22×C4—372no—
151C22×C4—372no—
152C22×C4—744no—
153C22×C4—1484no—
154C22×C4C22×D3712yes—
155C22×C4C22×D3712yes—
156C22×C4C22×D3711yescharacteristic
157C22×C4C22×D3711yescharacteristic
158C22×C4C22×D3711yescharacteristic
159C4⋊C4—744no—
160C4⋊C4—742no—
161C4⋊C4—742no—
162C4⋊C4—1484no—
163C22⋊C4—3716no—
164C22⋊C4—744no—
165C22⋊C4—1484no—
166C42—22no—
167C42—744no—
168C42C4×D3714yescomplemented
169C42D14814yes—
170C23—374no—
171C23—742no—
172C23—742no—
173C23—744no—
174C23—744no—
175C23C23×D3711yescharacteristic
176Q8—748no—
177D4—748no—
178C2×C4—22no—
179C2×C4—22no—
180C2×C4—22no—
181C2×C4—22no—
182C2×C4—22no—
183C2×C4—22no—
184C2×C4—22no—
185C2×C4—374no—
186C2×C4—742no—
187C2×C4—742no—
188C2×C4—742no—
189C2×C4—742no—
190C2×C4—744no—
191C2×C4—744no—
192C2×C4—1488no—
193C2×C4C2×C4×D3714yes—
194C2×C4C2×C4×D3712yes—
195C2×C4C2×D14812yes—
196C2×C4C2×D14812yes—
197C2×C4C2×D14812yes—
198C2×C4#1059696212yes—
199C22—744no—
200C22—748no—
201C22—1482no—
202C22—1482no—
203C22—1484no—
204C22C22×C4×D3711yescharacteristic
205C22C22×D14811yescharacteristic
206C22C2×#1059696211yescharacteristic
207C22#1192458212yes—
208C22#1192458312yes—
209C4—24no—
210C4—22no—
211C4—22no—
212C4—42no—
213C4—744no—
214C4—1482no—
215C4—1482no—
216C4C4×D14814yes—
217C2—1484no—
218C2C2×C4×D14811yescharacteristic
219C2#1455801312yes—
220C2#1455810211yescharacteristic
221C2#1455810911yescharacteristic
222C2#1455815412yes—
subgroupG/H
center, FrattiniC23C23×D37
commutatorC22×C37C23×C4
FittingC37×#376C2
socle, cosocleC23×C37C24
prime radicalC22×D148C22
hypercenter#376D37
hypocenterC37#2326
LCSG ▷ C22×C37 ▷ C37
UCSC1 ◁ C23 ◁ #376

Complements

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

NH ≅ G/Ntype
2. C2×C4×Dic74C2semidirect
4. #14557439C2semidirect
6. #14557425C2semidirect
7. #14557407C2semidirect
8. #14557403C2semidirect
9. C37×#376C2semidirect
13. #11924524C4semidirect
46. C2×Dic74C2×C4semidirect
48. C42×C37C2×C4semidirect
52. #10596938C2×C4semidirect
120. #376D37semidirect
121. C37#2326semidirect
168. C42C4×D37semidirect

Maximal subgroups

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

subgroupindex
C2×#119245242
C2×C4×Dic742
C2×C4×D1482
#145574392
#145574352
#145574252
#145574072
#145574032
C37×#3762
#232637