Finite Group Databasebeta

← #17994567

Subgroups — #17994567

Order 4736.

588 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 588 conjugacy classes of subgroups — too many to draw. Only its 97 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×#11924582C211yesmaximal, characteristic
2#14557549C212yesmaximal, complemented
3#14557539C212yesmaximal, complemented
4#14557507C211yesmaximal, characteristic, complemented
5C37×#413C211yesmaximal, characteristic, complemented
6#11924582—24no—
7#11924582—24no—
8C2×#10596966—21no—
9C2×#10596962C412yescomplemented
10C2×#10596962C2211yescharacteristic
11C22×D148C412yescomplemented
12C22×D148C2211yescharacteristic
13C2×D4×D37—22no—
14C2×D4×D37—21no—
15#11924495C2212yescomplemented
16#11924462—24no—
17#11924458—24no—
18#11924451—22no—
19C2×#41×C37C2211yescharacteristic
20C2×M16×C37C2212yescomplemented
21#101×C37—21no—
22#100×C37—21no—
23C23×D37—22no—
24C23×D37—21no—
25#41×C37—22no—
26#41×C37—22no—
27C2×Q8×C37D411yescharacteristic
28C2×D4×C37D412yes—
29C2×D4×C37D411yescharacteristic
30C22×C4×C37C2×C412yes—
31C22×C4×C37C2311yescharacteristic
32C2×C22⋊D37—22no—
33C2×C22⋊D37—21no—
34#10596966—28no—
35D4×D37—216no—
36D4×D37—28no—
37#10596962—22no—
38#10596962—24no—
39#10596962—24no—
40#10596962D414yes—
41C2×D148—22no—
42C2×D148—21no—
43C2×D148—22no—
44C2×D148—24no—
45C2×D148—24no—
46C2×D148C2×C411yescharacteristic
47C2×D148C2×C412yes—
48C2×D148D414yes—
49C2×C4×D37—22no—
50C2×C4×D37—21no—
51C2×Dic74C2×C411yescharacteristic
52M16×C37—22no—
53M16×C37—22no—
54C2×C8×C37—22no—
55C22⋊Dic37—22no—
56C4⋊Dic37—22no—
57C4×Dic37—22no—
58C23×C37C22⋊C412yes—
59C23×C37C2×D411yescharacteristic
60C22×D37—22no—
61C22×D37—21no—
62C22×D37—28no—
63C22×D37—24no—
64C22×D37—82no—
65C22×D37—84no—
66Q8×C37—22no—
67D4×C37—24no—
68D4×C37—22no—
69C2×C4×C37—22no—
70C2×C4×C37—22no—
71C2×C4×C37C22⋊C412yes—
72C2×C4×C37C22×C411yescharacteristic
73C2×C4×C37C2×D412yes—
74C2×C4×C37C2×D412yes—
75C2×C4×C37C2×D411yescharacteristic
76C22⋊D37—28no—
77C22⋊D37—24no—
78C2×Dic37—21no—
79C2×Dic37—42no—
80D148—28no—
81D148—21no—
82D148—22no—
83D148—24no—
84D148C22⋊C412yes—
85D148C22⋊C414yes—
86C4×D37—28no—
87C4×D37—24no—
88Dic74—21no—
89Dic74C22⋊C412yes—
90C8×C37—42no—
91C22×C37—22no—
92C22×C37—41no—
93C22×C37—42no—
94C22×C37C2×C22⋊C411yescharacteristic
95C22×C37#12012yes—
96C2×D37—24no—
97C2×D37—22no—
98C2×D37—44no—
99C2×D37—42no—
100C2×D37—84no—
101C2×D37—88no—
102C4×C37—22no—
103C4×C37C2×C22⋊C411yescharacteristic
104C4×C37C2×C22⋊C411yescharacteristic
105C4×C37#12012yes—
106Dic37—22no—
107Dic37—82no—
108#1795—371nomaximal, Sylow
109C2×C37—22no—
110C2×C37—21no—
111C2×C37—42no—
112C2×C37#37911yescharacteristic
113D37—44no—
114D37—42no—
115C2×H(5,𝔽2)—371no—
116#421—372no—
117#418—372no—
118#413D3711yescharacteristic, complemented
119#410—371no—
120C37#179511yescharacteristic, complemented, Sylow
121H(5,𝔽2)—744no—
122H(5,𝔽2)—744no—
123C2×#41—372no—
124C2×#41—371no—
125C2×#41—741no—
126C2×#41C2×D3711yescharacteristic
127C22×D4—372no—
128C22×D4—371no—
129C22×D4—742no—
130C22×D4—741no—
131C2×M16C2×D3712yescomplemented
132#117—372no—
133#104—744no—
134#102—744no—
135#101—21no—
136#100—21no—
137#99—742no—
138C24—742no—
139C24—741no—
140#41—22no—
141#41—22no—
142#41—374no—
143#41—744no—
144#41—744no—
145#41—748no—
146#41—742no—
147C2×Q8—371no—
148C2×Q8C22⋊D3711yescharacteristic
149C2×D4—372no—
150C2×D4—371no—
151C2×D4—374no—
152C2×D4—744no—
153C2×D4—744no—
154C2×D4—7416no—
155C2×D4—748no—
156C2×D4—742no—
157C2×D4—741no—
158C2×D4—742no—
159C2×D4—742no—
160C2×D4—741no—
161C2×D4D14812yes—
162C2×D4C22⋊D3711yescharacteristic
163C22×C4—742no—
164C22×C4—741no—
165C22×C4C4×D3712yes—
166C22×C4C22×D3711yescharacteristic
167M16—22no—
168M16—22no—
169C2×C8—22no—
170C4⋊C4—742no—
171C22⋊C4—742no—
172C42—742no—
173C23—2962no—
174C23—2964no—
175C23—748no—
176C23—744no—
177C23—742no—
178C23—741no—
179C23#1059693812yes—
180C23C2×D14811yescharacteristic
181Q8—22no—
182Q8—372no—
183Q8—741no—
184D4—24no—
185D4—22no—
186D4—374no—
187D4—372no—
188D4—748no—
189D4—748no—
190D4—744no—
191D4—741no—
192D4—742no—
193D4—744no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—748no—
197C2×C4—741no—
198C2×C4—744no—
199C2×C4—1482no—
200C2×C4#1059693812yes—
201C2×C4C2×C4×D3711yescharacteristic
202C2×C4D4×D3712yes—
203C2×C4D4×D3712yes—
204C2×C4C2×C22⋊D3711yescharacteristic
205C8—42no—
206C22—22no—
207C22—41no—
208C22—42no—
209C22—2964no—
210C22—2968no—
211C22—744no—
212C22—742no—
213C22—1484no—
214C22—1482no—
215C22C2×#1059693811yescharacteristic
216C22#1192450612yes—
217C4—22no—
218C4—2962no—
219C4—742no—
220C4C22⋊C4×D3711yescharacteristic
221C4C22⋊C4×D3711yescharacteristic
222C4#1192450412yes—
223C2—22no—
224C2—21no—
225C2—42no—
226C2—1484no—
227C2—1482no—
228C2#1455738811yescharacteristic
subgroupG/H
centerC2#14557388
commutatorC2×C4×C37C22×C4
FrattiniC22×C4C22×D37
FittingC37×#413C2
socleC2×C37#379
prime radicalC2×#11924582C2
cosocleC22×C4×C37C23
hypercenter#413D37
hypocenterC37#1795
LCSG ▷ C2×C4×C37 ▷ C2×C37 ▷ C37
UCSC1 ◁ C2 ◁ C22×C4 ◁ #413

Complements

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

NH ≅ G/Ntype
2. #14557549C2semidirect
3. #14557539C2semidirect
4. #14557507C2semidirect
5. C37×#413C2semidirect
9. C2×#10596962C4semidirect
11. C22×D148C4semidirect
15. #11924495C22semidirect
20. C2×M16×C37C22semidirect
118. #413D37semidirect
120. C37#1795semidirect
131. C2×M16C2×D37semidirect

Maximal subgroups

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

subgroupindex
C2×#119245822
#145575492
#145575392
#145575072
C37×#4132
#179537