Finite Group Databasebeta

← #17995098

Subgroups — #17995098

Order 4736.

332 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 332 conjugacy classes of subgroups — too many to draw. Only its 79 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
1#14558333C211yesmaximal, characteristic
2#14558001C211yesmaximal, characteristic, complemented
3#14557549C211yesmaximal, characteristic, complemented
4C2×#11924462C211yesmaximal, characteristic, complemented
5#14557506C211yesmaximal, characteristic, complemented
6#14557294C211yesmaximal, characteristic, complemented
7C37×#344C211yesmaximal, characteristic, complemented
8C2×#10596962C2211yescharacteristic
9C2×#10596962C2211yescharacteristic
10#11924569—21no—
11#11924569C412yescomplemented
12#11924564—21no—
13#11924564C412yescomplemented
14#11924558—22no—
15#11924501—21no—
16#11924498—21no—
17#11924496—22no—
18#11924495C2211yescharacteristic
19#11924493—21no—
20C2×C4×Dic37C2211yescharacteristic
21C42×D37—21no—
22#11924462—22no—
23#11924462—21no—
24#11924462—21no—
25#11924450—22no—
26C2×M16×C37C2211yescharacteristic
27C2×M16×C37C2211yescharacteristic
28#117×C37C2211yescharacteristic
29C22×C4×C37C2311yescharacteristic
30C2×C22⋊D37—21no—
31C2×C22⋊D37—21no—
32C22×Dic37—21no—
33#10596962—21no—
34#10596962—21no—
35#10596962—21no—
36#10596962—21no—
37#10596962—21no—
38#10596962—21no—
39#10596962C2×C412yes—
40#10596962D412yes—
41C2×D148C2×C411yescharacteristic, complemented
42C2×D148D411yescharacteristic
43C2×C4×D37—21no—
44C2×C4×D37—22no—
45C2×C4×D37—21no—
46C2×Dic74C2×C411yescharacteristic, complemented
47C2×Dic74D411yescharacteristic
48M16×C37—21no—
49M16×C37—21no—
50M16×C37—21no—
51M16×C37C2×C412yescomplemented
52C2×C8×C37—21no—
53C2×C8×C37—21no—
54C4⋊C4×C37—21no—
55C22⋊C4×C37D412yes—
56C42×C37—21no—
57C22⋊Dic37—21no—
58#10596938—22no—
59C4⋊Dic37—21no—
60#10596936—22no—
61C4×Dic37—22no—
62C4×Dic37—21no—
63C4×Dic37—21no—
64C4×Dic37D411yescharacteristic
65C4×Dic37D411yescharacteristic
66Dic148—22no—
67D296—22no—
68#10596930—22no—
69#10596930—22no—
70C23×C37C2×D411yescharacteristic
71C22×D37—21no—
72C22×D37—21no—
73C2×C4×C37—22no—
74C2×C4×C37C22×C411yescharacteristic
75C2×C4×C37C2×D411yescharacteristic
76C2×C4×C37C2×D411yescharacteristic
77C2×C4×C37C2×D411yescharacteristic
78C2×C4×C37#4111yescharacteristic
79C2×C4×C37#4111yescharacteristic
80C22⋊D37—22no—
81C22⋊D37—22no—
82C22⋊D37—41no—
83C22⋊D37—41no—
84C2×Dic37—21no—
85C2×Dic37—21no—
86C2×Dic37—21no—
87C2×Dic37—21no—
88C2×Dic37—42no—
89C2×Dic37—41no—
90D148—21no—
91D148—21no—
92D148—21no—
93D148C22⋊C412yes—
94C4×D37—22no—
95C4×D37—22no—
96C4×D37—44no—
97C4×D37—41no—
98C4×D37—41no—
99Dic74—21no—
100Dic74—21no—
101Dic74—21no—
102Dic74C22⋊C412yes—
103C8×C37—22no—
104C8×C37—41no—
105C22×C37—41no—
106C22×C37C4×D411yescharacteristic
107C22×C37#12011yescharacteristic
108C22×C37#12111yescharacteristic
109C2×D37—22no—
110C2×D37—42no—
111C2×D37—41no—
112C2×D37—81no—
113C4×C37—42no—
114C4×C37C2×C22⋊C411yescharacteristic
115C4×C37C4×D411yescharacteristic
116C4×C37#12111yescharacteristic
117C4×C37#12311yescharacteristic
118Dic37—22no—
119Dic37—22no—
120Dic37—41no—
121Dic37—41no—
122Dic37—81no—
123#1906—371nomaximal, Sylow
124C2×C37—21no—
125C2×C37—21no—
126C2×C37—21no—
127C2×C37#38611yescharacteristic
128D37—42no—
129D37—81no—
130#575—371no—
131C4×#41—371no—
132#421—371no—
133C2×#104—371no—
134#410—371no—
135#344D3711yescharacteristic, complemented
136#339—371no—
137C37#190611yescharacteristic, complemented, Sylow
138C2×#41—371no—
139C2×#41—371no—
140#137—372no—
141#137—741no—
142AGL(1,ℤ/8)—372no—
143AGL(1,ℤ/8)—741no—
144#135—742no—
145C2×M16C2×D3711yescharacteristic
146C2×M16C2×D3711yescharacteristic
147C4×Q8—741no—
148C4×D4—741no—
149C4×D4—742no—
150#117—371no—
151#117—741no—
152#117C2×D3711yescharacteristic
153C2×C42—371no—
154C2×C42—741no—
155#104—741no—
156#104—741no—
157#104—742no—
158#99—742no—
159#41—372no—
160#41—372no—
161#41—741no—
162#41—741no—
163#41—741no—
164#41—741no—
165#41—741no—
166#41—741no—
167C2×Q8—371no—
168C2×Q8—371no—
169C2×D4—371no—
170C2×D4—371no—
171C2×D4—741no—
172C2×D4—741no—
173C22×C4—742no—
174C22×C4—741no—
175C22×C4—741no—
176C22×C4—741no—
177C22×C4C22×D3711yescharacteristic
178Q16—742no—
179SD16—742no—
180SD16—742no—
181D8—742no—
182M16—21no—
183M16—21no—
184M16—21no—
185M16C4×D3712yescomplemented
186C2×C8—21no—
187C2×C8—21no—
188C4⋊C4—21no—
189C4⋊C4—742no—
190C4⋊C4—741no—
191C22⋊C4—742no—
192C22⋊C4—741no—
193C22⋊C4D14812yes—
194C42—21no—
195C42—371no—
196C42—371no—
197C42—742no—
198C42—741no—
199C42—741no—
200C23—741no—
201C23—741no—
202C23C2×D14811yescharacteristic
203Q8—372no—
204Q8—741no—
205Q8—741no—
206Q8—741no—
207D4—372no—
208D4—742no—
209D4—742no—
210D4—741no—
211D4—741no—
212D4—741no—
213D4—1481no—
214D4—1481no—
215C2×C4—22no—
216C2×C4—741no—
217C2×C4—741no—
218C2×C4—742no—
219C2×C4—742no—
220C2×C4—741no—
221C2×C4—741no—
222C2×C4—1484no—
223C2×C4—1481no—
224C2×C4—1482no—
225C2×C4—1481no—
226C2×C4—1481no—
227C2×C4C2×C4×D3711yescharacteristic
228C2×C4#1059696211yescharacteristic
229C2×C4D4×D3711yescharacteristic
230C2×C4D4×D3711yescharacteristic
231C2×C4D4×D3711yescharacteristic
232C2×C4#1059696611yescharacteristic
233C8—22no—
234C8—41no—
235C22—41no—
236C22—2961no—
237C22—742no—
238C22—1482no—
239C22—1481no—
240C22C4×D14811yescharacteristic
241C22#1192450611yescharacteristic
242C22#1192451111yescharacteristic
243C4—42no—
244C4—2961no—
245C4—742no—
246C4—742no—
247C4—1481no—
248C4—1481no—
249C4C22⋊C4×D3711yescharacteristic
250C4#1192449811yescharacteristic
251C4#1192450911yescharacteristic
252C4#1192452111yescharacteristic
253C2—21no—
254C2—21no—
255C2—21no—
256C2—2961no—
257C2—1482no—
258C2#1455741111yescharacteristic
subgroupG/H
centerC4#11924509
commutatorC2×C4×C37C22×C4
FrattiniC22×C4C22×D37
FittingC37×#344C2
socleC2×C37#386
prime radical#14558333C2
cosocleC22×C4×C37C23
hypercenter#344D37
hypocenterC37#1906
LCSG ▷ C2×C4×C37 ▷ C2×C37 ▷ C37
UCSC1 ◁ C4 ◁ C22×C4 ◁ #344

Complements

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

NH ≅ G/Ntype
2. #14558001C2semidirect
3. #14557549C2semidirect
4. C2×#11924462C2semidirect
5. #14557506C2semidirect
6. #14557294C2semidirect
7. C37×#344C2semidirect
11. #11924569C4semidirect
13. #11924564C4semidirect
41. C2×D148C2×C4semidirect
46. C2×Dic74C2×C4semidirect
51. M16×C37C2×C4semidirect
135. #344D37semidirect
137. C37#1906semidirect
185. M16C4×D37semidirect

Maximal subgroups

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

subgroupindex
#145583332
#145580012
#145575492
C2×#119244622
#145575062
#145572942
C37×#3442
#190637