Finite Group Databasebeta

← #17995125

Subgroups — #17995125

Order 4736.

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

normal subgroups only

This group has 362 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#14558097C211yesmaximal, characteristic
2#14558007C211yesmaximal, characteristic, complemented
3#14557550C211yesmaximal, characteristic, complemented
4C2×#11924452C211yesmaximal, characteristic, complemented
5#14557290C211yesmaximal, characteristic, complemented
6#14557289C211yesmaximal, characteristic, complemented
7C37×#421C211yesmaximal, characteristic, complemented
8C22×D148C2211yescharacteristic
9C2×#10596934C2211yescharacteristic
10#11924532C412yescomplemented
11#11924525C412yescomplemented
12#11924510—22no—
13#11924504—22no—
14C4×D148—22no—
15#11924498—22no—
16#11924495C2211yescharacteristic
17#11924495C2211yescharacteristic
18C22⋊C4×D37—22no—
19#11924463—22no—
20#11924452—22no—
21C2×#41×C37C2211yescharacteristic
22C2×M16×C37C2211yescharacteristic
23#117×C37C2211yescharacteristic
24#104×C37—22no—
25C23×D37D412yes—
26#41×C37—21no—
27#41×C37—21no—
28#41×C37—21no—
29#41×C37—21no—
30C2×Q8×C37C2×C411yescharacteristic, complemented
31C2×D4×C37—21no—
32C2×D4×C37C2×C411yescharacteristic, complemented
33C22×C4×C37—21no—
34C22×C4×C37C2311yescharacteristic
35C2×C22⋊D37—24no—
36C2×D148—21no—
37C2×D148—21no—
38C2×D148—21no—
39C2×D148—21no—
40C2×D148C2×C412yes—
41C2×D148D412yes—
42C2×C4×D37—24no—
43M16×C37—21no—
44M16×C37—21no—
45C2×C8×C37—21no—
46C4⋊C4×C37—21no—
47C22⋊C4×C37—21no—
48C42×C37D412yescomplemented
49C22⋊Dic37—21no—
50C22⋊Dic37—21no—
51#10596938—24no—
52#10596938—24no—
53C4⋊Dic37—21no—
54C4⋊Dic37—21no—
55C4×Dic37C2×C412yescomplemented
56C4×Dic37D412yescomplemented
57#10596934—21no—
58#10596934—21no—
59C2×#64436—21no—
60C23×C37—21no—
61C23×C37C2×D411yescharacteristic
62C22×D37—22no—
63C22×D37—22no—
64C22×D37—82no—
65C22×D37C22⋊C414yes—
66Q8×C37—21no—
67Q8×C37—21no—
68D4×C37—21no—
69D4×C37—21no—
70D4×C37—41no—
71D4×C37—41no—
72C2×C4×C37—21no—
73C2×C4×C37—22no—
74C2×C4×C37—41no—
75C2×C4×C37—41no—
76C2×C4×C37C22×C411yescharacteristic
77C2×C4×C37C2×D411yescharacteristic
78C2×C4×C37C2×D411yescharacteristic
79C2×C4×C37C2×D411yescharacteristic
80C2×C4×C37#4111yescharacteristic
81C2×C4×C37#4111yescharacteristic
82C22⋊D37—84no—
83C2×Dic37—22no—
84C2×Dic37—22no—
85D148—22no—
86D148—22no—
87D148—22no—
88D148—22no—
89C4×D37—84no—
90C8×C37—41no—
91#64436—41no—
92C22×C37—41no—
93C22×C37—42no—
94C22×C37—41no—
95C22×C37C2×C22⋊C411yescharacteristic
96C22×C37#12011yescharacteristic
97C22×C37#12311yescharacteristic
98C2×D37—24no—
99C2×D37—44no—
100C2×D37—44no—
101C2×D37—82no—
102C4×C37—42no—
103C4×C37—41no—
104C4×C37C4×D411yescharacteristic
105C4×C37C4×D411yescharacteristic
106C4×C37#12111yescharacteristic
107C4×C37#12111yescharacteristic
108Dic37—42no—
109Dic37—42no—
110#1911—371nomaximal, Sylow
111C2×C37—21no—
112C2×C37—21no—
113C2×C37—21no—
114C2×C37—81no—
115C2×C37#38611yescharacteristic
116D37—44no—
117#535—371no—
118#518—371no—
119#421—371no—
120#421D3711yescharacteristic, complemented
121C2×#100—371no—
122H(3,ℤ/4)—371no—
123H(3,ℤ/4)—371no—
124C37#191111yescharacteristic, complemented, Sylow
125C2×#41C2×D3711yescharacteristic
126C22×D4—371no—
127C2×M16—371no—
128C2×M16C2×D3711yescharacteristic
129#127—372no—
130#124—372no—
131#121—742no—
132#120—742no—
133C4×D4—742no—
134C4×D4—742no—
135#117—371no—
136#117—371no—
137#117C2×D3711yescharacteristic
138C2×C22⋊C4—742no—
139#104—22no—
140#104—742no—
141#100—742no—
142C24—372no—
143#41—21no—
144#41—21no—
145#41—21no—
146#41—21no—
147C2×Q8C4×D3711yescharacteristic, complemented
148C2×D4—21no—
149C2×D4—372no—
150C2×D4—372no—
151C2×D4—744no—
152C2×D4—741no—
153C2×D4—741no—
154C2×D4—741no—
155C2×D4—741no—
156C2×D4C4×D3711yescharacteristic, complemented
157C22×C4—21no—
158C22×C4—744no—
159C22×C4C22×D3711yescharacteristic
160M16—21no—
161M16—21no—
162M16—741no—
163M16—741no—
164C2×C8—21no—
165C2×C8—741no—
166C4⋊C4—21no—
167C4⋊C4—741no—
168C4⋊C4—741no—
169C22⋊C4—21no—
170C22⋊C4—744no—
171C22⋊C4—744no—
172C22⋊C4—741no—
173C22⋊C4—741no—
174C42—372no—
175C42—372no—
176C42C22⋊D3712yescomplemented
177C23—21no—
178C23—374no—
179C23—2962no—
180C23—742no—
181C23—742no—
182C23D4×D3711yescharacteristic
183Q8—21no—
184Q8—21no—
185D4—21no—
186D4—21no—
187D4—41no—
188D4—41no—
189D4—2964no—
190D4—742no—
191D4—742no—
192D4—742no—
193D4—742no—
194C2×C4—21no—
195C2×C4—22no—
196C2×C4—41no—
197C2×C4—41no—
198C2×C4—2964no—
199C2×C4—742no—
200C2×C4—742no—
201C2×C4C2×C4×D3711yescharacteristic
202C2×C4#1059696211yescharacteristic
203C2×C4D4×D3711yescharacteristic
204C2×C4D4×D3711yescharacteristic
205C2×C4#1059696411yescharacteristic
206C2×C4C2×C22⋊D3711yescharacteristic
207C8—41no—
208C8—1481no—
209C22—41no—
210C22—42no—
211C22—41no—
212C22—2962no—
213C22—744no—
214C22—1484no—
215C22—1484no—
216C22C22⋊C4×D3711yescharacteristic
217C22#1192450411yescharacteristic
218C22#1192451811yescharacteristic
219C4—42no—
220C4—41no—
221C4—1482no—
222C4—1482no—
223C4#1192449611yescharacteristic
224C4C4×C22⋊D3711yescharacteristic
225C4#1192450711yescharacteristic
226C4#1192450911yescharacteristic
227C2—21no—
228C2—21no—
229C2—21no—
230C2—81no—
231C2—1484no—
232C2#1455740911yescharacteristic
subgroupG/H
centerC2#14557409
commutatorC2×C4×C37C22×C4
FrattiniC22×C4C22×D37
FittingC37×#421C2
socleC2×C37#386
prime radical#14558097C2
cosocleC22×C4×C37C23
hypercenter#421D37
hypocenterC37#1911
LCSG ▷ C2×C4×C37 ▷ C2×C37 ▷ C37
UCSC1 ◁ C2 ◁ C22×C4 ◁ #421

Complements

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

NH ≅ G/Ntype
2. #14558007C2semidirect
3. #14557550C2semidirect
4. C2×#11924452C2semidirect
5. #14557290C2semidirect
6. #14557289C2semidirect
7. C37×#421C2semidirect
10. #11924532C4semidirect
11. #11924525C4semidirect
30. C2×Q8×C37C2×C4semidirect
32. C2×D4×C37C2×C4semidirect
48. C42×C37D4semidirect
55. C4×Dic37C2×C4semidirect
56. C4×Dic37D4semidirect
120. #421D37semidirect
124. C37#1911semidirect
147. C2×Q8C4×D37semidirect
156. C2×D4C4×D37semidirect
176. C42C22⋊D37semidirect

Maximal subgroups

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

subgroupindex
#145580972
#145580072
#145575502
C2×#119244522
#145572902
#145572892
C37×#4212
#191137