Finite Group Databasebeta

← #352×D37 #17996230

Subgroups — #352×D37 #17996230

Order 4736.

278 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 278 conjugacy classes of subgroups — too many to draw. Only its 57 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#123×D37C211yesmaximal, characteristic
2#100×D37C211yesmaximal, characteristic
3#99×D37C211yesmaximal, characteristic
4#14557324C211yesmaximal, characteristic, complemented
5#14557323C211yesmaximal, characteristic, complemented
6#14557322C211yesmaximal, characteristic, complemented
7C37×#352C211yesmaximal, characteristic, complemented
8C2×D4×D37C2211yescharacteristic
9C22×C4×D37C411yescharacteristic, complemented
10M16×D37—21no—
11#11924521—21no—
12#11924520C411yescharacteristic, complemented
13#11924519C411yescharacteristic, complemented
14#11924518—21no—
15#11924517C2211yescharacteristic
16C4⋊C4×D37—21no—
17C22⋊C4×D37—21no—
18C22⋊C4×D37—21no—
19C22⋊C4×D37C411yescharacteristic, complemented
20#11924453C2211yescharacteristic
21#11924452—21no—
22#11924451C2211yescharacteristic
23#11924450—21no—
24#123×C37C2211yescharacteristic
25#100×C37C2211yescharacteristic
26#99×C37C2211yescharacteristic
27C23×D37C2×C411yescharacteristic
28C23×D37D411yescharacteristic
29C2×D4×C37C2311yescharacteristic
30C22×C4×C37C2×C411yescharacteristic, complemented
31C2×C22⋊D37C2×C411yescharacteristic
32C2×C22⋊D37D411yescharacteristic
33C22×Dic37C2×C411yescharacteristic, complemented
34D4×D37—41no—
35D4×D37—41no—
36C2×D148D411yescharacteristic
37C2×C4×D37—21no—
38C2×C4×D37—21no—
39C2×C4×D37—21no—
40C2×C4×D37—41no—
41C2×C4×D37—41no—
42C2×C4×D37—41no—
43C2×C4×D37D411yescharacteristic
44M16×C37—21no—
45C4⋊C4×C37—21no—
46C22⋊C4×C37—21no—
47C22⋊C4×C37—21no—
48C22⋊C4×C37C2×C411yescharacteristic, complemented
49C22⋊Dic37—21no—
50C22⋊Dic37—21no—
51C22⋊Dic37C2×C411yescharacteristic, complemented
52#10596938—21no—
53#10596938—21no—
54#10596938—21no—
55#10596938—41no—
56C4⋊Dic37—21no—
57#10596936—21no—
58#10596936—21no—
59#10596934—21no—
60#10596929—41no—
61C8×D37—41no—
62C23×C37C22×C411yescharacteristic
63C23×C37C2×D411yescharacteristic
64C22×D37—21no—
65C22×D37—21no—
66C22×D37—21no—
67C22×D37—21no—
68C22×D37—42no—
69C22×D37—42no—
70C22×D37C22⋊C411yescharacteristic
71C22×D37C22⋊C411yescharacteristic
72C22×D37C22⋊C411yescharacteristic
73D4×C37—41no—
74C2×C4×C37—21no—
75C2×C4×C37—21no—
76C2×C4×C37—41no—
77C2×C4×C37—41no—
78C2×C4×C37C2×D411yescharacteristic
79C22⋊D37—41no—
80C22⋊D37—41no—
81C2×Dic37—21no—
82C2×Dic37—21no—
83C2×Dic37—41no—
84C2×Dic37—41no—
85C2×Dic37C22⋊C411yescharacteristic
86D148—41no—
87C4×D37—21no—
88C4×D37—21no—
89C4×D37—42no—
90C4×D37—42no—
91C4×D37—41no—
92C4×D37—41no—
93C4×D37—82no—
94C8×C37—41no—
95#64436—41no—
96C22×C37—21no—
97C22×C37—21no—
98C22×C37—41no—
99C22×C37—41no—
100C22×C37C2×C22⋊C411yescharacteristic
101C2×D37—22no—
102C2×D37—21no—
103C2×D37—21no—
104C2×D37—42no—
105C2×D37—42no—
106C2×D37—42no—
107C2×D37—41no—
108C2×D37—41no—
109C2×D37—81no—
110C2×D37#9911yescharacteristic
111C2×D37#9911yescharacteristic
112C4×C37—21no—
113C4×C37—41no—
114C4×C37—41no—
115C4×C37—81no—
116Dic37—21no—
117Dic37—41no—
118Dic37—41no—
119Dic37—81no—
120C2×#352—371nomaximal, Sylow
121C2×C37—21no—
122C2×C37—41no—
123C2×C37—41no—
124C2×C37C2×#9911yescharacteristic
125D37—21no—
126D37—41no—
127D37—41no—
128D37#35212yesdirect product factor
129C2×#123—371no—
130C2×#100—371no—
131C2×#99—371no—
132#352—371no—
133#352—371no—
134#352—371no—
135#352D3711yescharacteristic, direct product factor
136C37C2×#35211yescharacteristic, complemented, Sylow
137C22×D4—371no—
138C23×C4—371no—
139C2×M16—741no—
140#123—371no—
141#123—371no—
142#123—371no—
143#123—741no—
144#123—741no—
145#123C2×D3711yescharacteristic
146C2×C4⋊C4—741no—
147C2×C22⋊C4—371no—
148C2×C22⋊C4—741no—
149C2×C22⋊C4—741no—
150#100—371no—
151#100—741no—
152#100C2×D3711yescharacteristic
153#99—371no—
154#99—741no—
155#99C2×D3711yescharacteristic
156C24—371no—
157C24—371no—
158C2×D4—371no—
159C2×D4—371no—
160C2×D4—371no—
161C2×D4—1481no—
162C2×D4—1481no—
163C2×D4C22×D3711yescharacteristic
164C22×C4—371no—
165C22×C4—371no—
166C22×C4—741no—
167C22×C4—741no—
168C22×C4—741no—
169C22×C4—1481no—
170C22×C4—1481no—
171C22×C4—1481no—
172C22×C4C4×D3711yescharacteristic, complemented
173M16—21no—
174M16—741no—
175M16—1481no—
176C2×C8—1481no—
177C4⋊C4—21no—
178C4⋊C4—741no—
179C4⋊C4—741no—
180C4⋊C4—741no—
181C22⋊C4—21no—
182C22⋊C4—21no—
183C22⋊C4—371no—
184C22⋊C4—741no—
185C22⋊C4—741no—
186C22⋊C4—741no—
187C22⋊C4—741no—
188C22⋊C4—741no—
189C22⋊C4—1481no—
190C22⋊C4C4×D3711yescharacteristic, complemented
191C23—371no—
192C23—371no—
193C23—371no—
194C23—741no—
195C23—741no—
196C23—741no—
197C23—741no—
198C23—1482no—
199C23—1482no—
200C23C2×C4×D3711yescharacteristic
201C23D4×D3711yescharacteristic
202D4—41no—
203D4—1481no—
204D4—1481no—
205D4—1481no—
206C2×C4—21no—
207C2×C4—21no—
208C2×C4—41no—
209C2×C4—41no—
210C2×C4—371no—
211C2×C4—2962no—
212C2×C4—741no—
213C2×C4—741no—
214C2×C4—741no—
215C2×C4—741no—
216C2×C4—1481no—
217C2×C4—1481no—
218C2×C4—1482no—
219C2×C4—1482no—
220C2×C4—1481no—
221C2×C4—1481no—
222C2×C4D4×D3711yescharacteristic
223C8—41no—
224C8—1481no—
225C22—21no—
226C22—21no—
227C22—41no—
228C22—41no—
229C22—371no—
230C22—371no—
231C22—2961no—
232C22—742no—
233C22—741no—
234C22—741no—
235C22—1481no—
236C22—1481no—
237C22—1482no—
238C22—1482no—
239C22—1482no—
240C22C22⋊C4×D3711yescharacteristic
241C4—21no—
242C4—41no—
243C4—41no—
244C4—81no—
245C4—2961no—
246C4—741no—
247C4—1481no—
248C4—1481no—
249C2—21no—
250C2—41no—
251C2—41no—
252C2—372no—
253C2—741no—
254C2—1481no—
255C2—1481no—
256C2#99×D3711yescharacteristic
subgroupG/H
centerC2#99×D37
commutatorC23×C37C22×C4
FrattiniC2×D4C22×D37
FittingC37×#352C2
socleC2×C37C2×#99
prime radicalC2×D4×D37C22
cosocleC2×D4×C37C23
hypercenter#352D37
hypocenterC37C2×#352
LCSG ▷ C23×C37 ▷ C22×C37 ▷ C2×C37 ▷ C37
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #352

Complements

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

NH ≅ G/Ntype
4. #14557324C2semidirect
5. #14557323C2semidirect
6. #14557322C2semidirect
7. C37×#352C2semidirect
9. C22×C4×D37C4semidirect
12. #11924520C4semidirect
13. #11924519C4semidirect
19. C22⋊C4×D37C4semidirect
30. C22×C4×C37C2×C4semidirect
33. C22×Dic37C2×C4semidirect
48. C22⋊C4×C37C2×C4semidirect
51. C22⋊Dic37C2×C4semidirect
128. D37#352direct (N × H)
135. #352D37direct (N × H)
136. C37C2×#352semidirect
172. C22×C4C4×D37semidirect
190. C22⋊C4C4×D37semidirect

Maximal subgroups

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

subgroupindex
#123×D372
#100×D372
#99×D372
#145573242
#145573232
#145573222
C37×#3522
C2×#35237