Finite Group Databasebeta

← #18004003

Subgroups — #18004003

Order 4736.

716 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 716 conjugacy classes of subgroups — too many to draw. Only its 145 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#14558369C211yesmaximal, characteristic
2#14558109C212yesmaximal, complemented
3#14558104C211yesmaximal, characteristic, complemented
4#14557724C218yesmaximal, complemented
5#14557506C212yesmaximal, complemented
6C37×#410C211yesmaximal, characteristic, complemented
7#11924584—21no—
8#11924583—22no—
9#11924582—22no—
10#11924581—21no—
11#11924581C2214yes—
12C2×#10596962—22no—
13C2×#10596962C2212yes—
14#41×D37—24no—
15C22×Dic74C2211yescharacteristic, complemented
16#11924536—22no—
17#11924534—21no—
18#11924524—22no—
19#11924523—21no—
20#11924520C2218yescomplemented
21#11924517C2214yescomplemented
22#11924516—24no—
23#11924515—22no—
24#11924495C2211yescharacteristic, complemented
25#11924450C2218yescomplemented
26C2×#41×C37C2211yescharacteristic
27#117×C37C2212yescomplemented
28#99×C37C2214yescomplemented
29#41×C37—22no—
30#41×C37—22no—
31C2×Q8×C37D411yescharacteristic
32C2×D4×C37D411yescharacteristic
33C2×D4×C37C2312yes—
34C22×C4×C37D412yes—
35C22×C4×C37C2311yescharacteristic
36C2×C22⋊D37—24no—
37C2×C22⋊D37C2314yes—
38C22×Dic37C2312yes—
39#10596966—24no—
40Q8×D37—24no—
41#10596964—24no—
42#10596964—28no—
43D4×D37—24no—
44D4×D37—28no—
45#10596962—24no—
46#10596962—24no—
47#10596962—22no—
48#10596962—22no—
49#10596962—24no—
50#10596962—22no—
51#10596962—22no—
52C2×D148—22no—
53C2×D148D412yes—
54C2×C4×D37—24no—
55C2×C4×D37D414yes—
56C2×Dic74—22no—
57C2×Dic74—21no—
58C2×Dic74—22no—
59C2×Dic74—21no—
60C2×Dic74—22no—
61C2×Dic74D412yes—
62C4⋊C4×C37—22no—
63C22⋊C4×C37C2314yes—
64C42×C37—22no—
65C22⋊Dic37—24no—
66C22⋊Dic37C2312yes—
67#10596938—28no—
68C4⋊Dic37—21no—
69C4⋊Dic37—28no—
70#10596936—24no—
71C4×Dic37—21no—
72C23×C37C2×D412yes—
73C23×C37C2411yescharacteristic
74C22×D37—24no—
75C22×D37C2×D414yes—
76Q8×C37—22no—
77D4×C37—24no—
78D4×C37—22no—
79C2×C4×C37—22no—
80C2×C4×C37—22no—
81C2×C4×C37—24no—
82C2×C4×C37C2×D412yes—
83C2×C4×C37C2×D411yescharacteristic
84C2×C4×C37C2×D411yescharacteristic
85C2×C4×C37C2×D412yes—
86C2×C4×C37C2×D412yes—
87C22⋊D37—28no—
88C22⋊D37—24no—
89C22⋊D37—24no—
90C2×Dic37—24no—
91C2×Dic37—22no—
92C2×Dic37—22no—
93C2×Dic37—24no—
94C2×Dic37C2×D414yes—
95D148—24no—
96D148—22no—
97D148—22no—
98C4×D37—28no—
99C4×D37—24no—
100C4×D37—24no—
101Dic74—24no—
102Dic74—22no—
103Dic74—22no—
104Dic74—41no—
105Dic74—42no—
106Dic74—41no—
107C22×C37—22no—
108C22×C37—41no—
109C22×C37—42no—
110C22×C37C22×D412yes—
111C22×C37C22×D411yescharacteristic
112C2×D37—24no—
113C2×D37—44no—
114C2×D37—84no—
115C4×C37—22no—
116C4×C37—44no—
117C4×C37#12011yescharacteristic
118C4×C37#12012yes—
119C4×C37#12011yescharacteristic
120Dic37—24no—
121Dic37—42no—
122Dic37—42no—
123#3028—371nomaximal, Sylow
124C2×C37—21no—
125C2×C37—22no—
126C2×C37—42no—
127C2×C37C2×#12011yescharacteristic
128D37—44no—
129#585—371no—
130#536—371no—
131#536—372no—
132#458—378no—
133#410—372no—
134#410D3711yescharacteristic, complemented
135C37#302811yescharacteristic, complemented, Sylow
1362−1+4—742no—
1372−1+4—741no—
138H(5,𝔽2)—374no—
139H(5,𝔽2)—742no—
140H(5,𝔽2)—741no—
141C2×#41—372no—
142C2×#41—744no—
143C2×#41—742no—
144C2×#41C2×D3711yescharacteristic
145C22×Q8—371no—
146C4⋊Q8—742no—
147C4⋊Q8—741no—
148#124—742no—
149#124—741no—
150#123—374no—
151#123—378no—
152C22⋊Q8—744no—
153C22⋊Q8—742no—
154#117—371no—
155#117C2×D3712yescomplemented
156#99—378no—
157#99C2×D3714yescomplemented
158#41—22no—
159#41—22no—
160#41—744no—
161#41—744no—
162#41—748no—
163#41—744no—
164#41—744no—
165#41—742no—
166#41—742no—
167#41—744no—
168#41—742no—
169#41—742no—
170C2×Q8—372no—
171C2×Q8—742no—
172C2×Q8—744no—
173C2×Q8—741no—
174C2×Q8—741no—
175C2×Q8—742no—
176C2×Q8—742no—
177C2×Q8D14811yescharacteristic
178C2×D4—374no—
179C2×D4—372no—
180C2×D4—742no—
181C2×D4—748no—
182C2×D4—744no—
183C2×D4—744no—
184C2×D4D14811yescharacteristic
185C2×D4C22×D3712yes—
186C22×C4—374no—
187C22×C4—372no—
188C22×C4—744no—
189C22×C4D14812yes—
190C22×C4C22×D3711yescharacteristic
191C4⋊C4—22no—
192C4⋊C4—748no—
193C4⋊C4—744no—
194C4⋊C4—741no—
195C22⋊C4—372no—
196C22⋊C4—748no—
197C22⋊C4—744no—
198C22⋊C4C22×D3714yes—
199C42—22no—
200C42—741no—
201C23—374no—
202C23—744no—
203C23C2×D14812yes—
204C23C23×D3711yescharacteristic
205Q8—22no—
206Q8—744no—
207Q8—742no—
208Q8—742no—
209Q8—1481no—
210Q8—1481no—
211Q8—1482no—
212D4—24no—
213D4—22no—
214D4—744no—
215D4—748no—
216D4—744no—
217D4—744no—
218D4—742no—
219D4—742no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—24no—
223C2×C4—374no—
224C2×C4—748no—
225C2×C4—744no—
226C2×C4—744no—
227C2×C4—744no—
228C2×C4—744no—
229C2×C4—742no—
230C2×C4—742no—
231C2×C4C2×D14812yes—
232C2×C4C2×D14811yescharacteristic
233C2×C4C2×D14811yescharacteristic
234C2×C4D4×D3712yes—
235C2×C4D4×D3712yes—
236C22—22no—
237C22—41no—
238C22—42no—
239C22—2964no—
240C22—744no—
241C22—1484no—
242C22C2×D4×D3712yes—
243C22C22×D14811yescharacteristic
244C4—22no—
245C4—44no—
246C4—744no—
247C4—1482no—
248C4—1482no—
249C4#1192450611yescharacteristic
250C4#1192450611yescharacteristic
251C4#1192450612yes—
252C2—22no—
253C2—21no—
254C2—42no—
255C2—1484no—
256C2C2×#1192450611yescharacteristic
subgroupG/H
centerC2C2×#11924506
commutator, cosocleC23×C37C24
FrattiniC23C23×D37
FittingC37×#410C2
socleC2×C37C2×#120
prime radical#14558369C2
hypercenter#410D37
hypocenterC37#3028
LCSG ▷ C23×C37 ▷ C2×C37 ▷ C37
UCSC1 ◁ C2 ◁ C22×C4 ◁ #410

Complements

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

NH ≅ G/Ntype
2. #14558109C2semidirect
3. #14558104C2semidirect
4. #14557724C2semidirect
5. #14557506C2semidirect
6. C37×#410C2semidirect
15. C22×Dic74C22semidirect
20. #11924520C22semidirect
21. #11924517C22semidirect
24. #11924495C22semidirect
25. #11924450C22semidirect
27. #117×C37C22semidirect
28. #99×C37C22semidirect
134. #410D37semidirect
135. C37#3028semidirect
155. #117C2×D37semidirect
157. #99C2×D37semidirect

Maximal subgroups

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

subgroupindex
#145583692
#145581092
#145581042
#145577242
#145575062
C37×#4102
#302837