Finite Group Databasebeta

← #17995720

Subgroups — #17995720

Order 4736.

638 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 638 conjugacy classes of subgroups — too many to draw. Only its 75 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×#11924504C212yesmaximal, complemented
2C2×#11924451C212yesmaximal, complemented
3#14557388C211yesmaximal, characteristic, complemented
4#14557300C211yesmaximal, characteristic, complemented
5C37×#534C211yesmaximal, characteristic, complemented
6C24×D37C2211yescharacteristic
7C22×C22⋊D37—22no—
8#11924504—24no—
9#11924504—24no—
10C2×C22⋊Dic37C2211yescharacteristic
11C2×C22⋊Dic37C2212yes—
12C2×#10596938—22no—
13C2×#10596938—22no—
14#11924451—24no—
15C22×D4×C37C2212yes—
16#123×C37—22no—
17#121×C37—22no—
18#120×C37—22no—
19C2×C22⋊C4×C37C2211yescharacteristic
20C24×C37D412yescomplemented
21C24×C37C2311yescharacteristic
22C23×D37—21no—
23C23×D37—22no—
24C23×D37—22no—
25C23×D37—21no—
26C23×D37—44no—
27C23×D37D412yescomplemented
28C2×D4×C37—22no—
29C2×D4×C37—22no—
30C2×D4×C37—22no—
31C2×D4×C37—22no—
32C2×D4×C37—22no—
33C2×D4×C37—22no—
34C2×D4×C37—22no—
35C22×C4×C37D412yescomplemented
36C22×C4×C37D412yescomplemented
37C2×C22⋊D37—28no—
38C2×C22⋊D37—48no—
39C22×Dic37—22no—
40C22×Dic37—21no—
41C4⋊C4×C37—22no—
42C22⋊C4×C37—24no—
43C22⋊C4×C37—22no—
44C22⋊Dic37—22no—
45C22⋊Dic37—22no—
46C22⋊Dic37D414yescomplemented
47#10596938—28no—
48#10596938—44no—
49C23×C37—22no—
50C23×C37—22no—
51C23×C37—22no—
52C23×C37—42no—
53C23×C37—44no—
54C23×C37C2×D411yescharacteristic
55C23×C37C2×D412yes—
56C23×C37C2×D412yes—
57C23×C37C2×D411yescharacteristic
58C23×C37#4111yescharacteristic
59C22×D37—24no—
60C22×D37—28no—
61C22×D37—24no—
62C22×D37—22no—
63C22×D37—44no—
64C22×D37—48no—
65C22×D37—44no—
66C22×D37—42no—
67C22×D37—44no—
68C22×D37—41no—
69C22×D37—42no—
70C22×D37—41no—
71D4×C37—42no—
72D4×C37—42no—
73D4×C37—42no—
74D4×C37—42no—
75D4×C37—82no—
76C2×C4×C37—22no—
77C2×C4×C37—22no—
78C2×C4×C37—22no—
79C2×C4×C37—22no—
80C2×C4×C37—42no—
81C22⋊D37—416no—
82C2×Dic37—24no—
83C2×Dic37—44no—
84C2×Dic37—42no—
85C2×Dic37—41no—
86C22×C37—22no—
87C22×C37—24no—
88C22×C37—41no—
89C22×C37—42no—
90C22×C37—41no—
91C22×C37—44no—
92C22×C37—42no—
93C22×C37—44no—
94C22×C37—82no—
95C22×C37#12012yes—
96C22×C37#12011yescharacteristic
97C22×C37#12111yescharacteristic
98C22×C37#12112yes—
99C22×C37#12711yescharacteristic
100C2×D37—28no—
101C2×D37—24no—
102C2×D37—44no—
103C2×D37—44no—
104C2×D37—42no—
105C2×D37—48no—
106C2×D37—42no—
107C2×D37—44no—
108C4×C37—42no—
109C4×C37—42no—
110Dic37—44no—
111Dic37—81no—
112#2021—371nomaximal, Sylow
113C2×C37—22no—
114C2×C37—21no—
115C2×C37—21no—
116C2×C37—22no—
117C2×C37—82no—
118C2×C37#39211yescharacteristic
119C2×C37B(4,𝔽2)12yes—
120D37—44no—
121#534D3711yescharacteristic, complemented
122C2×#120—372no—
123C2×#99—372no—
124#379—371no—
125#342—371no—
126C37#202111yescharacteristic, complemented, Sylow
127C25—371no—
128C22×D4—742no—
129C22×D4C2×D3712yes—
130#123—22no—
131#121—22no—
132#120—22no—
133#120—744no—
134#120—744no—
135C2×C22⋊C4—372no—
136C2×C22⋊C4—371no—
137C2×C22⋊C4—742no—
138C2×C22⋊C4—742no—
139C2×C22⋊C4C2×D3711yescharacteristic
140#99—744no—
141C24—372no—
142C24—741no—
143C24—742no—
144C24—742no—
145C24—741no—
146C24—1484no—
147C24C22⋊D3712yescomplemented
148C24C22×D3711yescharacteristic
149C2×D4—22no—
150C2×D4—22no—
151C2×D4—22no—
152C2×D4—22no—
153C2×D4—22no—
154C2×D4—22no—
155C2×D4—22no—
156C2×D4—748no—
157C2×D4—1488no—
158C22×C4—742no—
159C22×C4—741no—
160C22×C4C22⋊D3712yescomplemented
161C22×C4C22⋊D3712yescomplemented
162C4⋊C4—22no—
163C22⋊C4—24no—
164C22⋊C4—22no—
165C22⋊C4—374no—
166C22⋊C4—748no—
167C22⋊C4—742no—
168C22⋊C4—742no—
169C22⋊C4—1484no—
170C23—22no—
171C23—22no—
172C23—22no—
173C23—42no—
174C23—44no—
175C23—744no—
176C23—748no—
177C23—744no—
178C23—742no—
179C23—1482no—
180C23—1484no—
181C23—1484no—
182C23—1488no—
183C23—1484no—
184C23—1481no—
185C23—1482no—
186C23—1481no—
187C23D4×D3711yescharacteristic
188C23D4×D3712yes—
189C23#1059696411yescharacteristic
190C23C2×C22⋊D3712yes—
191C23C2×C22⋊D3711yescharacteristic
192D4—42no—
193D4—42no—
194D4—42no—
195D4—42no—
196D4—82no—
197D4—14816no—
198C2×C4—22no—
199C2×C4—22no—
200C2×C4—22no—
201C2×C4—22no—
202C2×C4—42no—
203C2×C4—744no—
204C2×C4—1482no—
205C2×C4—1481no—
206C2×C4—1484no—
207C22—22no—
208C22—24no—
209C22—42no—
210C22—41no—
211C22—41no—
212C22—44no—
213C22—42no—
214C22—44no—
215C22—82no—
216C22—748no—
217C22—744no—
218C22—1484no—
219C22—1484no—
220C22—1488no—
221C22—1482no—
222C22—1482no—
223C22—1484no—
224C22#1192450412yes—
225C22#1192450511yescharacteristic
226C22#1192450711yescharacteristic
227C22#1192450812yes—
228C22#1192453211yescharacteristic
229C4—42no—
230C4—42no—
231C4—2961no—
232C4—1484no—
233C2—21no—
234C2—22no—
235C2—22no—
236C2—21no—
237C2—82no—
238C2—1484no—
239C2#1455744111yescharacteristic
240C2#1455772012yes—
subgroupG/H
centerC22#11924505
commutator, cosocleC24×C37C23
FrattiniC24C22×D37
FittingC37×#534C2
socleC22×C37#120
hypercenter#534D37
hypocenterC37#2021
LCSG ▷ C24×C37 ▷ C2×C37 ▷ C37
UCSC1 ◁ C22 ◁ C24 ◁ #534

Complements

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

NH ≅ G/Ntype
1. C2×#11924504C2semidirect
2. C2×#11924451C2semidirect
3. #14557388C2semidirect
4. #14557300C2semidirect
5. C37×#534C2semidirect
20. C24×C37D4semidirect
27. C23×D37D4semidirect
35. C22×C4×C37D4semidirect
36. C22×C4×C37D4semidirect
46. C22⋊Dic37D4semidirect
121. #534D37semidirect
126. C37#2021semidirect
147. C24C22⋊D37semidirect
160. C22×C4C22⋊D37semidirect
161. C22×C4C22⋊D37semidirect

Maximal subgroups

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

subgroupindex
C2×#119245042
C2×#119244512
#145573882
#145573002
C37×#5342
#202137