Finite Group Databasebeta

← #17994787

Subgroups — #17994787

Order 4736.

360 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 107 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×C4⋊C4×D37C211yesmaximal, characteristic
2C2×#11924467C211yesmaximal, characteristic, complemented
3C2×#11924448C211yesmaximal, characteristic
4#14557390C211yesmaximal, characteristic
5#14557296C211yesmaximal, characteristic, complemented
6#14557295C211yesmaximal, characteristic, complemented
7C2×#106×C37C211yesmaximal, characteristic, complemented
8C22×C4×D37—21no—
9C22×C4×D37C2211yescharacteristic
10C22×#64436C2211yescharacteristic
11C2×C4⋊Dic37C2211yescharacteristic
12C2×C4⋊Dic37C2211yescharacteristic
13C2×#10596936—21no—
14C4⋊C4×D37—22no—
15C4⋊C4×D37—22no—
16C2×#10596938—21no—
17#11924467—22no—
18#11924448C414yescomplemented
19#11924443—21no—
20C22×C8×C37C2211yescharacteristic
21C2×C4⋊C4×C37C2211yescharacteristic
22C2×C4⋊C4×C37C2211yescharacteristic
23#106×C37—22no—
24C23×D37D411yescharacteristic
25C22×C4×C37—21no—
26C22×C4×C37—21no—
27C22×C4×C37C2311yescharacteristic
28C22×Dic37—21no—
29C22×Dic37—21no—
30C22×Dic37D411yescharacteristic
31C2×C4×D37—21no—
32C2×C4×D37—21no—
33C2×C4×D37—21no—
34C2×C4×D37—21no—
35C2×C4×D37—24no—
36C2×C4×D37—42no—
37C2×C4×D37—42no—
38C2×C4×D37C2×C412yes—
39C2×C4×D37Q812yes—
40C2×C8×C37—21no—
41C2×C8×C37—21no—
42C2×C8×C37C2×C412yescomplemented
43C4⋊C4×C37—21no—
44C4⋊C4×C37—21no—
45C4⋊C4×C37—21no—
46C4⋊C4×C37D412yes—
47#10596938—42no—
48C4⋊Dic37—21no—
49C4⋊Dic37—21no—
50C4⋊Dic37—21no—
51C4⋊Dic37D412yes—
52#10596936—24no—
53C2×#64436—21no—
54C2×#64436—21no—
55C2×#64436C2×C412yescomplemented
56C23×C37C2×D411yescharacteristic
57C22×D37—21no—
58C22×D37—21no—
59C22×D37—21no—
60C22×D37—21no—
61C22×D37C4⋊C412yes—
62C22×D37SD1612yes—
63C22×D37SD1612yes—
64C2×C4×C37—22no—
65C2×C4×C37—41no—
66C2×C4×C37—41no—
67C2×C4×C37—41no—
68C2×C4×C37—41no—
69C2×C4×C37—41no—
70C2×C4×C37C22×C411yescharacteristic
71C2×C4×C37C2×D411yescharacteristic
72C2×C4×C37C2×D411yescharacteristic
73C2×C4×C37C2×Q811yescharacteristic
74C2×C4×C37#4111yescharacteristic
75C2×C4×C37#4111yescharacteristic
76C2×Dic37—21no—
77C2×Dic37—21no—
78C2×Dic37—22no—
79C2×Dic37—41no—
80C2×Dic37—41no—
81C2×Dic37—41no—
82C2×Dic37—41no—
83C2×Dic37—41no—
84C2×Dic37C4⋊C412yes—
85C4×D37—22no—
86C4×D37—22no—
87C4×D37—22no—
88C4×D37—22no—
89C4×D37—48no—
90C8×C37—22no—
91#64436—22no—
92C22×C37C2×C4⋊C411yescharacteristic
93C22×C37#12011yescharacteristic
94C22×C37#12311yescharacteristic
95C22×C37C2×SD1611yescharacteristic
96C22×C37C2×SD1611yescharacteristic
97C22×C37AGL(1,ℤ/8)11yescharacteristic
98C22×C37#13711yescharacteristic
99C2×D37—22no—
100C2×D37—22no—
101C2×D37—22no—
102C2×D37—22no—
103C2×D37—22no—
104C2×D37—22no—
105C2×D37#10614yescomplemented
106C4×C37—42no—
107C4×C37—81no—
108C4×C37C4×D411yescharacteristic
109C4×C37C4×D411yescharacteristic
110C4×C37C22⋊Q811yescharacteristic
111C4×C37C22⋊Q811yescharacteristic
112Dic37—22no—
113Dic37—42no—
114Dic37—81no—
115#1853—371nomaximal, Sylow
116C2×C37#38511yescharacteristic
117C2×C37C2×#10611yescharacteristic
118C2×C37#42811yescharacteristic
119C2×C37#44811yescharacteristic
120C2×C37#45011yescharacteristic
121C2×C37#48111yescharacteristic
122C2×C37#48311yescharacteristic
123D37—24no—
124C22×C4⋊C4—371no—
125C2×#106—371no—
126C2×#106D3711yescharacteristic, complemented
127C2×C22⋊C8—371no—
128#380—371no—
129#340—371no—
130#340—371no—
131C37#185311yescharacteristic, complemented, Sylow
132C23×C4—371no—
133C23×C4—741no—
134C22×C8—371no—
135C22×C8C2×D3711yescharacteristic
136C2×C4⋊C4—371no—
137C2×C4⋊C4—371no—
138C2×C4⋊C4—742no—
139C2×C4⋊C4—741no—
140C2×C4⋊C4—742no—
141C2×C4⋊C4C2×D3711yescharacteristic
142C2×C4⋊C4C2×D3711yescharacteristic
143C2×C22⋊C4—741no—
144#106—22no—
145#106—742no—
146C22⋊C8—374no—
147#95—741no—
148C24—371no—
149C22×C4—21no—
150C22×C4—21no—
151C22×C4—371no—
152C22×C4—372no—
153C22×C4—372no—
154C22×C4—744no—
155C22×C4—741no—
156C22×C4—741no—
157C22×C4—741no—
158C22×C4—741no—
159C22×C4—741no—
160C22×C4—741no—
161C22×C4—1482no—
162C22×C4—1482no—
163C22×C4C22×D3711yescharacteristic
164C2×C8—21no—
165C2×C8—21no—
166C2×C8—372no—
167C2×C8—741no—
168C2×C8—741no—
169C2×C8C4×D3712yescomplemented
170C4⋊C4—21no—
171C4⋊C4—21no—
172C4⋊C4—21no—
173C4⋊C4—372no—
174C4⋊C4—744no—
175C4⋊C4—741no—
176C4⋊C4—741no—
177C4⋊C4—741no—
178C4⋊C4C22⋊D3712yes—
179C22⋊C4—1482no—
180C23—372no—
181C23—372no—
182C23—372no—
183C23—741no—
184C23—741no—
185C23—741no—
186C23—741no—
187C23D4×D3711yescharacteristic
188C2×C4—22no—
189C2×C4—41no—
190C2×C4—41no—
191C2×C4—41no—
192C2×C4—41no—
193C2×C4—41no—
194C2×C4—372no—
195C2×C4—741no—
196C2×C4—741no—
197C2×C4—742no—
198C2×C4—742no—
199C2×C4—742no—
200C2×C4—742no—
201C2×C4—742no—
202C2×C4—1488no—
203C2×C4—1481no—
204C2×C4—1481no—
205C2×C4—1481no—
206C2×C4—1481no—
207C2×C4—1481no—
208C2×C4C2×C4×D3711yescharacteristic
209C2×C4#1059696211yescharacteristic
210C2×C4D4×D3711yescharacteristic
211C2×C4Q8×D3711yescharacteristic
212C2×C4#1059696611yescharacteristic
213C2×C4C2×C22⋊D3711yescharacteristic
214C8—22no—
215C8—742no—
216C22—374no—
217C22—742no—
218C22—742no—
219C22—742no—
220C22—742no—
221C22—742no—
222C22—742no—
223C22C4⋊C4×D3711yescharacteristic
224C22#1192450411yescharacteristic
225C22#1192452111yescharacteristic
226C22SD16×D3711yescharacteristic
227C22SD16×D3711yescharacteristic
228C22#1192456311yescharacteristic
229C22#1192456511yescharacteristic
230C4—42no—
231C4—81no—
232C4—2961no—
233C4—742no—
234C4—1482no—
235C4C4×C22⋊D3711yescharacteristic
236C4#1192449811yescharacteristic
237C4#1192451211yescharacteristic
238C4#1192451411yescharacteristic
239C2—744no—
240C2#1455740411yescharacteristic
241C2#106×D3711yescharacteristic
242C2#1455757211yescharacteristic
243C2#1455767011yescharacteristic
244C2#1455768211yescharacteristic
245C2#1455785311yescharacteristic
246C2#1455786511yescharacteristic
subgroupG/H
centerC23D4×D37
commutatorC2×C4×C37C22×C4
FrattiniC22×C4C22×D37
FittingC2×#106×C37C2
socleC23×C37C2×D4
prime radicalC23×D37D4
cosocleC22×C4×C37C23
hypercenterC2×#106D37
hypocenterC37#1853
LCSG ▷ C2×C4×C37 ▷ C2×C37 ▷ C37
UCSC1 ◁ C23 ◁ C22×C4 ◁ C2×#106

Complements

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

NH ≅ G/Ntype
2. C2×#11924467C2semidirect
5. #14557296C2semidirect
6. #14557295C2semidirect
7. C2×#106×C37C2semidirect
18. #11924448C4semidirect
42. C2×C8×C37C2×C4semidirect
55. C2×#64436C2×C4semidirect
105. C2×D37#106semidirect
126. C2×#106D37semidirect
131. C37#1853semidirect
169. C2×C8C4×D37semidirect

Maximal subgroups

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

subgroupindex
C2×C4⋊C4×D372
C2×#119244672
C2×#119244482
#145573902
#145572962
#145572952
C2×#106×C372
#185337