Finite Group Databasebeta

← #18010850

Subgroups — #18010850

Order 4736.

264 conjugacy classes of subgroups; only the 52 normal subgroups are shown.

normal subgroups only

This group has 264 conjugacy classes of subgroups — too many to draw. Only its 52 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#14558520C211yesmaximal, characteristic, complemented
2#14558513C211yesmaximal, characteristic, complemented
3#14558435C212yesmaximal, complemented
4#14558434C212yesmaximal, complemented
5#14558041C211yesmaximal, characteristic
6C2×#10596957—21no—
7#11924615—21no—
8#11924597—21no—
9#11924596C2212yes—
10#11924594—21no—
11#11924594C2212yes—
12C2×#10596964C2211yescharacteristic
13C22×C4×D37C411yescharacteristic, complemented
14#11924519C2211yescharacteristic
15#11924514—21no—
16#11924509—21no—
17#11924506C2211yescharacteristic
18#11924504C411yescharacteristic, complemented
19#11924498—21no—
20#11924496—21no—
21#11924492C411yescharacteristic, complemented
22#123×C37C411yescharacteristic, complemented
23C23×D37C2×C411yescharacteristic
24C2×D4×C37C2×C411yescharacteristic
25C22×C4×C37C2×C411yescharacteristic
26C2×C22⋊D37—21no—
27C2×C22⋊D37D411yescharacteristic
28C2×C22⋊D37C2311yescharacteristic
29C22×Dic37C2×C411yescharacteristic
30C22×Dic37D411yescharacteristic
31#10596964—41no—
32#10596964—41no—
33C2×D148—21no—
34C2×C4×D37—21no—
35C2×C4×D37—21no—
36C2×C4×D37—41no—
37C2×C4×D37—41no—
38C2×C4×D37D411yescharacteristic
39C2×Dic74D411yescharacteristic
40C22⋊#3866—22no—
41#10596957—22no—
42#10596957—41no—
43C2×#64438—41no—
44C4⋊#3866—41no—
45C4×#3866—41no—
46C4⋊C4×C37—21no—
47C22⋊C4×C37—21no—
48C22⋊C4×C37C2×C411yescharacteristic
49C22⋊Dic37C2×C411yescharacteristic
50#10596938—21no—
51#10596938—21no—
52#10596938—21no—
53#10596936—21no—
54#10596936—21no—
55C4×Dic37—21no—
56C23×C37C22⋊C411yescharacteristic
57C23×C37C22×C411yescharacteristic
58C22×D37—21no—
59C22×D37—42no—
60C22×D37—41no—
61C22×D37C2×D411yescharacteristic
62C2×#3866—42no—
63D4×C37—41no—
64C2×C4×C37—21no—
65C2×C4×C37—21no—
66C2×C4×C37—41no—
67C2×C4×C37C22⋊C411yescharacteristic
68C22⋊D37—41no—
69C22⋊D37—41no—
70C22⋊D37—42no—
71C2×Dic37—21no—
72C2×Dic37—21no—
73C2×Dic37—21no—
74C2×Dic37—21no—
75C2×Dic37—21no—
76C2×Dic37C22⋊C411yescharacteristic
77C2×Dic37C22⋊C411yescharacteristic
78C2×Dic37C2×D411yescharacteristic
79D148—42no—
80C4×D37—21no—
81C4×D37—21no—
82C4×D37—42no—
83C4×D37—41no—
84C4×D37—41no—
85Dic74—41no—
86#64438—42no—
87C22×C37—21no—
88C22×C37—21no—
89C22×C37—41no—
90C22×C37—41no—
91C22×C37C2×C22⋊C411yescharacteristic
92C2×D37—22no—
93C2×D37—21no—
94C2×D37—41no—
95C2×D37—42no—
96C2×D37—41no—
97C2×D37—41no—
98C2×D37—42no—
99#3866—82no—
100C4×C37—21no—
101C4×C37—41no—
102C4×C37—41no—
103Dic37—21no—
104Dic37—21no—
105Dic37—41no—
106Dic37#9911yescharacteristic
107Dic37#9911yescharacteristic
108#2120—371nomaximal, Sylow
109C2×C37—21no—
110C2×C37—41no—
111C2×C37—41no—
112C2×C37C2×#9911yescharacteristic
113D37—41no—
114D37—42no—
115#525—371no—
116#413—371no—
117#410—371no—
118#352—372no—
119AΣL(1,𝔽16)—372no—
120C37#212011yescharacteristic, complemented, Sylow
121C2×#41—371no—
122C23×C4—371no—
123C2×M16—741no—
124#123—371no—
125#123#386611yescharacteristic, complemented
126C22⋊Q8—741no—
127#121—741no—
128#120—371no—
129#120—371no—
130C4×D4—741no—
131C4×D4—741no—
132#117—371no—
133#117—741no—
134#101—741no—
135#100—372no—
136#99—372no—
137#99—741no—
138C24—371no—
139#41—1481no—
140#41—1481no—
141C2×Q8—371no—
142C2×D4—371no—
143C2×D4—371no—
144C2×D4—741no—
145C2×D4—741no—
146C2×D4C2×#386611yescharacteristic
147C22×C4—371no—
148C22×C4—371no—
149C22×C4—371no—
150C22×C4—741no—
151C22×C4—741no—
152C22×C4—1481no—
153C22×C4—1481no—
154C22×C4C2×#386611yescharacteristic
155M16—742no—
156M16—1481no—
157C2×C8—1481no—
158C4⋊C4—21no—
159C4⋊C4—741no—
160C4⋊C4—741no—
161C4⋊C4—1481no—
162C22⋊C4—21no—
163C22⋊C4—371no—
164C22⋊C4—741no—
165C22⋊C4—741no—
166C22⋊C4—741no—
167C22⋊C4—742no—
168C22⋊C4C2×#386611yescharacteristic
169C42—741no—
170C42—1481no—
171C23—371no—
172C23—741no—
173C23—1481no—
174C23—1482no—
175C23C22⋊#386611yescharacteristic
176C23C22×#386611yescharacteristic
177Q8—1481no—
178D4—41no—
179D4—1482no—
180D4—1481no—
181D4—1481no—
182D4—1482no—
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—41no—
186C2×C4—371no—
187C2×C4—371no—
188C2×C4—371no—
189C2×C4—741no—
190C2×C4—741no—
191C2×C4—741no—
192C2×C4—741no—
193C2×C4—741no—
194C2×C4—741no—
195C2×C4—741no—
196C2×C4—1481no—
197C2×C4—1481no—
198C2×C4—1482no—
199C2×C4—1482no—
200C2×C4C22⋊#386611yescharacteristic
201C8—1482no—
202C22—21no—
203C22—21no—
204C22—41no—
205C22—41no—
206C22—742no—
207C22—741no—
208C22—1481no—
209C22—1481no—
210C22—1482no—
211C22—1482no—
212C22—1481no—
213C22C2×C22⋊#386611yescharacteristic
214C4—21no—
215C4—41no—
216C4—41no—
217C4—371no—
218C4—371no—
219C4—2962no—
220C4—741no—
221C4—741no—
222C4—1481no—
223C2—21no—
224C2—41no—
225C2—41no—
226C2—1481no—
227C2—1482no—
228C2D37⋊#9911yescharacteristic
subgroupG/H
centerC2D37⋊#99
commutatorC23×C37C22×C4
FrattiniC23C22×#3866
Fitting#123×C37C4
socleC2×C37C2×#99
prime radical#14558041C2
cosocleC2×C22⋊D37C23
hypercenter#123#3866
hypocenterC37#2120
LCSG ▷ C23×C37 ▷ C22×C37 ▷ C2×C37 ▷ C37
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #123

Complements

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

NH ≅ G/Ntype
1. #14558520C2semidirect
2. #14558513C2semidirect
3. #14558435C2semidirect
4. #14558434C2semidirect
13. C22×C4×D37C4semidirect
18. #11924504C4semidirect
21. #11924492C4semidirect
22. #123×C37C4semidirect
120. C37#2120semidirect
125. #123#3866semidirect

Maximal subgroups

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

subgroupindex
#145585202
#145585132
#145584352
#145584342
#145580412
#212037