Finite Group Databasebeta

← #17996237

Subgroups — #17996237

Order 4736.

264 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 264 conjugacy classes of subgroups — too many to draw. Only its 73 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#14557517C211yesmaximal, characteristic, complemented
2#14557507C211yesmaximal, characteristic, complemented
3#14557323C212yesmaximal, complemented
4#14557320C212yesmaximal, complemented
5C37×#525C211yesmaximal, characteristic
6C2×#10596934—21no—
7#11924495—21no—
8#11924455—21no—
9#11924453C2212yes—
10#11924451—21no—
11#11924451C2212yes—
12C2×#41×C37C2211yescharacteristic
13C23×C4×C37C411yescharacteristic, complemented
14#123×C37C411yescharacteristic, complemented
15#123×C37C2211yescharacteristic
16C22⋊Q8×C37—21no—
17#121×C37—21no—
18#120×C37C411yescharacteristic, complemented
19#120×C37C2211yescharacteristic
20C4×D4×C37—21no—
21C4×D4×C37—21no—
22#117×C37C411yescharacteristic, complemented
23C24×C37C2×C411yescharacteristic
24#41×C37—41no—
25#41×C37—41no—
26C2×Q8×C37D411yescharacteristic
27C2×D4×C37—21no—
28C2×D4×C37—21no—
29C2×D4×C37C2×C411yescharacteristic
30C2×D4×C37D411yescharacteristic
31C2×D4×C37C2311yescharacteristic
32C22×C4×C37—21no—
33C22×C4×C37—21no—
34C22×C4×C37—41no—
35C22×C4×C37—41no—
36C22×C4×C37C2×C411yescharacteristic
37C22×C4×C37C2×C411yescharacteristic
38C22×C4×C37D411yescharacteristic
39C22×C4×C37D411yescharacteristic
40C4⋊C4×C37—21no—
41C4⋊C4×C37—21no—
42C4⋊C4×C37—21no—
43C22⋊C4×C37—21no—
44C22⋊C4×C37—21no—
45C22⋊C4×C37—21no—
46C22⋊C4×C37—21no—
47C22⋊C4×C37C2×C411yescharacteristic
48C22⋊C4×C37C2×C411yescharacteristic
49C42×C37—21no—
50C22⋊Dic37—22no—
51C4⋊Dic37—41no—
52C4×Dic37—41no—
53#10596934—22no—
54#10596934—41no—
55C2×#64436—41no—
56C23×C37—21no—
57C23×C37—41no—
58C23×C37—42no—
59C23×C37C22⋊C411yescharacteristic
60C23×C37C22×C411yescharacteristic
61C23×C37C2×D411yescharacteristic
62Q8×C37—41no—
63D4×C37—41no—
64D4×C37—41no—
65D4×C37—41no—
66D4×C37—42no—
67D4×C37—42no—
68C2×C4×C37—21no—
69C2×C4×C37—21no—
70C2×C4×C37—21no—
71C2×C4×C37—21no—
72C2×C4×C37—21no—
73C2×C4×C37—21no—
74C2×C4×C37—21no—
75C2×C4×C37—21no—
76C2×C4×C37—21no—
77C2×C4×C37—42no—
78C2×C4×C37—41no—
79C2×C4×C37—41no—
80C2×C4×C37—41no—
81C2×C4×C37C22⋊C411yescharacteristic
82C2×C4×C37C22⋊C411yescharacteristic
83C2×C4×C37C22⋊C411yescharacteristic
84C2×C4×C37C2×D411yescharacteristic
85C2×Dic37—42no—
86#64436—42no—
87C22×C37—22no—
88C22×C37—21no—
89C22×C37—21no—
90C22×C37—21no—
91C22×C37—41no—
92C22×C37—41no—
93C22×C37—41no—
94C22×C37—41no—
95C22×C37—42no—
96C22×C37—42no—
97C22×C37—41no—
98C22×C37C2×C22⋊C411yescharacteristic
99C4×C37—21no—
100C4×C37—21no—
101C4×C37—21no—
102C4×C37—41no—
103C4×C37—41no—
104C4×C37—41no—
105C4×C37#9911yescharacteristic
106C4×C37#9911yescharacteristic
107Dic37—82no—
108#2120—371nomaximal, Sylow
109C2×C37—21no—
110C2×C37—41no—
111C2×C37—41no—
112C2×C37—42no—
113C2×C37—41no—
114C2×C37C2×#9911yescharacteristic
115#525D3711yescharacteristic
116#413—371no—
117#410—371no—
118#352—372no—
119AΣL(1,𝔽16)—372no—
120C37#212011yescharacteristic, complemented, Sylow
121C2×#41C2×D3711yescharacteristic
122C23×C4Dic3711yescharacteristic, complemented
123C2×M16—741no—
124#123Dic3711yescharacteristic, complemented
125#123C2×D3711yescharacteristic
126C22⋊Q8—21no—
127#121—21no—
128#120Dic3711yescharacteristic, complemented
129#120C2×D3711yescharacteristic
130C4×D4—21no—
131C4×D4—21no—
132#117—741no—
133#117Dic3711yescharacteristic, complemented
134#101—741no—
135#100—372no—
136#99—372no—
137#99—741no—
138C24C2×Dic3711yescharacteristic
139#41—41no—
140#41—41no—
141C2×Q8C22⋊D3711yescharacteristic
142C2×D4—21no—
143C2×D4—21no—
144C2×D4C2×Dic3711yescharacteristic
145C2×D4C22⋊D3711yescharacteristic
146C2×D4C22×D3711yescharacteristic
147C22×C4—21no—
148C22×C4—21no—
149C22×C4—41no—
150C22×C4—41no—
151C22×C4C2×Dic3711yescharacteristic
152C22×C4C2×Dic3711yescharacteristic
153C22×C4C22⋊D3711yescharacteristic
154C22×C4C22⋊D3711yescharacteristic
155M16—742no—
156M16—1481no—
157C2×C8—1481no—
158C4⋊C4—21no—
159C4⋊C4—21no—
160C4⋊C4—21no—
161C4⋊C4—1481no—
162C22⋊C4—21no—
163C22⋊C4—21no—
164C22⋊C4—21no—
165C22⋊C4—21no—
166C22⋊C4—742no—
167C22⋊C4C2×Dic3711yescharacteristic
168C22⋊C4C2×Dic3711yescharacteristic
169C42—21no—
170C42—1481no—
171C23—21no—
172C23—41no—
173C23—42no—
174C23C22⋊Dic3711yescharacteristic
175C23C22×Dic3711yescharacteristic
176C23C2×C22⋊D3711yescharacteristic
177Q8—41no—
178D4—41no—
179D4—41no—
180D4—41no—
181D4—42no—
182D4—42no—
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—21no—
186C2×C4—21no—
187C2×C4—21no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—42no—
193C2×C4—41no—
194C2×C4—41no—
195C2×C4—41no—
196C2×C4—1482no—
197C2×C4C22⋊Dic3711yescharacteristic
198C2×C4C22⋊Dic3711yescharacteristic
199C2×C4C22⋊Dic3711yescharacteristic
200C2×C4C2×C22⋊D3711yescharacteristic
201C8—1482no—
202C22—22no—
203C22—21no—
204C22—21no—
205C22—21no—
206C22—41no—
207C22—41no—
208C22—41no—
209C22—41no—
210C22—42no—
211C22—42no—
212C22—41no—
213C22C2×C22⋊Dic3711yescharacteristic
214C4—21no—
215C4—21no—
216C4—21no—
217C4—41no—
218C4—41no—
219C4—41no—
220C4—2962no—
221C4#1192445111yescharacteristic
222C4#1192445111yescharacteristic
223C2—21no—
224C2—41no—
225C2—42no—
226C2—41no—
227C2—41no—
228C2C2×#1192445111yescharacteristic
subgroupG/H
centerC4#11924451
commutatorC23×C37C22×C4
FrattiniC2×D4C22×D37
Fitting, prime radicalC37×#525C2
socleC2×C37C2×#99
cosocleC2×D4×C37C23
hypercenter#525D37
hypocenterC37#2120
LCSG ▷ C23×C37 ▷ C22×C37 ▷ C2×C37 ▷ C37
UCSC1 ◁ C4 ◁ C2×C4 ◁ C2×#41 ◁ #525

Complements

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

NH ≅ G/Ntype
1. #14557517C2semidirect
2. #14557507C2semidirect
3. #14557323C2semidirect
4. #14557320C2semidirect
13. C23×C4×C37C4semidirect
14. #123×C37C4semidirect
18. #120×C37C4semidirect
22. #117×C37C4semidirect
120. C37#2120semidirect
122. C23×C4Dic37semidirect
124. #123Dic37semidirect
128. #120Dic37semidirect
133. #117Dic37semidirect

Maximal subgroups

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

subgroupindex
#145575172
#145575072
#145573232
#145573202
C37×#5252
#212037