Finite Group Databasebeta

← #17993034

Subgroups — #17993034

Order 4736.

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

normal subgroups only

This group has 266 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
1C4×#10596966C211yesmaximal, characteristic
2#14557555C211yesmaximal, characteristic, complemented
3C4×#10596929C211yesmaximal, characteristic, complemented
4Q8⋊#64436C211yesmaximal, characteristic, complemented
5#14557262C211yesmaximal, characteristic, complemented
6#14557260C211yesmaximal, characteristic, complemented
7C37×Q8⋊C8C211yesmaximal, characteristic, complemented
8C2×#10596966C411yescharacteristic
9C2×#10596929—21no—
10C2×#10596929—21no—
11Q8×Dic37C411yescharacteristic
12C4×D148—21no—
13C4×D148C2211yescharacteristic
14#11924498—21no—
15#11924498C411yescharacteristic
16#11924494—21no—
17#11924494C411yescharacteristic
18C42×D37—21no—
19C42×D37C2211yescharacteristic
20C4⋊#64436C2211yescharacteristic
21#11924465—21no—
22#11924446—21no—
23C4×#64436C2211yescharacteristic
24C4×Q8×C37C2211yescharacteristic
25C4⋊C8×C37C2211yescharacteristic
26C4×C8×C37C2211yescharacteristic
27C2×Q8×C37C2×C411yescharacteristic
28#10596966—22no—
29#10596966—22no—
30C2×D148—21no—
31C2×D148C2×C411yescharacteristic
32C2×C4×D37—21no—
33C2×C4×D37—21no—
34C2×C4×D37—21no—
35C2×C4×D37C2×C411yescharacteristic
36C2×C4×D37D411yescharacteristic
37C2×C4×D37D411yescharacteristic
38C2×C8×C37—21no—
39C2×C8×C37—21no—
40C4⋊C4×C37—21no—
41C4⋊C4×C37C2×C411yescharacteristic
42C42×C37—21no—
43C42×C37C2311yescharacteristic
44#10596938—21no—
45#10596938—21no—
46#10596938—21no—
47C4⋊Dic37—21no—
48C4⋊Dic37C2×C411yescharacteristic
49C4×Dic37—21no—
50C4×Dic37—21no—
51C4×Dic37C2×C411yescharacteristic
52C4×Dic37D411yescharacteristic
53C4×Dic37D411yescharacteristic
54C2×#64436—21no—
55C2×#64436—21no—
56#10596929—24no—
57#10596929—42no—
58C22×D37—21no—
59C22×D37C22⋊C411yescharacteristic
60Q8×C37—21no—
61Q8×C37M1612yescomplemented
62C2×C4×C37—21no—
63C2×C4×C37—21no—
64C2×C4×C37C22×C411yescharacteristic
65C2×C4×C37C2×D411yescharacteristic
66C2×C4×C37C2×D411yescharacteristic
67C2×Dic37—21no—
68C2×Dic37C22⋊C411yescharacteristic
69C2×Dic37C22⋊C411yescharacteristic
70C2×Dic37C22⋊C411yescharacteristic
71D148—24no—
72D148—21no—
73D148M1612yescomplemented
74C4×D37—24no—
75C4×D37—21no—
76C4×D37—21no—
77C4×D37—21no—
78C4×D37—21no—
79C4×D37—22no—
80C4×D37—41no—
81C4×D37—41no—
82C4×D37—42no—
83C8×C37—22no—
84C8×C37—41no—
85#64436—22no—
86#64436—41no—
87C22×C37C2×C22⋊C411yescharacteristic
88C2×D37—21no—
89C2×D37—22no—
90C2×D37—21no—
91C2×D37—21no—
92C2×D37—41no—
93C2×D37—41no—
94C4×C37—22no—
95C4×C37—21no—
96C4×C37—41no—
97C4×C37C2×M1611yescharacteristic
98C4×C37C2×M1611yescharacteristic
99C4×C37AGL(1,ℤ/8)11yescharacteristic
100C4×C37#13711yescharacteristic
101Dic37—21no—
102Dic37—21no—
103Dic37—41no—
104Dic37#10412yes—
105#1489—371nomaximal, Sylow
106C2×C37#40711yescharacteristic
107C2×C37#41711yescharacteristic
108C2×C37C2×#10411yescharacteristic
109D37—41no—
110D37—42no—
111C4×#41—371no—
112#423—371no—
113C4×M16—371no—
114Q8⋊C8—371no—
115Q8⋊C8D3711yescharacteristic, complemented
116D4⋊C8—371no—
117D4⋊C8—371no—
118C37#148911yescharacteristic, complemented, Sylow
119C2×#41—371no—
120C2×M16—741no—
121C2×M16—741no—
122C4×Q8—371no—
123C4×Q8C2×D3711yescharacteristic
124C4×D4—371no—
125C4×D4—371no—
126C4×D4—741no—
127C4×D4—741no—
128#117—371no—
129#117—741no—
130C2×C42—371no—
131C2×C42—741no—
132C4⋊C8—371no—
133C4⋊C8—741no—
134C4⋊C8C2×D3711yescharacteristic
135#97—741no—
136C4×C8—371no—
137C4×C8C2×D3711yescharacteristic
138#41—742no—
139#41—742no—
140C2×Q8C4×D3711yescharacteristic
141C2×D4—371no—
142C2×D4—741no—
143C22×C4—371no—
144C22×C4—371no—
145C22×C4—371no—
146C22×C4—741no—
147C22×C4—741no—
148C22×C4—741no—
149M16—744no—
150M16—1482no—
151C2×C8—21no—
152C2×C8—21no—
153C2×C8—741no—
154C2×C8—741no—
155C4⋊C4—21no—
156C4⋊C4—371no—
157C4⋊C4—741no—
158C4⋊C4C4×D3711yescharacteristic
159C22⋊C4—741no—
160C22⋊C4—741no—
161C22⋊C4—741no—
162C42—21no—
163C42—371no—
164C42—371no—
165C42—371no—
166C42—741no—
167C42—741no—
168C42C22×D3711yescharacteristic
169C23—371no—
170C23—741no—
171Q8—21no—
172Q8#1059692912yescomplemented
173D4—372no—
174D4—744no—
175D4—741no—
176C2×C4—21no—
177C2×C4—21no—
178C2×C4—371no—
179C2×C4—371no—
180C2×C4—371no—
181C2×C4—742no—
182C2×C4—744no—
183C2×C4—741no—
184C2×C4—741no—
185C2×C4—741no—
186C2×C4—741no—
187C2×C4—741no—
188C2×C4—1482no—
189C2×C4—1481no—
190C2×C4—1481no—
191C2×C4C2×C4×D3711yescharacteristic
192C2×C4D4×D3711yescharacteristic
193C2×C4D4×D3711yescharacteristic
194C8—22no—
195C8—41no—
196C8—742no—
197C8—1481no—
198C22—742no—
199C22—741no—
200C22—741no—
201C22—741no—
202C22—1481no—
203C22—1481no—
204C22C22⋊C4×D3711yescharacteristic
205C4—22no—
206C4—21no—
207C4—41no—
208C4—372no—
209C4—741no—
210C4—741no—
211C4—1481no—
212C4M16×D3711yescharacteristic
213C4C2×#1059692911yescharacteristic
214C4#1192456311yescharacteristic
215C4#1192456811yescharacteristic
216C2—1481no—
217C2—1482no—
218C2#1455749511yescharacteristic
219C2#1455753311yescharacteristic
220C2#104×D3711yescharacteristic
subgroupG/H
centerC2×C4D4×D37
commutatorC2×C4×C37C22×C4
FrattiniC42C22×D37
FittingC37×Q8⋊C8C2
socleC22×C37C2×C22⋊C4
prime radicalC2×#10596966C4
cosocleC42×C37C23
hypercenterQ8⋊C8D37
hypocenterC37#1489
LCSG ▷ C2×C4×C37 ▷ C2×C37 ▷ C37
UCSC1 ◁ C2×C4 ◁ C42 ◁ Q8⋊C8

Complements

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

NH ≅ G/Ntype
2. #14557555C2semidirect
3. C4×#10596929C2semidirect
4. Q8⋊#64436C2semidirect
5. #14557262C2semidirect
6. #14557260C2semidirect
7. C37×Q8⋊C8C2semidirect
61. Q8×C37M16semidirect
73. D148M16semidirect
115. Q8⋊C8D37semidirect
118. C37#1489semidirect
172. Q8#10596929semidirect

Maximal subgroups

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

subgroupindex
C4×#105969662
#145575552
C4×#105969292
Q8⋊#644362
#145572622
#145572602
C37×Q8⋊C82
#148937