Finite Group Databasebeta

← #16817981

Subgroups — #16817981

Order 3712.

304 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 304 conjugacy classes of subgroups — too many to draw. Only its 77 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#14271674C211yesmaximal, characteristic
2C2×C22⋊#6597C211yesmaximal, characteristic, complemented
3#14270899C212yesmaximal, complemented
4#14270896C212yesmaximal, complemented
5C4×C22⋊C4×C29C211yesmaximal, characteristic, complemented
6C23×C4×C29C2211yescharacteristic
7C2×#90805C411yescharacteristic, complemented
8C2×C22⋊C4×C29—21no—
9C2×C42×C29—21no—
10#11745315—21no—
11#11745292C411yescharacteristic, complemented
12#11745292C2211yescharacteristic
13#11745291—21no—
14C4×C22⋊D29—21no—
15C4×C22⋊D29—21no—
16#11745288C411yescharacteristic, complemented
17#11745286C411yescharacteristic, complemented
18#11745286C2211yescharacteristic
19C22×#6597—21no—
20C29×#95C2212yes—
21C22⋊#6597—21no—
22C22⋊#6597C2212yes—
23C24×C29D411yescharacteristic
24C22×C4×C29—22no—
25C22×C4×C29—42no—
26C22×C4×C29C2×C411yescharacteristic
27C22×C4×C29C2×C411yescharacteristic
28C22×C4×C29D411yescharacteristic
29C22×C4×C29D411yescharacteristic
30C22×C4×C29D411yescharacteristic
31C22×C4×C29C2311yescharacteristic
32C2×C22⋊D29—21no—
33C2×C22⋊D29—21no—
34C2×C22⋊D29—21no—
35#90805—42no—
36C2×D116C2×C411yescharacteristic
37C2×C4×D29—21no—
38C2×Dic58C2×C411yescharacteristic
39C22⋊C4×C29—24no—
40C42×C29—42no—
41C22⋊Dic29—21no—
42C22⋊Dic29—21no—
43C22⋊Dic29—21no—
44#90781—21no—
45#90781—21no—
46C4⋊Dic29C2×C411yescharacteristic
47C4⋊Dic29C2×C411yescharacteristic
48#90779—21no—
49#90779—21no—
50C4×Dic29—21no—
51C2×#6597—22no—
52C2×#6597—42no—
53C23×C29—22no—
54C23×C29—22no—
55C23×C29—41no—
56C23×C29C22⋊C411yescharacteristic
57C23×C29C22⋊C411yescharacteristic
58C23×C29C2×D411yescharacteristic
59C22×D29—21no—
60C2×C4×C29—22no—
61C2×C4×C29—22no—
62C2×C4×C29—22no—
63C2×C4×C29—24no—
64C2×C4×C29—41no—
65C2×C4×C29—41no—
66C2×C4×C29—41no—
67C2×C4×C29—42no—
68C2×C4×C29—42no—
69C2×C4×C29C22⋊C411yescharacteristic
70C2×C4×C29C22⋊C411yescharacteristic
71C2×C4×C29C22×C411yescharacteristic
72C2×C4×C29C2×D411yescharacteristic
73C22⋊D29—42no—
74C22⋊D29—42no—
75C22⋊D29—42no—
76C2×Dic29—21no—
77C2×Dic29—21no—
78C2×Dic29—21no—
79D116—41no—
80C4×D29—42no—
81Dic58—41no—
82#6597—42no—
83#1501—291nomaximal, Sylow
84C22×C29—21no—
85C22×C29—21no—
86C22×C29—21no—
87C22×C29—21no—
88C22×C29—21no—
89C22×C29—21no—
90C22×C29—21no—
91C22×C29—44no—
92C22×C29#10412yes—
93C22×C29C2×C22⋊C411yescharacteristic
94C22×C29#13512yes—
95C2×D29—41no—
96C2×D29—41no—
97C2×D29—41no—
98C4×C29—21no—
99C4×C29—21no—
100C4×C29—21no—
101C4×C29—44no—
102C4×C29#9912yes—
103Dic29—41no—
104Dic29—41no—
105Dic29—41no—
106#525—291no—
107C2×C22⋊C8—291no—
108C4×C22⋊C4D2911yescharacteristic, complemented
109#328—292no—
110#327—292no—
111C2×C29—22no—
112C2×C29—22no—
113C2×C29—41no—
114C2×C29C2×#9911yescharacteristic
115C2×C29#41611yescharacteristic
116C2×C29C2×#10411yescharacteristic
117D29—81no—
118C2×#41—291no—
119C23×C4C2×D2911yescharacteristic
120C22×C8—581no—
121#123—581no—
122C22⋊Q8—291no—
123C22⋊Q8—291no—
124#121—291no—
125#121—291no—
126#120—581no—
127C4×D4—581no—
128C4×D4—581no—
129#117—291no—
130C2×C22⋊C4—21no—
131C2×C42—21no—
132C22⋊C8—292no—
133C22⋊C8—581no—
134#95C2×D2912yes—
135C29#150111yescharacteristic, complemented, Sylow
136C24C22⋊D2911yescharacteristic
137#41—1162no—
138C2×Q8—291no—
139C2×D4—291no—
140C2×D4—581no—
141C2×D4—581no—
142C2×D4—581no—
143C22×C4—22no—
144C22×C4—42no—
145C22×C4—581no—
146C22×C4C4×D2911yescharacteristic
147C22×C4C4×D2911yescharacteristic
148C22×C4D11611yescharacteristic
149C22×C4D11611yescharacteristic
150C22×C4C22⋊D2911yescharacteristic
151C22×C4C22×D2911yescharacteristic
152C2×C8—582no—
153C2×C8—1162no—
154C4⋊C4—291no—
155C4⋊C4—291no—
156C4⋊C4—581no—
157C4⋊C4—581no—
158C22⋊C4—24no—
159C22⋊C4—581no—
160C22⋊C4—581no—
161C22⋊C4—581no—
162C22⋊C4—581no—
163C22⋊C4—581no—
164C42—42no—
165C42—581no—
166C23—22no—
167C23—22no—
168C23—41no—
169C23—581no—
170C23#9078111yescharacteristic
171C23#9078111yescharacteristic
172C23C2×C22⋊D2911yescharacteristic
173Q8—1161no—
174D4—1161no—
175D4—1162no—
176D4—1162no—
177D4—1162no—
178C2×C4—22no—
179C2×C4—22no—
180C2×C4—22no—
181C2×C4—24no—
182C2×C4—41no—
183C2×C4—41no—
184C2×C4—41no—
185C2×C4—42no—
186C2×C4—42no—
187C2×C4—581no—
188C2×C4—581no—
189C2×C4—581no—
190C2×C4—1162no—
191C2×C4#9078111yescharacteristic
192C2×C4#9078111yescharacteristic
193C2×C4C2×C4×D2911yescharacteristic
194C2×C4C2×D11611yescharacteristic
195C8—1162no—
196C22—21no—
197C22—21no—
198C22—21no—
199C22—21no—
200C22—21no—
201C22—21no—
202C22—21no—
203C22—44no—
204C22—1161no—
205C22—1161no—
206C22—1161no—
207C22#1174515312yes—
208C22C2×#9078111yescharacteristic
209C22#1174531212yes—
210C4—21no—
211C4—21no—
212C4—21no—
213C4—44no—
214C4—1161no—
215C4—1161no—
216C4—1161no—
217C4#1174515412yes—
218C2—22no—
219C2—22no—
220C2—41no—
221C2—2321no—
222C2C2×#1174515411yescharacteristic
223C2#1427115711yescharacteristic
224C2C2×#1174515311yescharacteristic
subgroupG/H
centerC2×C4#90781
commutatorC2×C4×C29C22×C4
FrattiniC22×C4C22×D29
FittingC4×C22⋊C4×C29C2
socleC22×C29C2×C22⋊C4
prime radical#14271674C2
cosocleC22×C4×C29C23
hypercenterC4×C22⋊C4D29
hypocenterC29#1501
LCSG ▷ C2×C4×C29 ▷ C22×C29 ▷ C29
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ C4×C22⋊C4

Complements

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

NH ≅ G/Ntype
2. C2×C22⋊#6597C2semidirect
3. #14270899C2semidirect
4. #14270896C2semidirect
5. C4×C22⋊C4×C29C2semidirect
7. C2×#90805C4semidirect
11. #11745292C4semidirect
16. #11745288C4semidirect
17. #11745286C4semidirect
108. C4×C22⋊C4D29semidirect
135. C29#1501semidirect

Maximal subgroups

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

subgroupindex
#142716742
C2×C22⋊#65972
#142708992
#142708962
C4×C22⋊C4×C292
#150129