Finite Group Databasebeta

← #16829025

Subgroups — #16829025

Order 3712.

458 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 458 conjugacy classes of subgroups — too many to draw. Only its 133 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×#11745313C211yesmaximal, characteristic
2C22×Dic116C211yesmaximal, characteristic, complemented
3#14271734C211yesmaximal, characteristic
4#14271640C211yesmaximal, characteristic, complemented
5#14271384C214yesmaximal, complemented
6#14271372C214yesmaximal, complemented
7#14271189C211yesmaximal, characteristic, complemented
8#14271171C211yesmaximal, characteristic, complemented
9C29×#416C211yesmaximal, characteristic
10C2×#41×C29C2211yescharacteristic
11C22×Dic58C2211yescharacteristic
12C22×Dic58C2211yescharacteristic
13C22×C8×C29C2211yescharacteristic
14C29×#117C2211yescharacteristic
15#11745313—24no—
16#11745306C2212yes—
17C2×#90785C2212yes—
18#11745298C2212yes—
19#11745297—22no—
20C2×#90783C2212yes—
21#11745288C2211yescharacteristic
22#11745286—22no—
23C2×#90777C2211yescharacteristic
24C2×Dic116—24no—
25C2×Dic116C2214yes—
26#11745249C2214yes—
27#11745239—22no—
28C4×Dic58—22no—
29C29×#103C2212yes—
30C29×#102C2212yes—
31#11745162C2214yes—
32#11745158C2214yes—
33#41×C29—22no—
34#41×C29—22no—
35C2×Q8×C29C2311yescharacteristic
36C2×D4×C29—21no—
37C2×D4×C29C2311yescharacteristic
38C22×C4×C29—21no—
39C22×C4×C29C2311yescharacteristic
40C22×Dic29—21no—
41C22×Dic29—21no—
42C2×Dic58—21no—
43C2×Dic58—22no—
44C2×Dic58—22no—
45C2×Dic58—22no—
46C2×Dic58—21no—
47C2×Dic58D414yes—
48C2×Dic58C2312yes—
49C2×Dic58C2312yes—
50C2×C8×C29—22no—
51C2×C8×C29C2312yes—
52C4⋊C4×C29C2312yes—
53C22⋊C4×C29—21no—
54C42×C29—21no—
55C22⋊Dic29—24no—
56C22⋊Dic29D412yes—
57#90785—24no—
58#90783—24no—
59C4⋊Dic29D412yes—
60#90779—24no—
61#90779—24no—
62C4×Dic29—24no—
63C4×Dic29C2312yes—
64#90777—22no—
65C2×#6597C2312yes—
66Dic116—28no—
67C23×C29—21no—
68C23×C29C2×D411yescharacteristic
69Q8×C29—21no—
70Q8×C29—21no—
71D4×C29—21no—
72D4×C29—21no—
73D4×C29—42no—
74C2×C4×C29—22no—
75C2×C4×C29—21no—
76C2×C4×C29—42no—
77C2×C4×C29C2×D411yescharacteristic
78C2×C4×C29C2×D412yes—
79C2×C4×C29#4112yes—
80C2×C4×C29C2411yescharacteristic
81C2×Dic29—24no—
82C2×Dic29—22no—
83C2×Dic29—22no—
84C2×Dic29—42no—
85C2×Dic29C2×D414yes—
86Dic58—24no—
87Dic58—22no—
88Dic58—22no—
89Dic58—22no—
90Dic58—42no—
91Dic58C2×D414yes—
92C8×C29—22no—
93#6597—22no—
94#3044—291nomaximal, Sylow
95C22×C29—22no—
96C22×C29—41no—
97C22×C29—42no—
98C22×C29#12112yes—
99C22×C29C22×D411yescharacteristic
100C4×C29—41no—
101C4×C29—42no—
102C4×C29#12112yes—
103C4×C29C22×D411yescharacteristic
104C4×C29C2×#4111yescharacteristic
105Dic29—24no—
106Dic29—24no—
107Dic29—42no—
108C2×#137—291no—
109C22×Q16—291no—
110#536—291no—
111#519—291no—
112#464—294no—
113#462—294no—
114#424—291no—
115#419—291no—
116#416D2911yescharacteristic
117C2×C29—22no—
118C2×C29—81no—
119C2×C29C2×#12111yescharacteristic
120C2×C29#57812yes—
121C2×#41C2×D2911yescharacteristic
122C22×Q8—291no—
123C22×Q8—291no—
124#137—584no—
125C2×Q16—292no—
126C2×Q16—294no—
127C2×Q16—584no—
128C2×SD16—292no—
129C2×M16—291no—
130C22×C8C2×D2911yescharacteristic
131C4⋊Q8—292no—
132#124—292no—
133#123—582no—
134C22⋊Q8—582no—
135C4×Q8—294no—
136C4×Q8—582no—
137#117—291no—
138#117—582no—
139#117C2×D2911yescharacteristic
140C4⋊C8—294no—
141#103—294no—
142#103C2×D2912yes—
143#102C2×D2912yes—
144C29#304411yescharacteristic, complemented, Sylow
145#41—22no—
146#41—22no—
147C2×Q8—294no—
148C2×Q8—292no—
149C2×Q8—292no—
150C2×Q8—582no—
151C2×Q8—581no—
152C2×Q8—581no—
153C2×Q8—582no—
154C2×Q8—582no—
155C2×Q8C22×D2911yescharacteristic
156C2×D4—21no—
157C2×D4C22×D2911yescharacteristic
158C22×C4—21no—
159C22×C4—581no—
160C22×C4—581no—
161C22×C4C22×D2911yescharacteristic
162Q16—588no—
163Q16—584no—
164SD16—584no—
165M16—582no—
166C2×C8—22no—
167C2×C8—292no—
168C2×C8C22×D2912yes—
169C4⋊C4—292no—
170C4⋊C4—584no—
171C4⋊C4—584no—
172C4⋊C4C22×D2912yes—
173C22⋊C4—21no—
174C22⋊C4—292no—
175C22⋊C4—584no—
176C42—21no—
177C42—292no—
178C42—584no—
179C23—21no—
180C23D4×D2911yescharacteristic
181Q8—21no—
182Q8—21no—
183Q8—294no—
184Q8—582no—
185Q8—584no—
186Q8—582no—
187Q8—582no—
188Q8—1162no—
189D4—21no—
190D4—21no—
191D4—42no—
192C2×C4—22no—
193C2×C4—21no—
194C2×C4—42no—
195C2×C4—294no—
196C2×C4—582no—
197C2×C4—584no—
198C2×C4—582no—
199C2×C4—1162no—
200C2×C4#9080512yes—
201C2×C4D4×D2912yes—
202C2×C4D4×D2911yescharacteristic
203C2×C4C23×D2911yescharacteristic
204C8—22no—
205C8—582no—
206C22—22no—
207C22—41no—
208C22—42no—
209C22#1174524612yes—
210C22C2×D4×D2911yescharacteristic
211C4—41no—
212C4—42no—
213C4—584no—
214C4—584no—
215C4—1162no—
216C4#1174524612yes—
217C4C2×#9080511yescharacteristic
218C4C2×D4×D2911yescharacteristic
219C2—22no—
220C2—81no—
221C2C2×#1174524611yescharacteristic
222C2#1427197712yes—
subgroupG/H
centerC22C2×D4×D29
commutator, cosocleC2×C4×C29C24
FrattiniC2×C4C23×D29
FittingC29×#416C2
socleC22×C29C22×D4
prime radicalC2×#41×C29C22
hypercenter#416D29
hypocenterC29#3044
LCSG ▷ C2×C4×C29 ▷ C2×C29 ▷ C29
UCSC1 ◁ C22 ◁ C22×C4 ◁ #416

Complements

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

NH ≅ G/Ntype
2. C22×Dic116C2semidirect
4. #14271640C2semidirect
5. #14271384C2semidirect
6. #14271372C2semidirect
7. #14271189C2semidirect
8. #14271171C2semidirect
144. C29#3044semidirect

Maximal subgroups

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

subgroupindex
C2×#117453132
C22×Dic1162
#142717342
#142716402
#142713842
#142713722
#142711892
#142711712
C29×#4162
#304429