Finite Group Databasebeta

← #16819463

Subgroups — #16819463

Order 3712.

588 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 588 conjugacy classes of subgroups — too many to draw. Only its 97 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×#11745364C211yesmaximal, characteristic
2#14271179C212yesmaximal, complemented
3#14271169C212yesmaximal, complemented
4#14271137C211yesmaximal, characteristic, complemented
5C29×#413C211yesmaximal, characteristic, complemented
6C2×#41×C29C2211yescharacteristic
7#11745364—24no—
8#11745364—24no—
9C2×#90809—21no—
10C2×D4×D29—22no—
11C2×D4×D29—21no—
12C2×#90805C412yescomplemented
13C2×#90805C2211yescharacteristic
14C22×D116C412yescomplemented
15C22×D116C2211yescharacteristic
16C2×M16×C29C2212yescomplemented
17#11745288C2212yescomplemented
18C29×#101—21no—
19C29×#100—21no—
20#11745182—22no—
21#11745173—24no—
22#11745169—24no—
23C23×D29—22no—
24C23×D29—21no—
25#41×C29—22no—
26#41×C29—22no—
27C2×Q8×C29D411yescharacteristic
28C2×D4×C29D412yes—
29C2×D4×C29D411yescharacteristic
30C22×C4×C29C2×C412yes—
31C22×C4×C29C2311yescharacteristic
32C2×C22⋊D29—22no—
33C2×C22⋊D29—21no—
34#90809—28no—
35D4×D29—216no—
36D4×D29—28no—
37#90805—22no—
38#90805—24no—
39#90805—24no—
40#90805D414yes—
41C2×D116—22no—
42C2×D116—21no—
43C2×D116—22no—
44C2×D116—24no—
45C2×D116—24no—
46C2×D116C2×C411yescharacteristic
47C2×D116C2×C412yes—
48C2×D116D414yes—
49C2×C4×D29—22no—
50C2×C4×D29—21no—
51C2×Dic58C2×C411yescharacteristic
52M16×C29—22no—
53M16×C29—22no—
54C2×C8×C29—22no—
55C22⋊Dic29—22no—
56C4⋊Dic29—22no—
57C4×Dic29—22no—
58C23×C29C22⋊C412yes—
59C23×C29C2×D411yescharacteristic
60C22×D29—22no—
61C22×D29—21no—
62C22×D29—28no—
63C22×D29—24no—
64C22×D29—82no—
65C22×D29—84no—
66Q8×C29—22no—
67D4×C29—24no—
68D4×C29—22no—
69C2×C4×C29—22no—
70C2×C4×C29—22no—
71C2×C4×C29C22⋊C412yes—
72C2×C4×C29C22×C411yescharacteristic
73C2×C4×C29C2×D412yes—
74C2×C4×C29C2×D412yes—
75C2×C4×C29C2×D411yescharacteristic
76C22⋊D29—28no—
77C22⋊D29—24no—
78C2×Dic29—21no—
79C2×Dic29—42no—
80D116—28no—
81D116—21no—
82D116—22no—
83D116—24no—
84D116C22⋊C412yes—
85D116C22⋊C414yes—
86C4×D29—28no—
87C4×D29—24no—
88Dic58—21no—
89Dic58C22⋊C412yes—
90C8×C29—42no—
91#1795—291nomaximal, Sylow
92C22×C29—22no—
93C22×C29—41no—
94C22×C29—42no—
95C22×C29C2×C22⋊C411yescharacteristic
96C22×C29#12012yes—
97C2×D29—24no—
98C2×D29—22no—
99C2×D29—44no—
100C2×D29—42no—
101C2×D29—84no—
102C2×D29—88no—
103C4×C29—22no—
104C4×C29C2×C22⋊C411yescharacteristic
105C4×C29C2×C22⋊C411yescharacteristic
106C4×C29#12012yes—
107Dic29—22no—
108Dic29—82no—
109C2×H(5,𝔽2)—291no—
110#421—292no—
111#418—292no—
112#413D2911yescharacteristic, complemented
113#410—291no—
114C2×C29—22no—
115C2×C29—21no—
116C2×C29—42no—
117C2×C29#37911yescharacteristic
118D29—44no—
119D29—42no—
120H(5,𝔽2)—584no—
121H(5,𝔽2)—584no—
122C2×#41—292no—
123C2×#41—291no—
124C2×#41—581no—
125C2×#41C2×D2911yescharacteristic
126C22×D4—292no—
127C22×D4—291no—
128C22×D4—582no—
129C22×D4—581no—
130C2×M16C2×D2912yescomplemented
131#117—292no—
132#104—584no—
133#102—584no—
134#101—21no—
135#100—21no—
136#99—582no—
137C29#179511yescharacteristic, complemented, Sylow
138C24—582no—
139C24—581no—
140#41—22no—
141#41—22no—
142#41—294no—
143#41—584no—
144#41—584no—
145#41—588no—
146#41—582no—
147C2×Q8—291no—
148C2×Q8C22⋊D2911yescharacteristic
149C2×D4—292no—
150C2×D4—291no—
151C2×D4—294no—
152C2×D4—584no—
153C2×D4—584no—
154C2×D4—5816no—
155C2×D4—588no—
156C2×D4—582no—
157C2×D4—581no—
158C2×D4—582no—
159C2×D4—582no—
160C2×D4—581no—
161C2×D4D11612yes—
162C2×D4C22⋊D2911yescharacteristic
163C22×C4—582no—
164C22×C4—581no—
165C22×C4C4×D2912yes—
166C22×C4C22×D2911yescharacteristic
167M16—22no—
168M16—22no—
169C2×C8—22no—
170C4⋊C4—582no—
171C22⋊C4—582no—
172C42—582no—
173C23—588no—
174C23—582no—
175C23—581no—
176C23—584no—
177C23—2322no—
178C23—2324no—
179C23#9078112yes—
180C23C2×D11611yescharacteristic
181Q8—22no—
182Q8—292no—
183Q8—581no—
184D4—24no—
185D4—22no—
186D4—294no—
187D4—292no—
188D4—588no—
189D4—588no—
190D4—584no—
191D4—584no—
192D4—581no—
193D4—582no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—588no—
197C2×C4—581no—
198C2×C4—584no—
199C2×C4—1162no—
200C2×C4#9078112yes—
201C2×C4C2×C4×D2911yescharacteristic
202C2×C4D4×D2912yes—
203C2×C4D4×D2912yes—
204C2×C4C2×C22⋊D2911yescharacteristic
205C8—42no—
206C22—22no—
207C22—41no—
208C22—42no—
209C22—584no—
210C22—582no—
211C22—1164no—
212C22—1162no—
213C22—2324no—
214C22—2328no—
215C22#1174524412yes—
216C22C2×#9078111yescharacteristic
217C4—22no—
218C4—582no—
219C4—2322no—
220C4C22⋊C4×D2911yescharacteristic
221C4C22⋊C4×D2911yescharacteristic
222C4#1174529912yes—
223C2—22no—
224C2—21no—
225C2—42no—
226C2—1164no—
227C2—1162no—
228C2#1427101811yescharacteristic
subgroupG/H
centerC2#14271018
commutatorC2×C4×C29C22×C4
FrattiniC22×C4C22×D29
FittingC29×#413C2
socleC2×C29#379
prime radicalC2×#11745364C2
cosocleC22×C4×C29C23
hypercenter#413D29
hypocenterC29#1795
LCSG ▷ C2×C4×C29 ▷ C2×C29 ▷ C29
UCSC1 ◁ C2 ◁ C22×C4 ◁ #413

Complements

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

NH ≅ G/Ntype
2. #14271179C2semidirect
3. #14271169C2semidirect
4. #14271137C2semidirect
5. C29×#413C2semidirect
12. C2×#90805C4semidirect
14. C22×D116C4semidirect
16. C2×M16×C29C22semidirect
17. #11745288C22semidirect
112. #413D29semidirect
130. C2×M16C2×D29semidirect
137. C29#1795semidirect

Maximal subgroups

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

subgroupindex
C2×#117453642
#142711792
#142711692
#142711372
C29×#4132
#179529