Finite Group Databasebeta

← #16819462

Subgroups — #16819462

Order 3712.

588 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 588 conjugacy classes of subgroups — too many to draw. Only its 121 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#14271180C212yesmaximal, complemented
2#14271168C212yesmaximal, complemented
3#14271147C211yesmaximal, characteristic, complemented
4#14271137C211yesmaximal, characteristic, complemented
5C2×C29×H(5,𝔽2)C211yesmaximal, characteristic
6C29×H(5,𝔽2)—24no—
7C29×H(5,𝔽2)—24no—
8C2×#41×C29—21no—
9C2×#41×C29C412yescomplemented
10C2×#41×C29C2211yescharacteristic
11C2×#41×C29C2211yescharacteristic
12C22×D4×C29—22no—
13C22×D4×C29—21no—
14C22×D4×C29C412yescomplemented
15C22×D4×C29C2211yescharacteristic
16#11745288C2212yescomplemented
17C2×#90777C2212yescomplemented
18#11745185—24no—
19#11745184—21no—
20#11745182—22no—
21#11745181—21no—
22#11745180—24no—
23C24×C29—22no—
24C24×C29—21no—
25#41×C29—28no—
26#41×C29—24no—
27#41×C29—24no—
28#41×C29—22no—
29#41×C29—22no—
30#41×C29—22no—
31#41×C29D414yes—
32C2×Q8×C29C2×C411yescharacteristic
33C2×Q8×C29D411yescharacteristic
34C2×D4×C29—216no—
35C2×D4×C29—28no—
36C2×D4×C29—22no—
37C2×D4×C29—21no—
38C2×D4×C29—24no—
39C2×D4×C29—24no—
40C2×D4×C29—22no—
41C2×D4×C29—21no—
42C2×D4×C29—22no—
43C2×D4×C29C2×C411yescharacteristic
44C2×D4×C29C2×C412yes—
45C2×D4×C29D412yes—
46C2×D4×C29D411yescharacteristic
47C2×D4×C29D414yes—
48C22×C4×C29—22no—
49C22×C4×C29—21no—
50C22×C4×C29C2×C412yes—
51C22×C4×C29C2311yescharacteristic
52C22⋊Dic29—22no—
53C4⋊Dic29—22no—
54C4×Dic29—22no—
55#90777—22no—
56#90777—22no—
57C2×#6597—22no—
58C23×C29—28no—
59C23×C29—22no—
60C23×C29—21no—
61C23×C29—24no—
62C23×C29—82no—
63C23×C29—84no—
64C23×C29C22⋊C412yes—
65C23×C29C2×D411yescharacteristic
66Q8×C29—21no—
67Q8×C29—22no—
68Q8×C29C22⋊C412yes—
69D4×C29—28no—
70D4×C29—28no—
71D4×C29—24no—
72D4×C29—24no—
73D4×C29—22no—
74D4×C29—21no—
75D4×C29—22no—
76D4×C29—24no—
77D4×C29C22⋊C412yes—
78D4×C29C22⋊C414yes—
79C2×C4×C29—28no—
80C2×C4×C29—21no—
81C2×C4×C29—22no—
82C2×C4×C29—22no—
83C2×C4×C29—24no—
84C2×C4×C29C22⋊C412yes—
85C2×C4×C29C22×C411yescharacteristic
86C2×C4×C29C2×D412yes—
87C2×C4×C29C2×D412yes—
88C2×C4×C29C2×D411yescharacteristic
89C2×Dic29—42no—
90#6597—42no—
91#1795—291nomaximal, Sylow
92C22×C29—22no—
93C22×C29—24no—
94C22×C29—22no—
95C22×C29—41no—
96C22×C29—42no—
97C22×C29—44no—
98C22×C29—42no—
99C22×C29—84no—
100C22×C29—88no—
101C22×C29C2×C22⋊C411yescharacteristic
102C22×C29#12012yes—
103C4×C29—22no—
104C4×C29—22no—
105C4×C29C2×C22⋊C411yescharacteristic
106C4×C29C2×C22⋊C411yescharacteristic
107C4×C29#12012yes—
108Dic29—82no—
109C2×H(5,𝔽2)D2911yescharacteristic
110#421—292no—
111#418—292no—
112#413—291no—
113#410—291no—
114C2×C29—22no—
115C2×C29—21no—
116C2×C29—44no—
117C2×C29—42no—
118C2×C29—42no—
119C2×C29#37911yescharacteristic
120H(5,𝔽2)—24no—
121H(5,𝔽2)—24no—
122C2×#41—21no—
123C2×#41Dic2912yescomplemented
124C2×#41C2×D2911yescharacteristic
125C2×#41C2×D2911yescharacteristic
126C22×D4—22no—
127C22×D4—21no—
128C22×D4Dic2912yescomplemented
129C22×D4C2×D2911yescharacteristic
130C2×M16—292no—
131#117—292no—
132#104—584no—
133#102—584no—
134#101—581no—
135#100—581no—
136#99—582no—
137C29#179511yescharacteristic, complemented, Sylow
138C24—22no—
139C24—21no—
140#41—28no—
141#41—24no—
142#41—24no—
143#41—22no—
144#41—22no—
145#41—22no—
146#41C22⋊D2914yes—
147C2×Q8C2×Dic2911yescharacteristic
148C2×Q8C22⋊D2911yescharacteristic
149C2×D4—216no—
150C2×D4—28no—
151C2×D4—22no—
152C2×D4—21no—
153C2×D4—24no—
154C2×D4—24no—
155C2×D4—22no—
156C2×D4—21no—
157C2×D4—22no—
158C2×D4C2×Dic2911yescharacteristic
159C2×D4C2×Dic2912yes—
160C2×D4C22⋊D2912yes—
161C2×D4C22⋊D2911yescharacteristic
162C2×D4C22⋊D2914yes—
163C22×C4—22no—
164C22×C4—21no—
165C22×C4C2×Dic2912yes—
166C22×C4C22×D2911yescharacteristic
167M16—582no—
168M16—582no—
169C2×C8—582no—
170C4⋊C4—582no—
171C22⋊C4—582no—
172C42—582no—
173C23—28no—
174C23—22no—
175C23—21no—
176C23—24no—
177C23—82no—
178C23—84no—
179C23C22⋊Dic2912yes—
180C23C2×C22⋊D2911yescharacteristic
181Q8—21no—
182Q8—22no—
183Q8C22⋊Dic2912yes—
184D4—28no—
185D4—28no—
186D4—24no—
187D4—24no—
188D4—22no—
189D4—21no—
190D4—22no—
191D4—24no—
192D4C22⋊Dic2912yes—
193D4C22⋊Dic2914yes—
194C2×C4—28no—
195C2×C4—21no—
196C2×C4—22no—
197C2×C4—22no—
198C2×C4—24no—
199C2×C4—1162no—
200C2×C4C22⋊Dic2912yes—
201C2×C4C22×Dic2911yescharacteristic
202C2×C4C2×C22⋊D2912yes—
203C2×C4C2×C22⋊D2912yes—
204C2×C4C2×C22⋊D2911yescharacteristic
205C8—1162no—
206C22—22no—
207C22—24no—
208C22—22no—
209C22—41no—
210C22—42no—
211C22—44no—
212C22—42no—
213C22—84no—
214C22—88no—
215C22C2×C22⋊Dic2911yescharacteristic
216C22#1174531512yes—
217C4—22no—
218C4—22no—
219C4—2322no—
220C4C2×C22⋊Dic2911yescharacteristic
221C4C2×C22⋊Dic2911yescharacteristic
222C4#1174531512yes—
223C2—22no—
224C2—21no—
225C2—44no—
226C2—42no—
227C2—42no—
228C2#1427101911yescharacteristic
subgroupG/H
centerC2#14271019
commutatorC2×C4×C29C22×C4
FrattiniC22×C4C22×D29
Fitting, prime radicalC2×C29×H(5,𝔽2)C2
socleC2×C29#379
cosocleC22×C4×C29C23
hypercenterC2×H(5,𝔽2)D29
hypocenterC29#1795
LCSG ▷ C2×C4×C29 ▷ C2×C29 ▷ C29
UCSC1 ◁ C2 ◁ C22×C4 ◁ C2×H(5,𝔽2)

Complements

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

NH ≅ G/Ntype
1. #14271180C2semidirect
2. #14271168C2semidirect
3. #14271147C2semidirect
4. #14271137C2semidirect
9. C2×#41×C29C4semidirect
14. C22×D4×C29C4semidirect
16. #11745288C22semidirect
17. C2×#90777C22semidirect
123. C2×#41Dic29semidirect
128. C22×D4Dic29semidirect
137. C29#1795semidirect

Maximal subgroups

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

subgroupindex
#142711802
#142711682
#142711472
#142711372
C2×C29×H(5,𝔽2)2
#179529