Finite Group Databasebeta

← #16821518

Subgroups — #16821518

Order 3712.

236 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 236 conjugacy classes of subgroups — too many to draw. Only its 45 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#14271345C211yesmaximal, characteristic, complemented
2#14271335C212yesmaximal, complemented
3#14270962C211yesmaximal, characteristic, complemented
4#14270960C212yesmaximal, complemented
5C29×#555C211yesmaximal, characteristic, complemented
6#11745362C2211yescharacteristic
7C29×C4⋊Q8C2211yescharacteristic
8C29×#124—21no—
9C29×#124C2212yes—
10C29×#121—22no—
11C4×D4×C29—21no—
12C29×#117—22no—
13#11745304—21no—
14#11745294—22no—
15#11745184C2212yes—
16#11745184C2211yescharacteristic
17#11745153—22no—
18#11745153—21no—
19C2×Q8×C29D411yescharacteristic
20C2×Q8×C29C2311yescharacteristic
21C2×D4×C29—21no—
22C2×D4×C29—21no—
23C2×D4×C29D412yes—
24C22×C4×C29—22no—
25#90809—41no—
26Q8×D29—41no—
27#90805—22no—
28#90805—21no—
29C4⋊C4×C29—21no—
30C4⋊C4×C29—22no—
31C22⋊C4×C29—22no—
32C22⋊C4×C29—22no—
33C22⋊C4×C29—22no—
34C42×C29—21no—
35C42×C29D412yes—
36C42×C29D411yescharacteristic
37#90785—41no—
38#90784—41no—
39#90783—42no—
40#90782—42no—
41#90777—22no—
42#90777—21no—
43C23×C29—22no—
44Q8×C29—41no—
45Q8×C29—41no—
46D4×C29—42no—
47D4×C29—42no—
48D4×C29—81no—
49C2×C4×C29—21no—
50C2×C4×C29—22no—
51C2×C4×C29—21no—
52C2×C4×C29—44no—
53C2×C4×C29C2×D411yescharacteristic
54C2×C4×C29C2×D412yes—
55C22⋊D29—41no—
56D116—22no—
57D116—21no—
58C4×D29—42no—
59C4×D29—41no—
60Dic58—22no—
61Dic58—21no—
62#6597—42no—
63#6597—41no—
64#2204—291nomaximal, Sylow
65C22×C29—42no—
66C22×C29—42no—
67C22×C29—42no—
68C22×C29#12011yescharacteristic
69C2×D29—41no—
70C4×C29—22no—
71C4×C29—22no—
72C4×C29—21no—
73C4×C29—42no—
74C4×C29—41no—
75Dic29—41no—
76#555D2911yescharacteristic, complemented
77#456—291no—
78#454—292no—
79#356—291no—
80#355—292no—
81C2×C29—21no—
82C2×C29—82no—
83C2×C29B(4,𝔽2)11yescharacteristic
84D29—81no—
852−1+4—291no—
86#137—581no—
87AGL(1,ℤ/8)—582no—
88C4⋊Q8C2×D2911yescharacteristic
89#124—21no—
90#124C2×D2912yes—
91#121—22no—
92C4×D4—21no—
93#117—22no—
94#104—582no—
95#104—581no—
96#101—291no—
97#101—292no—
98C29#220411yescharacteristic, complemented, Sylow
99#41—582no—
100#41—581no—
101#41—1161no—
102C2×Q8—1161no—
103C2×Q8C22⋊D2911yescharacteristic
104C2×Q8C22×D2911yescharacteristic
105C2×D4—21no—
106C2×D4—21no—
107C2×D4C22⋊D2912yes—
108C22×C4—22no—
109Q16—1161no—
110SD16—1162no—
111SD16—1161no—
112D8—1162no—
113M16—582no—
114M16—581no—
115C4⋊C4—21no—
116C4⋊C4—22no—
117C22⋊C4—22no—
118C22⋊C4—22no—
119C22⋊C4—22no—
120C42—21no—
121C42C22⋊D2912yes—
122C42C22⋊D2911yescharacteristic
123C23—22no—
124Q8—41no—
125Q8—41no—
126Q8—582no—
127Q8—581no—
128D4—42no—
129D4—42no—
130D4—81no—
131D4—582no—
132D4—581no—
133D4—1161no—
134C2×C4—21no—
135C2×C4—22no—
136C2×C4—21no—
137C2×C4—44no—
138C2×C4—1162no—
139C2×C4—1161no—
140C2×C4C2×C22⋊D2911yescharacteristic
141C2×C4C2×C22⋊D2912yes—
142C8—1162no—
143C8—1161no—
144C22—42no—
145C22—42no—
146C22—42no—
147C22—1161no—
148C22#1174531511yescharacteristic
149C4—22no—
150C4—22no—
151C4—21no—
152C4—42no—
153C4—41no—
154C4—1161no—
155C2—21no—
156C2—82no—
157C2—2321no—
158C2#1427135011yescharacteristic
subgroupG/H
centerC2#14271350
commutator, cosocleC2×Q8×C29C23
FrattiniC2×Q8C22×D29
FittingC29×#555C2
socleC2×C29B(4,𝔽2)
hypercenter#555D29
hypocenterC29#2204
LCSG ▷ C2×Q8×C29 ▷ C22×C29 ▷ C2×C29 ▷ C29
UCSC1 ◁ C2 ◁ C22 ◁ C2×Q8 ◁ #555

Complements

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

NH ≅ G/Ntype
1. #14271345C2semidirect
2. #14271335C2semidirect
3. #14270962C2semidirect
4. #14270960C2semidirect
5. C29×#555C2semidirect
76. #555D29semidirect
98. C29#2204semidirect

Maximal subgroups

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

subgroupindex
#142713452
#142713352
#142709622
#142709602
C29×#5552
#220429