Finite Group Databasebeta

← #16821505

Subgroups — #16821505

Order 3712.

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

normal subgroups only

This group has 226 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#14271352C212yesmaximal
2#14271340C211yesmaximal, characteristic
3#14270958C211yesmaximal, characteristic, complemented
4#14270953C212yesmaximal, complemented
5C29×#563C211yesmaximal, characteristic, complemented
6#11745359C2211yescharacteristic
7C29×C4⋊Q8—21no—
8C29×#126—22no—
9C29×#125—21no—
10C29×#124C2211yescharacteristic
11C29×#123C2212yes—
12C29×C22⋊Q8—22no—
13#11745304—21no—
14#11745291—22no—
15#11745182C2212yes—
16#11745181C2211yescharacteristic
17#11745154—22no—
18#11745153—21no—
19C2×Q8×C29—21no—
20C2×Q8×C29D411yescharacteristic
21C2×D4×C29C2311yescharacteristic
22C22×C4×C29D412yes—
23C2×C22⋊D29—22no—
24#90807—41no—
25D4×D29—41no—
26#90805—21no—
27C4⋊C4×C29—21no—
28C4⋊C4×C29—22no—
29C4⋊C4×C29—22no—
30C4⋊C4×C29—21no—
31C4⋊C4×C29—22no—
32C22⋊C4×C29—22no—
33C22⋊C4×C29—22no—
34C22⋊C4×C29D412yes—
35C42×C29—21no—
36C42×C29D411yescharacteristic
37C22⋊Dic29—22no—
38#90785—41no—
39#90784—41no—
40#90781—42no—
41#90779—42no—
42#90777—21no—
43C23×C29C2×D412yes—
44C22×D29—22no—
45Q8×C29—41no—
46Q8×C29—81no—
47D4×C29—41no—
48C2×C4×C29—21no—
49C2×C4×C29—22no—
50C2×C4×C29—21no—
51C2×C4×C29—22no—
52C2×C4×C29—42no—
53C2×C4×C29C2×D411yescharacteristic
54C22⋊D29—42no—
55C22⋊D29—41no—
56C2×Dic29—22no—
57C2×Dic29—42no—
58D116—21no—
59C4×D29—41no—
60Dic58—21no—
61#6597—41no—
62#2201—291nomaximal, Sylow
63C22×C29—22no—
64C22×C29—42no—
65C22×C29#12011yescharacteristic
66C2×D29—41no—
67C2×D29—82no—
68C4×C29—21no—
69C4×C29—42no—
70C4×C29—41no—
71C4×C29—42no—
72C4×C29—41no—
73Dic29—41no—
74Dic29—82no—
75#563D2911yescharacteristic, complemented
76#458—292no—
77#455—291no—
78#354—291no—
79#352—292no—
80C2×C29—21no—
81C2×C29—42no—
82C2×C29B(4,𝔽2)11yescharacteristic
83D29—81no—
84H(5,𝔽2)—291no—
85#137—581no—
86C4⋊Q8—21no—
87#126—22no—
88#125—21no—
89#124C2×D2911yescharacteristic
90#123—582no—
91#123C2×D2912yes—
92C22⋊Q8—22no—
93#104—581no—
94#100—291no—
95#99—292no—
96#99—582no—
97C29#220111yescharacteristic, complemented, Sylow
98#41—581no—
99#41—1161no—
100C2×Q8—21no—
101C2×Q8C22⋊D2911yescharacteristic
102C2×D4—582no—
103C2×D4—1161no—
104C2×D4C22×D2911yescharacteristic
105C22×C4C22⋊D2912yes—
106Q16—1161no—
107SD16—1161no—
108M16—581no—
109C4⋊C4—21no—
110C4⋊C4—22no—
111C4⋊C4—22no—
112C4⋊C4—21no—
113C4⋊C4—22no—
114C4⋊C4—1162no—
115C22⋊C4—22no—
116C22⋊C4—22no—
117C22⋊C4—582no—
118C22⋊C4—1162no—
119C22⋊C4C22⋊D2912yes—
120C42—21no—
121C42C22⋊D2911yescharacteristic
122C23—582no—
123C23C2×C22⋊D2912yes—
124Q8—41no—
125Q8—81no—
126Q8—581no—
127D4—41no—
128D4—581no—
129D4—1162no—
130D4—1161no—
131C2×C4—21no—
132C2×C4—22no—
133C2×C4—21no—
134C2×C4—22no—
135C2×C4—42no—
136C2×C4—582no—
137C2×C4—1161no—
138C2×C4—1162no—
139C2×C4C2×C22⋊D2911yescharacteristic
140C8—1161no—
141C22—22no—
142C22—42no—
143C22—1161no—
144C22—2322no—
145C22#1174531511yescharacteristic
146C4—21no—
147C4—42no—
148C4—41no—
149C4—42no—
150C4—41no—
151C4—1161no—
152C4—2322no—
153C2—21no—
154C2—42no—
155C2—2321no—
156C2#1427135011yescharacteristic
subgroupG/H
centerC2#14271350
commutator, cosocleC2×D4×C29C23
FrattiniC2×D4C22×D29
FittingC29×#563C2
socleC2×C29B(4,𝔽2)
prime radical#11745359C22
hypercenter#563D29
hypocenterC29#2201
LCSG ▷ C2×D4×C29 ▷ C22×C29 ▷ C2×C29 ▷ C29
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #563

Complements

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

NH ≅ G/Ntype
3. #14270958C2semidirect
4. #14270953C2semidirect
5. C29×#563C2semidirect
75. #563D29semidirect
97. C29#2201semidirect

Maximal subgroups

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

subgroupindex
#142713522
#142713402
#142709582
#142709532
C29×#5632
#220129