Finite Group Databasebeta

← #16818790

Subgroups — #16818790

Order 3712.

206 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 206 conjugacy classes of subgroups — too many to draw. Only its 49 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#14271553C211yesmaximal, characteristic
2#14271529C212yesmaximal, complemented
3#14270905C212yesmaximal, complemented
4#14270903C211yesmaximal, characteristic, complemented
5C29×#563C211yesmaximal, characteristic, complemented
6C29×C4⋊Q8—21no—
7C29×#126—22no—
8C29×#125C2212yes—
9C29×#124C2211yescharacteristic
10C29×#123—21no—
11C29×C22⋊Q8—22no—
12C2×#90784—21no—
13C2×#90782—21no—
14#11745236C2211yescharacteristic
15#11745157—22no—
16#11745157—22no—
17#11745151C2212yes—
18#11745151C2211yescharacteristic
19C2×Q8×C29—21no—
20C2×Q8×C29D411yescharacteristic
21C2×D4×C29D411yescharacteristic
22C22×C4×C29—21no—
23C2×D116—21no—
24C2×D116—22no—
25C4⋊C4×C29—22no—
26C4⋊C4×C29—22no—
27C4⋊C4×C29—22no—
28C4⋊C4×C29D412yes—
29C4⋊C4×C29D412yes—
30C22⋊C4×C29—22no—
31C22⋊C4×C29—21no—
32C22⋊C4×C29—22no—
33C42×C29—21no—
34C42×C29C2311yescharacteristic
35#90784—42no—
36#90782—42no—
37C2×#6597—21no—
38C2×#6597—22no—
39C23×C29—21no—
40C22×D29—41no—
41Q8×C29—41no—
42Q8×C29—81no—
43D4×C29—41no—
44C2×C4×C29—22no—
45C2×C4×C29—21no—
46C2×C4×C29—22no—
47C2×C4×C29—81no—
48C2×C4×C29C2×D412yes—
49C2×C4×C29C2×D411yescharacteristic
50D116—41no—
51D116—41no—
52D116—44no—
53#6597—22no—
54#6597—42no—
55#1657—291nomaximal, Sylow
56C22×C29—41no—
57C22×C29—42no—
58C22×C29#12011yescharacteristic
59C2×D29—81no—
60C2×D29—82no—
61C4×C29—21no—
62C4×C29—22no—
63C4×C29—42no—
64C4×C29—41no—
65C4×C29—42no—
66#563D2911yescharacteristic, complemented
67#496—291no—
68#490—292no—
69#330—292no—
70#329—291no—
71C2×C29—81no—
72C2×C29#45311yescharacteristic
73C2×C29#45412yes—
74D29—161no—
75C2×SD16—581no—
76C2×D8—581no—
77C4⋊Q8—21no—
78#127—291no—
79#126—22no—
80#125C2×D2912yes—
81#124C2×D2911yescharacteristic
82#123—21no—
83C22⋊Q8—22no—
84#102—582no—
85#102—582no—
86#97—292no—
87#97—291no—
88C29#165711yescharacteristic, complemented, Sylow
89C2×Q8—21no—
90C2×Q8C22⋊D2911yescharacteristic
91C2×D4—582no—
92C2×D4—581no—
93C2×D4C22⋊D2911yescharacteristic
94C22×C4—21no—
95SD16—1162no—
96D8—1162no—
97C2×C8—581no—
98C2×C8—582no—
99C4⋊C4—22no—
100C4⋊C4—22no—
101C4⋊C4—22no—
102C4⋊C4C22⋊D2912yes—
103C4⋊C4C22⋊D2912yes—
104C22⋊C4—22no—
105C22⋊C4—21no—
106C22⋊C4—22no—
107C42—21no—
108C42C22×D2911yescharacteristic
109C23—21no—
110C23—1161no—
111Q8—41no—
112Q8—81no—
113D4—41no—
114D4—1161no—
115D4—1161no—
116D4—1164no—
117C2×C4—22no—
118C2×C4—21no—
119C2×C4—22no—
120C2×C4—81no—
121C2×C4C2×C22⋊D2912yes—
122C2×C4C2×C22⋊D2911yescharacteristic
123C8—582no—
124C8—1162no—
125C22—41no—
126C22—42no—
127C22—2321no—
128C22—2322no—
129C22#1174531511yescharacteristic
130C4—21no—
131C4—22no—
132C4—42no—
133C4—41no—
134C4—42no—
135C2—81no—
136C2—4641no—
137C2#1427133111yescharacteristic
138C2#1427133612yes—
subgroupG/H
centerC22#11745315
commutator, cosocleC42×C29C23
FrattiniC42C22×D29
FittingC29×#563C2
socleC22×C29#120
prime radical#14271553C2
hypercenter#563D29
hypocenterC29#1657
LCSG ▷ C42×C29 ▷ C22×C29 ▷ C29
UCSC1 ◁ C22 ◁ C42 ◁ #563

Complements

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

NH ≅ G/Ntype
2. #14271529C2semidirect
3. #14270905C2semidirect
4. #14270903C2semidirect
5. C29×#563C2semidirect
66. #563D29semidirect
88. C29#1657semidirect

Maximal subgroups

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

subgroupindex
#142715532
#142715292
#142709052
#142709032
C29×#5632
#165729