Finite Group Databasebeta

← #16821515

Subgroups — #16821515

Order 3712.

216 conjugacy classes of subgroups; only the 41 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 41 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#14271913C211yesmaximal, characteristic, complemented
2#14271348C212yesmaximal, complemented
3#14271335C211yesmaximal, characteristic
4#14270961C212yesmaximal, complemented
5C29×#355C211yesmaximal, characteristic, complemented
6#11745362C2211yescharacteristic
7C29×#124C2211yescharacteristic
8#11745306C2212yes—
9#11745297—21no—
10#11745294—21no—
11#11745271—22no—
12#11745241—22no—
13#11745240—22no—
14#11745232—21no—
15C29×#101C2212yes—
16#11745184C2211yescharacteristic
17#11745173—22no—
18#11745153—21no—
19C2×Q8×C29C2311yescharacteristic
20C2×D4×C29D411yescharacteristic
21C22×Dic29—21no—
22#90809—41no—
23Q8×D29—41no—
24#90805—21no—
25#90805—22no—
26C2×Dic58D412yes—
27M16×C29—22no—
28C22⋊C4×C29—22no—
29C42×C29D411yescharacteristic
30C22⋊Dic29—22no—
31C22⋊Dic29—21no—
32#90783—41no—
33#90782—41no—
34C4⋊Dic29—21no—
35C4⋊Dic29—21no—
36#90779—22no—
37#90779—22no—
38#90779—22no—
39C4×Dic29D412yes—
40#90777—21no—
41Dic116—42no—
42#90773—42no—
43C23×C29—21no—
44Q8×C29—41no—
45D4×C29—41no—
46C2×C4×C29—21no—
47C2×C4×C29C2×D411yescharacteristic
48C2×C4×C29C2×D412yes—
49C22⋊D29—41no—
50C2×Dic29—22no—
51C2×Dic29—22no—
52C2×Dic29—81no—
53D116—22no—
54D116—21no—
55C4×D29—41no—
56C4×D29—42no—
57Dic58—22no—
58Dic58—21no—
59Dic58—42no—
60C8×C29—42no—
61#6597—41no—
62#2203—291nomaximal, Sylow
63C22×C29—42no—
64C22×C29—41no—
65C22×C29#12011yescharacteristic
66C2×D29—41no—
67C4×C29—21no—
68C4×C29—22no—
69C4×C29—41no—
70Dic29—42no—
71Dic29—41no—
72Dic29—42no—
73#563—291no—
74#456—292no—
75#454—291no—
76#356—292no—
77#355D2911yescharacteristic, complemented
78C2×C29—21no—
79C2×C29—81no—
80C2×C29B(4,𝔽2)11yescharacteristic
81D29—81no—
822−1+4—291no—
83#137—582no—
84AGL(1,ℤ/8)—581no—
85C4⋊Q8—292no—
86#126—582no—
87#125—581no—
88#124C2×D2911yescharacteristic
89#123—581no—
90C22⋊Q8—582no—
91#104—582no—
92#104—581no—
93#101—291no—
94#101C2×D2912yes—
95C29#220311yescharacteristic, complemented, Sylow
96#41—581no—
97#41—582no—
98#41—1161no—
99C2×Q8—292no—
100C2×Q8—1161no—
101C2×Q8C22×D2911yescharacteristic
102C2×D4D11611yescharacteristic
103C22×C4—581no—
104Q16—1162no—
105SD16—1162no—
106SD16—1161no—
107D8—1161no—
108M16—22no—
109M16—581no—
110C4⋊C4—581no—
111C4⋊C4—582no—
112C4⋊C4—581no—
113C4⋊C4—582no—
114C4⋊C4—582no—
115C22⋊C4—22no—
116C22⋊C4—582no—
117C22⋊C4—581no—
118C42—292no—
119C42D11611yescharacteristic
120C23—21no—
121Q8—41no—
122Q8—581no—
123Q8—582no—
124Q8—1162no—
125D4—41no—
126D4—581no—
127D4—582no—
128D4—1161no—
129C2×C4—21no—
130C2×C4—582no—
131C2×C4—582no—
132C2×C4—1162no—
133C2×C4—1161no—
134C2×C4—2321no—
135C2×C4C2×D11611yescharacteristic
136C2×C4D4×D2912yes—
137C8—42no—
138C8—1161no—
139C22—42no—
140C22—41no—
141C22—1161no—
142C22#1174524411yescharacteristic
143C4—22no—
144C4—21no—
145C4—41no—
146C4—1162no—
147C4—1162no—
148C4—1161no—
149C2—21no—
150C2—81no—
151C2—2321no—
152C2#1427135111yescharacteristic
subgroupG/H
centerC2#14271351
commutator, cosocleC2×Q8×C29C23
FrattiniC2×Q8C22×D29
FittingC29×#355C2
socleC2×C29B(4,𝔽2)
prime radical#14271335C2
hypercenter#355D29
hypocenterC29#2203
LCSG ▷ C2×Q8×C29 ▷ C22×C29 ▷ C2×C29 ▷ C29
UCSC1 ◁ C2 ◁ C22 ◁ C2×Q8 ◁ #355

Complements

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

NH ≅ G/Ntype
1. #14271913C2semidirect
2. #14271348C2semidirect
4. #14270961C2semidirect
5. C29×#355C2semidirect
77. #355D29semidirect
95. C29#2203semidirect

Maximal subgroups

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

subgroupindex
#142719132
#142713482
#142713352
#142709612
C29×#3552
#220329