Finite Group Databasebeta

← #17996625

Subgroups — #17996625

Order 4736.

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

normal subgroups only

This group has 224 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#14558263C211yesmaximal, characteristic, complemented
2#14557716C212yesmaximal
3#14557332C211yesmaximal, characteristic, complemented
4#14557331C212yesmaximal, complemented
5C37×#456C211yesmaximal, characteristic
6#11924566—22no—
7#11924535—22no—
8#11924534—21no—
9#11924534C2212yes—
10C4×Dic74—21no—
11#11924501—22no—
12#11924463—22no—
13#11924455C2212yes—
142−1+4×C37C2211yescharacteristic
15#137×C37—21no—
16C4⋊Q8×C37C2211yescharacteristic
17#104×C37—21no—
18#101×C37C2211yescharacteristic
19#41×C37—22no—
20#41×C37—21no—
21#41×C37—41no—
22C2×Q8×C37—41no—
23C2×Q8×C37D411yescharacteristic
24C2×Q8×C37C2311yescharacteristic
25C2×Dic74—21no—
26C2×Dic74—21no—
27C2×Dic74D412yes—
28Q16×C37—41no—
29SD16×C37—41no—
30M16×C37—21no—
31C4⋊C4×C37—22no—
32C42×C37D411yescharacteristic
33#10596942—42no—
34#10596940—42no—
35C4⋊Dic37—21no—
36#10596936—22no—
37#10596936—22no—
38#10596936—22no—
39C4×Dic37—21no—
40C4×Dic37—22no—
41C4×Dic37D412yes—
42#10596934—22no—
43Q8×C37—22no—
44Q8×C37—21no—
45Q8×C37—41no—
46Q8×C37—41no—
47D4×C37—22no—
48D4×C37—21no—
49D4×C37—41no—
50C2×C4×C37—21no—
51C2×C4×C37—21no—
52C2×C4×C37—42no—
53C2×C4×C37—41no—
54C2×C4×C37C2×D412yes—
55C2×C4×C37C2×D411yescharacteristic
56C2×Dic37—22no—
57C2×Dic37—22no—
58Dic74—24no—
59Dic74—42no—
60Dic74—81no—
61C8×C37—41no—
62#64436—42no—
63C22×C37—41no—
64C22×C37#12011yescharacteristic
65C4×C37—22no—
66C4×C37—22no—
67C4×C37—21no—
68C4×C37—41no—
69C4×C37—41no—
70Dic37—24no—
71Dic37—42no—
72#2205—371nomaximal, Sylow
73C2×C37—21no—
74C2×C37—81no—
75C2×C37B(4,𝔽2)11yescharacteristic
76Q82—371no—
77#456—372no—
78#456D3711yescharacteristic
79#356—371no—
80#356—372no—
81C37#220511yescharacteristic, complemented, Sylow
822−1+4C2×D3711yescharacteristic
83#137—21no—
84#137—742no—
85C4⋊Q8—372no—
86C4⋊Q8—742no—
87C4⋊Q8—741no—
88C4⋊Q8C2×D3711yescharacteristic
89C4×Q8—742no—
90C4×Q8—741no—
91#104—21no—
92#104—742no—
93#101—372no—
94#101C2×D3711yescharacteristic
95#41—22no—
96#41—21no—
97#41—41no—
98C2×Q8—41no—
99C2×Q8—372no—
100C2×Q8—741no—
101C2×Q8—741no—
102C2×Q8C22⋊D3711yescharacteristic
103C2×Q8C22×D3711yescharacteristic
104Q16—41no—
105Q16—1482no—
106SD16—41no—
107SD16—1482no—
108M16—21no—
109M16—742no—
110C4⋊C4—22no—
111C4⋊C4—742no—
112C4⋊C4—741no—
113C4⋊C4—742no—
114C4⋊C4—742no—
115C42—372no—
116C42—742no—
117C42—741no—
118C42C22⋊D3711yescharacteristic
119Q8—21no—
120Q8—22no—
121Q8—41no—
122Q8—41no—
123Q8—2961no—
124Q8—744no—
125Q8—1482no—
126D4—21no—
127D4—22no—
128D4—41no—
129C2×C4—21no—
130C2×C4—21no—
131C2×C4—41no—
132C2×C4—42no—
133C2×C4—742no—
134C2×C4—742no—
135C2×C4D4×D3712yes—
136C2×C4C2×C22⋊D3711yescharacteristic
137C8—41no—
138C8—1482no—
139C22—41no—
140C22#1192450411yescharacteristic
141C4—22no—
142C4—21no—
143C4—22no—
144C4—41no—
145C4—41no—
146C4—744no—
147C4—1482no—
148C2—21no—
149C2—81no—
150C2#1455771911yescharacteristic
subgroupG/H
centerC2#14557719
commutator, cosocleC2×Q8×C37C23
FrattiniC2×Q8C22×D37
FittingC37×#456C2
socleC2×C37B(4,𝔽2)
prime radical2−1+4×C37C22
hypercenter#456D37
hypocenterC37#2205
LCSG ▷ C2×Q8×C37 ▷ C22×C37 ▷ C2×C37 ▷ C37
UCSC1 ◁ C2 ◁ C22 ◁ C2×Q8 ◁ #456

Complements

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

NH ≅ G/Ntype
1. #14558263C2semidirect
3. #14557332C2semidirect
4. #14557331C2semidirect
81. C37#2205semidirect

Maximal subgroups

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

subgroupindex
#145582632
#145577162
#145573322
#145573312
C37×#4562
#220537