Finite Group Databasebeta

← #17996547

Subgroups — #17996547

Order 4736.

216 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 53 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#14557973C212yesmaximal, complemented
2#14557791C211yesmaximal, characteristic, complemented
3#14557343C212yesmaximal, complemented
4#14557315C211yesmaximal, characteristic, complemented
5C37×#575C211yesmaximal, characteristic
6C2×Dic148C2211yescharacteristic, complemented
7#11924515—21no—
8#11924482—22no—
9#11924480—22no—
10#11924476C2212yescomplemented
11#11924468C2211yescharacteristic, complemented
12#11924461—21no—
13C2×#41×C37—21no—
14#137×C37—22no—
15AGL(1,ℤ/8)×C37—22no—
16#135×C37—21no—
17#135×C37C2212yes—
18C2×M16×C37C2211yescharacteristic
19#41×C37—22no—
20#41×C37—22no—
21#41×C37—41no—
22#41×C37—41no—
23C2×Q8×C37—21no—
24C2×D4×C37—21no—
25C2×D4×C37—41no—
26C22×C4×C37—41no—
27C22×C4×C37D411yescharacteristic
28C2×Dic74—21no—
29Q16×C37—21no—
30Q16×C37D412yes—
31SD16×C37—22no—
32SD16×C37—22no—
33D8×C37—21no—
34D8×C37D412yes—
35M16×C37—22no—
36C2×C8×C37D411yescharacteristic
37C2×C8×C37C2311yescharacteristic
38C22⋊Dic37—41no—
39C4⋊Dic37—21no—
40#10596936—41no—
41Dic148—22no—
42#10596925—22no—
43C23×C37—41no—
44C23×C37D811yescharacteristic
45Q8×C37—22no—
46Q8×C37—41no—
47D4×C37—22no—
48D4×C37—41no—
49D4×C37—42no—
50D4×C37—42no—
51C2×C4×C37—21no—
52C2×C4×C37—21no—
53C2×C4×C37—42no—
54C2×C4×C37—41no—
55C2×C4×C37—42no—
56C2×C4×C37D811yescharacteristic
57C2×C4×C37C2×D411yescharacteristic
58C2×Dic37—41no—
59C2×Dic37—41no—
60Dic74—41no—
61Dic74—41no—
62C8×C37—21no—
63C8×C37C2×D412yes—
64C22×C37—21no—
65C22×C37—41no—
66C22×C37—42no—
67C22×C37—82no—
68C22×C37C2×D811yescharacteristic
69C4×C37—21no—
70C4×C37—42no—
71C4×C37#12011yescharacteristic
72C4×C37AGL(1,ℤ/8)11yescharacteristic
73Dic37—81no—
74Dic37—81no—
75#2191—371nomaximal, Sylow
76C2×C37—21no—
77C2×C37—41no—
78C2×C37—82no—
79C2×C37#44711yescharacteristic
80#575D3711yescharacteristic
81#510—372no—
82#470—371no—
83#360—372no—
84#349—371no—
85C37#219111yescharacteristic, complemented, Sylow
86C2×#41—21no—
87#137—22no—
88AGL(1,ℤ/8)—22no—
89#135—21no—
90#135C2×D3712yes—
91C2×Q16—371no—
92C2×M16C2×D3711yescharacteristic
93C22⋊Q8—741no—
94Q32—742no—
95SD32—742no—
96M32—372no—
97#106—371no—
98#103—741no—
99#41—22no—
100#41—22no—
101#41—41no—
102#41—41no—
103C2×Q8—21no—
104C2×Q8—741no—
105C2×D4—21no—
106C2×D4—41no—
107C22×C4—41no—
108C22×C4C22⋊D3711yescharacteristic
109Q16—21no—
110Q16—742no—
111Q16C22⋊D3712yes—
112SD16—22no—
113SD16—22no—
114D8—21no—
115D8C22⋊D3712yes—
116M16—22no—
117C2×C8C22⋊D3711yescharacteristic
118C2×C8C22×D3711yescharacteristic
119C4⋊C4—741no—
120C4⋊C4—1481no—
121C22⋊C4—1481no—
122C16—742no—
123C23—41no—
124C23#1059693911yescharacteristic
125Q8—22no—
126Q8—41no—
127Q8—1481no—
128Q8—1481no—
129D4—22no—
130D4—41no—
131D4—42no—
132D4—42no—
133C2×C4—21no—
134C2×C4—21no—
135C2×C4—42no—
136C2×C4—41no—
137C2×C4—42no—
138C2×C4—1481no—
139C2×C4—1481no—
140C2×C4#1059693911yescharacteristic
141C2×C4C2×C22⋊D3711yescharacteristic
142C8—21no—
143C8C2×C22⋊D3712yes—
144C22—21no—
145C22—41no—
146C22—42no—
147C22—82no—
148C22C2×#1059693911yescharacteristic
149C4—21no—
150C4—42no—
151C4—2961no—
152C4—2961no—
153C4#1192450511yescharacteristic
154C4#1192456211yescharacteristic
155C2—21no—
156C2—41no—
157C2—82no—
158C2#1455766711yescharacteristic
subgroupG/H
centerC2#14557667
commutator, cosocleC2×C8×C37C23
FrattiniC2×C8C22×D37
Fitting, prime radicalC37×#575C2
socleC2×C37#447
hypercenter#575D37
hypocenterC37#2191
LCSG ▷ C2×C8×C37 ▷ C4×C37 ▷ C2×C37 ▷ C37
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×M16 ◁ #575

Complements

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

NH ≅ G/Ntype
1. #14557973C2semidirect
2. #14557791C2semidirect
3. #14557343C2semidirect
4. #14557315C2semidirect
6. C2×Dic148C22semidirect
10. #11924476C22semidirect
11. #11924468C22semidirect
85. C37#2191semidirect

Maximal subgroups

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

subgroupindex
#145579732
#145577912
#145573432
#145573152
C37×#5752
#219137