Finite Group Databasebeta

← #31780067

Subgroups — #31780067

Order 17536.

194 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 194 conjugacy classes of subgroups — too many to draw. Only its 55 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#22578128C211yesmaximal, characteristic
2#22577799C211yesmaximal, characteristic
3#22577689C211yesmaximal, characteristic
4#22577614C211yesmaximal, characteristic
5#22577133C211yesmaximal, characteristic
6#22577133C211yesmaximal, characteristic
7#332×C137C211yesmaximal, characteristic
8C2×Dic548—21no—
9#17984088C2211yescharacteristic
10#17984086—21no—
11#17984079—21no—
12#17984078—22no—
13Q8×Dic137—21no—
14C4×Dic274C2211yescharacteristic
15C4×Dic274C2211yescharacteristic
16#17984053—21no—
17#17984020—21no—
18#17984019—22no—
19#17984013—21no—
20#17984013—21no—
21C4×#11764040C2211yescharacteristic
22C4⋊Q8×C137C2211yescharacteristic
23C4⋊C8×C137C2211yescharacteristic
24C4⋊C8×C137C2211yescharacteristic
25C2×Dic274—21no—
26C2×Dic274D411yescharacteristic
27C2×Dic274D411yescharacteristic
28Dic548—42no—
29C2×#11764040—21no—
30C4⋊Dic137—22no—
31C4⋊Dic137—21no—
32C4⋊Dic137—21no—
33C4⋊Dic137D411yescharacteristic
34C4⋊Dic137D411yescharacteristic
35#14555489—22no—
36#14555489—22no—
37#14555489—21no—
38#14555489—21no—
39C4×Dic137—22no—
40C4×Dic137—21no—
41C4×Dic137—21no—
42C2×Q8×C137—21no—
43C2×C8×C137—21no—
44C2×C8×C137—21no—
45C4⋊C4×C137—21no—
46C4⋊C4×C137D412yes—
47C42×C137C2311yescharacteristic
48Dic274—21no—
49Dic274—21no—
50Dic274—21no—
51Dic274—41no—
52Dic274—41no—
53Dic274SD1612yes—
54C2×Dic137—21no—
55C2×Dic137—21no—
56C2×Dic137—21no—
57C2×Dic137—21no—
58C2×Dic137—41no—
59#11764040—22no—
60Q8×C137—41no—
61Q8×C137—41no—
62C2×C4×C137—22no—
63C2×C4×C137C2×D411yescharacteristic
64C2×C4×C137C2×D411yescharacteristic
65C2×C4×C137C2×D411yescharacteristic
66C8×C137—41no—
67C8×C137—41no—
68C22×C137#12011yescharacteristic
69C4×C137—21no—
70C4×C137—42no—
71C4×C137C2×SD1611yescharacteristic
72C4×C137#13511yescharacteristic
73C4×C137#13711yescharacteristic
74C4×C137#13711yescharacteristic
75Dic137—22no—
76Dic137—41no—
77Dic137—41no—
78Dic137—41no—
79Dic137—81no—
80C2×C137#44811yescharacteristic
81C2×C137#45211yescharacteristic
82C2×C137#45611yescharacteristic
83C137#165311yescharacteristic, complemented, Sylow
84#1653—1371nomaximal, Sylow
85#557—1371no—
86#498—1371no—
87#475—1371no—
88#462—1371no—
89#332D13711yescharacteristic
90Q8⋊C8—1371no—
91Q8⋊C8—1371no—
92C2×Q16—2741no—
93C4⋊Q8—2741no—
94C4⋊Q8—1371no—
95C4⋊Q8C2×D13711yescharacteristic
96#125—2741no—
97#125—2742no—
98C4×Q8—2741no—
99C4×Q8—2741no—
100C4×Q8—1371no—
101C4×Q8—1371no—
102#106—2741no—
103#106—2742no—
104C4⋊C8C2×D13711yescharacteristic
105C4⋊C8C2×D13711yescharacteristic
106#103—2741no—
107#103—2741no—
108C4×C8—1371no—
109C2×Q8—21no—
110C2×Q8—2741no—
111C2×Q8—1371no—
112C2×Q8—1371no—
113Q16—5482no—
114C2×C8—21no—
115C2×C8—21no—
116C2×C8—2741no—
117C4⋊C4—21no—
118C4⋊C4—2742no—
119C4⋊C4—2742no—
120C4⋊C4—2742no—
121C4⋊C4—2741no—
122C4⋊C4—2741no—
123C4⋊C4—2741no—
124C4⋊C4—2741no—
125C4⋊C4—1371no—
126C4⋊C4—1371no—
127C4⋊C4D54812yes—
128C42—2742no—
129C42—2741no—
130C42—2741no—
131C42C22×D13711yescharacteristic
132Q8—41no—
133Q8—41no—
134Q8—2741no—
135Q8—2741no—
136Q8—2741no—
137Q8—5481no—
138Q8—5481no—
139Q8—1372no—
140C2×C4—22no—
141C2×C4—2741no—
142C2×C4—2741no—
143C2×C4—2741no—
144C2×C4—2741no—
145C2×C4—5481no—
146C2×C4D4×D13711yescharacteristic
147C2×C4D4×D13711yescharacteristic
148C2×C4C2×D54811yescharacteristic
149C8—41no—
150C8—41no—
151C8—2742no—
152C22#1798405811yescharacteristic
153C4—21no—
154C4—42no—
155C4—2742no—
156C4—5481no—
157C4—5481no—
158C4—5481no—
159C4—10961no—
160C4SD16×D13711yescharacteristic
161C4#1798410711yescharacteristic
162C4#1798411711yescharacteristic
163C4#1798412011yescharacteristic
164C2#2257754311yescharacteristic
165C2#2257756611yescharacteristic
166C2#2257758811yescharacteristic
subgroupG/H
centerC22#17984058
commutator, cosocleC42×C137C23
FrattiniC42C22×D137
Fitting#332×C137C2
socle, prime radicalC22×C137#120
hypercenter#332D137
hypocenterC137#1653
LCSG ▷ C42×C137 ▷ C22×C137 ▷ C137
UCSC1 ◁ C22 ◁ C42 ◁ #332

Complements

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

NH ≅ G/Ntype
83. C137#1653semidirect

Maximal subgroups

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

subgroupindex
#225781282
#225777992
#225776892
#225776142
#225771332
#332×C1372
#1653137