Finite Group Databasebeta

← #31780065

Subgroups — #31780065

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#22578131C211yesmaximal, characteristic
2#22577799C211yesmaximal, characteristic
3#22577610C211yesmaximal, characteristic
4#22577150C211yesmaximal, characteristic
5Q8⋊#11764040C211yesmaximal, characteristic
6#22577133C211yesmaximal, characteristic
7#475×C137C211yesmaximal, characteristic
8C2×#14555500—21no—
9#17984088C2211yescharacteristic
10#17984086—21no—
11#17984080—21no—
12#17984078—22no—
13C4×Dic274—21no—
14C4×Dic274C2211yescharacteristic
15#17984053—21no—
16#17984019—22no—
17C4⋊#11764040C2211yescharacteristic
18#17984012—21no—
19C4×#11764040C2211yescharacteristic
20C4⋊Q8×C137C2211yescharacteristic
21C4×Q8×C137C2211yescharacteristic
22#106×C137—21no—
23C4⋊C8×C137C2211yescharacteristic
24#103×C137—21no—
25C2×Dic274—21no—
26C2×Dic274D411yescharacteristic
27#14555500—42no—
28C2×#11764040—21no—
29C2×#11764040—21no—
30C4⋊Dic137—21no—
31C4⋊Dic137D411yescharacteristic
32C4⋊Dic137D412yes—
33#14555489—22no—
34#14555489—22no—
35#14555489—22no—
36#14555489—21no—
37C4×Dic137—22no—
38C4×Dic137—21no—
39C2×Q8×C137—21no—
40C2×Q8×C137D411yescharacteristic
41C2×C8×C137—21no—
42C4⋊C4×C137—21no—
43C4⋊C4×C137—21no—
44C4⋊C4×C137—21no—
45C4⋊C4×C137D411yescharacteristic
46C42×C137—21no—
47C42×C137C2311yescharacteristic
48Dic274—21no—
49Dic274—41no—
50Dic274—41no—
51Dic274SD1612yes—
52C2×Dic137—21no—
53C2×Dic137—21no—
54C2×Dic137—22no—
55#11764040—22no—
56#11764040—41no—
57Q8×C137—21no—
58Q8×C137—21no—
59Q8×C137—41no—
60Q8×C137—41no—
61C2×C4×C137—21no—
62C2×C4×C137—21no—
63C2×C4×C137—41no—
64C2×C4×C137C2×D411yescharacteristic
65C2×C4×C137C2×D411yescharacteristic
66C2×C4×C137C2×D411yescharacteristic
67C8×C137—41no—
68C22×C137#12011yescharacteristic
69C4×C137—21no—
70C4×C137—41no—
71C4×C137—41no—
72C4×C137—81no—
73C4×C137C2×SD1611yescharacteristic
74C4×C137#13511yescharacteristic
75C4×C137#13711yescharacteristic
76C4×C137#13711yescharacteristic
77Dic137—22no—
78Dic137—42no—
79Dic137—41no—
80C2×C137#44811yescharacteristic
81C2×C137#45211yescharacteristic
82C2×C137#45611yescharacteristic
83C137#165311yescharacteristic, complemented, Sylow
84#1653—1371nomaximal, Sylow
85#557—1371no—
86#498—1371no—
87#475D13711yescharacteristic
88#462—1371no—
89#332—1371no—
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×Q8C2×D13711yescharacteristic
102#106—21no—
103#106—2742no—
104C4⋊C8—1371no—
105C4⋊C8C2×D13711yescharacteristic
106#103—21no—
107#103—2741no—
108C4×C8—1371no—
109C2×Q8—21no—
110C2×Q8—2741no—
111C2×Q8—1371no—
112C2×Q8C22⋊D13711yescharacteristic
113Q16—5482no—
114C2×C8—21no—
115C2×C8—2741no—
116C2×C8—2741no—
117C4⋊C4—21no—
118C4⋊C4—21no—
119C4⋊C4—21no—
120C4⋊C4—2742no—
121C4⋊C4—2742no—
122C4⋊C4—2742no—
123C4⋊C4—2741no—
124C4⋊C4—2741no—
125C4⋊C4—1371no—
126C4⋊C4—1372no—
127C4⋊C4C22⋊D13711yescharacteristic
128C42—21no—
129C42—2742no—
130C42—2741no—
131C42C22×D13711yescharacteristic
132Q8—21no—
133Q8—21no—
134Q8—41no—
135Q8—41no—
136Q8—2741no—
137Q8—5481no—
138Q8—5481no—
139Q8—1372no—
140C2×C4—21no—
141C2×C4—21no—
142C2×C4—41no—
143C2×C4—2741no—
144C2×C4—2741no—
145C2×C4—2742no—
146C2×C4D4×D13711yescharacteristic
147C2×C4D4×D13711yescharacteristic
148C2×C4C2×C22⋊D13711yescharacteristic
149C8—41no—
150C8—2742no—
151C8—5481no—
152C22#1798405611yescharacteristic
153C4—21no—
154C4—41no—
155C4—41no—
156C4—81no—
157C4—2742no—
158C4—5482no—
159C4—5481no—
160C4SD16×D13711yescharacteristic
161C4#1798410811yescharacteristic
162C4#1798411711yescharacteristic
163C4#1798411911yescharacteristic
164C2#2257754011yescharacteristic
165C2#2257756311yescharacteristic
166C2#2257758411yescharacteristic
subgroupG/H
centerC22#17984056
commutator, cosocleC42×C137C23
FrattiniC42C22×D137
Fitting#475×C137C2
socle, prime radicalC22×C137#120
hypercenter#475D137
hypocenterC137#1653
LCSG ▷ C42×C137 ▷ C22×C137 ▷ C137
UCSC1 ◁ C22 ◁ C42 ◁ #475

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
#225781312
#225777992
#225776102
#225771502
Q8⋊#117640402
#225771332
#475×C1372
#1653137