Finite Group Databasebeta

← #31779896

Subgroups — #31779896

Order 17536.

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#22578127C211yesmaximal, characteristic, complemented
2#22577600C211yesmaximal, characteristic, complemented
3#22577595C211yesmaximal, characteristic
4#22577149C211yesmaximal, characteristic, complemented
5#22577135C211yesmaximal, characteristic, complemented
6#22577132C211yesmaximal, characteristic, complemented
7#486×C137C211yesmaximal, characteristic, complemented
8C2×#14555498—21no—
9C2×#14555495—21no—
10#17984083—22no—
11#17984077—21no—
12#17984068—21no—
13Q8×Dic137—21no—
14C4×Dic274C2211yescharacteristic
15C4×D548C2211yescharacteristic
16#17984046—21no—
17C4⋊#11764040C2211yescharacteristic
18C4⋊#11764040C2211yescharacteristic
19#17984009—21no—
20#17984009—21no—
21C4⋊Q8×C137C2211yescharacteristic
22#127×C137C2211yescharacteristic
23#102×C137—22no—
24C4×C8×C137C2211yescharacteristic
25C2×Dic274D411yescharacteristic
26C2×D548D411yescharacteristic
27C2×C4×D137—21no—
28#14555498—42no—
29#14555495—42no—
30C2×#11764040—21no—
31C2×#11764040—21no—
32C4⋊Dic137—22no—
33C4⋊Dic137D411yescharacteristic
34C4⋊Dic137D411yescharacteristic
35#14555489—21no—
36#14555487—22no—
37#14555487—22no—
38#14555487—21no—
39C4×Dic137—22no—
40C4×Dic137—21no—
41C2×Q8×C137—21no—
42C2×D4×C137—21no—
43C2×D4×C137—21no—
44C2×D4×C137—21no—
45C2×C8×C137—21no—
46C4⋊C4×C137—21no—
47C4⋊C4×C137D412yes—
48C42×C137C2311yescharacteristic
49C22×D137—21no—
50Dic274—21no—
51Dic274—21no—
52D548—21no—
53D548—21no—
54C2×Dic137—21no—
55C2×Dic137—21no—
56C2×Dic137—21no—
57C4×D137—41no—
58C4×D137—41no—
59#11764040—41no—
60#11764040—41no—
61C23×C137—41no—
62Q8×C137—41no—
63Q8×C137—41no—
64D4×C137—42no—
65D4×C137—41no—
66D4×C137—41no—
67D4×C137—41no—
68D4×C137—41no—
69C2×C4×C137—22no—
70C2×C4×C137C2×D411yescharacteristic
71C2×C4×C137C2×D411yescharacteristic
72C2×C4×C137C2×D411yescharacteristic
73C8×C137—41no—
74C22×C137—81no—
75C22×C137—81no—
76C22×C137—81no—
77C22×C137#12011yescharacteristic
78C2×D137—41no—
79C2×D137—41no—
80C2×D137—41no—
81C4×C137—21no—
82C4×C137—42no—
83C4×C137#13511yescharacteristic
84C4×C137#13511yescharacteristic
85C4×C137AGL(1,ℤ/8)11yescharacteristic
86C4×C137AGL(1,ℤ/8)11yescharacteristic
87Dic137—41no—
88Dic137—41no—
89Dic137—41no—
90C2×C137—161no—
91C2×C137#44911yescharacteristic
92C2×C137#44911yescharacteristic
93C2×C137#45411yescharacteristic
94D137—81no—
95C137#162811yescharacteristic, complemented, Sylow
96#1628—1371nomaximal, Sylow
97#556—1371no—
98#486D13711yescharacteristic, complemented
99#460—1371no—
100#459—1371no—
101#332—1371no—
102Q8⋊C8—1371no—
103D4⋊C8—1371no—
104C2×SD16—2741no—
105C2×D8—2741no—
106C4⋊Q8C2×D13711yescharacteristic
107#127C2×D13711yescharacteristic
108#126—2742no—
109#124—2741no—
110C22⋊Q8—2741no—
111C4×Q8—2741no—
112C4×Q8—1371no—
113C4×D4—1371no—
114#117—2741no—
115C4⋊C8—1371no—
116C4⋊C8—1371no—
117#102—22no—
118#102—2741no—
119#102—2741no—
120C4×C8C2×D13711yescharacteristic
121C2×Q8—21no—
122C2×Q8—1371no—
123C2×D4—21no—
124C2×D4—21no—
125C2×D4—21no—
126C2×D4—1371no—
127C22×C4—2741no—
128SD16—5482no—
129D8—5482no—
130C2×C8—21no—
131C2×C8—2741no—
132C2×C8—2741no—
133C4⋊C4—21no—
134C4⋊C4—2742no—
135C4⋊C4—2741no—
136C4⋊C4—1371no—
137C4⋊C4—1371no—
138C4⋊C4C22⋊D13712yes—
139C22⋊C4—2742no—
140C22⋊C4—2742no—
141C22⋊C4—2741no—
142C42—2742no—
143C42—2741no—
144C42C22×D13711yescharacteristic
145C23—41no—
146C23—2741no—
147Q8—41no—
148Q8—41no—
149Q8—2741no—
150Q8—2741no—
151D4—42no—
152D4—41no—
153D4—41no—
154D4—41no—
155D4—41no—
156D4—2741no—
157D4—2741no—
158C2×C4—22no—
159C2×C4—2741no—
160C2×C4—2741no—
161C2×C4—2741no—
162C2×C4—5481no—
163C2×C4—5481no—
164C2×C4D4×D13711yescharacteristic
165C2×C4D4×D13711yescharacteristic
166C2×C4C2×C22⋊D13711yescharacteristic
167C8—41no—
168C8—5481no—
169C8—5481no—
170C22—81no—
171C22—81no—
172C22—81no—
173C22—5481no—
174C22—5481no—
175C22—5481no—
176C22#1798405611yescharacteristic
177C4—21no—
178C4—42no—
179C4—5481no—
180C4—5481no—
181C4—5481no—
182C4#1798410711yescharacteristic
183C4#1798410911yescharacteristic
184C4#1798411211yescharacteristic
185C4#1798411511yescharacteristic
186C2—161no—
187C2—10961no—
188C2#2257754511yescharacteristic
189C2#2257754711yescharacteristic
190C2#2257757511yescharacteristic
subgroupG/H
centerC22#17984056
commutator, cosocleC42×C137C23
FrattiniC42C22×D137
Fitting#486×C137C2
socleC22×C137#120
prime radical#22577595C2
hypercenter#486D137
hypocenterC137#1628
LCSG ▷ C42×C137 ▷ C22×C137 ▷ C137
UCSC1 ◁ C22 ◁ C42 ◁ #486

Complements

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

NH ≅ G/Ntype
1. #22578127C2semidirect
2. #22577600C2semidirect
4. #22577149C2semidirect
5. #22577135C2semidirect
6. #22577132C2semidirect
7. #486×C137C2semidirect
95. C137#1628semidirect
98. #486D137semidirect

Maximal subgroups

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

subgroupindex
#225781272
#225776002
#225775952
#225771492
#225771352
#225771322
#486×C1372
#1628137