Finite Group Databasebeta

← #31779986

Subgroups — #31779986

Order 17536.

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

normal subgroups only

This group has 250 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#22578073C211yesmaximal, characteristic
2#22577685C211yesmaximal, characteristic, complemented
3#22577605C211yesmaximal, characteristic, complemented
4#22577147C211yesmaximal, characteristic, complemented
5#22577135C211yesmaximal, characteristic, complemented
6#22577132C211yesmaximal, characteristic, complemented
7#486×C137C211yesmaximal, characteristic, complemented
8C2×#14555509—21no—
9C2×#14555497—21no—
10#17984084—21no—
11#17984060—22no—
12#17984059—21no—
13C4×Dic274C2211yescharacteristic
14C4×D548C2211yescharacteristic
15D4×Dic137—21no—
16#17984019—21no—
17C4⋊#11764040C2211yescharacteristic
18C4⋊#11764040C2211yescharacteristic
19#17984012—21no—
20#17984011—21no—
21C4⋊Q8×C137C2211yescharacteristic
22#127×C137C2211yescharacteristic
23#102×C137—22no—
24C4×C8×C137C2211yescharacteristic
25#14555509—42no—
26#14555509—42no—
27C2×Dic274D411yescharacteristic
28C2×D548D411yescharacteristic
29C2×C22⋊D137—22no—
30C2×C22⋊D137—22no—
31C22×Dic137—22no—
32C2×C4×D137—21no—
33#14555497—42no—
34C2×#11764040—21no—
35C2×#11764040—21no—
36C4⋊Dic137D411yescharacteristic
37C4⋊Dic137D411yescharacteristic
38#14555489—21no—
39C22⋊Dic137—22no—
40#14555487—21no—
41C4×Dic137—21no—
42C2×Q8×C137—21no—
43C2×D4×C137—21no—
44C2×D4×C137—21no—
45C2×D4×C137D412yes—
46C2×C8×C137—21no—
47C4⋊C4×C137—21no—
48C4⋊C4×C137—21no—
49C42×C137C2311yescharacteristic
50C22×D137—21no—
51Dic274—21no—
52Dic274SD1612yescomplemented
53D548—21no—
54D548—21no—
55C22⋊D137—44no—
56C22⋊D137—82no—
57C2×Dic137—21no—
58C2×Dic137—21no—
59C2×Dic137—21no—
60C2×Dic137—44no—
61C4×D137—41no—
62C4×D137—41no—
63#11764040—41no—
64#11764040—41no—
65C23×C137—22no—
66Q8×C137—41no—
67Q8×C137—41no—
68D4×C137—42no—
69D4×C137—41no—
70D4×C137—41no—
71D4×C137—41no—
72D4×C137—41no—
73C2×C4×C137—41no—
74C2×C4×C137C2×D411yescharacteristic
75C2×C4×C137C2×D411yescharacteristic
76C2×C4×C137C2×D411yescharacteristic
77C8×C137—41no—
78C22×C137—42no—
79C22×C137—42no—
80C22×C137—42no—
81C22×C137#12011yescharacteristic
82C2×D137—41no—
83C2×D137—41no—
84C2×D137—41no—
85C4×C137—21no—
86C4×C137—81no—
87C4×C137C2×SD1611yescharacteristic
88C4×C137#13511yescharacteristic
89C4×C137AGL(1,ℤ/8)11yescharacteristic
90C4×C137AGL(1,ℤ/8)11yescharacteristic
91Dic137—22no—
92Dic137—41no—
93Dic137—41no—
94C2×C137—82no—
95C2×C137#44911yescharacteristic
96C2×C137#45011yescharacteristic
97C2×C137#45511yescharacteristic
98D137—81no—
99C137#164211yescharacteristic, complemented, Sylow
100#1642—1371nomaximal, Sylow
101#550—1371no—
102#486D13711yescharacteristic, complemented
103#475—1371no—
104#461—1371no—
105#331—1371no—
106Q8⋊C8—1371no—
107D4⋊C8—1371no—
108C2×#41—2741no—
109C2×SD16—2741no—
110C4⋊Q8C2×D13711yescharacteristic
111#127—2741no—
112#127C2×D13711yescharacteristic
113#121—2742no—
114#121—2741no—
115C4×Q8—1371no—
116C4×D4—2741no—
117C4×D4—1371no—
118#106—2741no—
119C4⋊C8—1371no—
120C4⋊C8—1371no—
121#103—2741no—
122#103—2741no—
123#102—22no—
124C4×C8C2×D13711yescharacteristic
125#41—5482no—
126#41—5482no—
127C2×Q8—21no—
128C2×Q8—1371no—
129C2×D4—21no—
130C2×D4—21no—
131C2×D4—2742no—
132C2×D4—2742no—
133C2×D4—1371no—
134C2×D4C22⋊D13712yes—
135C22×C4—2742no—
136C22×C4—2741no—
137SD16—5482no—
138C2×C8—21no—
139C2×C8—2741no—
140C2×C8—2741no—
141C4⋊C4—21no—
142C4⋊C4—21no—
143C4⋊C4—2741no—
144C4⋊C4—1371no—
145C4⋊C4—1371no—
146C22⋊C4—2742no—
147C22⋊C4—2741no—
148C42—2741no—
149C42C22×D13711yescharacteristic
150C23—22no—
151C23—2741no—
152Q8—41no—
153Q8—41no—
154Q8—2741no—
155Q8—1372no—
156D4—42no—
157D4—41no—
158D4—41no—
159D4—41no—
160D4—41no—
161D4—2741no—
162D4—2741no—
163D4—5484no—
164D4—10962no—
165C2×C4—41no—
166C2×C4—2741no—
167C2×C4—2741no—
168C2×C4—2741no—
169C2×C4—5484no—
170C2×C4—5481no—
171C2×C4—5481no—
172C2×C4D4×D13711yescharacteristic
173C2×C4D4×D13711yescharacteristic
174C2×C4C2×C22⋊D13711yescharacteristic
175C8—41no—
176C8—5481no—
177C8—5481no—
178C22—42no—
179C22—42no—
180C22—42no—
181C22—5481no—
182C22—5481no—
183C22—5481no—
184C22#1798405611yescharacteristic
185C4—21no—
186C4—81no—
187C4—2742no—
188C4—5481no—
189C4—5481no—
190C4SD16×D13711yescharacteristic
191C4#1798410711yescharacteristic
192C4#1798411211yescharacteristic
193C4#1798411511yescharacteristic
194C2—82no—
195C2—10961no—
196C2#2257754511yescharacteristic
197C2#2257755211yescharacteristic
198C2#2257758011yescharacteristic
subgroupG/H
centerC22#17984056
commutator, cosocleC42×C137C23
FrattiniC42C22×D137
Fitting#486×C137C2
socleC22×C137#120
prime radical#22578073C2
hypercenter#486D137
hypocenterC137#1642
LCSG ▷ C42×C137 ▷ C22×C137 ▷ C137
UCSC1 ◁ C22 ◁ C42 ◁ #486

Complements

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

NH ≅ G/Ntype
2. #22577685C2semidirect
3. #22577605C2semidirect
4. #22577147C2semidirect
5. #22577135C2semidirect
6. #22577132C2semidirect
7. #486×C137C2semidirect
52. Dic274SD16semidirect
99. C137#1642semidirect
102. #486D137semidirect

Maximal subgroups

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

subgroupindex
#225780732
#225776852
#225776052
#225771472
#225771352
#225771322
#486×C1372
#1642137