Finite Group Databasebeta

← #31780071

Subgroups — #31780071

Order 17536.

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

normal subgroups only

This group has 240 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#22578058C211yesmaximal, characteristic
2#22577799C211yesmaximal, characteristic, complemented
3#22577680C211yesmaximal, characteristic, complemented
4#22577148C211yesmaximal, characteristic, complemented
5D4⋊#11764040C211yesmaximal, characteristic, complemented
6#22577130C211yesmaximal, characteristic, complemented
7#461×C137C211yesmaximal, characteristic, complemented
8C2×#14555510—21no—
9#17984088C2211yescharacteristic
10#17984070—22no—
11#17984067—21no—
12#17984059—21no—
13C4×D548C2211yescharacteristic
14C4⋊C4×D137—21no—
15#17984021—21no—
16#17984019—22no—
17C4⋊#11764040C2211yescharacteristic
18#17984008—21no—
19C4×#11764040C2211yescharacteristic
20C2×SD16×C137—21no—
21C4⋊Q8×C137C2211yescharacteristic
22C4×D4×C137C2211yescharacteristic
23C4⋊C8×C137C2211yescharacteristic
24#103×C137—21no—
25#14555510—42no—
26#14555510—42no—
27C2×Dic274—21no—
28C2×D548D411yescharacteristic
29C2×C22⋊D137—22no—
30C2×C4×D137—22no—
31C2×C4×D137—21no—
32C2×#11764040—21no—
33C2×#11764040—21no—
34C4⋊Dic137—21no—
35C4⋊Dic137D412yes—
36C4⋊Dic137D411yescharacteristic
37#14555489—22no—
38C22⋊Dic137—22no—
39#14555487—21no—
40C2×Q8×C137—21no—
41C2×D4×C137D411yescharacteristic
42C22×C4×C137—21no—
43SD16×C137—42no—
44C2×C8×C137—21no—
45C4⋊C4×C137—21no—
46C4⋊C4×C137—21no—
47C4⋊C4×C137D411yescharacteristic
48C22⋊C4×C137—21no—
49C42×C137C2311yescharacteristic
50C22×D137—21no—
51Dic274—41no—
52Dic274—41no—
53D548—21no—
54D548SD1612yescomplemented
55C22⋊D137—44no—
56C2×Dic137—21no—
57C2×Dic137—22no—
58C4×D137—44no—
59C4×D137—41no—
60C4×D137—41no—
61#11764040—22no—
62#11764040—41no—
63C23×C137—21no—
64Q8×C137—41no—
65Q8×C137—41no—
66D4×C137—21no—
67D4×C137—21no—
68C2×C4×C137—21no—
69C2×C4×C137—41no—
70C2×C4×C137—41no—
71C2×C4×C137—41no—
72C2×C4×C137C2×D411yescharacteristic
73C2×C4×C137C2×D411yescharacteristic
74C2×C4×C137C2×D411yescharacteristic
75C8×C137—41no—
76C22×C137—41no—
77C22×C137—41no—
78C22×C137—41no—
79C22×C137#12011yescharacteristic
80C2×D137—22no—
81C2×D137—41no—
82C2×D137—41no—
83C4×C137—21no—
84C4×C137—41no—
85C4×C137—81no—
86C4×C137C2×SD1611yescharacteristic
87C4×C137#13511yescharacteristic
88C4×C137AGL(1,ℤ/8)11yescharacteristic
89C4×C137AGL(1,ℤ/8)11yescharacteristic
90Dic137—41no—
91Dic137—42no—
92C2×C137—81no—
93C2×C137#44911yescharacteristic
94C2×C137#45011yescharacteristic
95C2×C137#45611yescharacteristic
96D137—42no—
97C137#165411yescharacteristic, complemented, Sylow
98#1654—1371nomaximal, Sylow
99#547—1371no—
100#498—1371no—
101#474—1371no—
102#461D13711yescharacteristic, complemented
103#332—1371no—
104D4⋊C8—1371no—
105D4⋊C8—1371no—
106C2×#41—2741no—
107C2×SD16—21no—
108C4⋊Q8—1371no—
109C4⋊Q8C2×D13711yescharacteristic
110#123—2742no—
111C22⋊Q8—2741no—
112#121—2741no—
113C4×D4—1371no—
114C4×D4C2×D13711yescharacteristic
115C2×C4⋊C4—2741no—
116#107—2741no—
117#106—2742no—
118C4⋊C8—1371no—
119C4⋊C8C2×D13711yescharacteristic
120#103—21no—
121#102—2741no—
122C4×C8—1371no—
123#41—5482no—
124#41—5482no—
125C2×Q8—21no—
126C2×Q8—2741no—
127C2×D4—2742no—
128C2×D4—1371no—
129C2×D4C22⋊D13711yescharacteristic
130C22×C4—21no—
131C22×C4—2742no—
132C22×C4—2741no—
133SD16—42no—
134C2×C8—21no—
135C2×C8—2741no—
136C2×C8—2741no—
137C4⋊C4—21no—
138C4⋊C4—21no—
139C4⋊C4—2742no—
140C4⋊C4—2741no—
141C4⋊C4—1372no—
142C4⋊C4—1371no—
143C4⋊C4C22⋊D13711yescharacteristic
144C22⋊C4—21no—
145C22⋊C4—2742no—
146C22⋊C4—2741no—
147C42C22×D13711yescharacteristic
148C23—21no—
149C23—2741no—
150Q8—41no—
151Q8—41no—
152Q8—5481no—
153Q8—5481no—
154D4—21no—
155D4—21no—
156D4—2741no—
157D4—5484no—
158D4—1372no—
159C2×C4—21no—
160C2×C4—41no—
161C2×C4—41no—
162C2×C4—41no—
163C2×C4—2741no—
164C2×C4—2742no—
165C2×C4—5484no—
166C2×C4—5481no—
167C2×C4—5481no—
168C2×C4D4×D13711yescharacteristic
169C2×C4D4×D13711yescharacteristic
170C2×C4C2×C22⋊D13711yescharacteristic
171C8—41no—
172C8—2742no—
173C8—5481no—
174C22—41no—
175C22—41no—
176C22—41no—
177C22—2742no—
178C22—5481no—
179C22—5481no—
180C22#1798405611yescharacteristic
181C4—21no—
182C4—41no—
183C4—81no—
184C4—5481no—
185C4—5482no—
186C4SD16×D13711yescharacteristic
187C4#1798410811yescharacteristic
188C4#1798411411yescharacteristic
189C4#1798411511yescharacteristic
190C2—81no—
191C2—5482no—
192C2#2257754611yescharacteristic
193C2#2257755211yescharacteristic
194C2#2257758411yescharacteristic
subgroupG/H
centerC22#17984056
commutator, cosocleC42×C137C23
FrattiniC42C22×D137
Fitting#461×C137C2
socleC22×C137#120
prime radical#22578058C2
hypercenter#461D137
hypocenterC137#1654
LCSG ▷ C42×C137 ▷ C22×C137 ▷ C137
UCSC1 ◁ C22 ◁ C42 ◁ #461

Complements

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

NH ≅ G/Ntype
2. #22577799C2semidirect
3. #22577680C2semidirect
4. #22577148C2semidirect
5. D4⋊#11764040C2semidirect
6. #22577130C2semidirect
7. #461×C137C2semidirect
54. D548SD16semidirect
97. C137#1654semidirect
102. #461D137semidirect

Maximal subgroups

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

subgroupindex
#225780582
#225777992
#225776802
#225771482
D4⋊#117640402
#225771302
#461×C1372
#1654137