Finite Group Databasebeta

← #31782745

Subgroups — #31782745

Order 17536.

220 conjugacy classes of subgroups; only the 47 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 47 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#22578204C211yesmaximal, characteristic, complemented
2#22577839C211yesmaximal, characteristic, complemented
3#22577670C211yesmaximal, characteristic, complemented
4#22577216C211yesmaximal, characteristic, complemented
5#22577211C211yesmaximal, characteristic, complemented
6#22577186C211yesmaximal, characteristic, complemented
7C2×D16×C137C211yesmaximal, characteristic, complemented
8#17984133—21no—
9#17984114—21no—
10#17984114—21no—
11#17984112—22no—
12#17984110C2211yescharacteristic
13#17984107—21no—
14D8×D137—21no—
15#17984094C2211yescharacteristic
16#17984032—21no—
17#17984029—21no—
18#17984028C2211yescharacteristic, complemented
19#17984024C2211yescharacteristic, complemented
20#17984005—21no—
21C2×D8×C137C2211yescharacteristic
22C2×D8×C137C2211yescharacteristic, complemented
23D16×C137—22no—
24C2×C16×C137C2211yescharacteristic, complemented
25#14555510—21no—
26#14555510D411yescharacteristic
27#14555509—22no—
28#14555509—41no—
29C2×C22⋊D137—42no—
30D4×D137—22no—
31D4×D137—41no—
32Dic548D411yescharacteristic
33#14555499—21no—
34#14555498—22no—
35#14555498—41no—
36D1096D411yescharacteristic
37#14555495—22no—
38#14555495—41no—
39#14555494—21no—
40#14555494D411yescharacteristic
41#14555493—21no—
42C8×D137—21no—
43#14555485—21no—
44C2×D4×C137—21no—
45C2×D4×C137—21no—
46D8×C137—21no—
47D8×C137—21no—
48D8×C137—21no—
49D8×C137D412yes—
50C2×C8×C137C2311yescharacteristic
51C16×C137—21no—
52C22×D137—42no—
53Dic274—21no—
54Dic274D811yescharacteristic
55D548—21no—
56D548D811yescharacteristic
57C22⋊D137—21no—
58C22⋊D137—42no—
59C22⋊D137—42no—
60C22⋊D137—41no—
61C2×Dic137—42no—
62C4×D137—21no—
63C4×D137—41no—
64#11764040—21no—
65#11764040—41no—
66C23×C137—41no—
67C23×C137—41no—
68D4×C137—22no—
69D4×C137—41no—
70D4×C137—41no—
71D4×C137—41no—
72C2×C4×C137C2×D411yescharacteristic
73C8×C137C2×D411yescharacteristic
74C8×C137C2×D411yescharacteristic
75C22×C137—42no—
76C22×C137—81no—
77C22×C137—161no—
78C22×C137—161no—
79C22×C137AGL(1,ℤ/8)11yescharacteristic
80C2×D137—21no—
81C2×D137—41no—
82C2×D137—162no—
83C4×C137#12011yescharacteristic
84C4×C137C2×D811yescharacteristic
85Dic137—21no—
86Dic137—41no—
87C2×C137—21no—
88C2×C137—82no—
89C2×C137—161no—
90C2×C137#44711yescharacteristic
91D137—41no—
92D137—81no—
93C137#219211yescharacteristic, complemented, Sylow
94#2192—1371nomaximal, Sylow
95#576—1371no—
96#509—1371no—
97C2×D16D13711yescharacteristic, complemented
98#472—1371no—
99#361—1371no—
100#359—1371no—
101#350—1371no—
102H(5,𝔽2)—2741no—
103AGL(1,ℤ/8)—2742no—
104AGL(1,ℤ/8)—2741no—
105AGL(1,ℤ/8)—2741no—
106#135—2741no—
107#135—1371no—
108C2×D8—2741no—
109C2×D8C2×D13711yescharacteristic
110C2×D8C2×D13711yescharacteristic, complemented
111#131—1371no—
112SD32—2741no—
113D16—22no—
114D16—2741no—
115M32—1371no—
116C2×C16C2×D13711yescharacteristic, complemented
117#108—1371no—
118#100—2741no—
119#41—2742no—
120#41—2741no—
121#41—5481no—
122#41—1371no—
123C2×D4—21no—
124C2×D4—21no—
125C2×D4—2742no—
126C2×D4—5481no—
127C2×D4—5482no—
128Q16—1371no—
129SD16—2742no—
130SD16—2741no—
131SD16—5481no—
132D8—21no—
133D8—21no—
134D8—21no—
135D8—2742no—
136D8—5481no—
137D8—1371no—
138D8C22⋊D13712yes—
139M16—2741no—
140M16—2741no—
141M16—1371no—
142C2×C8—2741no—
143C2×C8C22×D13711yescharacteristic
144C16—21no—
145C16—2741no—
146C23—41no—
147C23—41no—
148C23—5482no—
149Q8—2741no—
150Q8—1371no—
151D4—22no—
152D4—41no—
153D4—41no—
154D4—41no—
155D4—2741no—
156D4—2741no—
157D4—5482no—
158D4—5482no—
159D4—5481no—
160D4—1371no—
161C2×C4—2741no—
162C2×C4—5482no—
163C2×C4—5481no—
164C2×C4D4×D13711yescharacteristic
165C8—2741no—
166C8—5481no—
167C8D4×D13711yescharacteristic
168C8C2×C22⋊D13711yescharacteristic
169C22—42no—
170C22—81no—
171C22—161no—
172C22—161no—
173C22—2741no—
174C22—5481no—
175C22—21922no—
176C22#1798411211yescharacteristic
177C4—2741no—
178C4—5481no—
179C4#1798405611yescharacteristic
180C4D8×D13711yescharacteristic
181C2—21no—
182C2—82no—
183C2—161no—
184C2—5481no—
185C2—10961no—
186C2#2257753511yescharacteristic
subgroupG/H
centerC2#22577535
commutator, cosocleC2×C8×C137C23
FrattiniC2×C8C22×D137
FittingC2×D16×C137C2
socleC2×C137#447
hypercenterC2×D16D137
hypocenterC137#2192
LCSG ▷ C2×C8×C137 ▷ C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ C2×D16

Complements

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

NH ≅ G/Ntype
1. #22578204C2semidirect
2. #22577839C2semidirect
3. #22577670C2semidirect
4. #22577216C2semidirect
5. #22577211C2semidirect
6. #22577186C2semidirect
7. C2×D16×C137C2semidirect
18. #17984028C22semidirect
19. #17984024C22semidirect
22. C2×D8×C137C22semidirect
24. C2×C16×C137C22semidirect
93. C137#2192semidirect
97. C2×D16D137semidirect
110. C2×D8C2×D137semidirect
116. C2×C16C2×D137semidirect

Maximal subgroups

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

subgroupindex
#225782042
#225778392
#225776702
#225772162
#225772112
#225771862
C2×D16×C1372
#2192137