Finite Group Databasebeta

← #31782749

Subgroups — #31782749

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#22577841C211yesmaximal, characteristic, complemented
3C2×#17984029C211yesmaximal, characteristic, complemented
4#22577674C211yesmaximal, characteristic, complemented
5#22577209C211yesmaximal, characteristic, complemented
6#22577188C211yesmaximal, characteristic, complemented
7#361×C137C211yesmaximal, characteristic, complemented
8#17984133—21no—
9#17984116—21no—
10#17984114—21no—
11#17984112—22no—
12#17984110C2211yescharacteristic
13#17984107—21no—
14C2×D1096C2211yescharacteristic, complemented
15D8×D137—21no—
16#17984094C2211yescharacteristic
17#17984033—21no—
18D2192—21no—
19#17984029—22no—
20C2×#14555485C2211yescharacteristic, complemented
21#17984004—21no—
22C2×D8×C137C2211yescharacteristic
23M32×C137C2211yescharacteristic, complemented
24#108×C137C2211yescharacteristic, complemented
25#14555510—21no—
26#14555510D411yescharacteristic
27#14555509—22no—
28#14555509—41no—
29C2×D548—21no—
30C2×C22⋊D137—42no—
31D4×D137—22no—
32D4×D137—41no—
33Dic548D411yescharacteristic
34#14555499—21no—
35#14555499—41no—
36#14555498—22no—
37D1096—21no—
38D1096—21no—
39D1096—41no—
40D1096D411yescharacteristic
41#14555495—22no—
42#14555494D411yescharacteristic
43#14555493—21no—
44C8×D137—21no—
45#14555485—21no—
46C2×D4×C137—21no—
47D8×C137—21no—
48D8×C137D412yes—
49M16×C137—21no—
50C2×C8×C137C2311yescharacteristic
51C16×C137—21no—
52C22×D137—42no—
53C22×D137—41no—
54Dic274—21no—
55Dic274D811yescharacteristic
56D548—21no—
57D548—41no—
58D548—41no—
59D548D811yescharacteristic
60C22⋊D137—21no—
61C22⋊D137—42no—
62C22⋊D137—42no—
63C22⋊D137—41no—
64C2×Dic137—42no—
65C4×D137—21no—
66C4×D137—41no—
67#11764040—21no—
68C23×C137—41no—
69D4×C137—22no—
70D4×C137—41no—
71C2×C4×C137C2×D411yescharacteristic
72C8×C137—41no—
73C8×C137C2×D411yescharacteristic
74C8×C137C2×D411yescharacteristic
75C22×C137—42no—
76C22×C137—161no—
77C22×C137AGL(1,ℤ/8)11yescharacteristic
78C2×D137—21no—
79C2×D137—41no—
80C2×D137—81no—
81C2×D137—162no—
82C2×D137—161no—
83C4×C137#12011yescharacteristic
84C4×C137C2×D811yescharacteristic
85Dic137—21no—
86Dic137—41no—
87C2×C137—21no—
88C2×C137—82no—
89C2×C137#44711yescharacteristic
90D137—41no—
91D137—81no—
92D137—161no—
93C137#219211yescharacteristic, complemented, Sylow
94#2192—1371nomaximal, Sylow
95#576—1371no—
96#509—1371no—
97C2×D16—1371no—
98#472—1371no—
99#361D13711yescharacteristic, complemented
100#359—1371no—
101#350—1371no—
102H(5,𝔽2)—2741no—
103AGL(1,ℤ/8)—2741no—
104AGL(1,ℤ/8)—2742no—
105AGL(1,ℤ/8)—2741no—
106#135—2741no—
107#135—1371no—
108C2×D8—2741no—
109C2×D8—1371no—
110C2×D8C2×D13711yescharacteristic
111#131—1371no—
112SD32—2741no—
113D16—2741no—
114D16—2742no—
115M32C2×D13711yescharacteristic, complemented
116C2×C16—1371no—
117#108C2×D13711yescharacteristic, complemented
118#100—2741no—
119#41—2742no—
120#41—2741no—
121#41—5481no—
122#41—1371no—
123C2×D4—21no—
124C2×D4—2742no—
125C2×D4—2741no—
126C2×D4—5481no—
127C2×D4—5482no—
128Q16—1371no—
129SD16—2742no—
130SD16—2741no—
131SD16—5481no—
132D8—21no—
133D8—2742no—
134D8—2741no—
135D8—2741no—
136D8—5481no—
137D8—1371no—
138D8D54812yes—
139M16—21no—
140M16—2741no—
141M16—1371no—
142C2×C8—2741no—
143C2×C8C22×D13711yescharacteristic
144C16—21no—
145C16—2741no—
146C23—41no—
147C23—5482no—
148C23—5481no—
149Q8—2741no—
150Q8—1371no—
151D4—22no—
152D4—41no—
153D4—2741no—
154D4—2741no—
155D4—5482no—
156D4—5482no—
157D4—5481no—
158D4—5481no—
159D4—5481no—
160D4—1371no—
161C2×C4—2741no—
162C2×C4—5482no—
163C2×C4—5481no—
164C2×C4D4×D13711yescharacteristic
165C8—41no—
166C8—2741no—
167C8D4×D13711yescharacteristic
168C8C2×D54811yescharacteristic
169C22—42no—
170C22—161no—
171C22—2741no—
172C22—5481no—
173C22—10961no—
174C22—21921no—
175C22—21922no—
176C22#1798411511yescharacteristic
177C4—2741no—
178C4—5481no—
179C4#1798405811yescharacteristic
180C4D8×D13711yescharacteristic
181C2—21no—
182C2—82no—
183C2—5481no—
184C2—10961no—
185C2—21921no—
186C2#2257753811yescharacteristic
subgroupG/H
centerC2#22577538
commutator, cosocleC2×C8×C137C23
FrattiniC2×C8C22×D137
Fitting#361×C137C2
socleC2×C137#447
hypercenter#361D137
hypocenterC137#2192
LCSG ▷ C2×C8×C137 ▷ C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ #361

Complements

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

NH ≅ G/Ntype
1. #22578204C2semidirect
2. #22577841C2semidirect
3. C2×#17984029C2semidirect
4. #22577674C2semidirect
5. #22577209C2semidirect
6. #22577188C2semidirect
7. #361×C137C2semidirect
14. C2×D1096C22semidirect
20. C2×#14555485C22semidirect
23. M32×C137C22semidirect
24. #108×C137C22semidirect
93. C137#2192semidirect
99. #361D137semidirect
115. M32C2×D137semidirect
117. #108C2×D137semidirect

Maximal subgroups

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

subgroupindex
#225782042
#225778412
C2×#179840292
#225776742
#225772092
#225771882
#361×C1372
#2192137