Finite Group Databasebeta

← #31782864

Subgroups — #31782864

Order 17536.

170 conjugacy classes of subgroups; only the 43 normal subgroups are shown.

normal subgroups only

This group has 170 conjugacy classes of subgroups — too many to draw. Only its 43 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#22577841C211yesmaximal, characteristic, complemented
2#22577839C211yesmaximal, characteristic, complemented
3#22577778C211yesmaximal, characteristic, complemented
4#22577519C211yesmaximal, characteristic, complemented
5#22577223C211yesmaximal, characteristic, complemented
6#22577215C211yesmaximal, characteristic, complemented
7#361×C137C211yesmaximal, characteristic, complemented
8#17984110C2211yescharacteristic
9C2×D1096C2211yescharacteristic
10C2×#14555495—21no—
11#17984096—21no—
12#17984084—21no—
13#17984033—21no—
14#17984032—21no—
15D2192—21no—
16#17984029—21no—
17#17984028C2211yescharacteristic
18#17984014—21no—
19C8×Dic137C2211yescharacteristic
20C2×D8×C137C2211yescharacteristic
21M32×C137C2211yescharacteristic
22#108×C137C2211yescharacteristic
23#14555510—21no—
24C2×D548—21no—
25C2×C22⋊D137—42no—
26Dic548D411yescharacteristic, complemented
27#14555499—21no—
28D1096—21no—
29D1096—21no—
30D1096D411yescharacteristic, complemented
31#14555495—41no—
32#14555495—41no—
33#14555493—41no—
34C2×#11764040D411yescharacteristic
35C8×D137—41no—
36C4×Dic137D411yescharacteristic
37#14555485—21no—
38C2×D4×C137—21no—
39D8×C137—21no—
40D8×C137—21no—
41M16×C137—21no—
42C2×C8×C137C2311yescharacteristic
43C16×C137—21no—
44C22×D137—41no—
45Dic274—21no—
46D548—21no—
47D548—41no—
48D548—41no—
49C22⋊D137—41no—
50C22⋊D137—84no—
51C2×Dic137—21no—
52C4×D137—41no—
53#11764040D811yescharacteristic
54#11764040D811yescharacteristic
55C23×C137—41no—
56D4×C137—41no—
57D4×C137—41no—
58C2×C4×C137C2×D411yescharacteristic
59C8×C137—41no—
60C8×C137C2×D411yescharacteristic
61C8×C137#4111yescharacteristic
62C22×C137—82no—
63C22×C137—81no—
64C22×C137AGL(1,ℤ/8)11yescharacteristic
65C2×D137—41no—
66C2×D137—82no—
67C2×D137—81no—
68C4×C137#12111yescharacteristic
69C4×C137C2×D811yescharacteristic
70Dic137—22no—
71Dic137—41no—
72C2×C137—21no—
73C2×C137—161no—
74C2×C137#45911yescharacteristic
75D137—81no—
76D137—161no—
77C137#221311yescharacteristic, complemented, Sylow
78#2213—1371nomaximal, Sylow
79#509—1371no—
80#509—1371no—
81#493—1371no—
82#443—1371no—
83#364—1371no—
84#361—1371no—
85#361D13711yescharacteristic, complemented
86#135—1371no—
87C2×D8—2741no—
88C2×D8—1371no—
89C2×D8C2×D13711yescharacteristic
90#131—2741no—
91#127—2741no—
92SD32—2741no—
93SD32—2741no—
94D16—2741no—
95D16—2741no—
96M32—1371no—
97M32C2×D13711yescharacteristic
98#108C2×D13711yescharacteristic
99#104—2741no—
100C4×C8—1371no—
101#41—2741no—
102C2×D4—21no—
103C2×D4—2741no—
104C2×D4—5482no—
105Q16—1371no—
106SD16—2741no—
107D8—21no—
108D8—21no—
109D8—2741no—
110D8—2741no—
111D8—5481no—
112D8—5481no—
113D8—1371no—
114M16—21no—
115M16—5481no—
116C2×C8—5481no—
117C2×C8—1371no—
118C2×C8C22×D13711yescharacteristic
119C42—1371no—
120C16—21no—
121C16—2741no—
122C23—41no—
123C23—5481no—
124Q8—2741no—
125D4—41no—
126D4—41no—
127D4—2741no—
128D4—5481no—
129D4—5481no—
130D4—5481no—
131D4—10964no—
132C2×C4—2741no—
133C2×C4—5481no—
134C2×C4D4×D13711yescharacteristic
135C8—41no—
136C8—1371no—
137C8—1371no—
138C8D4×D13711yescharacteristic
139C8#1455551011yescharacteristic
140C22—82no—
141C22—81no—
142C22—5481no—
143C22—10962no—
144C22—10961no—
145C22#1798411511yescharacteristic
146C4—2742no—
147C4—5481no—
148C4#1798406111yescharacteristic
149C4D8×D13711yescharacteristic
150C2—21no—
151C2—161no—
152C2—10961no—
153C2—21921no—
154C2#2257759711yescharacteristic
subgroupG/H
centerC2#22577597
commutator, cosocleC2×C8×C137C23
FrattiniC2×C8C22×D137
Fitting#361×C137C2
socleC2×C137#459
hypercenter#361D137
hypocenterC137#2213
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. #22577841C2semidirect
2. #22577839C2semidirect
3. #22577778C2semidirect
4. #22577519C2semidirect
5. #22577223C2semidirect
6. #22577215C2semidirect
7. #361×C137C2semidirect
26. Dic548D4semidirect
30. D1096D4semidirect
77. C137#2213semidirect
85. #361D137semidirect

Maximal subgroups

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

subgroupindex
#225778412
#225778392
#225777782
#225775192
#225772232
#225772152
#361×C1372
#2213137