Finite Group Databasebeta

← #31782766

Subgroups — #31782766

Order 17536.

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

normal subgroups only

This group has 206 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#22578216C211yesmaximal, characteristic
2#22577844C211yesmaximal, characteristic, complemented
3#22577675C211yesmaximal, characteristic, complemented
4#22577219C211yesmaximal, characteristic, complemented
5#22577211C211yesmaximal, characteristic, complemented
6#22577186C211yesmaximal, characteristic, complemented
7C2×Q32×C137C211yesmaximal, characteristic, complemented
8#17984136—21no—
9#17984120—22no—
10#17984119—21no—
11#17984119—21no—
12#17984111—21no—
13#17984110C2211yescharacteristic
14Q16×D137—21no—
15#17984094C2211yescharacteristic
16#17984034—21no—
17#17984031—21no—
18#17984028C2211yescharacteristic
19#17984024C2211yescharacteristic
20#17984007—21no—
21C2×Q16×C137C2211yescharacteristic
22C2×Q16×C137C2211yescharacteristic
23Q32×C137—22no—
24C2×C16×C137C2211yescharacteristic
25#14555511—22no—
26#14555511—41no—
27#14555510—21no—
28#14555510—42no—
29#14555510D411yescharacteristic
30Q8×D137—22no—
31Q8×D137—41no—
32Dic548D411yescharacteristic
33#14555500—22no—
34#14555500—41no—
35#14555499—21no—
36#14555497—22no—
37#14555497—41no—
38D1096D411yescharacteristic
39#14555494—21no—
40#14555494D411yescharacteristic
41#14555493—21no—
42C8×D137—21no—
43#14555485—21no—
44C2×Q8×C137—21no—
45C2×Q8×C137—21no—
46Q16×C137—21no—
47Q16×C137—21no—
48Q16×C137—21no—
49Q16×C137D412yes—
50C2×C8×C137C2311yescharacteristic
51C16×C137—21no—
52Dic274—21no—
53Dic274—42no—
54Dic274D811yescharacteristic
55D548—21no—
56D548—42no—
57D548D811yescharacteristic
58C22⋊D137—21no—
59C22⋊D137—41no—
60C4×D137—21no—
61C4×D137—42no—
62C4×D137—42no—
63C4×D137—41no—
64#11764040—21no—
65#11764040—41no—
66Q8×C137—22no—
67Q8×C137—41no—
68Q8×C137—41no—
69Q8×C137—41no—
70C2×C4×C137—41no—
71C2×C4×C137—41no—
72C2×C4×C137C2×D411yescharacteristic
73C8×C137C2×D411yescharacteristic
74C8×C137C2×D411yescharacteristic
75C22×C137AGL(1,ℤ/8)11yescharacteristic
76C2×D137—21no—
77C2×D137—41no—
78C4×C137—42no—
79C4×C137—81no—
80C4×C137#12011yescharacteristic
81C4×C137C2×D811yescharacteristic
82Dic137—21no—
83Dic137—41no—
84C2×C137—21no—
85C2×C137#44711yescharacteristic
86D137—41no—
87D137—81no—
88C137#219511yescharacteristic, complemented, Sylow
89#2195—1371nomaximal, Sylow
90#578—1371no—
91#510—1371no—
92C2×Q32D13711yescharacteristic, complemented
93#473—1371no—
94#362—1371no—
95#359—1371no—
96#350—1371no—
972−1+4—2741no—
98#137—2742no—
99#137—2741no—
100#137—2741no—
101#135—2741no—
102#135—1371no—
103C2×Q16—2741no—
104C2×Q16C2×D13711yescharacteristic
105C2×Q16C2×D13711yescharacteristic
106#131—1371no—
107Q32—22no—
108Q32—2741no—
109SD32—2741no—
110M32—1371no—
111C2×C16C2×D13711yescharacteristic
112#108—1371no—
113#101—2741no—
114#41—2742no—
115#41—2741no—
116#41—5482no—
117#41—5481no—
118#41—1371no—
119C2×Q8—21no—
120C2×Q8—21no—
121C2×Q8—2742no—
122C2×Q8—5481no—
123Q16—21no—
124Q16—21no—
125Q16—21no—
126Q16—2742no—
127Q16—5481no—
128Q16—1371no—
129Q16C22⋊D13712yes—
130SD16—2742no—
131SD16—2741no—
132SD16—5481no—
133D8—1371no—
134M16—2741no—
135M16—2741no—
136M16—1371no—
137C2×C8—2741no—
138C2×C8C22×D13711yescharacteristic
139C16—21no—
140C16—2741no—
141Q8—22no—
142Q8—41no—
143Q8—41no—
144Q8—41no—
145Q8—2741no—
146Q8—5482no—
147Q8—1371no—
148D4—2741no—
149D4—2741no—
150D4—5482no—
151D4—5481no—
152D4—1371no—
153C2×C4—41no—
154C2×C4—41no—
155C2×C4—2741no—
156C2×C4—5482no—
157C2×C4—5482no—
158C2×C4—5481no—
159C2×C4D4×D13711yescharacteristic
160C8—2741no—
161C8—5481no—
162C8D4×D13711yescharacteristic
163C8C2×C22⋊D13711yescharacteristic
164C22—2741no—
165C22—5481no—
166C22#1798411211yescharacteristic
167C4—42no—
168C4—81no—
169C4—2741no—
170C4—5481no—
171C4#1798405611yescharacteristic
172C4D8×D13711yescharacteristic
173C2—21no—
174C2—5481no—
175C2—10961no—
176C2#2257753511yescharacteristic
subgroupG/H
centerC2#22577535
commutator, cosocleC2×C8×C137C23
FrattiniC2×C8C22×D137
FittingC2×Q32×C137C2
socleC2×C137#447
prime radical#22578216C2
hypercenterC2×Q32D137
hypocenterC137#2195
LCSG ▷ C2×C8×C137 ▷ C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ C2×Q32

Complements

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

NH ≅ G/Ntype
2. #22577844C2semidirect
3. #22577675C2semidirect
4. #22577219C2semidirect
5. #22577211C2semidirect
6. #22577186C2semidirect
7. C2×Q32×C137C2semidirect
88. C137#2195semidirect
92. C2×Q32D137semidirect

Maximal subgroups

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

subgroupindex
#225782162
#225778442
#225776752
#225772192
#225772112
#225771862
C2×Q32×C1372
#2195137