Finite Group Databasebeta

← #31796111

Subgroups — #31796111

Order 17536.

218 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 218 conjugacy classes of subgroups — too many to draw. Only its 62 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#22578419C211yesmaximal, characteristic
2C4×#14555519C211yesmaximal, characteristic
3#22578253C212yesmaximal, complemented
4Q8⋊#11764047C212yesmaximal, complemented
5#22577955C211yesmaximal, characteristic
6C2×#14555519—21no—
7C4⋊#11764047C2212yes—
8C22⋊#11764047—21no—
9#17984143—21no—
10C4×#11764047—21no—
11C4×#11764047C2212yes—
12#17984087C411yescharacteristic
13#17984086C411yescharacteristic
14#17984078—21no—
15#17984067C411yescharacteristic
16#17984065—21no—
17Q8×Dic137C2211yescharacteristic
18#17984053—21no—
19#17984053C2211yescharacteristic
20#17984044—21no—
21C2×C4×Dic137C2211yescharacteristic
22C22⋊Q8×C137C411yescharacteristic
23#14555519—24no—
24C2×#11764047—22no—
25C2×#11764047—22no—
26C2×Dic274—21no—
27C2×Dic274C2×C411yescharacteristic
28C22×Dic137D411yescharacteristic
29C22×Dic137D411yescharacteristic
30C4⋊Dic137—21no—
31C4⋊Dic137C2×C411yescharacteristic
32#14555489—21no—
33#14555489—21no—
34#14555489—21no—
35#14555489—21no—
36#14555489—21no—
37C22⋊Dic137—21no—
38C4×Dic137—21no—
39C4×Dic137—21no—
40C4×Dic137C2×C411yescharacteristic
41C4×Dic137D411yescharacteristic
42C4×Dic137D411yescharacteristic
43C4×Dic137C2311yescharacteristic
44C2×Q8×C137C2×C411yescharacteristic
45C22×C4×C137C2×C411yescharacteristic
46C4⋊C4×C137—21no—
47C4⋊C4×C137C2×C411yescharacteristic
48C22⋊C4×C137—21no—
49#11764047—24no—
50#11764047—42no—
51Dic274—22no—
52Dic274—42no—
53C2×Dic137—22no—
54C2×Dic137—22no—
55C2×Dic137—21no—
56C2×Dic137—21no—
57C2×Dic137C22⋊C411yescharacteristic
58C2×Dic137C22⋊C411yescharacteristic
59C2×Dic137C2×D411yescharacteristic
60C2×Dic137C2×D411yescharacteristic
61C23×C137C22⋊C411yescharacteristic
62Q8×C137—22no—
63C2×C4×C137—22no—
64C2×C4×C137—21no—
65C2×C4×C137—21no—
66C2×C4×C137C22⋊C411yescharacteristic
67C2×C4×C137C22×C411yescharacteristic
68C22×C137—21no—
69C22×C137—21no—
70C22×C137—21no—
71C22×C137C2×C22⋊C411yescharacteristic
72C4×C137—21no—
73C4×C137—41no—
74C4×C137—41no—
75C4×C137#13112yes—
76Dic137—21no—
77Dic137—21no—
78Dic137—41no—
79Dic137—41no—
80Dic137#10412yes—
81Dic137#13712yes—
82C2×C137—41no—
83C2×C137#40811yescharacteristic
84C2×C137#41911yescharacteristic
85C2×C137C2×#10411yescharacteristic
86C137#148511yescharacteristic, complemented, Sylow
87#1485—1371nomaximal, Sylow
88#533—1371no—
89#431—1371no—
90C4×M16—1371no—
91Q8⋊C8—1372no—
92Q8⋊C8—1372no—
93C2×M16—2741no—
94C4⋊Q8—1371no—
95C4⋊Q8—1371no—
96#125—2741no—
97C22⋊Q8—2741no—
98C22⋊Q8—1371no—
99C22⋊Q8#1058773911yescharacteristic
100C4×Q8—2741no—
101C4×Q8—1371no—
102C4×Q8—1371no—
103#117—2741no—
104C2×C42—1371no—
105C4⋊C8—1372no—
106C22⋊C8—2741no—
107#97—2741no—
108C4×C8—2741no—
109C4×C8—1372no—
110C2×Q8—2741no—
111C2×Q8—1371no—
112C2×Q8C2×#1058773911yescharacteristic
113C22×C4—1371no—
114C22×C4—1371no—
115C22×C4C2×#1058773911yescharacteristic
116M16—2744no—
117C2×C8—2742no—
118C2×C8—2742no—
119C4⋊C4—21no—
120C4⋊C4—2741no—
121C4⋊C4—2741no—
122C4⋊C4—2741no—
123C4⋊C4—2741no—
124C4⋊C4—2741no—
125C4⋊C4—2741no—
126C4⋊C4—1371no—
127C4⋊C4C2×#1058773911yescharacteristic
128C22⋊C4—21no—
129C22⋊C4—2741no—
130C42—2741no—
131C42—2741no—
132C42—1371no—
133C42—1371no—
134C42—1371no—
135C42—1371no—
136C23C22⋊#1058773911yescharacteristic
137Q8—22no—
138Q8—2742no—
139Q8—5482no—
140C2×C4—22no—
141C2×C4—21no—
142C2×C4—21no—
143C2×C4—2742no—
144C2×C4—2742no—
145C2×C4—2741no—
146C2×C4—2741no—
147C2×C4—1371no—
148C2×C4—1371no—
149C2×C4—1371no—
150C2×C4—1371no—
151C2×C4C22⋊#1058773911yescharacteristic
152C2×C4C22×#1058773911yescharacteristic
153C8—2744no—
154C8—5482no—
155C22—21no—
156C22—21no—
157C22—21no—
158C22C2×C22⋊#1058773911yescharacteristic
159C4—21no—
160C4—41no—
161C4—41no—
162C4—2741no—
163C4—2741no—
164C4—5481no—
165C4—5481no—
166C4—1372no—
167C4—1372no—
168C4#1798417512yes—
169C2—41no—
170C2#2257837611yescharacteristic
171C2#2257839611yescharacteristic
172C2D137⋊#10411yescharacteristic
subgroupG/H
centerC22C2×C22⋊#10587739
commutatorC2×C4×C137C22×C4
FrattiniC2×C4C22×#10587739
FittingC22⋊Q8×C137C4
socleC22×C137C2×C22⋊C4
prime radicalC23×C137C22⋊C4
cosocleC4×Dic137C23
hypercenterC22⋊Q8#10587739
hypocenterC137#1485
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22⋊Q8

Complements

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

NH ≅ G/Ntype
3. #22578253C2semidirect
4. Q8⋊#11764047C2semidirect
86. C137#1485semidirect

Maximal subgroups

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

subgroupindex
#225784192
C4×#145555192
#225782532
Q8⋊#117640472
#225779552
#1485137