Finite Group Databasebeta

← #31796095

Subgroups — #31796095

Order 17536.

260 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 70 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#22578422C211yesmaximal, characteristic, complemented
2#22578404C211yesmaximal, characteristic, complemented
3D4⋊#11764047C214yesmaximal, complemented
4#22577951C211yesmaximal, characteristic
5C2×#14555518—21no—
6#17984155—21no—
7C4⋊#11764047—21no—
8C4⋊#11764047C2212yes—
9#17984145—21no—
10C4×#11764047C2212yes—
11C2×#14555509—21no—
12#17984088C411yescharacteristic
13#17984075—21no—
14#17984059—21no—
15#17984059C412yes—
16D4×Dic137—21no—
17D4×Dic137C2212yes—
18C42×D137C2211yescharacteristic
19#127×C137C411yescharacteristic
20#14555518—42no—
21C2×#11764047—22no—
22C2×#11764047—22no—
23#14555509—42no—
24#14555509—42no—
25C2×Dic274—21no—
26C2×C22⋊D137—22no—
27C22×Dic137—22no—
28C2×C4×D137C2×C411yescharacteristic
29C2×C4×D137D411yescharacteristic
30C2×C4×D137D411yescharacteristic
31C4⋊Dic137—21no—
32C4⋊Dic137C2×C412yes—
33C22⋊Dic137—22no—
34C22⋊Dic137—22no—
35C4×Dic137D411yescharacteristic
36C4×Dic137D411yescharacteristic
37C4×Dic137C2311yescharacteristic
38C2×D4×C137—21no—
39C2×D4×C137—21no—
40C2×D4×C137C2×C412yes—
41C42×C137C2×C411yescharacteristic
42#11764047—42no—
43#11764047—42no—
44C22×D137C22⋊C411yescharacteristic
45Dic274—42no—
46C22⋊D137—44no—
47C2×Dic137—22no—
48C2×Dic137—42no—
49C2×Dic137—42no—
50C2×Dic137C22⋊C411yescharacteristic
51C2×Dic137C2×D411yescharacteristic
52C2×Dic137C2×D411yescharacteristic
53C4×D137—22no—
54C4×D137—41no—
55C4×D137D812yes—
56C4×D137SD1612yes—
57C23×C137—22no—
58D4×C137—24no—
59D4×C137—42no—
60D4×C137—42no—
61C2×C4×C137C22⋊C411yescharacteristic
62C2×C4×C137C22⋊C411yescharacteristic
63C2×C4×C137C22×C411yescharacteristic
64C22×C137—42no—
65C22×C137—42no—
66C22×C137—42no—
67C22×C137C2×C22⋊C411yescharacteristic
68C2×D137—21no—
69C2×D137—21no—
70C2×D137#10212yes—
71C4×C137—21no—
72C4×C137#10212yes—
73C4×C137#13112yes—
74Dic137—21no—
75Dic137—21no—
76Dic137—42no—
77Dic137C2×D811yescharacteristic
78Dic137C2×SD1611yescharacteristic
79C2×C137—82no—
80C2×C137#40811yescharacteristic
81C2×C137C2×#10211yescharacteristic
82C2×C137#42111yescharacteristic
83D137—22no—
84C137#148011yescharacteristic, complemented, Sylow
85#1480—1371nomaximal, Sylow
86#532—1371no—
87#431—1371no—
88C2×C4⋊C8—1371no—
89D4⋊C8—1374no—
90C2×#41—2741no—
91C22×C8—2741no—
92C4⋊Q8—1371no—
93#127#1058773911yescharacteristic
94#124—2741no—
95#121—2741no—
96#121—1372no—
97C4×D4—2741no—
98C4×D4—1372no—
99C2×C42—1371no—
100C4⋊C8—2741no—
101C4⋊C8—2741no—
102C4⋊C8—1372no—
103C22⋊C8—2741no—
104C4×C8—1372no—
105#41—5482no—
106#41—5482no—
107C2×Q8—2741no—
108C2×D4—21no—
109C2×D4—21no—
110C2×D4—2742no—
111C2×D4C2×#1058773912yes—
112C22×C4—2742no—
113C22×C4—1371no—
114C22×C4—1371no—
115C22×C4—1371no—
116C2×C8—2742no—
117C2×C8—2742no—
118C2×C8—5482no—
119C4⋊C4—2741no—
120C4⋊C4—1372no—
121C22⋊C4—2742no—
122C22⋊C4—2742no—
123C42—1371no—
124C42—1371no—
125C42—1371no—
126C42C2×#1058773911yescharacteristic
127C23—22no—
128C23—1371no—
129Q8—5482no—
130D4—24no—
131D4—42no—
132D4—42no—
133D4—5484no—
134C2×C4—2742no—
135C2×C4—2742no—
136C2×C4—5481no—
137C2×C4—5482no—
138C2×C4—5482no—
139C2×C4—1371no—
140C2×C4—1371no—
141C2×C4—1371no—
142C2×C4—1372no—
143C2×C4—1372no—
144C2×C4C22⋊#1058773911yescharacteristic
145C2×C4C22⋊#1058773911yescharacteristic
146C2×C4C22×#1058773911yescharacteristic
147C8—5482no—
148C8—5482no—
149C22—42no—
150C22—42no—
151C22—42no—
152C22—2741no—
153C22—2741no—
154C22—1372no—
155C22C2×C22⋊#1058773911yescharacteristic
156C4—21no—
157C4—2741no—
158C4—2741no—
159C4—5482no—
160C4—1371no—
161C4—1371no—
162C4#1798415012yes—
163C4#1798417512yes—
164C2—82no—
165C2—2742no—
166C2#2257837711yescharacteristic
167C2C2×#1798415011yescharacteristic
168C2#2257840111yescharacteristic
subgroupG/H
centerC22C2×C22⋊#10587739
commutatorC2×C4×C137C22×C4
FrattiniC2×C4C22×#10587739
Fitting#127×C137C4
socleC22×C137C2×C22⋊C4
prime radical#22577951C2
cosocleC4×Dic137C23
hypercenter#127#10587739
hypocenterC137#1480
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C42 ◁ #127

Complements

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

NH ≅ G/Ntype
1. #22578422C2semidirect
2. #22578404C2semidirect
3. D4⋊#11764047C2semidirect
84. C137#1480semidirect

Maximal subgroups

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

subgroupindex
#225784222
#225784042
D4⋊#117640472
#225779512
#1480137