Finite Group Databasebeta

← #31796151

Subgroups — #31796151

Order 17536.

202 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 202 conjugacy classes of subgroups — too many to draw. Only its 60 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#22578424C211yesmaximal, characteristic
2#22578419C211yesmaximal, characteristic
3#22578253C212yesmaximal, complemented
4Q8⋊#11764047C212yesmaximal, complemented
5#22577955C211yesmaximal, characteristic
6C4⋊#11764047—21no—
7C4⋊#11764047C2212yes—
8C22⋊#11764047—21no—
9C22⋊#11764047—21no—
10#17984143—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
23C2×#11764047—22no—
24C2×#11764047—22no—
25C2×Dic274—21no—
26C2×Dic274C2×C411yescharacteristic
27C22×Dic137D411yescharacteristic
28C22×Dic137D411yescharacteristic
29C4⋊Dic137—21no—
30C4⋊Dic137C2×C411yescharacteristic
31#14555489—21no—
32#14555489—21no—
33#14555489—21no—
34#14555489—21no—
35#14555489—21no—
36C22⋊Dic137—21no—
37C4×Dic137—21no—
38C4×Dic137—21no—
39C4×Dic137C2×C411yescharacteristic
40C4×Dic137D411yescharacteristic
41C4×Dic137D411yescharacteristic
42C4×Dic137C2311yescharacteristic
43C2×Q8×C137C2×C411yescharacteristic
44C22×C4×C137C2×C411yescharacteristic
45C4⋊C4×C137—21no—
46C4⋊C4×C137C2×C411yescharacteristic
47C22⋊C4×C137—21no—
48#11764047—42no—
49#11764047—42no—
50Dic274—22no—
51Dic274—42no—
52C2×Dic137—22no—
53C2×Dic137—21no—
54C2×Dic137—21no—
55C2×Dic137—41no—
56C2×Dic137C22⋊C411yescharacteristic
57C2×Dic137C22⋊C411yescharacteristic
58C2×Dic137C2×D411yescharacteristic
59C2×Dic137C2×D411yescharacteristic
60C23×C137C22⋊C411yescharacteristic
61Q8×C137—22no—
62C2×C4×C137—22no—
63C2×C4×C137—21no—
64C2×C4×C137—21no—
65C2×C4×C137C22⋊C411yescharacteristic
66C2×C4×C137C22×C411yescharacteristic
67C22×C137—21no—
68C22×C137—21no—
69C22×C137—21no—
70C22×C137C2×C22⋊C411yescharacteristic
71C4×C137—21no—
72C4×C137—41no—
73C4×C137—41no—
74C4×C137#13112yes—
75Dic137—21no—
76Dic137—21no—
77Dic137—21no—
78Dic137—41no—
79Dic137—41no—
80Dic137#13712yes—
81C2×C137—41no—
82C2×C137#40811yescharacteristic
83C2×C137#41911yescharacteristic
84C2×C137#42111yescharacteristic
85C137#149711yescharacteristic, complemented, Sylow
86#1497—1371nomaximal, Sylow
87#533—1371no—
88#432—1371no—
89#431—1371no—
90Q8⋊C8—1372no—
91Q8⋊C8—1372no—
92C4⋊Q8—1371no—
93C4⋊Q8—1371no—
94#125—2741no—
95C22⋊Q8—2741no—
96C22⋊Q8—1371no—
97C22⋊Q8#1058773911yescharacteristic
98C4×Q8—2741no—
99C4×Q8—1371no—
100C4×Q8—1371no—
101#117—2741no—
102C2×C42—1371no—
103C4⋊C8—2741no—
104C4⋊C8—1372no—
105C22⋊C8—2741no—
106C22⋊C8—2741no—
107#97—2741no—
108C4×C8—1372no—
109C2×Q8—2741no—
110C2×Q8—1371no—
111C2×Q8C2×#1058773911yescharacteristic
112C22×C4—1371no—
113C22×C4—1371no—
114C22×C4C2×#1058773911yescharacteristic
115C2×C8—2742no—
116C2×C8—2742no—
117C4⋊C4—21no—
118C4⋊C4—2741no—
119C4⋊C4—2741no—
120C4⋊C4—2741no—
121C4⋊C4—2741no—
122C4⋊C4—2741no—
123C4⋊C4—2741no—
124C4⋊C4—1371no—
125C4⋊C4C2×#1058773911yescharacteristic
126C22⋊C4—21no—
127C22⋊C4—2741no—
128C42—2741no—
129C42—2741no—
130C42—1371no—
131C42—1371no—
132C42—1371no—
133C42—1371no—
134C23C22⋊#1058773911yescharacteristic
135Q8—22no—
136Q8—2742no—
137Q8—5482no—
138C2×C4—22no—
139C2×C4—21no—
140C2×C4—21no—
141C2×C4—2742no—
142C2×C4—2741no—
143C2×C4—2741no—
144C2×C4—5481no—
145C2×C4—1371no—
146C2×C4—1371no—
147C2×C4—1371no—
148C2×C4—1371no—
149C2×C4C22⋊#1058773911yescharacteristic
150C2×C4C22×#1058773911yescharacteristic
151C8—5482no—
152C8—5482no—
153C22—21no—
154C22—21no—
155C22—21no—
156C22C2×C22⋊#1058773911yescharacteristic
157C4—21no—
158C4—41no—
159C4—41no—
160C4—2741no—
161C4—2741no—
162C4—2741no—
163C4—5481no—
164C4—5481no—
165C4—1372no—
166C4#1798417512yes—
167C2—41no—
168C2#2257837611yescharacteristic
169C2#2257839611yescharacteristic
170C2#2257840011yescharacteristic
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#1497
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
85. C137#1497semidirect

Maximal subgroups

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

subgroupindex
#225784242
#225784192
#225782532
Q8⋊#117640472
#225779552
#1497137