Finite Group Databasebeta

← #31796182

Subgroups — #31796182

Order 17536.

228 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 58 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#22578426C211yesmaximal, characteristic
2#22578339C211yesmaximal, characteristic
3#22578257C212yesmaximal
4#22578256C212yesmaximal
5#22577957C211yesmaximal, characteristic
6C2×C4×#10587739—21no—
7#17984155—21no—
8#17984145C2212yes—
9#17984143—21no—
10(C2×#10587739)⋊C4—21no—
11(C2×#10587739)⋊C4C2212yes—
12#17984087C411yescharacteristic, complemented
13#17984086C411yescharacteristic, complemented
14#17984078—21no—
15#17984068C2211yescharacteristic
16#17984066—21no—
17#17984064C2211yescharacteristic
18C4×Dic274—21no—
19C4×Dic274C411yescharacteristic, complemented
20#17984045—21no—
21C42×D137C2211yescharacteristic
22C4×Q8×C137C411yescharacteristic, complemented
23C22×#10587739—22no—
24C2×#11764047—22no—
25C4×#10587739—24no—
26C2×Dic274—21no—
27C2×Dic274C2×C411yescharacteristic
28C2×C4×D137D411yescharacteristic
29C2×C4×D137D411yescharacteristic
30C2×C4×D137C2311yescharacteristic
31C4⋊Dic137—21no—
32C4⋊Dic137C2×C411yescharacteristic
33#14555489—21no—
34#14555489—21no—
35#14555489—21no—
36#14555489—21no—
37#14555489—21no—
38#14555487—21no—
39#14555487—21no—
40C4×Dic137—21no—
41C4×Dic137C2×C411yescharacteristic
42C4×Dic137D411yescharacteristic
43C4×Dic137D411yescharacteristic
44C2×Q8×C137C2×C411yescharacteristic
45C4⋊C4×C137—21no—
46C4⋊C4×C137C2×C411yescharacteristic
47C42×C137—21no—
48C42×C137C2×C411yescharacteristic
49C2×#10587739—24no—
50C2×#10587739—42no—
51C2×#10587739—42no—
52#11764047—42no—
53C22×D137C2×D411yescharacteristic
54Dic274—42no—
55Dic274—41no—
56C2×Dic137—21no—
57C2×Dic137—21no—
58C2×Dic137C22⋊C411yescharacteristic
59C2×Dic137C22⋊C411yescharacteristic
60C2×Dic137C2×D411yescharacteristic
61C4×D137—22no—
62C4×D137—22no—
63C4×D137—41no—
64Q8×C137—41no—
65C2×C4×C137—21no—
66C2×C4×C137—21no—
67C2×C4×C137C22⋊C411yescharacteristic
68C2×C4×C137C22⋊C411yescharacteristic
69C2×C4×C137C22×C411yescharacteristic
70C22×C137C2×C22⋊C411yescharacteristic
71C2×D137—21no—
72C2×D137—21no—
73C2×D137#13712yes—
74#10587739—44no—
75C4×C137—21no—
76C4×C137—21no—
77C4×C137—21no—
78C4×C137—41no—
79C4×C137—41no—
80Dic137—21no—
81Dic137—21no—
82Dic137—41no—
83Dic137—41no—
84Dic137#10412yes—
85C2×C137#41011yescharacteristic
86C2×C137#41911yescharacteristic
87C2×C137C2×#10411yescharacteristic
88D137—22no—
89C137#150611yescharacteristic, complemented, Sylow
90#1506—1371nomaximal, Sylow
91#533—1371no—
92#432—1371no—
93#376—1371no—
94#328—1372no—
95#328—1372no—
96C4⋊Q8—1371no—
97C4⋊Q8—1371no—
98#125—2741no—
99C22⋊Q8—2741no—
100C22⋊Q8—1371no—
101C22⋊Q8—1371no—
102C4×Q8—2741no—
103C4×Q8—1371no—
104C4×Q8#1058773911yescharacteristic, complemented
105#117—2741no—
106C2×C42—2741no—
107C2×C42—1371no—
108C4⋊C8—2741no—
109C22⋊C8—1372no—
110#97—2741no—
111#95—2741no—
112#95—1372no—
113C2×Q8—2741no—
114C2×Q8—1371no—
115C2×Q8C2×#1058773911yescharacteristic
116C22×C4—2742no—
117C22×C4—1371no—
118C22×C4—1371no—
119C22×C4—1371no—
120C2×C8—2742no—
121C4⋊C4—21no—
122C4⋊C4—2741no—
123C4⋊C4—2741no—
124C4⋊C4—2741no—
125C4⋊C4—2741no—
126C4⋊C4—2741no—
127C4⋊C4—2741no—
128C4⋊C4—1371no—
129C4⋊C4C2×#1058773911yescharacteristic
130C22⋊C4—2741no—
131C22⋊C4—2741no—
132C42—21no—
133C42—2744no—
134C42—2741no—
135C42—1371no—
136C42—1371no—
137C42—1371no—
138C42C2×#1058773911yescharacteristic
139C23—1371no—
140Q8—41no—
141Q8—5481no—
142Q8—5482no—
143C2×C4—21no—
144C2×C4—21no—
145C2×C4—2742no—
146C2×C4—2742no—
147C2×C4—2744no—
148C2×C4—2741no—
149C2×C4—2741no—
150C2×C4—5481no—
151C2×C4—5482no—
152C2×C4—5482no—
153C2×C4—1371no—
154C2×C4—1371no—
155C2×C4—1371no—
156C2×C4C22⋊#1058773911yescharacteristic
157C2×C4C22⋊#1058773911yescharacteristic
158C2×C4C22×#1058773911yescharacteristic
159C8—5482no—
160C22—2741no—
161C22—2741no—
162C22—1372no—
163C22C2×C22⋊#1058773911yescharacteristic
164C4—21no—
165C4—21no—
166C4—21no—
167C4—41no—
168C4—41no—
169C4—2741no—
170C4—2741no—
171C4—5484no—
172C4—5481no—
173C4—5481no—
174C4—1372no—
175C2—2742no—
176C2#2257838211yescharacteristic
177C2#2257839611yescharacteristic
178C2D137⋊#10411yescharacteristic
subgroupG/H
centerC22C2×C22⋊#10587739
commutatorC2×C4×C137C22×C4
FrattiniC2×C4C22×#10587739
FittingC4×Q8×C137C4
socleC22×C137C2×C22⋊C4
prime radicalC22×D137C2×D4
cosocleC2×C4×D137C23
hypercenterC4×Q8#10587739
hypocenterC137#1506
LCSG ▷ C2×C4×C137 ▷ C22×C137 ▷ C137
UCSC1 ◁ C22 ◁ C42 ◁ C4×Q8

Complements

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

NH ≅ G/Ntype
12. #17984087C4semidirect
13. #17984086C4semidirect
19. C4×Dic274C4semidirect
22. C4×Q8×C137C4semidirect
89. C137#1506semidirect
104. C4×Q8#10587739semidirect

Maximal subgroups

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

subgroupindex
#225784262
#225783392
#225782572
#225782562
#225779572
#1506137