Finite Group Databasebeta

← #31796232

Subgroups — #31796232

Order 17536.

218 conjugacy classes of subgroups; only the 56 normal subgroups are shown.

normal subgroups only

This group has 218 conjugacy classes of subgroups — too many to draw. Only its 56 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#22578348C211yesmaximal, characteristic, complemented
3#22578256C212yesmaximal, complemented
4#22578255C212yesmaximal
5#22577938C211yesmaximal, characteristic
6C4⋊#11764047—21no—
7#17984145C2212yes—
8C4×#11764047—21no—
9(C2×#10587739)⋊C4—21no—
10(C2×#10587739)⋊C4—21no—
11(C2×#10587739)⋊C4C2212yes—
12#17984083—21no—
13#17984076C411yescharacteristic, complemented
14#17984073—21no—
15#17984068C2211yescharacteristic
16#17984063C2211yescharacteristic
17#17984053C411yescharacteristic, complemented
18#17984050—21no—
19#17984050C411yescharacteristic, complemented
20#17984046—21no—
21C42×D137C2211yescharacteristic
22#125×C137C411yescharacteristic, complemented
23C22×#10587739—22no—
24C2×#11764047—22no—
25C2×Dic274C2×C411yescharacteristic
26C2×D548—21no—
27C2×D548C2×C411yescharacteristic
28C2×C4×D137—21no—
29C2×C4×D137D411yescharacteristic
30C2×C4×D137D411yescharacteristic
31C2×C4×D137C2311yescharacteristic
32C4⋊Dic137—21no—
33#14555489—21no—
34#14555487—21no—
35#14555487—21no—
36#14555487—21no—
37#14555487—21no—
38#14555487—21no—
39C4×Dic137—21no—
40C4×Dic137C2×C411yescharacteristic
41C4×Dic137D411yescharacteristic
42C4×Dic137D411yescharacteristic
43C4⋊C4×C137—21no—
44C4⋊C4×C137—21no—
45C4⋊C4×C137C2×C411yescharacteristic
46C4⋊C4×C137C2×C411yescharacteristic
47C42×C137C2×C411yescharacteristic
48C2×#10587739—42no—
49C2×#10587739—42no—
50C2×#10587739—42no—
51#11764047—42no—
52C22×D137—21no—
53C22×D137C2×D411yescharacteristic
54Dic274—41no—
55D548—41no—
56D548—42no—
57C2×Dic137—21no—
58C2×Dic137C22⋊C411yescharacteristic
59C2×Dic137C22⋊C411yescharacteristic
60C2×Dic137C2×D411yescharacteristic
61C4×D137—22no—
62C4×D137—41no—
63C4×D137—41no—
64C4×D137—42no—
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—41no—
74C2×D137—41no—
75C2×D137—41no—
76C2×D137#13512yes—
77#10587739—82no—
78C4×C137—21no—
79C4×C137—21no—
80C4×C137—21no—
81C4×C137—41no—
82C4×C137—41no—
83Dic137—21no—
84Dic137—21no—
85Dic137—21no—
86Dic137—41no—
87C2×C137#41011yescharacteristic
88C2×C137#41611yescharacteristic
89C2×C137#42111yescharacteristic
90D137—22no—
91D137—81no—
92C137#152111yescharacteristic, complemented, Sylow
93#1521—1371nomaximal, Sylow
94#529—1371no—
95#431—1371no—
96#383—1371no—
97#328—1372no—
98#327—1372no—
99#126—2741no—
100#125#1058773911yescharacteristic, complemented
101#124—1371no—
102#123—2741no—
103C22⋊Q8—1371no—
104#121—1371no—
105C4×Q8—1371no—
106C4×D4—2741no—
107C4×D4—1371no—
108#117—2741no—
109C2×C42—1371no—
110C4⋊C8—2741no—
111C22⋊C8—1372no—
112C4×C8—2741no—
113#95—2741no—
114#95—2741no—
115#95—1372no—
116C2×Q8—1371no—
117C2×D4—2741no—
118C2×D4—1371no—
119C22×C4—2741no—
120C22×C4—2742no—
121C22×C4—1371no—
122C22×C4—1371no—
123C22×C4—1371no—
124C2×C8—2742no—
125C4⋊C4—21no—
126C4⋊C4—21no—
127C4⋊C4—2741no—
128C4⋊C4—2741no—
129C4⋊C4C2×#1058773911yescharacteristic
130C4⋊C4C2×#1058773911yescharacteristic
131C22⋊C4—2741no—
132C22⋊C4—2741no—
133C22⋊C4—2741no—
134C22⋊C4—2741no—
135C22⋊C4—2741no—
136C42—2741no—
137C42—1371no—
138C42—1371no—
139C42—1371no—
140C42C2×#1058773911yescharacteristic
141C23—2741no—
142C23—1371no—
143Q8—5481no—
144D4—5481no—
145D4—5482no—
146C2×C4—21no—
147C2×C4—21no—
148C2×C4—2742no—
149C2×C4—2741no—
150C2×C4—5481no—
151C2×C4—5481no—
152C2×C4—5482no—
153C2×C4—5482no—
154C2×C4—5482no—
155C2×C4—5482no—
156C2×C4—1371no—
157C2×C4—1371no—
158C2×C4—1371no—
159C2×C4C22⋊#1058773911yescharacteristic
160C2×C4C22⋊#1058773911yescharacteristic
161C2×C4C22×#1058773911yescharacteristic
162C8—5482no—
163C22—2741no—
164C22—2741no—
165C22—5481no—
166C22—5481no—
167C22—5481no—
168C22—1372no—
169C22C2×C22⋊#1058773911yescharacteristic
170C4—21no—
171C4—21no—
172C4—21no—
173C4—41no—
174C4—41no—
175C4—2741no—
176C4—2741no—
177C4—2741no—
178C4—5481no—
179C4—10962no—
180C2—2742no—
181C2—10961no—
182C2#2257838411yescharacteristic
183C2#2257839311yescharacteristic
184C2#2257840011yescharacteristic
subgroupG/H
centerC22C2×C22⋊#10587739
commutatorC2×C4×C137C22×C4
FrattiniC2×C4C22×#10587739
Fitting#125×C137C4
socleC22×C137C2×C22⋊C4
prime radical#17984063C22
cosocleC2×C4×D137C23
hypercenter#125#10587739
hypocenterC137#1521
LCSG ▷ C2×C4×C137 ▷ C22×C137 ▷ C137
UCSC1 ◁ C22 ◁ C42 ◁ #125

Complements

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

NH ≅ G/Ntype
1. #22578422C2semidirect
2. #22578348C2semidirect
3. #22578256C2semidirect
13. #17984076C4semidirect
17. #17984053C4semidirect
19. #17984050C4semidirect
22. #125×C137C4semidirect
92. C137#1521semidirect
100. #125#10587739semidirect

Maximal subgroups

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

subgroupindex
#225784222
#225783482
#225782562
#225782552
#225779382
#1521137