Finite Group Databasebeta

← #31789249

Subgroups — #31789249

Order 17536.

800 conjugacy classes of subgroups; only the 249 normal subgroups are shown.

normal subgroups only

This group has 800 conjugacy classes of subgroups — too many to draw. Only its 249 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
1C2×#17984133C211yesmaximal, characteristic
2#22577878C211yesmaximal, characteristic, complemented
3#22577376C214yesmaximal, complemented
4C2×#17984002C218yesmaximal, complemented
5#518×C137C211yesmaximal, characteristic, complemented
6#17984133—24no—
7#17984133—24no—
8C2×#14555510C2212yes—
9C2×#14555509C414yescomplemented
10C22×C22⋊D137C2214yes—
11C2×D4×D137C414yescomplemented
12D4×Dic137—24no—
13#17984047C2212yescomplemented
14C2×C22⋊Dic137C2214yescomplemented
15#17984002C22116yescomplemented
16C22×D4×C137C2211yescharacteristic
17C4×D4×C137—24no—
18#117×C137C2212yescomplemented
19C2×C22⋊C4×C137C2214yescomplemented
20C23×D137C2×C414yescomplemented
21#14555510—24no—
22#14555510—24no—
23#14555509—216no—
24C2×Dic274C2×C412yescomplemented
25C2×D548C2×C412yescomplemented
26C2×C22⋊D137—28no—
27C2×C22⋊D137—28no—
28C2×C22⋊D137C2×C418yescomplemented
29C2×C22⋊D137C2314yes—
30D4×D137—216no—
31C22×Dic137—24no—
32C22×Dic137C2×C414yescomplemented
33C2×C4×D137C2×C414yescomplemented
34C4⋊Dic137—22no—
35C22⋊Dic137—24no—
36C22⋊Dic137C2314yes—
37C4×Dic137—22no—
38C24×C137C2312yes—
39C2×D4×C137C2×C414yes—
40C2×D4×C137D414yes—
41C2×D4×C137D414yes—
42C22×C4×C137—24no—
43C22×C4×C137C2311yescharacteristic
44C4⋊C4×C137—22no—
45C22⋊C4×C137—24no—
46C22⋊C4×C137C2314yes—
47C42×C137—22no—
48C22×D137—28no—
49C22×D137—84no—
50C22×D137C22×C414yes—
51Dic274—24no—
52D548—24no—
53C22⋊D137—216no—
54C22⋊D137—24no—
55C22⋊D137—24no—
56C2×Dic137—28no—
57C2×Dic137—24no—
58C2×Dic137—84no—
59C2×Dic137C22×C414yes—
60C4×D137—28no—
61C23×C137—82no—
62C23×C137C22×C414yes—
63C23×C137C2×D414yes—
64C23×C137C2×D414yes—
65C23×C137C2411yescharacteristic
66D4×C137C22⋊C4116yes—
67C2×C4×C137—24no—
68C2×C4×C137—84no—
69C2×C4×C137C22×C412yes—
70C2×C4×C137C2×D412yes—
71C2×C4×C137C2×D412yes—
72C22×C137—44no—
73C22×C137—41no—
74C22×C137—44no—
75C22×C137—44no—
76C22×C137C2×C22⋊C418yes—
77C22×C137C23×C411yescharacteristic
78C22×C137C22×D411yescharacteristic
79C22×C137C22×D411yescharacteristic
80C2×D137—24no—
81C2×D137—44no—
82C2×D137—88no—
83C4×C137—44no—
84C4×C137C2×C22⋊C414yes—
85Dic137—24no—
86Dic137—44no—
87C2×C137—21no—
88C2×C137—21no—
89C2×C137—21no—
90C2×C137—28no—
91C2×C137C22×C22⋊C411yescharacteristic
92D137—44no—
93C137#288311yescharacteristic, complemented, Sylow
94#2883—1371nomaximal, Sylow
95C2×H(5,𝔽2)—1371no—
96#518—1371no—
97#518D13711yescharacteristic, complemented
98#410—1374no—
99C2×#99—1378no—
100H(5,𝔽2)—2744no—
101H(5,𝔽2)—2744no—
102C2×#41—1374no—
103C2×#41—1372no—
104C22×D4—1374no—
105C22×D4—1374no—
106C22×D4C2×D13711yescharacteristic
107C4×D4—24no—
108C4×D4—2744no—
109#117—1372no—
110#117C2×D13712yescomplemented
111C2×C22⋊C4—1374no—
112C2×C22⋊C4C2×D13714yescomplemented
113#99—13716no—
114C24—1374no—
115C24C22×D13712yes—
116#41—27416no—
117#41—2744no—
118#41—2744no—
119C2×Q8—1372no—
120C2×D4—2748no—
121C2×D4—2748no—
122C2×D4—27416no—
123C2×D4—1378no—
124C2×D4—1374no—
125C2×D4—1372no—
126C2×D4C4×D13714yes—
127C2×D4C22⋊D13714yes—
128C2×D4D54814yes—
129C22×C4—24no—
130C22×C4—2744no—
131C22×C4—1374no—
132C22×C4—1374no—
133C22×C4C22×D13711yescharacteristic
134C4⋊C4—22no—
135C4⋊C4—2742no—
136C22⋊C4—24no—
137C22⋊C4—2744no—
138C22⋊C4—1374no—
139C22⋊C4C22×D13714yes—
140C42—22no—
141C42—2742no—
142C23—82no—
143C23—2748no—
144C23—10964no—
145C23—1374no—
146C23C2×C4×D13714yes—
147C23C2×C22⋊D13714yes—
148C23C2×D54814yes—
149C23C23×D13711yescharacteristic
150Q8—2744no—
151D4—27416no—
152D4—2744no—
153D4—2744no—
154D4—2744no—
155D4#14555487116yes—
156C2×C4—24no—
157C2×C4—84no—
158C2×C4—2748no—
159C2×C4—2748no—
160C2×C4—2744no—
161C2×C4—10964no—
162C2×C4—1374no—
163C2×C4C2×C4×D13712yes—
164C2×C4C2×C22⋊D13712yes—
165C2×C4C2×D54812yes—
166C22—44no—
167C22—44no—
168C22—41no—
169C22—44no—
170C22—2744no—
171C22—5484no—
172C22—10968no—
173C22C2×#1455548718yes—
174C22C22×C4×D13711yescharacteristic
175C22C22×C22⋊D13711yescharacteristic
176C22C22×D54811yescharacteristic
177C4—44no—
178C4—2744no—
179C4—5484no—
180C4C2×#1455548714yes—
181C2—21no—
182C2—21no—
183C2—21no—
184C2—28no—
185C2—5484no—
186C2C22×#1455548711yescharacteristic
subgroupG/H
centerC2C22×#14555487
commutatorC22×C137C23×C4
FrattiniC23C23×D137
Fitting#518×C137C2
socleC2×C137C22×C22⋊C4
prime radicalC2×#17984133C2
cosocleC23×C137C24
hypercenter#518D137
hypocenterC137#2883
LCSG ▷ C22×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C2 ◁ C22×D4 ◁ #518

Complements

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

NH ≅ G/Ntype
2. #22577878C2semidirect
3. #22577376C2semidirect
4. C2×#17984002C2semidirect
5. #518×C137C2semidirect
9. C2×#14555509C4semidirect
11. C2×D4×D137C4semidirect
13. #17984047C22semidirect
14. C2×C22⋊Dic137C22semidirect
15. #17984002C22semidirect
18. #117×C137C22semidirect
19. C2×C22⋊C4×C137C22semidirect
20. C23×D137C2×C4semidirect
24. C2×Dic274C2×C4semidirect
25. C2×D548C2×C4semidirect
28. C2×C22⋊D137C2×C4semidirect
32. C22×Dic137C2×C4semidirect
33. C2×C4×D137C2×C4semidirect
93. C137#2883semidirect
97. #518D137semidirect
110. #117C2×D137semidirect
112. C2×C22⋊C4C2×D137semidirect

Maximal subgroups

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

subgroupindex
C2×#179841332
#225778782
#225773762
C2×#179840022
#518×C1372
#2883137