Finite Group Databasebeta

← C2×#22578145 #31794909

Subgroups — C2×#22578145 #31794909

Order 17536.

1408 conjugacy classes of subgroups; only the 469 normal subgroups are shown.

normal subgroups only

This group has 1408 conjugacy classes of subgroups — too many to draw. Only its 469 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#22578145C2116yesmaximal, direct product factor
2C2×#17984077C212yesmaximal, complemented
3C2×#17984076C212yesmaximal, complemented
4C2×#17984074C214yesmaximal, complemented
5C2×#17984058C214yesmaximal, complemented
6C2×#17984056C212yesmaximal, complemented
7C2×#124×C137C211yesmaximal, characteristic, complemented
8C24×D137C2212yescomplemented
9C22×Dic274C2212yescomplemented
10C22×D548C2212yes—
11C22×D548C2212yes—
12C22×C22⋊D137C2214yes—
13#17984077C22116yescomplemented
14#17984076C22116yescomplemented
15#17984074C22132yescomplemented
16#17984058C22132yescomplemented
17#17984056C22116yescomplemented
18C2×C22⋊Dic137C2212yescomplemented
19C2×#14555487C2214yescomplemented
20C2×#14555487C2214yescomplemented
21C2×#14555487C2214yescomplemented
22C2×C4×Dic137C2212yescomplemented
23C22×Q8×C137C2211yescharacteristic, complemented
24C22×D4×C137C2211yescharacteristic
25#124×C137C2218yescomplemented
26C2×C22⋊C4×C137C2214yescomplemented
27C2×C42×C137C2211yescharacteristic, complemented
28C23×D137—44no—
29C23×D137—48no—
30C23×D137C2314yes—
31C23×D137C2318yescomplemented
32C2×Dic274—44no—
33C2×Dic274C2318yescomplemented
34C2×D548—44no—
35C2×D548—44no—
36C2×D548C2318yes—
37C2×D548C2318yes—
38C2×C22⋊D137—48no—
39C2×C22⋊D137C23116yes—
40C22×Dic137C2314yes—
41C22⋊Dic137C2318yescomplemented
42#14555487C23116yescomplemented
43#14555487C23116yescomplemented
44#14555487C23116yescomplemented
45C4×Dic137C2318yescomplemented
46C24×C137C2312yes—
47C2×Q8×C137—42no—
48C2×Q8×C137C2314yescomplemented
49C2×D4×C137—42no—
50C2×D4×C137C2314yes—
51C22×C4×C137C2312yes—
52C22×C4×C137C2311yescharacteristic
53C22×C4×C137C2312yes—
54C22⋊C4×C137C23116yescomplemented
55C42×C137C2314yescomplemented
56C22×D137—28no—
57C22×D137—28no—
58C22×D137—28no—
59C22×D137—416no—
60C22×D137—48no—
61C22×D137—432no—
62C22×D137C2418yes—
63Dic274—48no—
64D548—48no—
65D548—48no—
66C22⋊D137—416no—
67C2×Dic137—28no—
68C2×Dic137C2418yes—
69C23×C137—24no—
70C23×C137—28no—
71C23×C137C2414yes—
72C23×C137C2411yescharacteristic
73Q8×C137—44no—
74D4×C137—44no—
75C2×C4×C137—22no—
76C2×C4×C137—24no—
77C2×C4×C137—24no—
78C2×C4×C137C2414yes—
79C2×C4×C137C2412yes—
80C2×C4×C137C2414yes—
81C22×C137—24no—
82C22×C137—28no—
83C22×C137—48no—
84C22×C137H(5,𝔽2)12yes—
85C22×C137H(5,𝔽2)14yes—
86C22×C137C2511yescharacteristic
87C2×D137—28no—
88C2×D137—28no—
89C2×D137—28no—
90C2×D137—432no—
91C2×D137—416no—
92C4×C137—22no—
93C4×C137—24no—
94C4×C137—24no—
95Dic137—28no—
96C2×C137—44no—
97C2×C137H(3,𝔽4)14yescomplemented
98C2×C137C2×H(5,𝔽2)11yescharacteristic
99C2×C137C2×H(5,𝔽2)12yes—
100D137—48no—
101C137C2×H(3,𝔽4)11yescharacteristic, complemented, Sylow
102C2×H(3,𝔽4)—1371nomaximal, Sylow
103H(3,𝔽4)—13716no—
104C2×#124—1372no—
105C2×#124—1374no—
106C2×#124—1372no—
107C2×#124D13711yescharacteristic, complemented
108C2×#120—1372no—
109C2×#120—1374no—
110C25—1372no—
111C22×Q8—1372no—
112C22×Q8C2×D13711yescharacteristic, complemented
113C22×D4—1374no—
114C22×D4—1372no—
115C22×D4—1372no—
116C22×D4C2×D13711yescharacteristic
117#124—13716no—
118#124—13732no—
119#124—13716no—
120#124C2×D13718yescomplemented
121#120—13716no—
122#120—13732no—
123C2×C22⋊C4—1372no—
124C2×C22⋊C4—1374no—
125C2×C22⋊C4—1374no—
126C2×C22⋊C4—1374no—
127C2×C22⋊C4C2×D13714yescomplemented
128C2×C42—1372no—
129C2×C42C2×D13711yescharacteristic, complemented
130C24—5484no—
131C24—5488no—
132C24—1374no—
133C24—1378no—
134C24C22×D13712yes—
135C2×Q8—42no—
136C2×Q8—5484no—
137C2×Q8—1378no—
138C2×Q8C22×D13714yescomplemented
139C2×D4—42no—
140C2×D4—5484no—
141C2×D4—5484no—
142C2×D4—5488no—
143C2×D4—13716no—
144C2×D4—1378no—
145C2×D4—1378no—
146C2×D4C22×D13714yes—
147C22×C4—1374no—
148C22×C4C22×D13712yes—
149C22×C4C22×D13711yescharacteristic
150C22×C4C22×D13712yes—
151C22⋊C4—1378no—
152C22⋊C4—13716no—
153C22⋊C4—13716no—
154C22⋊C4—13716no—
155C22⋊C4C22×D137116yescomplemented
156C42—1378no—
157C42C22×D13714yescomplemented
158C23—24no—
159C23—28no—
160C23—2748no—
161C23—2748no—
162C23—2748no—
163C23—5488no—
164C23—54816no—
165C23—54832no—
166C23—1378no—
167C23C23×D13714yes—
168C23C23×D13711yescharacteristic
169Q8—44no—
170Q8—5488no—
171D4—44no—
172D4—5488no—
173D4—5488no—
174D4—54816no—
175C2×C4—22no—
176C2×C4—24no—
177C2×C4—24no—
178C2×C4—2748no—
179C2×C4—1378no—
180C2×C4C23×D13714yes—
181C2×C4C23×D13712yes—
182C2×C4C23×D13714yes—
183C22—24no—
184C22—28no—
185C22—48no—
186C22—2748no—
187C22—2748no—
188C22—2748no—
189C22—54832no—
190C22—54816no—
191C22#1798413312yes—
192C22#1798413414yes—
193C22C24×D13711yescharacteristic
194C4—22no—
195C4—24no—
196C4—24no—
197C4—2748no—
198C2—44no—
199C2—5488no—
200C2#2257814514yesdirect product factor
201C2C2×#1798413311yescharacteristic
202C2C2×#1798413412yes—
subgroupG/H
centerC23C23×D137
commutator, cosocleC22×C137C25
FrattiniC22C24×D137
FittingC2×#124×C137C2
socleC23×C137C24
hypercenterC2×#124D137
hypocenterC137C2×H(3,𝔽4)
LCSG ▷ C22×C137 ▷ C137
UCSC1 ◁ C23 ◁ C2×#124

Complements

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

NH ≅ G/Ntype
1. #22578145C2direct (N × H)
2. C2×#17984077C2semidirect
3. C2×#17984076C2semidirect
4. C2×#17984074C2semidirect
5. C2×#17984058C2semidirect
6. C2×#17984056C2semidirect
7. C2×#124×C137C2semidirect
8. C24×D137C22semidirect
9. C22×Dic274C22semidirect
13. #17984077C22semidirect
14. #17984076C22semidirect
15. #17984074C22semidirect
16. #17984058C22semidirect
17. #17984056C22semidirect
18. C2×C22⋊Dic137C22semidirect
19. C2×#14555487C22semidirect
20. C2×#14555487C22semidirect
21. C2×#14555487C22semidirect
22. C2×C4×Dic137C22semidirect
23. C22×Q8×C137C22semidirect
25. #124×C137C22semidirect
26. C2×C22⋊C4×C137C22semidirect
27. C2×C42×C137C22semidirect
31. C23×D137C23semidirect
33. C2×Dic274C23semidirect
41. C22⋊Dic137C23semidirect
42. #14555487C23semidirect
43. #14555487C23semidirect
44. #14555487C23semidirect
45. C4×Dic137C23semidirect
48. C2×Q8×C137C23semidirect
54. C22⋊C4×C137C23semidirect
55. C42×C137C23semidirect
97. C2×C137H(3,𝔽4)semidirect
101. C137C2×H(3,𝔽4)semidirect
107. C2×#124D137semidirect
112. C22×Q8C2×D137semidirect
120. #124C2×D137semidirect
127. C2×C22⋊C4C2×D137semidirect
129. C2×C42C2×D137semidirect
138. C2×Q8C22×D137semidirect
155. C22⋊C4C22×D137semidirect
157. C42C22×D137semidirect
200. C2#22578145direct (N × H)

Maximal subgroups

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

subgroupindex
#225781452
C2×#179840772
C2×#179840762
C2×#179840742
C2×#179840582
C2×#179840562
C2×#124×C1372
C2×H(3,𝔽4)137