Finite Group Databasebeta

← C2×#22577837 #31793910

Subgroups — C2×#22577837 #31793910

Order 17536.

400 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 400 conjugacy classes of subgroups — too many to draw. Only its 117 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×#17984107C211yesmaximal, characteristic, complemented
2C2×D8×D137C211yesmaximal, characteristic, complemented
3#22577837C218yesmaximal, direct product factor
4C2×#17984033C211yesmaximal, characteristic, complemented
5C2×#17984032C211yesmaximal, characteristic, complemented
6C2×#17984029C211yesmaximal, characteristic, complemented
7C2×#17984027C211yesmaximal, characteristic, complemented
8C2×D16×C137C211yesmaximal, characteristic, complemented
9C2×#14555509—21no—
10C2×D4×D137—21no—
11#17984107—22no—
12#17984107C2214yescomplemented
13C2×Dic548C2211yescharacteristic, complemented
14C2×#14555498—21no—
15C2×D1096C2211yescharacteristic
16C2×#14555495—21no—
17D8×D137—22no—
18D8×D137C2214yescomplemented
19C2×C8×D137C2211yescharacteristic
20#17984033C2214yescomplemented
21#17984032C2214yescomplemented
22#17984029C2214yescomplemented
23#17984027C2214yescomplemented
24C2×#14555485C2211yescharacteristic, complemented
25C2×D8×C137C2211yescharacteristic
26C2×D8×C137C2211yescharacteristic, complemented
27D16×C137C2214yescomplemented
28C2×C16×C137C2211yescharacteristic, complemented
29C23×D137—41no—
30#14555509—24no—
31#14555509—42no—
32C2×Dic274—21no—
33C2×D548—21no—
34C2×C22⋊D137—41no—
35C2×C22⋊D137—41no—
36D4×D137—24no—
37D4×D137—42no—
38C22×Dic137—41no—
39C2×C4×D137D411yescharacteristic
40Dic548—21no—
41Dic548C2312yescomplemented
42#14555498—24no—
43D1096—21no—
44D1096C2312yes—
45#14555495—24no—
46C2×#11764040D411yescharacteristic
47C8×D137D412yes—
48C8×D137C2312yes—
49#14555485C2312yescomplemented
50C2×D4×C137—21no—
51C2×D4×C137—21no—
52D8×C137—21no—
53D8×C137—21no—
54D8×C137C2312yes—
55D8×C137C2312yescomplemented
56C2×C8×C137C2311yescharacteristic
57C16×C137C2312yescomplemented
58C22×D137—41no—
59C22×D137—44no—
60C22×D137—84no—
61C22×D137D811yescharacteristic
62Dic274—22no—
63Dic274—41no—
64D548—22no—
65D548—41no—
66C22⋊D137—44no—
67C22⋊D137—44no—
68C2×Dic137—41no—
69C2×Dic137—44no—
70C2×Dic137D811yescharacteristic
71C4×D137D812yes—
72C4×D137C2×D412yes—
73#11764040C2×D412yes—
74C23×C137—41no—
75C23×C137—41no—
76D4×C137—22no—
77D4×C137—22no—
78D4×C137—41no—
79D4×C137—41no—
80C2×C4×C137C2×D411yescharacteristic
81C8×C137C2×D411yescharacteristic
82C8×C137C2411yescharacteristic
83C22×C137—42no—
84C22×C137—42no—
85C22×C137—82no—
86C22×C137—82no—
87C22×C137C2×D811yescharacteristic
88C2×D137—22no—
89C2×D137—42no—
90C2×D137—88no—
91C2×D137—82no—
92C2×D137C2×D812yes—
93C4×C137C2×D811yescharacteristic
94C4×C137C22×D411yescharacteristic
95Dic137—42no—
96Dic137C2×D812yes—
97C2×C137—82no—
98C2×C137—82no—
99C2×C137#50912yescomplemented
100C2×C137C22×D811yescharacteristic
101D137—22no—
102D137—82no—
103C137C2×#50911yescharacteristic, complemented, Sylow
104C2×#509—1371nomaximal, Sylow
105C2×#135—1371no—
106C22×D8—1371no—
107#509—1378no—
108C2×SD32—1371no—
109C2×SD32—1371no—
110C2×D16—1371no—
111C2×D16D13711yescharacteristic, complemented
112C2×M32—1371no—
113C2×#41—2741no—
114C22×D4—2741no—
115#135—2742no—
116#135—1374no—
117C2×Q16—1371no—
118C2×SD16—2741no—
119C2×D8—2742no—
120C2×D8—2741no—
121C2×D8—1374no—
122C2×D8—1371no—
123C2×D8C2×D13711yescharacteristic
124C2×D8C2×D13711yescharacteristic, complemented
125C22×C8—1371no—
126SD32—1374no—
127SD32—1374no—
128D16—1374no—
129D16C2×D13714yescomplemented
130M32—1374no—
131C2×C16—1371no—
132C2×C16C2×D13711yescharacteristic, complemented
133C24—5481no—
134#41—2744no—
135#41—5482no—
136C2×Q8—2741no—
137C2×D4—21no—
138C2×D4—21no—
139C2×D4—2744no—
140C2×D4—2741no—
141C2×D4—5481no—
142C2×D4—5482no—
143C2×D4—5481no—
144C22×C4—5481no—
145C22×C4—1371no—
146Q16—2741no—
147Q16—1372no—
148SD16—2744no—
149D8—21no—
150D8—21no—
151D8—2744no—
152D8—2741no—
153D8—1372no—
154D8C22×D13712yes—
155D8C22×D13712yescomplemented
156C2×C8—1371no—
157C2×C8—1372no—
158C2×C8—1372no—
159C2×C8C22×D13711yescharacteristic
160C16—1372no—
161C16C22×D13712yescomplemented
162C23—41no—
163C23—41no—
164C23—5484no—
165C23—5481no—
166C23—10964no—
167C23—1371no—
168Q8—2742no—
169Q8—5481no—
170D4—22no—
171D4—22no—
172D4—41no—
173D4—41no—
174D4—2742no—
175D4—5484no—
176D4—5484no—
177D4—5481no—
178C2×C4—5481no—
179C2×C4—5484no—
180C2×C4—1371no—
181C2×C4—1372no—
182C2×C4—1372no—
183C2×C4D4×D13711yescharacteristic
184C8—1372no—
185C8D4×D13711yescharacteristic
186C8C23×D13711yescharacteristic
187C22—42no—
188C22—42no—
189C22—82no—
190C22—82no—
191C22—2742no—
192C22—5482no—
193C22—10968no—
194C22—10962no—
195C22—1372no—
196C22D8×D13711yescharacteristic
197C4—5482no—
198C4—1372no—
199C4D8×D13711yescharacteristic
200C4C2×D4×D13711yescharacteristic
201C2—82no—
202C2—82no—
203C2—2742no—
204C2—10962no—
205C2#2257783712yesdirect product factor
206C2C2×D8×D13711yescharacteristic
subgroupG/H
centerC22D8×D137
commutator, cosocleC8×C137C24
FrattiniC8C23×D137
FittingC2×D16×C137C2
socleC22×C137C2×D8
hypercenterC2×D16D137
hypocenterC137C2×#509
LCSG ▷ C8×C137 ▷ C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×C8 ◁ C2×D16

Complements

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

NH ≅ G/Ntype
1. C2×#17984107C2semidirect
2. C2×D8×D137C2semidirect
3. #22577837C2direct (N × H)
4. C2×#17984033C2semidirect
5. C2×#17984032C2semidirect
6. C2×#17984029C2semidirect
7. C2×#17984027C2semidirect
8. C2×D16×C137C2semidirect
12. #17984107C22semidirect
13. C2×Dic548C22semidirect
18. D8×D137C22semidirect
20. #17984033C22semidirect
21. #17984032C22semidirect
22. #17984029C22semidirect
23. #17984027C22semidirect
24. C2×#14555485C22semidirect
26. C2×D8×C137C22semidirect
27. D16×C137C22semidirect
28. C2×C16×C137C22semidirect
41. Dic548C23semidirect
49. #14555485C23semidirect
55. D8×C137C23semidirect
57. C16×C137C23semidirect
99. C2×C137#509semidirect
103. C137C2×#509semidirect
111. C2×D16D137semidirect
124. C2×D8C2×D137semidirect
129. D16C2×D137semidirect
132. C2×C16C2×D137semidirect
155. D8C22×D137semidirect
161. C16C22×D137semidirect
205. C2#22577837direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#179841072
C2×D8×D1372
#225778372
C2×#179840332
C2×#179840322
C2×#179840292
C2×#179840272
C2×D16×C1372
C2×#509137