Finite Group Databasebeta

← C2×#22578444 #31797542

Subgroups — C2×#22578444 #31797542

Order 17536.

608 conjugacy classes of subgroups; only the 170 normal subgroups are shown.

normal subgroups only

This group has 608 conjugacy classes of subgroups — too many to draw. Only its 170 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×D4×#10587739C212yesmaximal, complemented
2#22578444C218yesmaximal, direct product factor
3C2×#17984156C211yesmaximal, characteristic, complemented
4C2×#17984150C212yesmaximal, complemented
5C2×#17984142C211yesmaximal, characteristic, complemented
6C2×D8×D137C211yesmaximal, characteristic
7C23×#10587739—22no—
8C2×#14555518C2211yescharacteristic
9D4×#10587739—24no—
10D4×#10587739C2218yescomplemented
11C2×C4⋊#10587739C2212yes—
12C2×C22⋊#10587739—22no—
13C2×C4×#10587739C2211yescharacteristic
14#17984156C2214yescomplemented
15#17984150C2218yescomplemented
16#17984142C2214yescomplemented
17C2×D4×D137C2212yes—
18C2×D1096C411yescharacteristic, complemented
19C2×#14555495C412yescomplemented
20D8×D137C414yescomplemented
21D8×D137C2214yes—
22C2×C8×D137C2211yescharacteristic
23C2×D8×C137C411yescharacteristic, complemented
24C22×#10587739—28no—
25C22×#10587739—22no—
26C22×#10587739—48no—
27C22×#10587739D412yes—
28#14555518—21no—
29#14555518C2312yes—
30C2×#11764047—21no—
31C4⋊#10587739—22no—
32C4⋊#10587739C2314yes—
33C22⋊#10587739—28no—
34C4×#10587739D412yes—
35C4×#10587739C2312yes—
36C23×D137—22no—
37C2×D548C2×C412yescomplemented
38C2×C22⋊D137—22no—
39D4×D137—22no—
40D4×D137—22no—
41D4×D137C2×C414yescomplemented
42D4×D137C2314yes—
43C2×C4×D137C2311yescharacteristic
44D1096C2×C414yescomplemented
45#14555495C2×C418yescomplemented
46C2×#11764040C2×C411yescharacteristic, complemented
47C8×D137C2×C412yescomplemented
48C8×D137C2312yes—
49C2×D4×C137C2×C412yescomplemented
50D8×C137C2×C414yescomplemented
51C2×C8×C137C2×C411yescharacteristic, complemented
52C2×#10587739—24no—
53C2×#10587739—24no—
54C2×#10587739—416no—
55C2×#10587739—44no—
56C2×#10587739C2×D414yes—
57#11764047—22no—
58C22×D137—22no—
59C22×D137—24no—
60C22×D137—24no—
61C22×D137—44no—
62C22×D137—44no—
63C22×D137C2×D411yescharacteristic
64D548—22no—
65D548C22×C414yescomplemented
66C22⋊D137—24no—
67C22⋊D137—24no—
68C2×Dic137#4111yescharacteristic
69C4×D137C22×C411yescharacteristic
70C4×D137C2×D411yescharacteristic
71C4×D137#4111yescharacteristic
72C4×D137C2411yescharacteristic
73#11764040C22×C412yescomplemented
74C23×C137—22no—
75D4×C137—22no—
76D4×C137C22×C414yescomplemented
77C2×C4×C137C22×C411yescharacteristic
78C8×C137C22×C412yescomplemented
79C22×C137—24no—
80C22×C137—44no—
81C22×C137C4×D411yescharacteristic
82C2×D137—24no—
83C2×D137—44no—
84C2×D137—44no—
85C2×D137—44no—
86C2×D137—48no—
87C2×D137C4×D411yescharacteristic
88C2×D137AGL(1,ℤ/8)14yes—
89C2×D137C22×D411yescharacteristic
90#10587739—24no—
91#10587739—44no—
92C4×C137C4×D411yescharacteristic
93C4×C137C23×C411yescharacteristic
94Dic137C4×D411yescharacteristic
95Dic137C2×#4111yescharacteristic
96C2×C137—44no—
97C2×C137#44212yescomplemented
98C2×C137C2×C4×D411yescharacteristic
99D137—44no—
100D137#44212yescomplemented
101D137C2×AGL(1,ℤ/8)12yes—
102C137C2×#44211yescharacteristic, complemented, Sylow
103C2×#442—1371nomaximal, Sylow
104C22×D8—1371no—
105C2×C4×D4—1372no—
106#442—1378no—
107C2×#106—1371no—
108C2×#102—1372no—
109C2×#97—1371no—
110C22×D4—1372no—
111C23×C4—2742no—
112C2×D8—1372no—
113C2×D8—1374no—
114C2×D8—1374no—
115C2×D8—1371no—
116C2×D8#1058773911yescharacteristic, complemented
117C22×C8—1371no—
118C22×C8—1371no—
119C4×D4—2744no—
120C4×D4—1378no—
121C2×C4⋊C4—1372no—
122C2×C22⋊C4—2742no—
123C2×C42—1371no—
124#106—1374no—
125#102—1378no—
126#97—1374no—
127C24—2742no—
128C2×D4—2742no—
129C2×D4—2742no—
130C2×D4—2742no—
131C2×D4—1374no—
132C2×D4—1374no—
133C2×D4—1372no—
134C2×D4C2×#1058773912yescomplemented
135C22×C4—2742no—
136C22×C4—2748no—
137C22×C4—5488no—
138C22×C4—1372no—
139C22×C4—1371no—
140D8—1378no—
141D8—1374no—
142D8C2×#1058773914yescomplemented
143C2×C8—2741no—
144C2×C8—2741no—
145C2×C8—1372no—
146C2×C8—1371no—
147C2×C8—1372no—
148C2×C8—1372no—
149C2×C8C2×#1058773911yescharacteristic, complemented
150C4⋊C4—2742no—
151C4⋊C4—1374no—
152C22⋊C4—2748no—
153C42—1372no—
154C42—1372no—
155C23—22no—
156C23—2744no—
157C23—2744no—
158C23—2742no—
159C23—5484no—
160C23—5484no—
161C23—1371no—
162D4—22no—
163D4—2744no—
164D4—2744no—
165D4—2742no—
166D4—1374no—
167D4C22×#1058773914yescomplemented
168C2×C4—2744no—
169C2×C4—2744no—
170C2×C4—54816no—
171C2×C4—5484no—
172C2×C4—1374no—
173C2×C4—1371no—
174C2×C4—1371no—
175C2×C4—1371no—
176C2×C4—1371no—
177C2×C4—1371no—
178C2×C4C22×#1058773911yescharacteristic
179C8—2742no—
180C8—1372no—
181C8C22×#1058773912yescomplemented
182C22—24no—
183C22—44no—
184C22—2744no—
185C22—5484no—
186C22—5484no—
187C22—5488no—
188C22—5484no—
189C22—1374no—
190C22—1371no—
191C22—1371no—
192C22D4×#1058773911yescharacteristic
193C4—2744no—
194C4—5484no—
195C4—1371no—
196C4—1371no—
197C4D4×#1058773911yescharacteristic
198C4C23×#1058773911yescharacteristic
199C2—44no—
200C2—5484no—
201C2—1372no—
202C2—1372no—
203C2#2257844412yesdirect product factor
204C2C2×D4×#1058773911yescharacteristic
subgroupG/H
centerC22D4×#10587739
commutatorC4×C137C23×C4
FrattiniC4C23×#10587739
FittingC2×D8×C137C4
socleC22×C137C4×D4
prime radicalC2×D8×D137C2
cosocleC4×D137C24
hypercenterC2×D8#10587739
hypocenterC137C2×#442
LCSG ▷ C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×D8

Complements

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

NH ≅ G/Ntype
1. C2×D4×#10587739C2semidirect
2. #22578444C2direct (N × H)
3. C2×#17984156C2semidirect
4. C2×#17984150C2semidirect
5. C2×#17984142C2semidirect
10. D4×#10587739C22semidirect
14. #17984156C22semidirect
15. #17984150C22semidirect
16. #17984142C22semidirect
18. C2×D1096C4semidirect
19. C2×#14555495C4semidirect
20. D8×D137C4semidirect
23. C2×D8×C137C4semidirect
37. C2×D548C2×C4semidirect
41. D4×D137C2×C4semidirect
44. D1096C2×C4semidirect
45. #14555495C2×C4semidirect
46. C2×#11764040C2×C4semidirect
47. C8×D137C2×C4semidirect
49. C2×D4×C137C2×C4semidirect
50. D8×C137C2×C4semidirect
51. C2×C8×C137C2×C4semidirect
65. D548C22×C4semidirect
73. #11764040C22×C4semidirect
76. D4×C137C22×C4semidirect
78. C8×C137C22×C4semidirect
97. C2×C137#442semidirect
100. D137#442semidirect
102. C137C2×#442semidirect
116. C2×D8#10587739semidirect
134. C2×D4C2×#10587739semidirect
142. D8C2×#10587739semidirect
149. C2×C8C2×#10587739semidirect
167. D4C22×#10587739semidirect
181. C8C22×#10587739semidirect
203. C2#22578444direct (N × H)

Maximal subgroups

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

subgroupindex
C2×D4×#105877392
#225784442
C2×#179841562
C2×#179841502
C2×#179841422
C2×D8×D1372
C2×#442137