Finite Group Databasebeta

← #31797548

Subgroups — #31797548

Order 17536.

528 conjugacy classes of subgroups; only the 160 normal subgroups are shown.

normal subgroups only

This group has 528 conjugacy classes of subgroups — too many to draw. Only its 160 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#22578470C212yesmaximal, complemented
2#22578444C214yesmaximal, complemented
3D8×#10587739C214yesmaximal, complemented
4#22578413C211yesmaximal, characteristic, complemented
5#22578395C212yesmaximal, complemented
6#22578366C211yesmaximal, characteristic, complemented
7C2×D8×D137C211yesmaximal, characteristic
8D137⋊M16C2211yescharacteristic
9D4×#10587739—24no—
10D4×#10587739C2218yescomplemented
11#17984167C2212yes—
12#17984167C2211yescharacteristic
13C2×C22⋊#10587739—24no—
14#17984157C2212yescomplemented
15#17984156C2212yescomplemented
16#17984150C2218yescomplemented
17#17984142C2212yescomplemented
18C8×#10587739C2212yescomplemented
19C2×D4×D137C2212yes—
20C2×D1096C411yescharacteristic, complemented
21C2×#14555495C412yescomplemented
22D8×D137C412yescomplemented
23D8×D137C412yescomplemented
24D8×D137C2212yes—
25D8×D137C2212yes—
26C2×C8×D137C2211yescharacteristic
27C2×D8×C137C411yescharacteristic, complemented
28C22×#10587739—28no—
29#14555520—21no—
30#14555519—21no—
31#14555518C2312yes—
32C4⋊#10587739D412yes—
33C4⋊#10587739C2314yes—
34C22⋊#10587739—28no—
35C22⋊#10587739—22no—
36C22⋊#10587739D412yes—
37C4×#10587739—22no—
38C4×#10587739C2312yes—
39C23×D137—22no—
40C2×D548C2×C412yescomplemented
41C2×C22⋊D137—22no—
42D4×D137—22no—
43D4×D137—22no—
44D4×D137C2×C414yescomplemented
45D4×D137C2314yes—
46C2×C4×D137C2311yescharacteristic
47D1096C2×C412yescomplemented
48D1096C2×C412yescomplemented
49#14555495C2×C414yescomplemented
50#14555495C2×C414yescomplemented
51C2×#11764040C2×C411yescharacteristic, complemented
52C8×D137C2×C411yescharacteristic, complemented
53C8×D137C2×C411yescharacteristic, complemented
54C8×D137C2311yescharacteristic
55C8×D137C2311yescharacteristic
56C2×D4×C137C2×C412yescomplemented
57D8×C137C2×C412yescomplemented
58D8×C137C2×C412yescomplemented
59C2×C8×C137C2×C411yescharacteristic, complemented
60C2×#10587739—24no—
61C2×#10587739—88no—
62C2×#10587739C2×D414yes—
63#11764047—22no—
64C22×D137—22no—
65C22×D137—24no—
66C22×D137—24no—
67C22×D137—84no—
68C22×D137C2×D411yescharacteristic
69D548—22no—
70D548C22×C414yes—
71C22⋊D137—24no—
72C22⋊D137—24no—
73C2×Dic137#4111yescharacteristic
74C4×D137C22×C411yescharacteristic
75C4×D137C2×D411yescharacteristic
76C4×D137#4111yescharacteristic
77C4×D137C2411yescharacteristic
78#11764040C22×C411yescharacteristic
79#11764040C22×C411yescharacteristic
80C23×C137—22no—
81D4×C137—22no—
82D4×C137C22×C414yes—
83C2×C4×C137C22×C411yescharacteristic
84C8×C137C22×C411yescharacteristic
85C8×C137C22×C411yescharacteristic
86C22×C137—24no—
87C22×C137—82no—
88C22×C137C4×D411yescharacteristic
89C2×D137—22no—
90C2×D137—24no—
91C2×D137—48no—
92C2×D137—84no—
93C2×D137—82no—
94C2×D137C4×D411yescharacteristic
95C2×D137C22×D411yescharacteristic
96#10587739—24no—
97#10587739—44no—
98C4×C137C4×D411yescharacteristic
99C4×C137C23×C411yescharacteristic
100Dic137C4×D411yescharacteristic
101Dic137C2×#4111yescharacteristic
102C2×C137—21no—
103C2×C137—44no—
104C2×C137C2×C4×D411yescharacteristic
105D137—21no—
106D137—44no—
107D137#57612yes—
108C137#294811yescharacteristic, complemented, Sylow
109#2948—1371nomaximal, Sylow
110C22×D8—1371no—
111#518—1372no—
112#442—1374no—
113C4×D8—1374no—
114#427—1371no—
115#418—1372no—
116#405—1371no—
117C22×D4—1372no—
118C2×D8—1372no—
119C2×D8—1372no—
120C2×D8—1372no—
121C2×D8—1372no—
122C2×D8—1372no—
123C2×D8—1371no—
124C2×D8#1058773911yescharacteristic, complemented
125C2×M16—1371no—
126C22×C8—1371no—
127C4×D4—2744no—
128C4×D4—1378no—
129#117—1372no—
130#117—1371no—
131C2×C22⋊C4—2744no—
132#107—1372no—
133#106—1372no—
134#102—1378no—
135#97—1372no—
136C4×C8—1372no—
137C24—2742no—
138C2×D4—2742no—
139C2×D4—2742no—
140C2×D4—2742no—
141C2×D4—1374no—
142C2×D4—1374no—
143C2×D4—1372no—
144C2×D4C2×#1058773912yescomplemented
145C22×C4—2748no—
146C22×C4—1371no—
147D8—1374no—
148D8—1374no—
149D8—1372no—
150D8—1372no—
151D8C2×#1058773912yescomplemented
152D8C2×#1058773912yescomplemented
153M16—2741no—
154M16—2741no—
155C2×C8—1372no—
156C2×C8—1371no—
157C2×C8—1371no—
158C2×C8—1371no—
159C2×C8—1371no—
160C2×C8—1371no—
161C2×C8C2×#1058773911yescharacteristic, complemented
162C4⋊C4—1374no—
163C4⋊C4—1372no—
164C22⋊C4—2742no—
165C22⋊C4—2748no—
166C22⋊C4—1372no—
167C42—2742no—
168C42—1372no—
169C23—22no—
170C23—2744no—
171C23—2744no—
172C23—2742no—
173C23—10964no—
174C23—1371no—
175D4—22no—
176D4—2744no—
177D4—2744no—
178D4—2742no—
179D4—1374no—
180D4C22×#1058773914yes—
181C2×C4—2744no—
182C2×C4—10968no—
183C2×C4—1374no—
184C2×C4—1371no—
185C2×C4—1371no—
186C2×C4—1371no—
187C2×C4—1371no—
188C2×C4—1371no—
189C2×C4C22×#1058773911yescharacteristic
190C8—2742no—
191C8—1371no—
192C8—1371no—
193C8C22×#1058773911yescharacteristic
194C8C22×#1058773911yescharacteristic
195C22—24no—
196C22—82no—
197C22—2742no—
198C22—2744no—
199C22—5488no—
200C22—10962no—
201C22—10964no—
202C22—1371no—
203C22—1371no—
204C22D4×#1058773911yescharacteristic
205C4—2744no—
206C4—5484no—
207C4—1371no—
208C4—1371no—
209C4D4×#1058773911yescharacteristic
210C4C23×#1058773911yescharacteristic
211C2—21no—
212C2—44no—
213C2—2741no—
214C2—5484no—
215C2—1372no—
216C2C2×D4×#1058773911yescharacteristic
subgroupG/H
centerC2C2×D4×#10587739
commutatorC4×C137C23×C4
FrattiniC4C23×#10587739
FittingC2×D8×C137C4
socleC2×C137C2×C4×D4
prime radicalC2×D8×D137C2
cosocleC4×D137C24
hypercenterC2×D8#10587739
hypocenterC137#2948
LCSG ▷ C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×D8

Complements

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

NH ≅ G/Ntype
1. #22578470C2semidirect
2. #22578444C2semidirect
3. D8×#10587739C2semidirect
4. #22578413C2semidirect
5. #22578395C2semidirect
6. #22578366C2semidirect
10. D4×#10587739C22semidirect
14. #17984157C22semidirect
15. #17984156C22semidirect
16. #17984150C22semidirect
17. #17984142C22semidirect
18. C8×#10587739C22semidirect
20. C2×D1096C4semidirect
21. C2×#14555495C4semidirect
22. D8×D137C4semidirect
23. D8×D137C4semidirect
27. C2×D8×C137C4semidirect
40. C2×D548C2×C4semidirect
44. D4×D137C2×C4semidirect
47. D1096C2×C4semidirect
48. D1096C2×C4semidirect
49. #14555495C2×C4semidirect
50. #14555495C2×C4semidirect
51. C2×#11764040C2×C4semidirect
52. C8×D137C2×C4semidirect
53. C8×D137C2×C4semidirect
56. C2×D4×C137C2×C4semidirect
57. D8×C137C2×C4semidirect
58. D8×C137C2×C4semidirect
59. C2×C8×C137C2×C4semidirect
108. C137#2948semidirect
124. C2×D8#10587739semidirect
144. C2×D4C2×#10587739semidirect
151. D8C2×#10587739semidirect
152. D8C2×#10587739semidirect
161. C2×C8C2×#10587739semidirect

Maximal subgroups

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

subgroupindex
#225784702
#225784442
D8×#105877392
#225784132
#225783952
#225783662
C2×D8×D1372
#2948137