Finite Group Databasebeta

← #31797388

Subgroups — #31797388

Order 17536.

800 conjugacy classes of subgroups; only the 198 normal subgroups are shown.

normal subgroups only

This group has 800 conjugacy classes of subgroups — too many to draw. Only its 198 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#22578383C212yesmaximal, complemented
3#22578382C212yesmaximal, complemented
4D137⋊#99C214yesmaximal, complemented
5C2×#17984146C214yesmaximal, complemented
6C2×#17984133C211yesmaximal, characteristic
7D4×#10587739—28no—
8#17984167C2214yescomplemented
9C2×C22⋊#10587739C2218yescomplemented
10#17984147C2214yescomplemented
11#17984146C2214yescomplemented
12#17984146C2218yescomplemented
13#17984133—28no—
14C2×#14555510C411yescharacteristic, complemented
15C2×#14555510C2211yescharacteristic
16C2×#14555509C411yescharacteristic, complemented
17C2×#14555509C412yescomplemented
18C2×#14555509C2211yescharacteristic
19C22×C22⋊D137C412yescomplemented
20C22×C22⋊D137C2212yes—
21C2×D4×D137C411yescharacteristic, complemented
22C2×D4×D137C2211yescharacteristic
23C2×D4×D137C2212yes—
24C22×D4×C137C411yescharacteristic, complemented
25C22×#10587739—28no—
26C4⋊#10587739—24no—
27C22⋊#10587739—28no—
28C22⋊#10587739C2318yes—
29C4×#10587739—24no—
30C23×D137C2×C412yescomplemented
31C23×D137C2312yes—
32#14555510—24no—
33#14555510—24no—
34#14555509—28no—
35#14555509—24no—
36#14555509—24no—
37C2×Dic274C2×C411yescharacteristic, complemented
38C2×Dic274C2×C411yescharacteristic, complemented
39C2×D548C2×C411yescharacteristic
40C2×D548C2311yescharacteristic
41C2×C22⋊D137—28no—
42C2×C22⋊D137—28no—
43C2×C22⋊D137C2×C412yes—
44C2×C22⋊D137C2×C412yescomplemented
45C2×C22⋊D137C2×C412yescomplemented
46C2×C22⋊D137C2×C411yescharacteristic, complemented
47C2×C22⋊D137C2×C412yescomplemented
48C2×C22⋊D137C2312yes—
49C2×C22⋊D137C2311yescharacteristic
50D4×D137—28no—
51D4×D137—24no—
52D4×D137D418yes—
53C22×Dic137C2×C412yescomplemented
54C22×Dic137C2×C412yescomplemented
55C2×C4×D137C2×C411yescharacteristic, complemented
56C2×C4×D137C2×C412yescomplemented
57C2×C4×D137C2311yescharacteristic
58C24×C137C2×C412yescomplemented
59C2×D4×C137—24no—
60C2×D4×C137C2×C411yescharacteristic
61C2×D4×C137C2×C411yescharacteristic, complemented
62C2×D4×C137C2×C412yescomplemented
63C22×C4×C137C2×C411yescharacteristic, complemented
64C2×#10587739—28no—
65C2×#10587739—88no—
66C22×D137—24no—
67C22×D137—82no—
68C22×D137—82no—
69C22×D137C22×C411yescharacteristic
70C22×D137C22×C412yes—
71C22×D137C2×D418yes—
72C22×D137C2411yescharacteristic
73Dic274—22no—
74Dic274—22no—
75D548—22no—
76D548C22⋊C414yes—
77C22⋊D137—24no—
78C22⋊D137—24no—
79C22⋊D137—24no—
80C22⋊D137—24no—
81C22⋊D137—22no—
82C22⋊D137—22no—
83C22⋊D137C22⋊C418yes—
84C2×Dic137—24no—
85C2×Dic137—24no—
86C2×Dic137C22×C411yescharacteristic
87C2×Dic137C22×C412yes—
88C2×Dic137C22×C411yescharacteristic
89C4×D137—24no—
90C4×D137—22no—
91C4×D137C2×D414yes—
92C23×C137—24no—
93C23×C137—82no—
94C23×C137C22×C411yescharacteristic
95C23×C137C22×C412yes—
96C23×C137C22×C412yes—
97D4×C137—24no—
98D4×C137—22no—
99D4×C137C22⋊C414yes—
100C2×C4×C137—22no—
101C2×C4×C137C22×C411yescharacteristic
102C2×C4×C137C22×C411yescharacteristic
103C22×C137—21no—
104C22×C137—22no—
105C22×C137—42no—
106C22×C137—41no—
107C22×C137—42no—
108C22×C137—84no—
109C22×C137C2×C22⋊C414yes—
110C22×C137C23×C411yescharacteristic
111C2×D137—21no—
112C2×D137—42no—
113C2×D137—41no—
114C2×D137—48no—
115C2×D137—41no—
116C2×D137—84no—
117C2×D137C2×C22⋊C414yes—
118C2×D137C22×D412yes—
119#10587739—48no—
120C4×C137—21no—
121C4×C137C2×C22⋊C412yes—
122Dic137—22no—
123Dic137—21no—
124Dic137C2×C22⋊C412yes—
125C2×C137—21no—
126C2×C137—24no—
127C2×C137—42no—
128C2×C137—41no—
129C2×C137C22×C22⋊C411yescharacteristic
130D137—22no—
131D137—24no—
132D137—41no—
133C137#288311yescharacteristic, complemented, Sylow
134#2883—1371nomaximal, Sylow
135C2×H(5,𝔽2)—1371no—
136#518—1372no—
137#410—1372no—
138#410—1372no—
139C2×#99—1374no—
140C2×#99—1374no—
141H(5,𝔽2)—2748no—
142C2×#41—1371no—
143C2×#41—1372no—
144C2×#41—1371no—
145C2×#41—1371no—
146C2×#41—1371no—
147C22×D4—1372no—
148C22×D4—1372no—
149C22×D4—1371no—
150C22×D4—1372no—
151C22×D4—1371no—
152C22×D4#1058773911yescharacteristic, complemented
153C4×D4—2748no—
154#117—1374no—
155C2×C22⋊C4—1378no—
156#99—1374no—
157#99—1378no—
158#99—1374no—
159C24—1372no—
160C24—1372no—
161C24C2×#1058773912yescomplemented
162#41—2744no—
163#41—2744no—
164#41—2748no—
165#41—2744no—
166#41—2744no—
167C2×Q8—1371no—
168C2×Q8—1371no—
169C2×D4—24no—
170C2×D4—2748no—
171C2×D4—2744no—
172C2×D4—2748no—
173C2×D4—2748no—
174C2×D4—1372no—
175C2×D4—1371no—
176C2×D4—1372no—
177C2×D4—1372no—
178C2×D4—1372no—
179C2×D4—1371no—
180C2×D4—1372no—
181C2×D4—1378no—
182C2×D4—1371no—
183C2×D4—1371no—
184C2×D4C2×#1058773911yescharacteristic
185C2×D4C2×#1058773912yescomplemented
186C2×D4C2×#1058773911yescharacteristic, complemented
187C22×C4—2748no—
188C22×C4—1372no—
189C22×C4—1372no—
190C22×C4—1372no—
191C22×C4—1371no—
192C22×C4—1371no—
193C22×C4C2×#1058773911yescharacteristic, complemented
194C4⋊C4—2744no—
195C22⋊C4—2748no—
196C22⋊C4—1378no—
197C42—2744no—
198C23—24no—
199C23—82no—
200C23—2744no—
201C23—10962no—
202C23—10962no—
203C23—1371no—
204C23—1378no—
205C23—1371no—
206C23—1372no—
207C23C22×#1058773912yes—
208C23C22×#1058773911yescharacteristic
209C23C22×#1058773912yes—
210Q8—2742no—
211Q8—2742no—
212D4—24no—
213D4—22no—
214D4—2742no—
215D4—2742no—
216D4—2744no—
217D4—2744no—
218D4—2744no—
219D4—2744no—
220D4—2742no—
221D4—1378no—
222D4—1374no—
223D4C22⋊#1058773914yes—
224C2×C4—22no—
225C2×C4—2744no—
226C2×C4—2744no—
227C2×C4—2744no—
228C2×C4—2742no—
229C2×C4—2748no—
230C2×C4—10968no—
231C2×C4—1372no—
232C2×C4—1371no—
233C2×C4—1371no—
234C2×C4—1374no—
235C2×C4C22×#1058773911yescharacteristic
236C2×C4C22×#1058773911yescharacteristic
237C22—21no—
238C22—22no—
239C22—42no—
240C22—41no—
241C22—42no—
242C22—84no—
243C22—2741no—
244C22—5488no—
245C22—5481no—
246C22—5482no—
247C22—5481no—
248C22—10964no—
249C22—1372no—
250C22—1374no—
251C22C2×C22⋊#1058773914yes—
252C22C23×#1058773911yescharacteristic
253C4—21no—
254C4—2742no—
255C4—2741no—
256C4—5488no—
257C4—1372no—
258C4C2×C22⋊#1058773912yes—
259C2—21no—
260C2—24no—
261C2—41no—
262C2—42no—
263C2—2742no—
264C2—2744no—
265C2—5481no—
266C2C22×C22⋊#1058773911yescharacteristic
subgroupG/H
centerC2C22×C22⋊#10587739
commutatorC22×C137C23×C4
FrattiniC22C23×#10587739
FittingC22×D4×C137C4
socleC2×C137C22×C22⋊C4
prime radicalC2×#17984133C2
cosocleC22×D137C24
hypercenterC22×D4#10587739
hypocenterC137#2883
LCSG ▷ C22×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C2 ◁ C2×D4 ◁ C22×D4

Complements

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

NH ≅ G/Ntype
1. #22578470C2semidirect
2. #22578383C2semidirect
3. #22578382C2semidirect
4. D137⋊#99C2semidirect
5. C2×#17984146C2semidirect
8. #17984167C22semidirect
9. C2×C22⋊#10587739C22semidirect
10. #17984147C22semidirect
11. #17984146C22semidirect
12. #17984146C22semidirect
14. C2×#14555510C4semidirect
16. C2×#14555509C4semidirect
17. C2×#14555509C4semidirect
19. C22×C22⋊D137C4semidirect
21. C2×D4×D137C4semidirect
24. C22×D4×C137C4semidirect
30. C23×D137C2×C4semidirect
37. C2×Dic274C2×C4semidirect
38. C2×Dic274C2×C4semidirect
44. C2×C22⋊D137C2×C4semidirect
45. C2×C22⋊D137C2×C4semidirect
46. C2×C22⋊D137C2×C4semidirect
47. C2×C22⋊D137C2×C4semidirect
53. C22×Dic137C2×C4semidirect
54. C22×Dic137C2×C4semidirect
55. C2×C4×D137C2×C4semidirect
56. C2×C4×D137C2×C4semidirect
58. C24×C137C2×C4semidirect
61. C2×D4×C137C2×C4semidirect
62. C2×D4×C137C2×C4semidirect
63. C22×C4×C137C2×C4semidirect
133. C137#2883semidirect
152. C22×D4#10587739semidirect
161. C24C2×#10587739semidirect
185. C2×D4C2×#10587739semidirect
186. C2×D4C2×#10587739semidirect
193. C22×C4C2×#10587739semidirect

Maximal subgroups

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

subgroupindex
#225784702
#225783832
#225783822
D137⋊#992
C2×#179841462
C2×#179841332
#2883137