Finite Group Databasebeta

← #31789248

Subgroups — #31789248

Order 17536.

800 conjugacy classes of subgroups; only the 209 normal subgroups are shown.

normal subgroups only

This group has 800 conjugacy classes of subgroups — too many to draw. Only its 209 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×#17984134C211yesmaximal, characteristic
2#22577877C212yesmaximal, complemented
3#22577377C211yesmaximal, characteristic, complemented
4#22577376C212yesmaximal, complemented
5#99×D137C218yesmaximal, complemented
6#410×C137C211yesmaximal, characteristic, complemented
7#17984134—24no—
8#17984134—24no—
9C2×#14555511C412yescomplemented
10C2×#14555510C412yescomplemented
11C2×#14555510C2211yescharacteristic
12C22×D548C412yescomplemented
13C22×D548C2211yescharacteristic
14C2×D4×D137C412yescomplemented
15C2×D4×D137C2214yes—
16C4×D548—24no—
17#17984050—24no—
18#17984047C2212yescomplemented
19C22⋊C4×D137C2218yescomplemented
20#17984003C2214yescomplemented
21#17984002C2218yescomplemented
22C2×#41×C137C2211yescharacteristic
23#117×C137C2212yescomplemented
24#99×C137C2214yescomplemented
25C23×D137C2×C414yescomplemented
26C23×D137C2312yes—
27#14555511—28no—
28#14555510—24no—
29#14555510—24no—
30#14555510—22no—
31#14555510—22no—
32C2×Dic274C2×C411yescharacteristic, complemented
33C2×D548—24no—
34C2×D548—24no—
35C2×D548C2×C412yes—
36C2×D548C2×C412yes—
37C2×D548C2×C414yescomplemented
38C2×D548C2×C411yescharacteristic, complemented
39C2×D548D414yes—
40C2×D548D414yes—
41C2×C22⋊D137C2×C414yescomplemented
42C2×C22⋊D137C2312yes—
43D4×D137—216no—
44D4×D137—28no—
45C2×C4×D137—28no—
46C2×C4×D137C2×C412yescomplemented
47C2×C4×D137C2×C414yescomplemented
48C4⋊Dic137—22no—
49C22⋊Dic137C2314yes—
50#14555487—28no—
51C4×Dic137—22no—
52#41×C137—22no—
53#41×C137—22no—
54C2×Q8×C137C2×C411yescharacteristic, complemented
55C2×D4×C137C2×C411yescharacteristic, complemented
56C2×D4×C137C2312yes—
57C22×C4×C137C2×C412yescomplemented
58C22×C4×C137C2311yescharacteristic
59C4⋊C4×C137—22no—
60C22⋊C4×C137C2314yes—
61C42×C137—22no—
62C22×D137—28no—
63C22×D137—82no—
64C22×D137—84no—
65C22×D137C22×C412yes—
66C22×D137C22×C414yes—
67C22×D137C2×D418yes—
68Dic274—21no—
69Dic274—21no—
70D548—28no—
71D548—21no—
72D548—21no—
73D548C22⋊C418yes—
74D548C22⋊C414yes—
75D548C22⋊C414yes—
76C22⋊D137—24no—
77C22⋊D137—28no—
78C2×Dic137—24no—
79C2×Dic137C22×C412yes—
80C4×D137—28no—
81C4×D137—24no—
82C4×D137—88no—
83C23×C137C22×C412yes—
84C23×C137C2411yescharacteristic
85Q8×C137—22no—
86D4×C137—24no—
87D4×C137—22no—
88C2×C4×C137—22no—
89C2×C4×C137—22no—
90C2×C4×C137—24no—
91C2×C4×C137C22×C412yes—
92C2×C4×C137C22×C411yescharacteristic
93C2×C4×C137C22×C411yescharacteristic
94C2×C4×C137C2×D412yes—
95C2×C4×C137C2×D412yes—
96C22×C137—22no—
97C22×C137—41no—
98C22×C137—42no—
99C22×C137C23×C411yescharacteristic
100C22×C137C22×D412yes—
101C2×D137—22no—
102C2×D137—48no—
103C2×D137—44no—
104C2×D137—42no—
105C2×D137—88no—
106C2×D137C2×C22⋊C418yes—
107C4×C137—22no—
108C4×C137—44no—
109C4×C137C2×C22⋊C411yescharacteristic
110C4×C137C2×C22⋊C411yescharacteristic
111C4×C137C2×C22⋊C412yes—
112Dic137—22no—
113Dic137—44no—
114C2×C137—22no—
115C2×C137—21no—
116C2×C137—42no—
117C2×C137C22×C22⋊C411yescharacteristic
118D137—28no—
119D137—42no—
120C137#288311yescharacteristic, complemented, Sylow
121#2883—1371nomaximal, Sylow
122C2×H(5,𝔽2)—1371no—
123#518—1372no—
124#410—1371no—
125#410—1372no—
126#410D13711yescharacteristic, complemented
127C2×#99—1378no—
128H(5,𝔽2)—2744no—
129H(5,𝔽2)—2744no—
130C2×#41—1372no—
131C2×#41—1372no—
132C2×#41—1371no—
133C2×#41C2×D13711yescharacteristic
134C22×D4—1372no—
135C22×D4—1374no—
136C22×D4—1372no—
137C22×D4—1371no—
138C4×D4—2744no—
139C4×D4—2744no—
140#117—1372no—
141#117C2×D13712yescomplemented
142C2×C22⋊C4—1378no—
143#99—1374no—
144#99—1378no—
145#99C2×D13714yescomplemented
146C24—1374no—
147C24—1372no—
148#41—22no—
149#41—22no—
150#41—2744no—
151#41—2744no—
152#41—2748no—
153#41—2742no—
154#41—2742no—
155C2×Q8—1371no—
156C2×Q8C4×D13711yescharacteristic, complemented
157C2×D4—2744no—
158C2×D4—2744no—
159C2×D4—27416no—
160C2×D4—2748no—
161C2×D4—1374no—
162C2×D4—1372no—
163C2×D4—1374no—
164C2×D4—1371no—
165C2×D4—1372no—
166C2×D4—1372no—
167C2×D4—1374no—
168C2×D4—1374no—
169C2×D4C4×D13711yescharacteristic, complemented
170C2×D4C22×D13712yes—
171C22×C4—2748no—
172C22×C4—1374no—
173C22×C4—1372no—
174C22×C4C4×D13712yescomplemented
175C22×C4C22×D13711yescharacteristic
176C4⋊C4—22no—
177C4⋊C4—2742no—
178C22⋊C4—2748no—
179C22⋊C4—1374no—
180C22⋊C4C22×D13714yes—
181C42—22no—
182C42—2742no—
183C23—2748no—
184C23—10962no—
185C23—10964no—
186C23—1372no—
187C23—1374no—
188C23—1378no—
189C23C2×C4×D13712yes—
190C23C23×D13711yescharacteristic
191Q8—22no—
192Q8—2741no—
193Q8—2741no—
194D4—24no—
195D4—22no—
196D4—2748no—
197D4—2748no—
198D4—2744no—
199D4—2741no—
200D4—2741no—
201D4—1374no—
202D4—1374no—
203D4—1378no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—24no—
207C2×C4—2748no—
208C2×C4—2744no—
209C2×C4—2744no—
210C2×C4—10968no—
211C2×C4—1372no—
212C2×C4C2×C4×D13712yes—
213C2×C4C2×C4×D13711yescharacteristic
214C2×C4C2×C4×D13711yescharacteristic
215C2×C4D4×D13712yes—
216C2×C4D4×D13712yes—
217C22—22no—
218C22—41no—
219C22—42no—
220C22—2742no—
221C22—5488no—
222C22—5484no—
223C22—5482no—
224C22—10968no—
225C22—1378no—
226C22C22×C4×D13711yescharacteristic
227C22C2×D4×D13712yes—
228C4—22no—
229C4—44no—
230C4—2742no—
231C4—5484no—
232C4C22⋊C4×D13711yescharacteristic
233C4C22⋊C4×D13711yescharacteristic
234C4C22⋊C4×D13712yes—
235C2—22no—
236C2—21no—
237C2—42no—
238C2—2748no—
239C2—5482no—
240C2C2×C22⋊C4×D13711yescharacteristic
subgroupG/H
centerC2C2×C22⋊C4×D137
commutatorC22×C137C23×C4
FrattiniC23C23×D137
Fitting#410×C137C2
socleC2×C137C22×C22⋊C4
prime radicalC2×#17984134C2
cosocleC23×C137C24
hypercenter#410D137
hypocenterC137#2883
LCSG ▷ C22×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C2 ◁ C22×C4 ◁ #410

Complements

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

NH ≅ G/Ntype
2. #22577877C2semidirect
3. #22577377C2semidirect
4. #22577376C2semidirect
5. #99×D137C2semidirect
6. #410×C137C2semidirect
9. C2×#14555511C4semidirect
10. C2×#14555510C4semidirect
12. C22×D548C4semidirect
14. C2×D4×D137C4semidirect
18. #17984047C22semidirect
19. C22⋊C4×D137C22semidirect
20. #17984003C22semidirect
21. #17984002C22semidirect
23. #117×C137C22semidirect
24. #99×C137C22semidirect
25. C23×D137C2×C4semidirect
32. C2×Dic274C2×C4semidirect
37. C2×D548C2×C4semidirect
38. C2×D548C2×C4semidirect
41. C2×C22⋊D137C2×C4semidirect
46. C2×C4×D137C2×C4semidirect
47. C2×C4×D137C2×C4semidirect
54. C2×Q8×C137C2×C4semidirect
55. C2×D4×C137C2×C4semidirect
57. C22×C4×C137C2×C4semidirect
120. C137#2883semidirect
126. #410D137semidirect
141. #117C2×D137semidirect
145. #99C2×D137semidirect
156. C2×Q8C4×D137semidirect
169. C2×D4C4×D137semidirect
174. C22×C4C4×D137semidirect

Maximal subgroups

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

subgroupindex
C2×#179841342
#225778772
#225773772
#225773762
#99×D1372
#410×C1372
#2883137