Finite Group Databasebeta

← #497×D137 #31791225

Subgroups — #497×D137 #31791225

Order 17536.

564 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 564 conjugacy classes of subgroups — too many to draw. Only its 145 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×Q16×D137C212yesmaximal, complemented
2C2×SD16×D137C212yesmaximal
3C4⋊Q8×D137C211yesmaximal, characteristic, complemented
4#124×D137C211yesmaximal, characteristic
5#22577798C211yesmaximal, characteristic, complemented
6#22577797C211yesmaximal, characteristic, complemented
7#22577796C212yesmaximal, complemented
8#22577795C211yesmaximal, characteristic, complemented
9#22577794C212yesmaximal, complemented
10#97×D137C211yesmaximal, characteristic, complemented
11#497×C137C211yesmaximal, characteristic, complemented
12C2×Q8×D137C2211yescharacteristic
13C2×Q8×D137C2212yes—
14C2×D4×D137C2211yescharacteristic
15C2×Dic548C2212yescomplemented
16C2×#14555500C2212yescomplemented
17C2×#14555500C2212yescomplemented
18Q16×D137—28no—
19C2×#14555499C2212yes—
20C2×#14555498C2212yes—
21C2×#14555497C2212yes—
22SD16×D137—28no—
23C2×C8×D137C2212yes—
24#17984088C2211yescharacteristic, complemented
25#17984087—22no—
26#17984086C2212yescomplemented
27#17984077C2211yescharacteristic
28#17984076C2211yescharacteristic
29#17984075C2211yescharacteristic
30#17984074—22no—
31C4⋊C4×D137—22no—
32C22⋊C4×D137—22no—
33C42×D137C2211yescharacteristic
34#17983999C2211yescharacteristic, complemented
35#17983998C2212yescomplemented
36C2×Q16×C137C2212yescomplemented
37C2×SD16×C137C2212yes—
38C4⋊Q8×C137C2211yescharacteristic, complemented
39#124×C137C2211yescharacteristic
40#97×C137C2211yescharacteristic, complemented
41C23×D137—21no—
42C2×Dic274—21no—
43C2×Dic274—22no—
44C2×Dic274C2311yescharacteristic
45C2×Dic274C2312yes—
46Q8×D137—24no—
47Q8×D137—24no—
48Q8×D137—24no—
49C2×D548C2311yescharacteristic
50C2×C22⋊D137—21no—
51D4×D137—24no—
52C2×C4×D137—21no—
53C2×C4×D137—22no—
54C2×C4×D137D412yes—
55C2×C4×D137C2311yescharacteristic
56Dic548—24no—
57#14555500—24no—
58#14555500—24no—
59#14555499—24no—
60#14555498—24no—
61#14555497—24no—
62C2×#11764040C2312yes—
63C8×D137D418yes—
64C4⋊Dic137—22no—
65#14555489—24no—
66C22⋊Dic137—22no—
67#14555487—22no—
68#14555487—22no—
69C4×Dic137D412yes—
70C4×Dic137C2311yescharacteristic
71C2×Q8×C137C2311yescharacteristic
72C2×Q8×C137C2312yes—
73C2×D4×C137C2311yescharacteristic
74Q16×C137—24no—
75SD16×C137—24no—
76C2×C8×C137C2312yes—
77C4⋊C4×C137—22no—
78C22⋊C4×C137—22no—
79C42×C137C2311yescharacteristic
80C22×D137—21no—
81C22×D137—42no—
82C22×D137—44no—
83C22×D137C2×D411yescharacteristic
84Dic274—22no—
85Dic274—22no—
86Dic274—22no—
87Dic274—44no—
88Dic274—42no—
89D548—22no—
90C22⋊D137—42no—
91C2×Dic137—21no—
92C2×Dic137—22no—
93C2×Dic137C2×D412yes—
94C2×Dic137C2×D411yescharacteristic
95C4×D137—24no—
96C4×D137—42no—
97C4×D137—44no—
98C4×D137C2×D414yes—
99#11764040C2×D414yes—
100C23×C137—21no—
101Q8×C137—22no—
102Q8×C137—22no—
103Q8×C137—22no—
104D4×C137—22no—
105C2×C4×C137—21no—
106C2×C4×C137—22no—
107C2×C4×C137C2×D412yes—
108C2×C4×C137C2411yescharacteristic
109C8×C137C2×D414yes—
110C22×C137—41no—
111C22×C137—42no—
112C22×C137C22×D411yescharacteristic
113C2×D137—41no—
114C2×D137—42no—
115C2×D137—84no—
116C2×D137#12712yes—
117C2×D137#13714yes—
118C4×C137—22no—
119C4×C137—41no—
120C4×C137—42no—
121C4×C137C22×D412yes—
122Dic137—22no—
123Dic137—41no—
124Dic137—42no—
125Dic137#12712yes—
126C2×C137—81no—
127C2×C137C2×#12711yescharacteristic
128C2×C137C2×#13712yes—
129D137—81no—
130D137#49714yesdirect product factor
131C137C2×#49711yescharacteristic, complemented, Sylow
132C2×#497—1371nomaximal, Sylow
133C22×Q16—1372no—
134C22×SD16—1372no—
135C2×C4⋊Q8—1371no—
136C2×#124—1371no—
137#497—1372no—
138#497—1372no—
139#497—1371no—
140#497—1371no—
141#497—1371no—
142#497D13711yescharacteristic, direct product factor
143C2×#97—1371no—
144C22×Q8—1371no—
145C22×Q8—1372no—
146C22×D4—1371no—
147C2×Q16—2748no—
148C2×Q16—1372no—
149C2×Q16—1372no—
150C2×Q16—1372no—
151C2×Q16C2×D13712yescomplemented
152C2×SD16—2748no—
153C2×SD16—1372no—
154C2×SD16—1372no—
155C2×SD16—1372no—
156C2×SD16C2×D13712yes—
157C22×C8—1372no—
158C4⋊Q8—2742no—
159C4⋊Q8—1372no—
160C4⋊Q8—1371no—
161C4⋊Q8C2×D13711yescharacteristic, complemented
162#124—2742no—
163#124—1371no—
164#124—1371no—
165#124—1371no—
166#124C2×D13711yescharacteristic
167C2×C4⋊C4—2742no—
168C2×C22⋊C4—2742no—
169C2×C42—1371no—
170#97—1372no—
171#97—1371no—
172#97C2×D13711yescharacteristic, complemented
173C24—2741no—
174C2×Q8—2741no—
175C2×Q8—2742no—
176C2×Q8—2744no—
177C2×Q8—2744no—
178C2×Q8—2744no—
179C2×Q8—1371no—
180C2×Q8—1372no—
181C2×Q8C22×D13711yescharacteristic
182C2×Q8C22×D13712yes—
183C2×D4—2744no—
184C2×D4—2741no—
185C2×D4—1371no—
186C2×D4C22×D13711yescharacteristic
187C22×C4—2741no—
188C22×C4—2742no—
189C22×C4—1372no—
190C22×C4—1371no—
191Q16—24no—
192Q16—2744no—
193Q16—2744no—
194Q16—2744no—
195SD16—24no—
196SD16—2744no—
197SD16—2744no—
198SD16—2744no—
199C2×C8—1372no—
200C2×C8—1378no—
201C2×C8C22×D13712yes—
202C4⋊C4—22no—
203C4⋊C4—2744no—
204C4⋊C4—2742no—
205C22⋊C4—22no—
206C22⋊C4—2742no—
207C22⋊C4—2742no—
208C22⋊C4—2742no—
209C42—1372no—
210C42—1371no—
211C42C22×D13711yescharacteristic
212C23—21no—
213C23—2741no—
214C23—5482no—
215C23—5484no—
216C23—1371no—
217Q8—22no—
218Q8—22no—
219Q8—22no—
220Q8—2742no—
221Q8—2742no—
222Q8—2742no—
223Q8—5482no—
224Q8—5484no—
225D4—22no—
226D4—2742no—
227D4—5482no—
228C2×C4—21no—
229C2×C4—22no—
230C2×C4—2744no—
231C2×C4—2741no—
232C2×C4—2742no—
233C2×C4—5482no—
234C2×C4—5484no—
235C2×C4—1372no—
236C2×C4—1371no—
237C2×C4—1374no—
238C2×C4D4×D13712yes—
239C2×C4C23×D13711yescharacteristic
240C8—1374no—
241C8D4×D13714yes—
242C22—42no—
243C22—41no—
244C22—5482no—
245C22—5481no—
246C22—10964no—
247C22—1374no—
248C22—1372no—
249C22C2×D4×D13711yescharacteristic
250C4—22no—
251C4—41no—
252C4—42no—
253C4—2742no—
254C4—5481no—
255C4—5482no—
256C4—1372no—
257C4C2×D4×D13712yes—
258C2—81no—
259C2—10961no—
260C2—1374no—
261C2#127×D13711yescharacteristic
262C2#137×D13712yes—
subgroupG/H
centerC22C2×D4×D137
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
Fitting#497×C137C2
socleC22×C137C22×D4
prime radicalC2×D4×D137C22
hypercenter#497D137
hypocenterC137C2×#497
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C42 ◁ #497

Complements

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

NH ≅ G/Ntype
1. C2×Q16×D137C2semidirect
3. C4⋊Q8×D137C2semidirect
5. #22577798C2semidirect
6. #22577797C2semidirect
7. #22577796C2semidirect
8. #22577795C2semidirect
9. #22577794C2semidirect
10. #97×D137C2semidirect
11. #497×C137C2semidirect
15. C2×Dic548C22semidirect
16. C2×#14555500C22semidirect
17. C2×#14555500C22semidirect
24. #17984088C22semidirect
26. #17984086C22semidirect
34. #17983999C22semidirect
35. #17983998C22semidirect
36. C2×Q16×C137C22semidirect
38. C4⋊Q8×C137C22semidirect
40. #97×C137C22semidirect
130. D137#497direct (N × H)
131. C137C2×#497semidirect
142. #497D137direct (N × H)
151. C2×Q16C2×D137semidirect
161. C4⋊Q8C2×D137semidirect
172. #97C2×D137semidirect

Maximal subgroups

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

subgroupindex
C2×Q16×D1372
C2×SD16×D1372
C4⋊Q8×D1372
#124×D1372
#225777982
#225777972
#225777962
#225777952
#225777942
#97×D1372
#497×C1372
C2×#497137