Finite Group Databasebeta

← #31792607

Subgroups — #31792607

Order 17536.

462 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 462 conjugacy classes of subgroups — too many to draw. Only its 139 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
1D4×D548C211yesmaximal, characteristic
2#22577751C211yesmaximal, characteristic, complemented
3#22577721C212yesmaximal, complemented
4#22577690C211yesmaximal, characteristic, complemented
5#22577634C212yesmaximal, complemented
6#22577603C211yesmaximal, characteristic, complemented
7#22577553C212yesmaximal, complemented
8C4×#14555497C211yesmaximal, characteristic, complemented
9D4×#11764040C211yesmaximal, characteristic, complemented
10C2×#17984008C212yesmaximal, complemented
11#554×C137C211yesmaximal, characteristic, complemented
12C22×D548C2212yes—
13C2×D4×D137—21no—
14C2×#14555497C2211yescharacteristic
15C2×#14555497C2212yescomplemented
16C22×#11764040C2212yescomplemented
17#17984085C2211yescharacteristic
18#17984063—21no—
19#17984062C2212yes—
20#17984058—22no—
21C4×D548C2211yescharacteristic
22#17984019C2212yescomplemented
23#17984019C2211yescharacteristic, complemented
24C4⋊#11764040C2211yescharacteristic, complemented
25#17984012C2211yescharacteristic, complemented
26#17984008C2212yes—
27#17984008C2211yescharacteristic
28#17984008C2212yescomplemented
29#17984008C2212yescomplemented
30#17984008C2212yescomplemented
31C22⋊#11764040C2212yescomplemented
32C4×#11764040C2211yescharacteristic, complemented
33C4⋊Q8×C137C2211yescharacteristic, complemented
34#125×C137—21no—
35C22⋊Q8×C137—22no—
36C22⋊Q8×C137C2212yescomplemented
37C4×Q8×C137C2211yescharacteristic, complemented
38C4×D4×C137—21no—
39C4×D4×C137C2211yescharacteristic
40C2×C4⋊C4×C137C2212yescomplemented
41C23×D137—22no—
42C2×D548—24no—
43C2×D548—22no—
44C2×D548—21no—
45C2×D548—21no—
46C2×D548—42no—
47C2×D548C2312yes—
48C2×D548C2311yescharacteristic
49C2×C22⋊D137—22no—
50D4×D137—44no—
51C2×C4×D137—21no—
52#14555497—22no—
53#14555497—42no—
54C2×#11764040—42no—
55C2×#11764040C2311yescharacteristic
56C2×#11764040C2312yes—
57C2×#11764040C2311yescharacteristic
58C4⋊Dic137C2311yescharacteristic
59#14555487—21no—
60#14555487—22no—
61C2×Q8×C137—21no—
62C2×Q8×C137C2311yescharacteristic
63C2×D4×C137D411yescharacteristic
64C22×C4×C137—22no—
65C22×C4×C137C2312yes—
66C4⋊C4×C137—22no—
67C4⋊C4×C137—21no—
68C4⋊C4×C137—21no—
69C4⋊C4×C137—22no—
70C4⋊C4×C137D411yescharacteristic
71C4⋊C4×C137C2311yescharacteristic
72C4⋊C4×C137C2312yes—
73C22⋊C4×C137—22no—
74C22⋊C4×C137D412yes—
75C42×C137—21no—
76C42×C137C2311yescharacteristic
77C22×D137—21no—
78C22×D137—22no—
79C22×D137—44no—
80C22×D137—44no—
81C22×D137—82no—
82D548—22no—
83D548—21no—
84D548—22no—
85D548—44no—
86D548—42no—
87D548—81no—
88D548#4112yes—
89C22⋊D137—44no—
90C2×Dic137—21no—
91C4×D137—42no—
92#11764040—21no—
93#11764040—21no—
94#11764040—22no—
95C23×C137C2×D412yes—
96Q8×C137—21no—
97Q8×C137—21no—
98Q8×C137—81no—
99D4×C137SD1614yescomplemented
100C2×C4×C137—22no—
101C2×C4×C137—22no—
102C2×C4×C137—21no—
103C2×C4×C137—21no—
104C2×C4×C137—22no—
105C2×C4×C137—82no—
106C2×C4×C137C2×D411yescharacteristic
107C2×C4×C137C2×D412yes—
108C2×C4×C137C2×D411yescharacteristic
109C2×C4×C137C2411yescharacteristic
110C22×C137—22no—
111C22×C137—22no—
112C22×C137C2×SD1614yes—
113C22×C137C22×D411yescharacteristic
114C2×D137—22no—
115C2×D137—41no—
116C2×D137—41no—
117C2×D137—42no—
118C2×D137—42no—
119C2×D137—42no—
120C2×D137—88no—
121C4×C137—22no—
122C4×C137—21no—
123C4×C137—41no—
124C4×C137—41no—
125C4×C137—42no—
126C4×C137C2×SD1612yes—
127C4×C137C2×#4111yescharacteristic
128C4×C137H(5,𝔽2)11yescharacteristic
129Dic137—41no—
130C2×C137—24no—
131C2×C137#54611yescharacteristic
132C2×C137C22×SD1611yescharacteristic
133C2×C137#57611yescharacteristic
134D137—42no—
135D137—82no—
136C137#329511yescharacteristic, complemented, Sylow
137#3295—1371nomaximal, Sylow
138#554D13711yescharacteristic, complemented
139D42—1371no—
140#486—1371no—
141#481—1372no—
142#476—1371no—
143#465—1372no—
144#460—1371no—
145#450—1372no—
146C4×SD16—1371no—
147C8×D4—1371no—
148C2×#102—1372no—
149C22×D4—2741no—
150C22×D4—1372no—
151C2×SD16—1371no—
152C2×SD16—1372no—
153C22×C8—1372no—
154C4⋊Q8C2×D13711yescharacteristic, complemented
155#127—1371no—
156#125—21no—
157C22⋊Q8—22no—
158C22⋊Q8C2×D13712yescomplemented
159#121—2741no—
160#121—1372no—
161#120—2742no—
162C4×Q8C2×D13711yescharacteristic, complemented
163C4×D4—21no—
164C4×D4—1371no—
165C4×D4C2×D13711yescharacteristic
166C2×C4⋊C4C2×D13712yescomplemented
167#106—1371no—
168#106—1372no—
169C4⋊C8—1371no—
170#103—1371no—
171#102—1371no—
172#102—1372no—
173#102—1372no—
174#102—1372no—
175#102—1372no—
176C22⋊C8—1372no—
177C4×C8—1371no—
178C24—2742no—
179C2×Q8—21no—
180C2×Q8C22×D13711yescharacteristic
181C2×D4—2744no—
182C2×D4—2742no—
183C2×D4—2742no—
184C2×D4—2741no—
185C2×D4—2741no—
186C2×D4—5482no—
187C2×D4—5484no—
188C2×D4—1372no—
189C2×D4—1371no—
190C2×D4C22⋊D13711yescharacteristic
191C22×C4—22no—
192C22×C4—2741no—
193C22×C4C22×D13712yes—
194SD16—2742no—
195SD16—5482no—
196C2×C8—5482no—
197C2×C8—1371no—
198C2×C8—1371no—
199C2×C8—1372no—
200C4⋊C4—22no—
201C4⋊C4—21no—
202C4⋊C4—21no—
203C4⋊C4—22no—
204C4⋊C4—1371no—
205C4⋊C4C22⋊D13711yescharacteristic
206C4⋊C4C22×D13711yescharacteristic
207C4⋊C4C22×D13712yes—
208C22⋊C4—22no—
209C22⋊C4—2741no—
210C22⋊C4—2742no—
211C22⋊C4C22⋊D13712yes—
212C42—21no—
213C42C22×D13711yescharacteristic
214C23—2741no—
215C23—2742no—
216C23—5484no—
217C23—5484no—
218C23—10962no—
219C23C2×C22⋊D13712yes—
220Q8—21no—
221Q8—21no—
222Q8—81no—
223D4—2742no—
224D4—2741no—
225D4—2742no—
226D4—5484no—
227D4—5484no—
228D4—5482no—
229D4—10961no—
230D4—1372no—
231D4#1455549714yescomplemented
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—22no—
237C2×C4—82no—
238C2×C4—2741no—
239C2×C4—5482no—
240C2×C4C2×C22⋊D13711yescharacteristic
241C2×C4C2×C22⋊D13712yes—
242C2×C4C2×C22⋊D13711yescharacteristic
243C2×C4C23×D13711yescharacteristic
244C8—2741no—
245C8—2741no—
246C8—2742no—
247C22—22no—
248C22—22no—
249C22—2742no—
250C22—5481no—
251C22—5481no—
252C22—5482no—
253C22—5482no—
254C22—5482no—
255C22—10968no—
256C22C2×#1455549714yes—
257C22C22×C22⋊D13711yescharacteristic
258C4—21no—
259C4—22no—
260C4—41no—
261C4—42no—
262C4—41no—
263C4—5481no—
264C4C2×#1455549712yes—
265C4#41×D13711yescharacteristic
266C4#1798413411yescharacteristic
267C2—24no—
268C2—5482no—
269C2—10962no—
270C2#2257804111yescharacteristic
271C2C22×#1455549711yescharacteristic
272C2#2257820611yescharacteristic
subgroupG/H
centerC22C22×C22⋊D137
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
Fitting#554×C137C2
socleC22×C137C22×D4
prime radicalD4×D548C2
hypercenter#554D137
hypocenterC137#3295
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C4×D4 ◁ #554

Complements

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

NH ≅ G/Ntype
2. #22577751C2semidirect
3. #22577721C2semidirect
4. #22577690C2semidirect
5. #22577634C2semidirect
6. #22577603C2semidirect
7. #22577553C2semidirect
8. C4×#14555497C2semidirect
9. D4×#11764040C2semidirect
10. C2×#17984008C2semidirect
11. #554×C137C2semidirect
15. C2×#14555497C22semidirect
16. C22×#11764040C22semidirect
22. #17984019C22semidirect
23. #17984019C22semidirect
24. C4⋊#11764040C22semidirect
25. #17984012C22semidirect
28. #17984008C22semidirect
29. #17984008C22semidirect
30. #17984008C22semidirect
31. C22⋊#11764040C22semidirect
32. C4×#11764040C22semidirect
33. C4⋊Q8×C137C22semidirect
36. C22⋊Q8×C137C22semidirect
37. C4×Q8×C137C22semidirect
40. C2×C4⋊C4×C137C22semidirect
99. D4×C137SD16semidirect
136. C137#3295semidirect
138. #554D137semidirect
154. C4⋊Q8C2×D137semidirect
158. C22⋊Q8C2×D137semidirect
162. C4×Q8C2×D137semidirect
166. C2×C4⋊C4C2×D137semidirect
231. D4#14555497semidirect

Maximal subgroups

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

subgroupindex
D4×D5482
#225777512
#225777212
#225776902
#225776342
#225776032
#225775532
C4×#145554972
D4×#117640402
C2×#179840082
#554×C1372
#3295137