Finite Group Databasebeta

← #31792613

Subgroups — #31792613

Order 17536.

462 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 462 conjugacy classes of subgroups — too many to draw. Only its 141 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#22578110C211yesmaximal, characteristic, complemented
2#22577749C211yesmaximal, characteristic, complemented
3#22577719C212yesmaximal, complemented
4#22577693C211yesmaximal, characteristic, complemented
5#22577630C212yesmaximal, complemented
6#22577600C211yesmaximal, characteristic, complemented
7#22577554C212yesmaximal, complemented
8C4×#14555498C211yesmaximal, characteristic, complemented
9D4×#11764040C211yesmaximal, characteristic, complemented
10C2×#17984009C212yesmaximal, complemented
11D42×C137C211yesmaximal, characteristic
12C2×#14555498C2211yescharacteristic
13C2×#14555498C2212yescomplemented
14C22×#11764040C2212yescomplemented
15#17984088C2211yescharacteristic, complemented
16#17984079—21no—
17#17984067C2212yescomplemented
18#17984065—22no—
19C4×Dic274C2211yescharacteristic, complemented
20D4×Dic137—21no—
21C2×C4⋊Dic137C2212yescomplemented
22#17984019C2212yescomplemented
23#17984019C2211yescharacteristic, complemented
24C4⋊#11764040C2211yescharacteristic, complemented
25#17984011C2211yescharacteristic, complemented
26#17984009C2212yes—
27#17984009C2211yescharacteristic
28#17984009C2212yescomplemented
29#17984009C2212yescomplemented
30#17984009C2212yescomplemented
31C22⋊#11764040C2212yescomplemented
32C4×#11764040C2211yescharacteristic, complemented
33C22×D4×C137—21no—
34C22×D4×C137C2212yes—
35#127×C137C2211yescharacteristic
36#121×C137—21no—
37#121×C137C2212yes—
38#120×C137—22no—
39C4×D4×C137C2211yescharacteristic
40C4×D4×C137C2211yescharacteristic
41C2×Dic274—21no—
42C2×Dic274C2311yescharacteristic
43C22×Dic137—22no—
44#14555498—22no—
45#14555498—42no—
46C2×#11764040—42no—
47C2×#11764040C2311yescharacteristic
48C2×#11764040C2312yes—
49C2×#11764040C2311yescharacteristic
50C4⋊Dic137—22no—
51C4⋊Dic137—21no—
52C4⋊Dic137C2311yescharacteristic
53C4⋊Dic137C2312yes—
54#14555489—21no—
55#14555489—22no—
56C22⋊Dic137—22no—
57C4×Dic137—21no—
58C24×C137—22no—
59C2×D4×C137—24no—
60C2×D4×C137—22no—
61C2×D4×C137—22no—
62C2×D4×C137—21no—
63C2×D4×C137—21no—
64C2×D4×C137—42no—
65C2×D4×C137—44no—
66C2×D4×C137D411yescharacteristic
67C2×D4×C137C2312yes—
68C2×D4×C137C2311yescharacteristic
69C22×C4×C137—21no—
70C22×C4×C137C2312yes—
71C4⋊C4×C137D411yescharacteristic
72C4⋊C4×C137C2311yescharacteristic
73C22⋊C4×C137—21no—
74C22⋊C4×C137—22no—
75C22⋊C4×C137D412yes—
76C42×C137C2311yescharacteristic
77Dic274—21no—
78Dic274—21no—
79Dic274—81no—
80C2×Dic137—21no—
81C2×Dic137—21no—
82C2×Dic137—22no—
83C2×Dic137—82no—
84#11764040—21no—
85#11764040—21no—
86#11764040—22no—
87C23×C137—21no—
88C23×C137—22no—
89C23×C137—44no—
90C23×C137—44no—
91C23×C137—82no—
92C23×C137C2×D412yes—
93D4×C137—22no—
94D4×C137—21no—
95D4×C137—22no—
96D4×C137—44no—
97D4×C137—44no—
98D4×C137—42no—
99D4×C137—81no—
100D4×C137SD1614yescomplemented
101D4×C137#4112yes—
102C2×C4×C137—22no—
103C2×C4×C137—22no—
104C2×C4×C137—21no—
105C2×C4×C137—42no—
106C2×C4×C137C2×D411yescharacteristic
107C2×C4×C137C2×D412yes—
108C2×C4×C137C2×D411yescharacteristic
109C2×C4×C137C2411yescharacteristic
110C22×C137—22no—
111C22×C137—22no—
112C22×C137—22no—
113C22×C137—41no—
114C22×C137—41no—
115C22×C137—42no—
116C22×C137—42no—
117C22×C137—42no—
118C22×C137—88no—
119C22×C137C2×SD1614yes—
120C22×C137C22×D411yescharacteristic
121C4×C137—22no—
122C4×C137—21no—
123C4×C137—41no—
124C4×C137C2×SD1612yes—
125C4×C137C2×#4111yescharacteristic
126C4×C137H(5,𝔽2)11yescharacteristic
127Dic137—41no—
128Dic137—41no—
129Dic137—42no—
130C2×C137—24no—
131C2×C137—42no—
132C2×C137—82no—
133C2×C137#54611yescharacteristic
134C2×C137C22×SD1611yescharacteristic
135C2×C137#57611yescharacteristic
136C137#329511yescharacteristic, complemented, Sylow
137#3295—1371nomaximal, Sylow
138#554—1371no—
139D42D13711yescharacteristic
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—21no—
150C22×D4C2×D13712yes—
151C2×SD16—1371no—
152C2×SD16—1372no—
153C22×C8—1372no—
154C4⋊Q8—1371no—
155#127C2×D13711yescharacteristic
156#125—2741no—
157C22⋊Q8—2742no—
158C22⋊Q8—1372no—
159#121—21no—
160#121C2×D13712yes—
161#120—22no—
162C4×Q8—1371no—
163C4×D4—2741no—
164C4×D4C2×D13711yescharacteristic
165C4×D4C2×D13711yescharacteristic
166C2×C4⋊C4—1372no—
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—22no—
179C2×Q8—2741no—
180C2×Q8—1371no—
181C2×D4—24no—
182C2×D4—22no—
183C2×D4—22no—
184C2×D4—21no—
185C2×D4—21no—
186C2×D4—42no—
187C2×D4—44no—
188C2×D4C22⋊D13711yescharacteristic
189C2×D4C22×D13712yes—
190C2×D4C22×D13711yescharacteristic
191C22×C4—21no—
192C22×C4—2742no—
193C22×C4C22×D13712yes—
194SD16—2742no—
195SD16—5482no—
196C2×C8—5482no—
197C2×C8—1371no—
198C2×C8—1371no—
199C2×C8—1372no—
200C4⋊C4—2741no—
201C4⋊C4—2742no—
202C4⋊C4—2742no—
203C4⋊C4—2741no—
204C4⋊C4—1371no—
205C4⋊C4—1372no—
206C4⋊C4C22⋊D13711yescharacteristic
207C4⋊C4C22×D13711yescharacteristic
208C22⋊C4—21no—
209C22⋊C4—22no—
210C22⋊C4—2742no—
211C22⋊C4C22⋊D13712yes—
212C42—2741no—
213C42C22×D13711yescharacteristic
214C23—21no—
215C23—22no—
216C23—44no—
217C23—44no—
218C23—82no—
219C23C2×C22⋊D13712yes—
220Q8—2741no—
221Q8—2741no—
222Q8—10961no—
223D4—22no—
224D4—21no—
225D4—22no—
226D4—44no—
227D4—44no—
228D4—42no—
229D4—81no—
230D4#1455549814yescomplemented
231D4#1455550912yes—
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—21no—
235C2×C4—42no—
236C2×C4—2741no—
237C2×C4—2741no—
238C2×C4—2742no—
239C2×C4—10962no—
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—22no—
250C22—41no—
251C22—41no—
252C22—42no—
253C22—42no—
254C22—42no—
255C22—88no—
256C22C2×#1455549814yes—
257C22C22×C22⋊D13711yescharacteristic
258C4—21no—
259C4—22no—
260C4—41no—
261C4—5481no—
262C4—5482no—
263C4—5481no—
264C4C2×#1455549812yes—
265C4C2×#1455550911yescharacteristic
266C4#1798413311yescharacteristic
267C2—24no—
268C2—42no—
269C2—82no—
270C2#2257804711yescharacteristic
271C2C22×#1455549811yescharacteristic
272C2#2257820611yescharacteristic
subgroupG/H
centerC22C22×C22⋊D137
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
Fitting, prime radicalD42×C137C2
socleC22×C137C22×D4
hypercenterD42D137
hypocenterC137#3295
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C4×D4 ◁ D42

Complements

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

NH ≅ G/Ntype
1. #22578110C2semidirect
2. #22577749C2semidirect
3. #22577719C2semidirect
4. #22577693C2semidirect
5. #22577630C2semidirect
6. #22577600C2semidirect
7. #22577554C2semidirect
8. C4×#14555498C2semidirect
9. D4×#11764040C2semidirect
10. C2×#17984009C2semidirect
13. C2×#14555498C22semidirect
14. C22×#11764040C22semidirect
15. #17984088C22semidirect
17. #17984067C22semidirect
19. C4×Dic274C22semidirect
21. C2×C4⋊Dic137C22semidirect
22. #17984019C22semidirect
23. #17984019C22semidirect
24. C4⋊#11764040C22semidirect
25. #17984011C22semidirect
28. #17984009C22semidirect
29. #17984009C22semidirect
30. #17984009C22semidirect
31. C22⋊#11764040C22semidirect
32. C4×#11764040C22semidirect
100. D4×C137SD16semidirect
136. C137#3295semidirect
230. D4#14555498semidirect

Maximal subgroups

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

subgroupindex
#225781102
#225777492
#225777192
#225776932
#225776302
#225776002
#225775542
C4×#145554982
D4×#117640402
C2×#179840092
D42×C1372
#3295137