Finite Group Databasebeta

← #31790332

Subgroups — #31790332

Order 17536.

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

normal subgroups only

This group has 504 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
1C2×#17984108C211yesmaximal, characteristic, complemented
2#22577970C211yesmaximal, characteristic, complemented
3#22577886C211yesmaximal, characteristic, complemented
4#22577623C212yesmaximal, complemented
5#22577616C212yesmaximal, complemented
6#22577601C212yesmaximal, complemented
7#22577596C212yesmaximal, complemented
8#22577431C211yesmaximal, characteristic, complemented
9#22577405C211yesmaximal, characteristic, complemented
10C2×#17984009C211yesmaximal, characteristic, complemented
11C2×AGL(1,ℤ/8)×C137C211yesmaximal, characteristic, complemented
12C2×#14555510C2211yescharacteristic
13#17984108—24no—
14C2×#14555500C2211yescharacteristic, complemented
15C2×#14555498C2211yescharacteristic, complemented
16C2×#14555497C2211yescharacteristic, complemented
17C2×#14555495C2211yescharacteristic, complemented
18C22×#11764040C2211yescharacteristic
19#17984084C2212yes—
20#17984075C2212yes—
21#17984059—21no—
22#17984059—21no—
23#17984057—22no—
24Q8×Dic137C2212yescomplemented
25D4×Dic137—22no—
26D4×Dic137C2212yescomplemented
27#17984047—21no—
28#17984047C2211yescharacteristic
29C2×C4⋊Dic137—21no—
30C2×C4⋊Dic137C2211yescharacteristic
31#17984017C2212yescomplemented
32#17984017C2212yescomplemented
33#17984013C2212yescomplemented
34#17984010C2212yescomplemented
35#17984009C2212yes—
36#17984009C2212yes—
37C2×#41×C137C2211yescharacteristic
38C22×D4×C137C2211yescharacteristic
39AGL(1,ℤ/8)×C137—24no—
40C2×SD16×C137C2212yes—
41C2×D8×C137C2212yes—
42C2×M16×C137C2211yescharacteristic
43#14555510—22no—
44#14555510—22no—
45C2×Dic274C2311yescharacteristic
46C2×D548C2311yescharacteristic
47C2×C22⋊D137—22no—
48C2×C22⋊D137—22no—
49C2×C22⋊D137—21no—
50C22×Dic137—22no—
51C22×Dic137—21no—
52C2×C4×D137—21no—
53#14555500—22no—
54#14555498—22no—
55#14555497—22no—
56#14555495—22no—
57C2×#11764040—22no—
58C2×#11764040C2311yescharacteristic
59C2×#11764040C2311yescharacteristic
60C4⋊Dic137—22no—
61C4⋊Dic137—21no—
62C4⋊Dic137D411yescharacteristic
63C4⋊Dic137D411yescharacteristic
64C4⋊Dic137C2312yes—
65C22⋊Dic137—22no—
66C22⋊Dic137—22no—
67C22⋊Dic137—22no—
68C22⋊Dic137D412yes—
69C4×Dic137—22no—
70C4×Dic137C2312yes—
71C24×C137—21no—
72#41×C137—22no—
73#41×C137D414yescomplemented
74C2×Q8×C137C2311yescharacteristic
75C2×D4×C137—21no—
76C2×D4×C137—22no—
77C2×D4×C137—21no—
78C2×D4×C137—22no—
79C2×D4×C137C2312yes—
80C2×D4×C137C2311yescharacteristic
81C22×C4×C137—21no—
82C22×C4×C137C2311yescharacteristic
83SD16×C137—24no—
84D8×C137—24no—
85M16×C137—22no—
86C2×C8×C137C2312yes—
87C22×D137—21no—
88Dic274—21no—
89Dic274—21no—
90D548—21no—
91D548—21no—
92C22⋊D137—42no—
93C22⋊D137—82no—
94C22⋊D137—82no—
95C2×Dic137—22no—
96C2×Dic137—21no—
97C2×Dic137—82no—
98C2×Dic137—81no—
99C2×Dic137C2×D412yes—
100C2×Dic137C2×D412yes—
101C4×D137—42no—
102#11764040—21no—
103#11764040—21no—
104C23×C137—21no—
105C23×C137—22no—
106C23×C137—42no—
107C23×C137—81no—
108C23×C137—81no—
109C23×C137C2×D411yescharacteristic
110Q8×C137—21no—
111Q8×C137C2×D412yes—
112D4×C137—24no—
113D4×C137—22no—
114D4×C137—21no—
115D4×C137—22no—
116D4×C137—42no—
117D4×C137C2×D412yes—
118C2×C4×C137—21no—
119C2×C4×C137—24no—
120C2×C4×C137C2×D411yescharacteristic
121C2×C4×C137C2×D412yes—
122C2×C4×C137#4112yes—
123C2×C4×C137C2411yescharacteristic
124C8×C137—22no—
125C22×C137—21no—
126C22×C137—21no—
127C22×C137—22no—
128C22×C137—41no—
129C22×C137—41no—
130C22×C137—42no—
131C22×C137—42no—
132C22×C137—42no—
133C22×C137—84no—
134C22×C137#12112yes—
135C22×C137C22×D411yescharacteristic
136C2×D137—41no—
137C2×D137—41no—
138C2×D137—41no—
139C4×C137—22no—
140C4×C137#12112yes—
141C4×C137C22×D411yescharacteristic
142C4×C137C2×#4111yescharacteristic
143Dic137—22no—
144Dic137—22no—
145Dic137—41no—
146Dic137—42no—
147C2×C137—22no—
148C2×C137—42no—
149C2×C137—82no—
150C2×C137C2×#12111yescharacteristic
151C2×C137#57611yescharacteristic
152C2×C137#57711yescharacteristic
153D137—81no—
154C137#304511yescharacteristic, complemented, Sylow
155#3045—1371nomaximal, Sylow
156C2×AGL(1,ℤ/8)D13711yescharacteristic, complemented
157C2×#135—1371no—
158#535—1371no—
159#520—1371no—
160#464—1372no—
161#463—1372no—
162#460—1372no—
163#459—1372no—
164#424—1371no—
165#417—1371no—
166C2×#102—1371no—
167C2×#41—1371no—
168C2×#41C2×D13711yescharacteristic
169C22×D4C2×D13711yescharacteristic
170AGL(1,ℤ/8)—24no—
171#135—2744no—
172C2×Q16—1371no—
173C2×SD16—1371no—
174C2×SD16—1371no—
175C2×SD16C2×D13712yes—
176C2×D8—1371no—
177C2×D8C2×D13712yes—
178C2×M16C2×D13711yescharacteristic
179C22×C8—1371no—
180#127—1372no—
181#124—1372no—
182#121—2741no—
183#121—2741no—
184#120—2742no—
185C4×Q8—1372no—
186C4×D4—2742no—
187C4×D4—1372no—
188#117—2741no—
189#117—1371no—
190C2×C4⋊C4—2741no—
191C2×C4⋊C4—1371no—
192C4⋊C8—1372no—
193C4⋊C8—1372no—
194#103—1372no—
195#102—1372no—
196#102—1372no—
197#102—1372no—
198C24—21no—
199#41—22no—
200#41—2742no—
201#41—2742no—
202#41C22⋊D13714yescomplemented
203C2×Q8—1371no—
204C2×Q8C22×D13711yescharacteristic
205C2×D4—21no—
206C2×D4—21no—
207C2×D4—22no—
208C2×D4—22no—
209C2×D4—2741no—
210C2×D4—2742no—
211C2×D4—2742no—
212C2×D4—1371no—
213C2×D4C22×D13712yes—
214C2×D4C22×D13711yescharacteristic
215C22×C4—21no—
216C22×C4—2741no—
217C22×C4—2741no—
218C22×C4—2742no—
219C22×C4C22×D13711yescharacteristic
220Q16—2742no—
221SD16—24no—
222SD16—2742no—
223SD16—2742no—
224D8—24no—
225D8—2742no—
226M16—22no—
227C2×C8—2742no—
228C2×C8—1371no—
229C2×C8—1371no—
230C2×C8C22×D13712yes—
231C4⋊C4—2742no—
232C4⋊C4—2741no—
233C4⋊C4—1372no—
234C4⋊C4—1371no—
235C4⋊C4—1371no—
236C22⋊C4—2742no—
237C22⋊C4—2742no—
238C22⋊C4—2742no—
239C22⋊C4—1372no—
240C42—2742no—
241C42—1372no—
242C23—21no—
243C23—22no—
244C23—42no—
245C23—81no—
246C23—81no—
247C23—2741no—
248C23D4×D13711yescharacteristic
249Q8—21no—
250Q8—2741no—
251Q8—2741no—
252Q8C2×C22⋊D13712yes—
253D4—24no—
254D4—22no—
255D4—21no—
256D4—22no—
257D4—42no—
258D4—2741no—
259D4—2741no—
260D4—5482no—
261D4—10962no—
262D4—10962no—
263D4C2×C22⋊D13712yes—
264C2×C4—21no—
265C2×C4—24no—
266C2×C4—2742no—
267C2×C4—2741no—
268C2×C4—5482no—
269C2×C4—10962no—
270C2×C4—10961no—
271C2×C4—1372no—
272C2×C4—1372no—
273C2×C4D4×D13711yescharacteristic
274C2×C4C2×C22⋊D13712yes—
275C2×C4#1455550912yes—
276C2×C4C23×D13711yescharacteristic
277C8—22no—
278C8—2741no—
279C8—2741no—
280C22—21no—
281C22—21no—
282C22—22no—
283C22—41no—
284C22—41no—
285C22—42no—
286C22—42no—
287C22—42no—
288C22—84no—
289C22—5481no—
290C22—5481no—
291C22—5481no—
292C22#1798406012yes—
293C22C2×D4×D13711yescharacteristic
294C4—22no—
295C4—2742no—
296C4—2742no—
297C4—5482no—
298C4—5481no—
299C4#1798406012yes—
300C4C22×C22⋊D13711yescharacteristic
301C4C2×#1455550911yescharacteristic
302C2—22no—
303C2—42no—
304C2—82no—
305C2—10961no—
306C2C2×#1798406011yescharacteristic
307C2#2257820711yescharacteristic
308C2#2257821311yescharacteristic
subgroupG/H
centerC22C2×D4×D137
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
FittingC2×AGL(1,ℤ/8)×C137C2
socleC22×C137C22×D4
hypercenterC2×AGL(1,ℤ/8)D137
hypocenterC137#3045
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. C2×#17984108C2semidirect
2. #22577970C2semidirect
3. #22577886C2semidirect
4. #22577623C2semidirect
5. #22577616C2semidirect
6. #22577601C2semidirect
7. #22577596C2semidirect
8. #22577431C2semidirect
9. #22577405C2semidirect
10. C2×#17984009C2semidirect
11. C2×AGL(1,ℤ/8)×C137C2semidirect
14. C2×#14555500C22semidirect
15. C2×#14555498C22semidirect
16. C2×#14555497C22semidirect
17. C2×#14555495C22semidirect
24. Q8×Dic137C22semidirect
26. D4×Dic137C22semidirect
31. #17984017C22semidirect
32. #17984017C22semidirect
33. #17984013C22semidirect
34. #17984010C22semidirect
73. #41×C137D4semidirect
154. C137#3045semidirect
156. C2×AGL(1,ℤ/8)D137semidirect
202. #41C22⋊D137semidirect

Maximal subgroups

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

subgroupindex
C2×#179841082
#225779702
#225778862
#225776232
#225776162
#225776012
#225775962
#225774312
#225774052
C2×#179840092
C2×AGL(1,ℤ/8)×C1372
#3045137