Finite Group Databasebeta

← #31790250

Subgroups — #31790250

Order 17536.

548 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 548 conjugacy classes of subgroups — too many to draw. Only its 143 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×#17984114C211yesmaximal, characteristic, complemented
2#22577949C211yesmaximal, characteristic, complemented
3C2×D4×Dic137C211yesmaximal, characteristic, complemented
4#22577620C212yesmaximal, complemented
5#22577616C212yesmaximal, complemented
6#22577606C212yesmaximal, complemented
7#22577596C212yesmaximal, complemented
8#22577426C211yesmaximal, characteristic, complemented
9#22577405C211yesmaximal, characteristic, complemented
10#22577404C211yesmaximal, characteristic, complemented
11C2×AGL(1,ℤ/8)×C137C211yesmaximal, characteristic, complemented
12C2×#14555510C2211yescharacteristic
13C23×Dic137—21no—
14#17984114—24no—
15C2×#14555498C2212yes—
16C2×#14555495C2212yes—
17C2×#14555494C2211yescharacteristic
18#17984086C2211yescharacteristic, complemented
19#17984084C2211yescharacteristic, complemented
20#17984077C2211yescharacteristic, complemented
21#17984075C2211yescharacteristic, complemented
22#17984060—21no—
23#17984060—21no—
24D4×Dic137—21no—
25D4×Dic137—21no—
26D4×Dic137C2212yes—
27D4×Dic137C2212yes—
28C4×C22⋊D137—21no—
29C4×C22⋊D137—21no—
30C2×C4⋊Dic137C2211yescharacteristic
31C2×C22⋊Dic137—21no—
32C2×C4×Dic137C2211yescharacteristic
33#17984017C2212yescomplemented
34#17984017C2212yescomplemented
35#17984013C2212yescomplemented
36#17984012C2212yescomplemented
37#17984010C2212yescomplemented
38#17984009C2212yescomplemented
39C2×#41×C137C2211yescharacteristic
40C22×D4×C137C2211yescharacteristic
41AGL(1,ℤ/8)×C137—24no—
42C2×SD16×C137C2212yes—
43C2×D8×C137C2212yes—
44C2×M16×C137C2211yescharacteristic
45#14555510—22no—
46#14555510—22no—
47C2×Dic274C2311yescharacteristic
48C2×D548C2311yescharacteristic
49C2×C22⋊D137—21no—
50C2×C22⋊D137—21no—
51C2×C22⋊D137—21no—
52C22×Dic137—22no—
53C22×Dic137—22no—
54C22×Dic137—21no—
55C22×Dic137—44no—
56C22×Dic137D411yescharacteristic
57C22×Dic137D411yescharacteristic
58C2×C4×D137—21no—
59#14555498—24no—
60#14555495—24no—
61#14555494—22no—
62C2×#11764040C2312yes—
63C4⋊Dic137—21no—
64C4⋊Dic137C2312yes—
65#14555489—21no—
66#14555489—21no—
67C22⋊Dic137—22no—
68C22⋊Dic137—22no—
69C22⋊Dic137—21no—
70C22⋊Dic137—21no—
71#14555487—21no—
72#14555487—21no—
73C4×Dic137D411yescharacteristic
74C4×Dic137D411yescharacteristic
75C4×Dic137C2311yescharacteristic
76C4×Dic137C2311yescharacteristic
77C24×C137—21no—
78#41×C137—22no—
79#41×C137—22no—
80C2×Q8×C137C2311yescharacteristic
81C2×D4×C137—21no—
82C2×D4×C137—21no—
83C2×D4×C137—22no—
84C2×D4×C137D414yescomplemented
85C2×D4×C137C2311yescharacteristic
86C2×D4×C137C2312yes—
87C22×C4×C137—21no—
88C22×C4×C137C2311yescharacteristic
89SD16×C137—24no—
90D8×C137—24no—
91M16×C137—22no—
92C2×C8×C137C2312yes—
93C22×D137—21no—
94Dic274—21no—
95Dic274—21no—
96D548—21no—
97D548—21no—
98C22⋊D137—42no—
99C22⋊D137—42no—
100C22⋊D137—42no—
101C2×Dic137—22no—
102C2×Dic137—21no—
103C2×Dic137—22no—
104C2×Dic137—41no—
105C2×Dic137—44no—
106C2×Dic137—42no—
107C2×Dic137—44no—
108C2×Dic137C2×D411yescharacteristic
109C2×Dic137C2×D411yescharacteristic
110C2×Dic137C2×D411yescharacteristic
111C2×Dic137C2×D411yescharacteristic
112C4×D137—42no—
113#11764040—22no—
114C23×C137—24no—
115C23×C137—22no—
116C23×C137—21no—
117C23×C137—42no—
118C23×C137—42no—
119C23×C137C2×D411yescharacteristic
120Q8×C137—21no—
121Q8×C137—21no—
122D4×C137—24no—
123D4×C137—21no—
124D4×C137—22no—
125D4×C137—21no—
126D4×C137—42no—
127D4×C137C2×D414yes—
128C2×C4×C137—21no—
129C2×C4×C137—42no—
130C2×C4×C137C2×D411yescharacteristic
131C2×C4×C137C2×D412yes—
132C2×C4×C137#4112yes—
133C2×C4×C137C2411yescharacteristic
134C8×C137—22no—
135C22×C137—21no—
136C22×C137—21no—
137C22×C137—24no—
138C22×C137—48no—
139C22×C137—42no—
140C22×C137—42no—
141C22×C137—41no—
142C22×C137—41no—
143C22×C137—41no—
144C22×C137#12112yes—
145C22×C137C22×D411yescharacteristic
146C2×D137—41no—
147C2×D137—41no—
148C2×D137—41no—
149C4×C137—41no—
150C4×C137#12112yes—
151C4×C137C22×D411yescharacteristic
152C4×C137C2×#4111yescharacteristic
153Dic137—21no—
154Dic137—21no—
155Dic137—21no—
156Dic137—41no—
157Dic137—42no—
158Dic137AGL(1,ℤ/8)12yescomplemented
159C2×C137—22no—
160C2×C137—44no—
161C2×C137—81no—
162C2×C137C2×#12111yescharacteristic
163C2×C137C2×AGL(1,ℤ/8)11yescharacteristic
164C2×C137#57511yescharacteristic
165D137—81no—
166C137#303611yescharacteristic, complemented, Sylow
167#3036—1371nomaximal, Sylow
168C2×AGL(1,ℤ/8)—1371no—
169C2×AGL(1,ℤ/8)D13711yescharacteristic, complemented
170#532—1371no—
171C2×C4×D4—1371no—
172#463—1372no—
173#463—1372no—
174#461—1372no—
175#459—1372no—
176#423—1371no—
177#417—1371no—
178#417—1371no—
179C2×#41—1371no—
180C2×#41C2×D13711yescharacteristic
181C22×D4C2×D13711yescharacteristic
182C23×C4—2741no—
183AGL(1,ℤ/8)—24no—
184AGL(1,ℤ/8)—2744no—
185C2×SD16—1372no—
186C2×SD16C2×D13712yes—
187C2×D8—1372no—
188C2×D8C2×D13712yes—
189C2×M16—1371no—
190C2×M16C2×D13711yescharacteristic
191C4⋊Q8—1371no—
192#127—1371no—
193#124—1371no—
194#124—1371no—
195#121—2741no—
196#121—2741no—
197C4×D4—2741no—
198C4×D4—2741no—
199C4×D4—2741no—
200C4×D4—2741no—
201C4×D4—1372no—
202C4×D4—1372no—
203C2×C4⋊C4—1371no—
204C2×C22⋊C4—2741no—
205C2×C42—1371no—
206C4⋊C8—1372no—
207C4⋊C8—1372no—
208#103—1372no—
209#103—1372no—
210#102—1372no—
211#102—1372no—
212C24—21no—
213#41—22no—
214#41—22no—
215#41—2742no—
216#41—2742no—
217C2×Q8—1371no—
218C2×Q8C22×D13711yescharacteristic
219C2×D4—21no—
220C2×D4—21no—
221C2×D4—22no—
222C2×D4—2741no—
223C2×D4—2741no—
224C2×D4—2741no—
225C2×D4—1371no—
226C2×D4C22⋊D13714yescomplemented
227C2×D4C22×D13711yescharacteristic
228C2×D4C22×D13712yes—
229C22×C4—21no—
230C22×C4—2741no—
231C22×C4—2742no—
232C22×C4—2742no—
233C22×C4—2741no—
234C22×C4—5484no—
235C22×C4—1371no—
236C22×C4—1371no—
237C22×C4C22×D13711yescharacteristic
238SD16—24no—
239SD16—2744no—
240D8—24no—
241D8—2744no—
242M16—22no—
243M16—2742no—
244C2×C8—1372no—
245C2×C8C22×D13712yes—
246C4⋊C4—2741no—
247C4⋊C4—2741no—
248C4⋊C4—2741no—
249C4⋊C4—1372no—
250C22⋊C4—2741no—
251C22⋊C4—2741no—
252C22⋊C4—2742no—
253C22⋊C4—2742no—
254C22⋊C4—2741no—
255C22⋊C4—2741no—
256C42—1371no—
257C42—1371no—
258C42—1371no—
259C42—1371no—
260C23—24no—
261C23—22no—
262C23—21no—
263C23—42no—
264C23—42no—
265C23—2741no—
266C23D4×D13711yescharacteristic
267Q8—21no—
268Q8—21no—
269Q8—2741no—
270Q8—2741no—
271D4—24no—
272D4—21no—
273D4—22no—
274D4—21no—
275D4—42no—
276D4—2741no—
277D4—2741no—
278D4—5482no—
279D4—5482no—
280D4—5482no—
281D4C2×C22⋊D13714yes—
282C2×C4—21no—
283C2×C4—42no—
284C2×C4—2742no—
285C2×C4—2741no—
286C2×C4—2742no—
287C2×C4—5481no—
288C2×C4—5484no—
289C2×C4—5482no—
290C2×C4—5484no—
291C2×C4—5482no—
292C2×C4—1371no—
293C2×C4—1371no—
294C2×C4—1371no—
295C2×C4—1371no—
296C2×C4D4×D13711yescharacteristic
297C2×C4C2×C22⋊D13712yes—
298C2×C4#1455550912yes—
299C2×C4C23×D13711yescharacteristic
300C8—22no—
301C8—2742no—
302C22—21no—
303C22—21no—
304C22—24no—
305C22—48no—
306C22—42no—
307C22—42no—
308C22—41no—
309C22—41no—
310C22—41no—
311C22—5481no—
312C22—5481no—
313C22—5481no—
314C22#1798406012yes—
315C22C2×D4×D13711yescharacteristic
316C4—41no—
317C4—2741no—
318C4—2741no—
319C4—2741no—
320C4—5482no—
321C4—5481no—
322C4—1372no—
323C4#1798406012yes—
324C4C22×C22⋊D13711yescharacteristic
325C4C2×#1455550911yescharacteristic
326C2—22no—
327C2—44no—
328C2—81no—
329C2—10961no—
330C2C2×#1798406011yescharacteristic
331C2AGL(1,ℤ/8)×D13711yescharacteristic
332C2#2257819911yescharacteristic
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#3036
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×#17984114C2semidirect
2. #22577949C2semidirect
3. C2×D4×Dic137C2semidirect
4. #22577620C2semidirect
5. #22577616C2semidirect
6. #22577606C2semidirect
7. #22577596C2semidirect
8. #22577426C2semidirect
9. #22577405C2semidirect
10. #22577404C2semidirect
11. C2×AGL(1,ℤ/8)×C137C2semidirect
18. #17984086C22semidirect
19. #17984084C22semidirect
20. #17984077C22semidirect
21. #17984075C22semidirect
33. #17984017C22semidirect
34. #17984017C22semidirect
35. #17984013C22semidirect
36. #17984012C22semidirect
37. #17984010C22semidirect
38. #17984009C22semidirect
84. C2×D4×C137D4semidirect
158. Dic137AGL(1,ℤ/8)semidirect
166. C137#3036semidirect
169. C2×AGL(1,ℤ/8)D137semidirect
226. C2×D4C22⋊D137semidirect

Maximal subgroups

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

subgroupindex
C2×#179841142
#225779492
C2×D4×Dic1372
#225776202
#225776162
#225776062
#225775962
#225774262
#225774052
#225774042
C2×AGL(1,ℤ/8)×C1372
#3036137