Finite Group Databasebeta

← #31790470

Subgroups — #31790470

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×#17984116C211yesmaximal, characteristic, complemented
2C2×#17984096C211yesmaximal, characteristic, complemented
3#22577984C211yesmaximal, characteristic, complemented
4#22577972C211yesmaximal, characteristic, complemented
5#22577657C212yesmaximal, complemented
6#22577650C212yesmaximal, complemented
7#22577640C212yesmaximal, complemented
8#22577633C212yesmaximal, complemented
9#22577397C211yesmaximal, characteristic, complemented
10C2×#17984008C211yesmaximal, characteristic, complemented
11#428×C137C211yesmaximal, characteristic, complemented
12C2×#14555510C2211yescharacteristic
13C2×#14555510C2211yescharacteristic
14C22×D548C2211yescharacteristic
15#17984116—24no—
16C2×#14555499C2212yes—
17C2×D1096C2212yes—
18#17984096—24no—
19C2×#14555493C2212yescomplemented
20C22×#11764040C2211yescharacteristic
21C2×C8×D137C2212yescomplemented
22#17984085—21no—
23#17984076—21no—
24#17984068—21no—
25#17984068C2212yescomplemented
26#17984063—21no—
27#17984063C2212yes—
28#17984063C2212yes—
29#17984063C2212yescomplemented
30#17984062—22no—
31#17984058—22no—
32#17984011C2211yescharacteristic, complemented
33#17984011C2211yescharacteristic, complemented
34#17984008C2212yes—
35#17984008C2212yes—
36#17984008C2211yescharacteristic, complemented
37#17984008C2211yescharacteristic, complemented
38C2×M16×C137C2211yescharacteristic
39#117×C137C2211yescharacteristic
40C2×C4⋊C4×C137C2211yescharacteristic
41#107×C137C2212yescomplemented
42#106×C137C2212yescomplemented
43C23×D137—21no—
44#14555510—22no—
45#14555510—22no—
46#14555510—22no—
47#14555510—22no—
48C2×Dic274D411yescharacteristic
49C2×Dic274C2311yescharacteristic
50C2×D548—22no—
51C2×D548—22no—
52C2×D548—22no—
53C2×D548—22no—
54C2×D548—21no—
55C2×D548—22no—
56C2×D548D411yescharacteristic
57C2×D548C2312yes—
58C2×D548C2311yescharacteristic
59C2×C22⋊D137—21no—
60C2×C22⋊D137D412yes—
61C2×C4×D137—21no—
62C2×C4×D137C2312yes—
63#14555499—24no—
64D1096—24no—
65#14555493—24no—
66C2×#11764040—22no—
67C2×#11764040C2311yescharacteristic
68C2×#11764040C2311yescharacteristic
69C8×D137—24no—
70C4⋊Dic137—22no—
71#14555487—22no—
72#14555487—22no—
73#14555487—22no—
74#14555487—22no—
75C22×C4×C137—21no—
76C22×C4×C137C2311yescharacteristic
77M16×C137D414yescomplemented
78C2×C8×C137C2312yes—
79C4⋊C4×C137—21no—
80C4⋊C4×C137C2312yes—
81C4⋊C4×C137C2311yescharacteristic
82C4⋊C4×C137C2311yescharacteristic
83C22⋊C4×C137—21no—
84C42×C137—21no—
85C22×D137—21no—
86C22×D137—22no—
87C22×D137—42no—
88C22×D137—81no—
89C22×D137—81no—
90C22×D137C2×D412yes—
91Dic274—22no—
92Dic274—21no—
93Dic274—21no—
94D548—22no—
95D548—21no—
96D548—22no—
97D548—22no—
98D548—21no—
99D548—42no—
100D548—82no—
101D548—82no—
102D548—82no—
103C22⋊D137—24no—
104C22⋊D137—42no—
105C2×Dic137—21no—
106C2×Dic137C2×D412yes—
107C4×D137—22no—
108C4×D137—22no—
109C4×D137—42no—
110#11764040—21no—
111#11764040—21no—
112C23×C137C2×D411yescharacteristic
113C2×C4×C137—21no—
114C2×C4×C137—21no—
115C2×C4×C137—22no—
116C2×C4×C137—81no—
117C2×C4×C137C2×D411yescharacteristic
118C2×C4×C137C2×D412yes—
119C2×C4×C137#4112yes—
120C2×C4×C137C2411yescharacteristic
121C8×C137C2×D414yes—
122C22×C137—21no—
123C22×C137—21no—
124C22×C137#12112yes—
125C22×C137C22×D411yescharacteristic
126C2×D137—22no—
127C2×D137—22no—
128C2×D137—22no—
129C2×D137—41no—
130C2×D137—42no—
131C2×D137—41no—
132C2×D137—42no—
133C2×D137—42no—
134C2×D137—41no—
135C2×D137—84no—
136C4×C137—42no—
137C4×C137—41no—
138C4×C137—41no—
139C4×C137#12112yes—
140C4×C137C22×D411yescharacteristic
141C4×C137C2×#4111yescharacteristic
142Dic137—22no—
143Dic137—41no—
144C2×C137—22no—
145C2×C137C2×#12111yescharacteristic
146C2×C137#57611yescharacteristic
147C2×C137#57711yescharacteristic
148D137—42no—
149D137—81no—
150D137—82no—
151C137#306211yescharacteristic, complemented, Sylow
152#3062—1371nomaximal, Sylow
153C2×AGL(1,ℤ/8)—1371no—
154C2×#131—1371no—
155#537—1371no—
156#535—1371no—
157#469—1372no—
158#468—1372no—
159#466—1372no—
160#465—1372no—
161#428D13711yescharacteristic, complemented
162#416—1371no—
163C2×#102—1371no—
164C2×#41—1371no—
165C2×#41—1371no—
166C22×D4—1371no—
167AGL(1,ℤ/8)—2744no—
168C2×SD16—1372no—
169C2×D8—1372no—
170#131—2744no—
171C2×M16—1372no—
172C2×M16C2×D13711yescharacteristic
173C22×C8—1371no—
174C22×C8—1372no—
175#127—2741no—
176#124—2741no—
177C22⋊Q8—2741no—
178C22⋊Q8—1372no—
179#121—2741no—
180#121—2742no—
181#121—1372no—
182#121—1372no—
183#121—1372no—
184#120—2742no—
185#117C2×D13711yescharacteristic
186C2×C4⋊C4C2×D13711yescharacteristic
187#107C2×D13712yescomplemented
188#106C2×D13712yescomplemented
189#103—1371no—
190#103—1371no—
191#102—1372no—
192#102—1372no—
193#102—1371no—
194#102—1371no—
195C24—2741no—
196#41—2742no—
197#41—2742no—
198#41—2742no—
199#41—2742no—
200C2×Q8—1371no—
201C2×Q8—1371no—
202C2×D4—2742no—
203C2×D4—2742no—
204C2×D4—2742no—
205C2×D4—2741no—
206C2×D4—2741no—
207C2×D4—2742no—
208C2×D4—2742no—
209C2×D4—1372no—
210C2×D4—1371no—
211C2×D4—1371no—
212C2×D4—1372no—
213C22×C4—21no—
214C22×C4—2741no—
215C22×C4—1372no—
216C22×C4C22×D13711yescharacteristic
217SD16—2744no—
218D8—2744no—
219M16—2744no—
220M16D54814yescomplemented
221C2×C8—2742no—
222C2×C8—2744no—
223C2×C8—1371no—
224C2×C8—1371no—
225C2×C8C22×D13712yes—
226C4⋊C4—21no—
227C4⋊C4—2742no—
228C4⋊C4C22×D13712yes—
229C4⋊C4C22×D13711yescharacteristic
230C4⋊C4C22×D13711yescharacteristic
231C22⋊C4—21no—
232C22⋊C4—2742no—
233C22⋊C4—2742no—
234C22⋊C4—2742no—
235C22⋊C4—2742no—
236C42—21no—
237C23—2741no—
238C23—2742no—
239C23—5482no—
240C23—10961no—
241C23—10961no—
242C23—1372no—
243C23D4×D13711yescharacteristic
244Q8—2742no—
245Q8—2741no—
246Q8—2741no—
247D4—2744no—
248D4—2742no—
249D4—2741no—
250D4—2742no—
251D4—2742no—
252D4—2741no—
253D4—5482no—
254D4—5482no—
255D4—10962no—
256D4—10962no—
257D4—10962no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—22no—
261C2×C4—81no—
262C2×C4—2742no—
263C2×C4—2742no—
264C2×C4—2741no—
265C2×C4—5482no—
266C2×C4—1372no—
267C2×C4D4×D13711yescharacteristic
268C2×C4C2×D54812yes—
269C2×C4#1455551112yes—
270C2×C4C23×D13711yescharacteristic
271C8—2741no—
272C8—2741no—
273C8C2×D54814yes—
274C22—21no—
275C22—21no—
276C22—2742no—
277C22—2742no—
278C22—2742no—
279C22—5481no—
280C22—5482no—
281C22—5482no—
282C22—5481no—
283C22—5481no—
284C22—5482no—
285C22—10964no—
286C22#1798406312yes—
287C22C2×D4×D13711yescharacteristic
288C4—42no—
289C4—41no—
290C4—41no—
291C4—2742no—
292C4—5481no—
293C4#1798406312yes—
294C4C22×D54811yescharacteristic
295C4C2×#1455551111yescharacteristic
296C2—22no—
297C2—5482no—
298C2—10962no—
299C2—10961no—
300C2C2×#1798406311yescharacteristic
301C2#2257820711yescharacteristic
302C2#2257821211yescharacteristic
subgroupG/H
centerC22C2×D4×D137
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
Fitting#428×C137C2
socleC22×C137C22×D4
hypercenter#428D137
hypocenterC137#3062
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C22×C4 ◁ #428

Complements

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

NH ≅ G/Ntype
1. C2×#17984116C2semidirect
2. C2×#17984096C2semidirect
3. #22577984C2semidirect
4. #22577972C2semidirect
5. #22577657C2semidirect
6. #22577650C2semidirect
7. #22577640C2semidirect
8. #22577633C2semidirect
9. #22577397C2semidirect
10. C2×#17984008C2semidirect
11. #428×C137C2semidirect
19. C2×#14555493C22semidirect
21. C2×C8×D137C22semidirect
25. #17984068C22semidirect
29. #17984063C22semidirect
32. #17984011C22semidirect
33. #17984011C22semidirect
36. #17984008C22semidirect
37. #17984008C22semidirect
41. #107×C137C22semidirect
42. #106×C137C22semidirect
77. M16×C137D4semidirect
151. C137#3062semidirect
161. #428D137semidirect
187. #107C2×D137semidirect
188. #106C2×D137semidirect
220. M16D548semidirect

Maximal subgroups

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

subgroupindex
C2×#179841162
C2×#179840962
#225779842
#225779722
#225776572
#225776502
#225776402
#225776332
#225773972
C2×#179840082
#428×C1372
#3062137