Finite Group Databasebeta

← #31790190

Subgroups — #31790190

Order 17536.

772 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 772 conjugacy classes of subgroups — too many to draw. Only its 145 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#22578239C211yesmaximal, characteristic, complemented
2#22577970C212yesmaximal, complemented
3#22577589C218yesmaximal, complemented
4#22577377C211yesmaximal, characteristic, complemented
5#22577376C212yesmaximal, complemented
6#535×C137C211yesmaximal, characteristic, complemented
7#17984136—21no—
8#17984135—22no—
9#17984134—22no—
10#17984133—21no—
11#17984133C2214yes—
12C2×#14555510—22no—
13C2×#14555510C2212yes—
14#41×D137—24no—
15#17984084—22no—
16#17984075—22no—
17#17984059—24no—
18#17984057C2218yescomplemented
19#17984047C2212yescomplemented
20#17984003C2214yescomplemented
21#17984002C2218yescomplemented
22C2×#41×C137C2211yescharacteristic
23C22×D4×C137C2211yescharacteristic, complemented
24#127×C137—21no—
25#124×C137—21no—
26#121×C137—22no—
27#120×C137C2214yescomplemented
28#117×C137C2211yescharacteristic, complemented
29#14555511—24no—
30#14555510—24no—
31#14555510—24no—
32#14555510—22no—
33#14555510—22no—
34#14555510—24no—
35#14555510—22no—
36#14555510—22no—
37#14555509—24no—
38#14555509—28no—
39C2×Dic274—22no—
40C2×Dic274D412yescomplemented
41Q8×D137—24no—
42C2×D548—22no—
43C2×D548D412yescomplemented
44C2×C22⋊D137—28no—
45C2×C22⋊D137—24no—
46C2×C22⋊D137C2314yes—
47D4×D137—24no—
48D4×D137—28no—
49C2×C4×D137—24no—
50C2×C4×D137D414yescomplemented
51C4⋊Dic137—22no—
52C22⋊Dic137—28no—
53C22⋊Dic137C2314yes—
54C4×Dic137—22no—
55C24×C137C2312yes—
56#41×C137—22no—
57#41×C137—22no—
58C2×Q8×C137D411yescharacteristic, complemented
59C2×D4×C137—24no—
60C2×D4×C137—22no—
61C2×D4×C137—22no—
62C2×D4×C137—21no—
63C2×D4×C137—21no—
64C2×D4×C137D411yescharacteristic, complemented
65C2×D4×C137C2312yes—
66C22×C4×C137D412yescomplemented
67C22×C4×C137C2311yescharacteristic
68C4⋊C4×C137—21no—
69C22⋊C4×C137—24no—
70C22⋊C4×C137C2312yes—
71C42×C137—21no—
72C22×D137—24no—
73C22×D137C2×D414yes—
74Dic274—24no—
75Dic274—22no—
76Dic274—22no—
77D548—24no—
78D548—22no—
79D548—22no—
80C22⋊D137—28no—
81C22⋊D137—24no—
82C22⋊D137—24no—
83C22⋊D137—88no—
84C2×Dic137—24no—
85C2×Dic137—24no—
86C2×Dic137C2×D414yes—
87C4×D137—28no—
88C4×D137—24no—
89C4×D137—24no—
90C23×C137—22no—
91C23×C137—24no—
92C23×C137—82no—
93C23×C137C2×D412yes—
94C23×C137C2411yescharacteristic
95Q8×C137—22no—
96D4×C137—24no—
97D4×C137—22no—
98D4×C137—41no—
99D4×C137—41no—
100D4×C137—42no—
101D4×C137—84no—
102C2×C4×C137—22no—
103C2×C4×C137—22no—
104C2×C4×C137—22no—
105C2×C4×C137C2×D412yes—
106C2×C4×C137C2×D412yes—
107C2×C4×C137C2×D412yes—
108C2×C4×C137C2×D411yescharacteristic
109C2×C4×C137C2×D411yescharacteristic
110C22×C137—22no—
111C22×C137—41no—
112C22×C137—42no—
113C22×C137—44no—
114C22×C137—48no—
115C22×C137—42no—
116C22×C137C22×D412yes—
117C22×C137C22×D411yescharacteristic
118C2×D137—24no—
119C2×D137—44no—
120C2×D137—84no—
121C4×C137—22no—
122C4×C137—42no—
123C4×C137#12011yescharacteristic
124C4×C137#12011yescharacteristic
125C4×C137#12012yes—
126Dic137—24no—
127Dic137—44no—
128C2×C137—22no—
129C2×C137—21no—
130C2×C137—42no—
131C2×C137—82no—
132C2×C137C2×#12011yescharacteristic
133D137—44no—
134C137#302711yescharacteristic, complemented, Sylow
135#3027—1371nomaximal, Sylow
136#585—1371no—
137#535—1372no—
138#535D13711yescharacteristic, complemented
139B(4,𝔽2)—1378no—
140#410—1371no—
141#410—1372no—
1422−1+4—2742no—
1432−1+4—2741no—
144H(5,𝔽2)—2742no—
145H(5,𝔽2)—2741no—
146H(5,𝔽2)—1374no—
147C2×#41—2744no—
148C2×#41—2742no—
149C2×#41—1372no—
150C2×#41C2×D13711yescharacteristic
151C22×D4C2×D13711yescharacteristic, complemented
152#127—21no—
153#127—2742no—
154#124—21no—
155#124—2742no—
156#121—22no—
157#121—2744no—
158#120—1378no—
159#120C2×D13714yescomplemented
160#117—1372no—
161#117C2×D13711yescharacteristic, complemented
162#99—1374no—
163#99—1378no—
164C24C22×D13712yes—
165#41—22no—
166#41—22no—
167#41—2744no—
168#41—2744no—
169#41—2744no—
170#41—2744no—
171#41—2748no—
172#41—2742no—
173#41—2742no—
174#41—2742no—
175#41—2744no—
176#41—2742no—
177C2×Q8—2742no—
178C2×Q8—2744no—
179C2×Q8—1372no—
180C2×Q8C22⋊D13711yescharacteristic, complemented
181C2×D4—24no—
182C2×D4—22no—
183C2×D4—22no—
184C2×D4—21no—
185C2×D4—21no—
186C2×D4—2742no—
187C2×D4—2744no—
188C2×D4—2748no—
189C2×D4—2744no—
190C2×D4—2748no—
191C2×D4—1374no—
192C2×D4—1372no—
193C2×D4C22⋊D13711yescharacteristic, complemented
194C2×D4C22×D13712yes—
195C22×C4—2744no—
196C22×C4—1374no—
197C22×C4C22⋊D13712yescomplemented
198C22×C4C22×D13711yescharacteristic
199C4⋊C4—21no—
200C4⋊C4—2742no—
201C22⋊C4—24no—
202C22⋊C4—2748no—
203C22⋊C4—1374no—
204C22⋊C4C22×D13712yes—
205C42—21no—
206C42—2742no—
207C23—24no—
208C23—22no—
209C23—82no—
210C23—2744no—
211C23—1374no—
212C23C2×C22⋊D13712yes—
213C23C23×D13711yescharacteristic
214Q8—22no—
215Q8—2744no—
216Q8—2742no—
217Q8—2742no—
218D4—24no—
219D4—22no—
220D4—41no—
221D4—41no—
222D4—42no—
223D4—84no—
224D4—2744no—
225D4—2748no—
226D4—2744no—
227D4—2744no—
228D4—2742no—
229D4—2742no—
230D4—10968no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—2748no—
235C2×C4—2744no—
236C2×C4—2744no—
237C2×C4—2744no—
238C2×C4—2744no—
239C2×C4—1374no—
240C2×C4D4×D13712yes—
241C2×C4D4×D13712yes—
242C2×C4C2×C22⋊D13712yes—
243C2×C4C2×C22⋊D13711yescharacteristic
244C2×C4C2×C22⋊D13711yescharacteristic
245C22—22no—
246C22—41no—
247C22—42no—
248C22—44no—
249C22—48no—
250C22—42no—
251C22—2744no—
252C22—5484no—
253C22—10964no—
254C22C2×D4×D13712yes—
255C22C22×C22⋊D13711yescharacteristic
256C4—22no—
257C4—42no—
258C4—2744no—
259C4—5484no—
260C4#1798405611yescharacteristic
261C4#1798405611yescharacteristic
262C4#1798405612yes—
263C2—21no—
264C2—22no—
265C2—42no—
266C2—82no—
267C2—5484no—
268C2C2×#1798405611yescharacteristic
subgroupG/H
centerC2C2×#17984056
commutator, cosocleC23×C137C24
FrattiniC23C23×D137
Fitting#535×C137C2
socleC2×C137C2×#120
hypercenter#535D137
hypocenterC137#3027
LCSG ▷ C23×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C2 ◁ C22×C4 ◁ #535

Complements

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

NH ≅ G/Ntype
1. #22578239C2semidirect
2. #22577970C2semidirect
3. #22577589C2semidirect
4. #22577377C2semidirect
5. #22577376C2semidirect
6. #535×C137C2semidirect
18. #17984057C22semidirect
19. #17984047C22semidirect
20. #17984003C22semidirect
21. #17984002C22semidirect
23. C22×D4×C137C22semidirect
27. #120×C137C22semidirect
28. #117×C137C22semidirect
40. C2×Dic274D4semidirect
43. C2×D548D4semidirect
50. C2×C4×D137D4semidirect
58. C2×Q8×C137D4semidirect
64. C2×D4×C137D4semidirect
66. C22×C4×C137D4semidirect
134. C137#3027semidirect
138. #535D137semidirect
151. C22×D4C2×D137semidirect
159. #120C2×D137semidirect
161. #117C2×D137semidirect
180. C2×Q8C22⋊D137semidirect
193. C2×D4C22⋊D137semidirect
197. C22×C4C22⋊D137semidirect

Maximal subgroups

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

subgroupindex
#225782392
#225779702
#225775892
#225773772
#225773762
#535×C1372
#3027137