Finite Group Databasebeta

← #31791174

Subgroups — #31791174

Order 17536.

400 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 400 conjugacy classes of subgroups — too many to draw. Only its 117 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#22577958C211yesmaximal, characteristic, complemented
2#124×D137C211yesmaximal, characteristic
3#22577765C211yesmaximal, characteristic, complemented
4#22577763C211yesmaximal, characteristic, complemented
5#22577762C211yesmaximal, characteristic, complemented
6#22577759C212yesmaximal, complemented
7#22577756C212yesmaximal, complemented
8#22577410C212yesmaximal, complemented
9#22577406C212yesmaximal
10#22577355C211yesmaximal, characteristic, complemented
11#489×C137C211yesmaximal, characteristic, complemented
12C2×Q8×D137C2211yescharacteristic
13C2×D4×D137C2211yescharacteristic
14C2×#14555493C2212yes—
15#17984088C2211yescharacteristic, complemented
16#17984079C2212yescomplemented
17#17984077C2211yescharacteristic
18#17984076C2211yescharacteristic
19#17984075C2211yescharacteristic
20#17984074—21no—
21#17984074—21no—
22#17984068—21no—
23#17984064—21no—
24Q8×Dic137—21no—
25#17984053—21no—
26#17984046C2212yes—
27C22⋊C4×D137—21no—
28C22⋊C4×D137—21no—
29C42×D137C2211yescharacteristic
30#17984013C2212yescomplemented
31#17984012C2212yescomplemented
32#17984011C2212yescomplemented
33#17984010C2212yes—
34#17984009C2212yes—
35#17984008C2212yes—
36#17983999C2211yescharacteristic, complemented
37C8×Dic137C2212yescomplemented
38C4⋊Q8×C137C2211yescharacteristic, complemented
39#124×C137C2211yescharacteristic
40#103×C137C2212yescomplemented
41#102×C137C2212yes—
42#97×C137C2211yescharacteristic, complemented
43C23×D137—21no—
44C2×Dic274—21no—
45C2×Dic274—21no—
46C2×Dic274C2311yescharacteristic
47Q8×D137—24no—
48C2×D548C2311yescharacteristic
49C2×C22⋊D137—21no—
50D4×D137—24no—
51C2×C4×D137—21no—
52C2×C4×D137D411yescharacteristic
53C2×C4×D137D411yescharacteristic
54C2×C4×D137C2311yescharacteristic
55#14555493—24no—
56C2×#11764040C2312yes—
57C4⋊Dic137—22no—
58C4⋊Dic137—21no—
59C4⋊Dic137C2312yes—
60#14555489—22no—
61C22⋊Dic137—21no—
62C22⋊Dic137—21no—
63#14555487—22no—
64#14555487—21no—
65#14555487—21no—
66#14555487—21no—
67#14555487—21no—
68C4×Dic137—22no—
69C4×Dic137D411yescharacteristic
70C4×Dic137D411yescharacteristic
71C4×Dic137C2311yescharacteristic
72C2×Q8×C137—21no—
73C2×Q8×C137C2311yescharacteristic
74C2×D4×C137C2311yescharacteristic
75C2×C8×C137C2312yes—
76C4⋊C4×C137—21no—
77C4⋊C4×C137C2312yes—
78C22⋊C4×C137—21no—
79C22⋊C4×C137—21no—
80C42×C137C2311yescharacteristic
81C22×D137—21no—
82C22×D137—42no—
83C22×D137—42no—
84C22×D137—42no—
85C22×D137C2×D411yescharacteristic
86Dic274—22no—
87Dic274—42no—
88Dic274—42no—
89D548—22no—
90C22⋊D137—42no—
91C2×Dic137—22no—
92C2×Dic137—21no—
93C2×Dic137C2×D411yescharacteristic
94C2×Dic137C2×D411yescharacteristic
95C2×Dic137C2×D411yescharacteristic
96C4×D137—41no—
97C4×D137—41no—
98C4×D137—42no—
99C4×D137#4114yes—
100#11764040—22no—
101C23×C137—21no—
102Q8×C137—22no—
103Q8×C137—42no—
104D4×C137—22no—
105C2×C4×C137—21no—
106C2×C4×C137—22no—
107C2×C4×C137C2×D411yescharacteristic
108C2×C4×C137C2×D411yescharacteristic
109C2×C4×C137C2411yescharacteristic
110C8×C137—22no—
111C22×C137—41no—
112C22×C137—41no—
113C22×C137—41no—
114C22×C137C22×D411yescharacteristic
115C2×D137—21no—
116C2×D137—21no—
117C2×D137—41no—
118C2×D137—41no—
119C2×D137—41no—
120C2×D137—84no—
121C2×D137#12412yes—
122C4×C137—21no—
123C4×C137—21no—
124C4×C137—41no—
125C4×C137—42no—
126C4×C137C2×#4112yes—
127Dic137—21no—
128Dic137—21no—
129Dic137—41no—
130Dic137—42no—
131Dic137#12412yes—
132C2×C137—81no—
133C2×C137C2×#12411yescharacteristic
134C2×C137#57511yescharacteristic
135C2×C137#57511yescharacteristic
136D137—22no—
137D137—81no—
138C137#314011yescharacteristic, complemented, Sylow
139#3140—1371nomaximal, Sylow
140#533—1371no—
141C2×#124—1371no—
142#489—1371no—
143#489—1371no—
144#489—1371no—
145#489D13711yescharacteristic, complemented
146#488—1372no—
147#488—1372no—
148#419—1372no—
149#418—1372no—
150C4×M16—1371no—
151C22×Q8—1371no—
152C22×D4—1371no—
153C2×M16—1372no—
154C4⋊Q8—1371no—
155C4⋊Q8C2×D13711yescharacteristic, complemented
156#125—1372no—
157#124—2741no—
158#124—2741no—
159#124—1371no—
160#124—1371no—
161#124—1371no—
162#124C2×D13711yescharacteristic
163C22⋊Q8—2741no—
164C22⋊Q8—2741no—
165C4×Q8—2741no—
166C4×Q8—2741no—
167#117—1372no—
168C2×C22⋊C4—2741no—
169C2×C22⋊C4—2741no—
170C2×C42—1371no—
171#103—1372no—
172#103—1372no—
173#103—1372no—
174#103C2×D13712yescomplemented
175#102—1372no—
176#102—1372no—
177#102—1372no—
178#102C2×D13712yes—
179#97—1371no—
180#97C2×D13711yescharacteristic, complemented
181C4×C8—1372no—
182C24—2741no—
183C2×Q8—21no—
184C2×Q8—2741no—
185C2×Q8—2741no—
186C2×Q8—2744no—
187C2×Q8—1371no—
188C2×Q8C22×D13711yescharacteristic
189C2×D4—2744no—
190C2×D4—2741no—
191C2×D4—1371no—
192C2×D4C22×D13711yescharacteristic
193C22×C4—2741no—
194C22×C4—1371no—
195C22×C4—1371no—
196C22×C4—1371no—
197M16—2744no—
198C2×C8—1372no—
199C2×C8C22×D13712yes—
200C4⋊C4—21no—
201C4⋊C4—2742no—
202C4⋊C4—2742no—
203C4⋊C4—2741no—
204C4⋊C4—1372no—
205C4⋊C4C22×D13712yes—
206C22⋊C4—21no—
207C22⋊C4—21no—
208C22⋊C4—2742no—
209C22⋊C4—2741no—
210C22⋊C4—2741no—
211C22⋊C4—2741no—
212C22⋊C4—2741no—
213C22⋊C4—2741no—
214C22⋊C4—2741no—
215C42—2742no—
216C42—1371no—
217C42—1371no—
218C42—1371no—
219C42C22×D13711yescharacteristic
220C23—21no—
221C23—2741no—
222C23—5482no—
223C23—5482no—
224C23—5482no—
225C23—1371no—
226Q8—22no—
227Q8—42no—
228Q8—2742no—
229Q8—5482no—
230Q8—5482no—
231D4—22no—
232D4—2742no—
233D4—5482no—
234C2×C4—22no—
235C2×C4—21no—
236C2×C4—2742no—
237C2×C4—2741no—
238C2×C4—5481no—
239C2×C4—5481no—
240C2×C4—5482no—
241C2×C4—1371no—
242C2×C4—1371no—
243C2×C4—1371no—
244C2×C4—1374no—
245C2×C4D4×D13711yescharacteristic
246C2×C4D4×D13711yescharacteristic
247C2×C4C23×D13711yescharacteristic
248C8—22no—
249C8—2742no—
250C22—41no—
251C22—41no—
252C22—41no—
253C22—2741no—
254C22—2741no—
255C22—5481no—
256C22—5481no—
257C22—5481no—
258C22—10964no—
259C22—1372no—
260C22C2×D4×D13711yescharacteristic
261C4—21no—
262C4—21no—
263C4—42no—
264C4—41no—
265C4—2741no—
266C4—2741no—
267C4—5482no—
268C4—5481no—
269C4—1372no—
270C4#41×D13712yes—
271C2—81no—
272C2—2742no—
273C2—10961no—
274C2#124×D13711yescharacteristic
275C2#2257819911yescharacteristic
276C2#2257820211yescharacteristic
subgroupG/H
centerC22C2×D4×D137
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
Fitting#489×C137C2
socleC22×C137C22×D4
prime radicalC2×D4×D137C22
hypercenter#489D137
hypocenterC137#3140
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C42 ◁ #489

Complements

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

NH ≅ G/Ntype
1. #22577958C2semidirect
3. #22577765C2semidirect
4. #22577763C2semidirect
5. #22577762C2semidirect
6. #22577759C2semidirect
7. #22577756C2semidirect
8. #22577410C2semidirect
10. #22577355C2semidirect
11. #489×C137C2semidirect
15. #17984088C22semidirect
16. #17984079C22semidirect
30. #17984013C22semidirect
31. #17984012C22semidirect
32. #17984011C22semidirect
36. #17983999C22semidirect
37. C8×Dic137C22semidirect
38. C4⋊Q8×C137C22semidirect
40. #103×C137C22semidirect
42. #97×C137C22semidirect
138. C137#3140semidirect
145. #489D137semidirect
155. C4⋊Q8C2×D137semidirect
174. #103C2×D137semidirect
180. #97C2×D137semidirect

Maximal subgroups

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

subgroupindex
#225779582
#124×D1372
#225777652
#225777632
#225777622
#225777592
#225777562
#225774102
#225774062
#225773552
#489×C1372
#3140137