Finite Group Databasebeta

← #31790748

Subgroups — #31790748

Order 17536.

418 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 418 conjugacy classes of subgroups — too many to draw. Only its 133 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#22577967C211yesmaximal, characteristic
2#22577886C211yesmaximal, characteristic, complemented
3#22577728C212yesmaximal, complemented
4#22577728C212yesmaximal, complemented
5#22577722C212yesmaximal, complemented
6#22577717C212yesmaximal, complemented
7#22577445C211yesmaximal, characteristic, complemented
8#22577441C211yesmaximal, characteristic, complemented
9#22577409C211yesmaximal, characteristic, complemented
10C2×#17984010C211yesmaximal, characteristic, complemented
11#408×C137C211yesmaximal, characteristic, complemented
12C22×D548C2211yescharacteristic
13#17984084—21no—
14#17984077—21no—
15#17984062C2212yes—
16#17984062C2212yes—
17#17984056—22no—
18Q8×Dic137—21no—
19D4×Dic137—22no—
20D4×Dic137—21no—
21#17984047C2211yescharacteristic
22#17984047C2211yescharacteristic
23#17984047C2212yescomplemented
24C2×C4⋊Dic137C2211yescharacteristic
25C2×C4⋊Dic137C2212yescomplemented
26#17984022C2211yescharacteristic, complemented
27#17984022C2211yescharacteristic, complemented
28#17984022C2212yescomplemented
29#17984020C2211yescharacteristic, complemented
30#17984020C2211yescharacteristic, complemented
31#17984020C2212yescomplemented
32#17984010C2212yes—
33#17984010C2212yes—
34#17984010C2212yes—
35#17984010C2212yes—
36C2×#41×C137C2211yescharacteristic
37C2×M16×C137C2211yescharacteristic
38C22×C8×C137C2211yescharacteristic
39C22⋊C8×C137C2212yescomplemented
40C22⋊C8×C137C2212yescomplemented
41C23×D137—21no—
42C2×D548—21no—
43C2×D548—22no—
44C2×D548—22no—
45C2×D548C2312yes—
46C2×C22⋊D137—22no—
47C2×C22⋊D137—22no—
48C22×Dic137—21no—
49C22×Dic137—22no—
50C4⋊Dic137—21no—
51C4⋊Dic137—22no—
52C4⋊Dic137C2312yes—
53C4⋊Dic137C2311yescharacteristic
54C4⋊Dic137C2311yescharacteristic
55C4⋊Dic137C2311yescharacteristic
56C4⋊Dic137C2311yescharacteristic
57C22⋊Dic137—21no—
58C22⋊Dic137—21no—
59C22⋊Dic137—22no—
60#14555487—22no—
61#14555487—22no—
62C4×Dic137—21no—
63C4×Dic137—21no—
64C4×Dic137—22no—
65#41×C137—22no—
66#41×C137—22no—
67C2×Q8×C137D411yescharacteristic
68C2×D4×C137D412yes—
69C2×D4×C137D411yescharacteristic
70C22×C4×C137C2312yes—
71C22×C4×C137C2311yescharacteristic
72M16×C137—22no—
73C2×C8×C137—22no—
74C2×C8×C137C2311yescharacteristic
75C2×C8×C137C2312yes—
76C2×C8×C137C2311yescharacteristic
77C22×D137—22no—
78C22×D137—42no—
79C22×D137—81no—
80C22×D137—81no—
81D548—22no—
82D548—22no—
83D548—42no—
84C22⋊D137—82no—
85C22⋊D137—82no—
86C2×Dic137—21no—
87C2×Dic137—21no—
88C2×Dic137—21no—
89C2×Dic137—21no—
90C2×Dic137—22no—
91C2×Dic137—81no—
92C2×Dic137—82no—
93C23×C137C2×D412yes—
94C23×C137C2×D411yescharacteristic
95Q8×C137—22no—
96D4×C137—24no—
97D4×C137—22no—
98C2×C4×C137—22no—
99C2×C4×C137—22no—
100C2×C4×C137C2×D412yes—
101C2×C4×C137C2×D411yescharacteristic
102C2×C4×C137#4112yes—
103C2×C4×C137#4112yes—
104C2×C4×C137C2411yescharacteristic
105C8×C137—21no—
106C8×C137—21no—
107C8×C137—22no—
108C22×C137—21no—
109C22×C137—21no—
110C22×C137—22no—
111C22×C137—22no—
112C22×C137—22no—
113C22×C137#12312yes—
114C22×C137C22×D411yescharacteristic
115C2×D137—42no—
116C2×D137—42no—
117C2×D137—42no—
118C2×D137—84no—
119C4×C137—22no—
120C4×C137#12312yes—
121C4×C137C2×#4111yescharacteristic
122C4×C137C2×#4111yescharacteristic
123Dic137—41no—
124Dic137—42no—
125Dic137—41no—
126Dic137—41no—
127Dic137—41no—
128C2×C137—22no—
129C2×C137—42no—
130C2×C137C2×#12311yescharacteristic
131C2×C137#57611yescharacteristic
132C2×C137#57711yescharacteristic
133D137—82no—
134C137#309611yescharacteristic, complemented, Sylow
135#3096—1371nomaximal, Sylow
136#535—1371no—
137#520—1371no—
138#482—1372no—
139#482—1372no—
140#481—1372no—
141#480—1372no—
142#428—1371no—
143#427—1371no—
144#418—1371no—
145C2×#102—1371no—
146#408D13711yescharacteristic, complemented
147C2×#41C2×D13711yescharacteristic
148C22×D4—1371no—
149C2×M16C2×D13711yescharacteristic
150C22×C8C2×D13711yescharacteristic
151#127—2741no—
152#124—2741no—
153#121—1372no—
154#121—1372no—
155#120—2742no—
156C4×Q8—2741no—
157C4×D4—2742no—
158C4×D4—2741no—
159#117—1371no—
160#117—1372no—
161#117—1371no—
162C2×C4⋊C4—1371no—
163C2×C4⋊C4—1372no—
164#107—1371no—
165#107—1371no—
166#107—1372no—
167#106—1371no—
168#106—1371no—
169#106—1372no—
170#102—1372no—
171#102—1372no—
172#102—1372no—
173#102—1372no—
174C22⋊C8C2×D13712yescomplemented
175C22⋊C8C2×D13712yescomplemented
176C24—2741no—
177#41—22no—
178#41—22no—
179C2×Q8D54811yescharacteristic
180C2×D4—2742no—
181C2×D4—2742no—
182C2×D4—2741no—
183C2×D4—2742no—
184C2×D4—2742no—
185C2×D4—1372no—
186C2×D4D54812yes—
187C2×D4D54811yescharacteristic
188C22×C4—2742no—
189C22×C4—2741no—
190C22×C4C22×D13712yes—
191C22×C4C22×D13711yescharacteristic
192M16—22no—
193C2×C8—22no—
194C2×C8C22×D13711yescharacteristic
195C2×C8C22×D13712yes—
196C2×C8C22×D13711yescharacteristic
197C4⋊C4—2742no—
198C4⋊C4—2741no—
199C4⋊C4—1372no—
200C4⋊C4—1371no—
201C4⋊C4—1371no—
202C4⋊C4—1371no—
203C4⋊C4—1371no—
204C22⋊C4—2742no—
205C22⋊C4—2741no—
206C22⋊C4—2741no—
207C22⋊C4—2742no—
208C22⋊C4—2742no—
209C42—2742no—
210C42—2741no—
211C42—2741no—
212C23—2742no—
213C23—5482no—
214C23—10961no—
215C23—10961no—
216C23C2×D54812yes—
217C23C2×D54811yescharacteristic
218Q8—22no—
219D4—24no—
220D4—22no—
221D4—2742no—
222D4—2742no—
223D4—5482no—
224D4—10962no—
225D4—10962no—
226C2×C4—22no—
227C2×C4—22no—
228C2×C4—2741no—
229C2×C4—2741no—
230C2×C4—2741no—
231C2×C4—2742no—
232C2×C4—2741no—
233C2×C4—10961no—
234C2×C4—10962no—
235C2×C4C2×D54812yes—
236C2×C4C2×D54811yescharacteristic
237C2×C4#1455550912yes—
238C2×C4#1455550912yes—
239C2×C4C23×D13711yescharacteristic
240C8—21no—
241C8—21no—
242C8—22no—
243C22—21no—
244C22—21no—
245C22—22no—
246C22—22no—
247C22—22no—
248C22—5482no—
249C22—5482no—
250C22—5482no—
251C22—10964no—
252C22#1798407212yes—
253C22C22×D54811yescharacteristic
254C4—22no—
255C4—5482no—
256C4—5481no—
257C4—5481no—
258C4—5481no—
259C4—5481no—
260C4#1798407212yes—
261C4C2×#1455550911yescharacteristic
262C4C2×#1455550911yescharacteristic
263C2—22no—
264C2—42no—
265C2—10962no—
266C2C2×#1798407211yescharacteristic
267C2#2257820811yescharacteristic
268C2#2257821511yescharacteristic
subgroupG/H
centerC22C22×D548
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
Fitting#408×C137C2
socleC22×C137C22×D4
prime radical#22577967C2
hypercenter#408D137
hypocenterC137#3096
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C2×#41 ◁ #408

Complements

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

NH ≅ G/Ntype
2. #22577886C2semidirect
3. #22577728C2semidirect
4. #22577728C2semidirect
5. #22577722C2semidirect
6. #22577717C2semidirect
7. #22577445C2semidirect
8. #22577441C2semidirect
9. #22577409C2semidirect
10. C2×#17984010C2semidirect
11. #408×C137C2semidirect
23. #17984047C22semidirect
25. C2×C4⋊Dic137C22semidirect
26. #17984022C22semidirect
27. #17984022C22semidirect
28. #17984022C22semidirect
29. #17984020C22semidirect
30. #17984020C22semidirect
31. #17984020C22semidirect
39. C22⋊C8×C137C22semidirect
40. C22⋊C8×C137C22semidirect
134. C137#3096semidirect
146. #408D137semidirect
174. C22⋊C8C2×D137semidirect
175. C22⋊C8C2×D137semidirect

Maximal subgroups

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

subgroupindex
#225779672
#225778862
#225777282
#225777222
#225777172
#225774452
#225774412
#225774092
C2×#179840102
#408×C1372
#3096137