Finite Group Databasebeta

← #31792716

Subgroups — #31792716

Order 17536.

398 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 398 conjugacy classes of subgroups — too many to draw. Only its 115 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#22578119C211yesmaximal, characteristic, complemented
2#22578056C211yesmaximal, characteristic, complemented
3#22577743C212yesmaximal, complemented
4#22577690C211yesmaximal, characteristic, complemented
5#22577635C212yesmaximal, complemented
6#22577605C211yesmaximal, characteristic, complemented
7#22577551C212yesmaximal, complemented
8C4×D1096C211yesmaximal, characteristic, complemented
9C8×D548C211yesmaximal, characteristic, complemented
10#22577398C212yesmaximal, complemented
11#487×C137C211yesmaximal, characteristic, complemented
12C2×#14555511C2212yes—
13C2×#14555497C2212yes—
14C2×D1096C2211yescharacteristic
15C2×C8×D137C2212yes—
16#17984077—22no—
17#17984073—22no—
18#17984064C2212yes—
19#17984063—21no—
20#17984063—21no—
21#17984063C2212yes—
22C4×D548C2211yescharacteristic
23C4×D548C2211yescharacteristic
24C4×D548C2211yescharacteristic
25#17984046—21no—
26#17984046C2212yes—
27C4⋊C4×D137—21no—
28#17984022C2211yescharacteristic
29#17984019C2212yes—
30C4⋊#11764040C2211yescharacteristic
31#17984012C2212yes—
32#17984010C2211yescharacteristic
33#17984010C2211yescharacteristic
34#17984008C2212yes—
35#17984000C2212yes—
36C4⋊Q8×C137C2211yescharacteristic
37C4⋊Q8×C137C2211yescharacteristic
38#103×C137C2212yes—
39#103×C137C2212yes—
40C4×C8×C137C2211yescharacteristic
41#14555511—24no—
42#14555511—42no—
43C2×D548—22no—
44C2×D548—22no—
45C2×D548D411yescharacteristic
46C2×D548C2311yescharacteristic
47C2×D548C2311yescharacteristic
48C2×C4×D137—22no—
49C2×C4×D137—21no—
50C2×C4×D137—21no—
51C2×C4×D137C2312yes—
52#14555497—42no—
53D1096—22no—
54C2×#11764040C2312yes—
55C8×D137—42no—
56C4⋊Dic137D411yescharacteristic
57C4⋊Dic137C2311yescharacteristic
58C4⋊Dic137C2311yescharacteristic
59#14555489—22no—
60#14555487—22no—
61#14555487—22no—
62#14555487—22no—
63#14555487—21no—
64#14555487—21no—
65#14555487D412yes—
66C4×Dic137—22no—
67C2×Q8×C137—21no—
68C2×Q8×C137C2312yes—
69C2×C8×C137C2311yescharacteristic
70C2×C8×C137C2311yescharacteristic
71C4⋊C4×C137—21no—
72C4⋊C4×C137—21no—
73C4⋊C4×C137—21no—
74C4⋊C4×C137C2312yes—
75C42×C137C2311yescharacteristic
76C22×D137—21no—
77C22×D137—21no—
78C22×D137C2×D412yes—
79D548—22no—
80D548—21no—
81D548—21no—
82D548—21no—
83D548—44no—
84D548—82no—
85D548#4112yes—
86C2×Dic137—21no—
87C2×Dic137—21no—
88C2×Dic137C2×D412yes—
89C4×D137—22no—
90C4×D137—22no—
91C4×D137—44no—
92C4×D137—42no—
93C4×D137—42no—
94#11764040—22no—
95Q8×C137—22no—
96Q8×C137—22no—
97Q8×C137—81no—
98C2×C4×C137—22no—
99C2×C4×C137—22no—
100C2×C4×C137C2×D411yescharacteristic
101C2×C4×C137C2×D411yescharacteristic
102C2×C4×C137C2411yescharacteristic
103C8×C137—21no—
104C8×C137—21no—
105C22×C137C22×D411yescharacteristic
106C2×D137—22no—
107C2×D137—22no—
108C2×D137—22no—
109C2×D137—22no—
110C2×D137—41no—
111C2×D137—41no—
112C2×D137—41no—
113C2×D137—41no—
114C2×D137—41no—
115C4×C137—21no—
116C4×C137—42no—
117C4×C137—42no—
118C4×C137#13512yes—
119C4×C137C2×#4111yescharacteristic
120C4×C137H(5,𝔽2)11yescharacteristic
121Dic137—22no—
122Dic137—41no—
123Dic137—41no—
124C2×C137#54611yescharacteristic
125C2×C137C2×#13511yescharacteristic
126C2×C137#57711yescharacteristic
127D137—42no—
128D137—42no—
129D137—81no—
130C137#330311yescharacteristic, complemented, Sylow
131#3303—1371nomaximal, Sylow
132#555—1371no—
133#547—1371no—
134#487D13711yescharacteristic, complemented
135#485—1372no—
136#476—1371no—
137#465—1372no—
138#461—1371no—
139#449—1372no—
140C4×D8—1371no—
141C8×D4—1371no—
142#416—1372no—
143C2×#41—1372no—
144C2×SD16—1372no—
145C2×D8—1371no—
146C22×C8—1372no—
147C4⋊Q8C2×D13711yescharacteristic
148C4⋊Q8C2×D13711yescharacteristic
149#124—2742no—
150#123—2742no—
151C22⋊Q8—1372no—
152#121—2741no—
153#121—2741no—
154#121—1372no—
155C4×D4—1371no—
156C4×D4—1371no—
157C4×D4—1371no—
158#117—2741no—
159#117—1372no—
160C2×C4⋊C4—2741no—
161#107—1371no—
162#106—1372no—
163C4⋊C8—1371no—
164#103—1372no—
165#103C2×D13712yes—
166#103C2×D13712yes—
167#102—1371no—
168#102—1371no—
169#102—1372no—
170C22⋊C8—1372no—
171C4×C8C2×D13711yescharacteristic
172#41—2744no—
173#41—5482no—
174C2×Q8—21no—
175C2×Q8C22×D13712yes—
176C2×D4—2742no—
177C2×D4—2742no—
178C2×D4—1371no—
179C2×D4—1371no—
180C2×D4—1371no—
181C22×C4—2742no—
182C22×C4—2741no—
183C22×C4—2741no—
184C22×C4—1372no—
185SD16—5482no—
186D8—2742no—
187C2×C8—5482no—
188C2×C8—1372no—
189C2×C8C22×D13711yescharacteristic
190C2×C8C22×D13711yescharacteristic
191C4⋊C4—21no—
192C4⋊C4—21no—
193C4⋊C4—21no—
194C4⋊C4—2742no—
195C4⋊C4—1371no—
196C4⋊C4—1371no—
197C4⋊C4—1371no—
198C4⋊C4C22×D13712yes—
199C22⋊C4—2742no—
200C22⋊C4—2742no—
201C22⋊C4—2742no—
202C22⋊C4—2741no—
203C22⋊C4—2741no—
204C22⋊C4—1372no—
205C42—2742no—
206C42C22×D13711yescharacteristic
207C23—2741no—
208C23—2741no—
209C23—1372no—
210Q8—22no—
211Q8—22no—
212Q8—81no—
213D4—2742no—
214D4—2741no—
215D4—2741no—
216D4—2741no—
217D4—5484no—
218D4—10962no—
219D4—1372no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—2742no—
223C2×C4—2742no—
224C2×C4—2741no—
225C2×C4—2741no—
226C2×C4—5484no—
227C2×C4—5482no—
228C2×C4—5482no—
229C2×C4—1372no—
230C2×C4D4×D13711yescharacteristic
231C2×C4D4×D13711yescharacteristic
232C2×C4C23×D13711yescharacteristic
233C8—21no—
234C8—21no—
235C8—2742no—
236C22—2742no—
237C22—2742no—
238C22—2742no—
239C22—2742no—
240C22—5481no—
241C22—5481no—
242C22—5481no—
243C22—5481no—
244C22—5481no—
245C22C2×D4×D13711yescharacteristic
246C4—21no—
247C4—42no—
248C4—42no—
249C4—2742no—
250C4—5481no—
251C4—5481no—
252C4#1798411112yes—
253C4#41×D13711yescharacteristic
254C4#1798413411yescharacteristic
255C2—5482no—
256C2—5482no—
257C2—10961no—
258C2#2257804611yescharacteristic
259C2C2×#1798411111yescharacteristic
260C2#2257820911yescharacteristic
subgroupG/H
centerC22C2×D4×D137
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
Fitting#487×C137C2
socleC22×C137C22×D4
hypercenter#487D137
hypocenterC137#3303
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C42 ◁ #487

Complements

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

NH ≅ G/Ntype
1. #22578119C2semidirect
2. #22578056C2semidirect
3. #22577743C2semidirect
4. #22577690C2semidirect
5. #22577635C2semidirect
6. #22577605C2semidirect
7. #22577551C2semidirect
8. C4×D1096C2semidirect
9. C8×D548C2semidirect
10. #22577398C2semidirect
11. #487×C137C2semidirect
130. C137#3303semidirect
134. #487D137semidirect

Maximal subgroups

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

subgroupindex
#225781192
#225780562
#225777432
#225776902
#225776352
#225776052
#225775512
C4×D10962
C8×D5482
#225773982
#487×C1372
#3303137