Finite Group Databasebeta

← #31792604

Subgroups — #31792604

Order 17536.

494 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 494 conjugacy classes of subgroups — too many to draw. Only its 125 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#22578058C211yesmaximal, characteristic, complemented
2D4×D548C211yesmaximal, characteristic, complemented
3#22577750C211yesmaximal, characteristic, complemented
4#22577718C212yesmaximal, complemented
5#22577683C211yesmaximal, characteristic, complemented
6#22577638C212yesmaximal, complemented
7#22577598C211yesmaximal, characteristic, complemented
8#22577535C212yesmaximal, complemented
9C8×D548C211yesmaximal, characteristic, complemented
10#102×D137C212yesmaximal, complemented
11C4×D8×C137C211yesmaximal, characteristic, complemented
12C2×#14555510—21no—
13C22×D548—21no—
14C2×D4×D137C2212yes—
15C2×#14555495C2212yescomplemented
16C2×C8×D137C2212yescomplemented
17#17984088C2211yescharacteristic, complemented
18#17984085C2211yescharacteristic
19#17984070—22no—
20#17984067—21no—
21#17984063C2212yes—
22#17984062—21no—
23#17984059C2212yescomplemented
24#17984058—22no—
25C4×D548C2211yescharacteristic
26C4⋊C4×D137C2212yescomplemented
27#17984021C2212yescomplemented
28C4⋊#11764040C2211yescharacteristic, complemented
29#17984010C2212yescomplemented
30#17984010C2212yescomplemented
31#17984009C2212yes—
32#17984008C2212yescomplemented
33#17984000C2212yescomplemented
34C2×D8×C137C2211yescharacteristic
35C4×D4×C137C2211yescharacteristic
36C4×D4×C137C2211yescharacteristic, complemented
37#107×C137C2211yescharacteristic, complemented
38#102×C137C2211yescharacteristic
39#102×C137C2211yescharacteristic, complemented
40C4×C8×C137C2211yescharacteristic, complemented
41C23×D137—22no—
42#14555510—44no—
43C2×Dic274—21no—
44C2×D548—22no—
45C2×D548—21no—
46C2×D548—21no—
47C2×D548—44no—
48C2×D548D411yescharacteristic
49C2×D548C2312yes—
50C2×C22⋊D137—22no—
51C2×C22⋊D137—22no—
52D4×D137—24no—
53D4×D137—42no—
54C2×C4×D137—22no—
55C2×C4×D137C2312yes—
56#14555495—42no—
57C2×#11764040C2312yes—
58C8×D137—42no—
59C4⋊Dic137—21no—
60C4⋊Dic137D411yescharacteristic
61C4⋊Dic137C2312yes—
62#14555489—22no—
63C22⋊Dic137—22no—
64#14555487—22no—
65#14555487D412yes—
66C2×D4×C137C2311yescharacteristic
67C2×D4×C137C2311yescharacteristic
68C22×C4×C137—21no—
69C22×C4×C137—21no—
70D8×C137—22no—
71C2×C8×C137C2311yescharacteristic
72C2×C8×C137C2311yescharacteristic
73C4⋊C4×C137C2311yescharacteristic
74C4⋊C4×C137C2311yescharacteristic
75C22⋊C4×C137—21no—
76C22⋊C4×C137—21no—
77C42×C137C2311yescharacteristic
78C22×D137—22no—
79C22×D137—44no—
80C22×D137—44no—
81C22×D137—82no—
82C22×D137C2×D412yes—
83Dic274—42no—
84D548—22no—
85D548—22no—
86D548—44no—
87D548—42no—
88D548—81no—
89D548D814yescomplemented
90C22⋊D137—44no—
91C22⋊D137—44no—
92C2×Dic137—22no—
93C2×Dic137C2×D412yes—
94C4×D137—22no—
95C4×D137—22no—
96C4×D137—44no—
97#11764040—22no—
98C23×C137—21no—
99C23×C137—21no—
100D4×C137—21no—
101D4×C137—21no—
102D4×C137—21no—
103D4×C137#4112yescomplemented
104C2×C4×C137—21no—
105C2×C4×C137—21no—
106C2×C4×C137—42no—
107C2×C4×C137—42no—
108C2×C4×C137C2×D411yescharacteristic
109C2×C4×C137C2×D411yescharacteristic
110C2×C4×C137C2411yescharacteristic
111C8×C137—21no—
112C8×C137—21no—
113C22×C137—22no—
114C22×C137—41no—
115C22×C137—41no—
116C22×C137—41no—
117C22×C137—41no—
118C22×C137—41no—
119C22×C137C22×D411yescharacteristic
120C2×D137—22no—
121C2×D137—22no—
122C2×D137—42no—
123C2×D137—42no—
124C2×D137—42no—
125C2×D137—88no—
126C2×D137C2×D814yes—
127C4×C137—21no—
128C4×C137—41no—
129C4×C137—41no—
130C4×C137C2×D812yes—
131C4×C137C2×#4111yescharacteristic
132C4×C137H(5,𝔽2)11yescharacteristic
133Dic137—22no—
134Dic137—42no—
135C2×C137—42no—
136C2×C137—81no—
137C2×C137#54611yescharacteristic
138C2×C137C22×D811yescharacteristic
139C2×C137#57711yescharacteristic
140D137—24no—
141D137—82no—
142C137#329411yescharacteristic, complemented, Sylow
143#3294—1371nomaximal, Sylow
144#547—1371no—
145D42—1371no—
146#486—1371no—
147#480—1372no—
148#474—1371no—
149#466—1372no—
150#459—1371no—
151#447—1372no—
152C4×D8D13711yescharacteristic, complemented
153C8×D4—1371no—
154C2×#102—1372no—
155C2×#41—2741no—
156C22×D4—2741no—
157C22×D4—1372no—
158C2×D8—1372no—
159C2×D8C2×D13711yescharacteristic
160C22×C8—1372no—
161C4⋊Q8—1371no—
162#127—1371no—
163#123—2742no—
164C22⋊Q8—2741no—
165#121—2741no—
166#121—1372no—
167#121—1372no—
168#120—2742no—
169C4×D4—1371no—
170C4×D4C2×D13711yescharacteristic
171C4×D4C2×D13711yescharacteristic, complemented
172C2×C4⋊C4—1372no—
173#107—1372no—
174#107C2×D13711yescharacteristic, complemented
175C4⋊C8—1371no—
176#102—1372no—
177#102—1372no—
178#102—1372no—
179#102—1372no—
180#102C2×D13711yescharacteristic
181#102C2×D13711yescharacteristic, complemented
182C22⋊C8—1372no—
183C4×C8C2×D13711yescharacteristic, complemented
184C24—2742no—
185#41—5484no—
186C2×Q8—2741no—
187C2×D4—2742no—
188C2×D4—2744no—
189C2×D4—2742no—
190C2×D4—2742no—
191C2×D4—2741no—
192C2×D4—2741no—
193C2×D4—5484no—
194C2×D4—5482no—
195C2×D4—1371no—
196C2×D4—1372no—
197C2×D4C22×D13711yescharacteristic
198C2×D4C22×D13711yescharacteristic
199C22×C4—21no—
200C22×C4—21no—
201C22×C4—2742no—
202C22×C4—1372no—
203D8—22no—
204D8—5482no—
205C2×C8—5482no—
206C2×C8—1372no—
207C2×C8C22×D13711yescharacteristic
208C2×C8C22×D13711yescharacteristic
209C4⋊C4—2742no—
210C4⋊C4—2741no—
211C4⋊C4—1372no—
212C4⋊C4—1371no—
213C4⋊C4C22×D13711yescharacteristic
214C4⋊C4C22×D13711yescharacteristic
215C22⋊C4—21no—
216C22⋊C4—21no—
217C22⋊C4—2742no—
218C22⋊C4—2742no—
219C22⋊C4—1372no—
220C42C22×D13711yescharacteristic
221C23—21no—
222C23—21no—
223C23—2742no—
224C23—5484no—
225C23—5484no—
226C23—10962no—
227C23—1372no—
228Q8—5482no—
229D4—21no—
230D4—21no—
231D4—21no—
232D4—2742no—
233D4—2742no—
234D4—5484no—
235D4—5484no—
236D4—5484no—
237D4—5482no—
238D4—10961no—
239D4—1374no—
240D4#1455551012yescomplemented
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—42no—
244C2×C4—42no—
245C2×C4—2742no—
246C2×C4—2742no—
247C2×C4—2742no—
248C2×C4—5484no—
249C2×C4—1372no—
250C2×C4D4×D13711yescharacteristic
251C2×C4D4×D13711yescharacteristic
252C2×C4C23×D13711yescharacteristic
253C8—21no—
254C8—21no—
255C8—2742no—
256C22—22no—
257C22—41no—
258C22—41no—
259C22—41no—
260C22—41no—
261C22—41no—
262C22—2742no—
263C22—2742no—
264C22—5482no—
265C22—5482no—
266C22—5482no—
267C22—10968no—
268C22—1374no—
269C22C2×D4×D13711yescharacteristic
270C4—21no—
271C4—41no—
272C4—41no—
273C4—2742no—
274C4—5482no—
275C4D8×D13712yes—
276C4C2×#1455551011yescharacteristic
277C4#1798413411yescharacteristic
278C2—42no—
279C2—81no—
280C2—2744no—
281C2—10962no—
282C2#2257804911yescharacteristic
283C2C2×D8×D13711yescharacteristic
284C2#2257821011yescharacteristic
subgroupG/H
centerC22C2×D4×D137
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
FittingC4×D8×C137C2
socleC22×C137C22×D4
hypercenterC4×D8D137
hypocenterC137#3294
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C42 ◁ C4×D8

Complements

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

NH ≅ G/Ntype
1. #22578058C2semidirect
2. D4×D548C2semidirect
3. #22577750C2semidirect
4. #22577718C2semidirect
5. #22577683C2semidirect
6. #22577638C2semidirect
7. #22577598C2semidirect
8. #22577535C2semidirect
9. C8×D548C2semidirect
10. #102×D137C2semidirect
11. C4×D8×C137C2semidirect
15. C2×#14555495C22semidirect
16. C2×C8×D137C22semidirect
17. #17984088C22semidirect
23. #17984059C22semidirect
26. C4⋊C4×D137C22semidirect
27. #17984021C22semidirect
28. C4⋊#11764040C22semidirect
29. #17984010C22semidirect
30. #17984010C22semidirect
32. #17984008C22semidirect
33. #17984000C22semidirect
36. C4×D4×C137C22semidirect
37. #107×C137C22semidirect
39. #102×C137C22semidirect
40. C4×C8×C137C22semidirect
89. D548D8semidirect
103. D4×C137#41semidirect
142. C137#3294semidirect
152. C4×D8D137semidirect
171. C4×D4C2×D137semidirect
174. #107C2×D137semidirect
181. #102C2×D137semidirect
183. C4×C8C2×D137semidirect
240. D4#14555510semidirect

Maximal subgroups

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

subgroupindex
#225780582
D4×D5482
#225777502
#225777182
#225776832
#225776382
#225775982
#225775352
C8×D5482
#102×D1372
C4×D8×C1372
#3294137