Finite Group Databasebeta

← #31792602

Subgroups — #31792602

Order 17536.

494 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 494 conjugacy classes of subgroups — too many to draw. Only its 141 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#22578057C211yesmaximal, characteristic, complemented
2#22577749C211yesmaximal, characteristic, complemented
3#22577714C212yesmaximal, complemented
4#22577683C211yesmaximal, characteristic, complemented
5#22577637C212yesmaximal, complemented
6#22577595C211yesmaximal, characteristic, complemented
7#22577537C212yesmaximal, complemented
8C4×#14555495C211yesmaximal, characteristic, complemented
9D4×#11764040C211yesmaximal, characteristic, complemented
10C2×#17984009C212yesmaximal, complemented
11D42×C137C211yesmaximal, characteristic, complemented
12C2×#14555509—21no—
13C2×#14555495C2211yescharacteristic
14C2×#14555495C2212yescomplemented
15C22×#11764040C2212yescomplemented
16#17984088C2211yescharacteristic, complemented
17#17984072—22no—
18#17984068—21no—
19#17984062C2212yescomplemented
20C4×D548C2211yescharacteristic, complemented
21C2×C4⋊Dic137C2212yescomplemented
22#17984021C2212yescomplemented
23#17984021C2211yescharacteristic, complemented
24C4⋊#11764040C2211yescharacteristic, complemented
25#17984009C2212yes—
26#17984009C2211yescharacteristic
27#17984009C2212yescomplemented
28#17984009C2212yescomplemented
29#17984009C2212yescomplemented
30#17984008C2211yescharacteristic, complemented
31C22⋊#11764040C2212yescomplemented
32C4×#11764040C2211yescharacteristic, complemented
33C22×D4×C137—21no—
34C22×D4×C137C2212yes—
35#127×C137C2211yescharacteristic
36#121×C137—21no—
37#121×C137C2212yes—
38#120×C137—22no—
39C4×D4×C137C2211yescharacteristic
40C4×D4×C137C2211yescharacteristic
41#14555509—44no—
42C2×Dic274—21no—
43C2×D548C2311yescharacteristic
44C2×C22⋊D137—22no—
45C22×Dic137—22no—
46C2×C4×D137—21no—
47#14555495—22no—
48#14555495—42no—
49C2×#11764040—42no—
50C2×#11764040C2311yescharacteristic
51C2×#11764040C2312yes—
52C2×#11764040C2311yescharacteristic
53C4⋊Dic137—22no—
54C4⋊Dic137—21no—
55C4⋊Dic137C2311yescharacteristic
56C4⋊Dic137C2312yes—
57#14555487—21no—
58#14555487—22no—
59C24×C137—22no—
60C2×D4×C137—24no—
61C2×D4×C137—22no—
62C2×D4×C137—22no—
63C2×D4×C137—21no—
64C2×D4×C137—21no—
65C2×D4×C137—42no—
66C2×D4×C137—44no—
67C2×D4×C137D411yescharacteristic
68C2×D4×C137C2312yes—
69C2×D4×C137C2311yescharacteristic
70C22×C4×C137—21no—
71C22×C4×C137C2312yes—
72C4⋊C4×C137D411yescharacteristic
73C4⋊C4×C137C2311yescharacteristic
74C22⋊C4×C137—21no—
75C22⋊C4×C137—22no—
76C22⋊C4×C137D412yes—
77C42×C137C2311yescharacteristic
78C22×D137—21no—
79Dic274—42no—
80D548—21no—
81D548—21no—
82C22⋊D137—44no—
83C2×Dic137—21no—
84C2×Dic137—22no—
85C2×Dic137—44no—
86C4×D137—42no—
87#11764040—21no—
88#11764040—21no—
89#11764040—22no—
90C23×C137—21no—
91C23×C137—22no—
92C23×C137—44no—
93C23×C137—44no—
94C23×C137—82no—
95C23×C137C2×D412yes—
96D4×C137—22no—
97D4×C137—21no—
98D4×C137—22no—
99D4×C137—44no—
100D4×C137—44no—
101D4×C137—42no—
102D4×C137—81no—
103D4×C137D814yescomplemented
104D4×C137#4112yescomplemented
105C2×C4×C137—22no—
106C2×C4×C137—22no—
107C2×C4×C137—21no—
108C2×C4×C137—42no—
109C2×C4×C137C2×D411yescharacteristic
110C2×C4×C137C2×D412yes—
111C2×C4×C137C2×D411yescharacteristic
112C2×C4×C137C2411yescharacteristic
113C22×C137—22no—
114C22×C137—22no—
115C22×C137—22no—
116C22×C137—41no—
117C22×C137—41no—
118C22×C137—42no—
119C22×C137—42no—
120C22×C137—42no—
121C22×C137—88no—
122C22×C137C2×D814yes—
123C22×C137C22×D411yescharacteristic
124C2×D137—41no—
125C2×D137—41no—
126C2×D137—41no—
127C4×C137—22no—
128C4×C137—21no—
129C4×C137—41no—
130C4×C137C2×D812yes—
131C4×C137C2×#4111yescharacteristic
132C4×C137H(5,𝔽2)11yescharacteristic
133Dic137—41no—
134Dic137—42no—
135C2×C137—24no—
136C2×C137—42no—
137C2×C137—82no—
138C2×C137#54611yescharacteristic
139C2×C137C22×D811yescharacteristic
140C2×C137#57711yescharacteristic
141D137—81no—
142C137#329411yescharacteristic, complemented, Sylow
143#3294—1371nomaximal, Sylow
144#547—1371no—
145D42D13711yescharacteristic, complemented
146#486—1371no—
147#480—1372no—
148#474—1371no—
149#466—1372no—
150#459—1371no—
151#447—1372no—
152C4×D8—1371no—
153C8×D4—1371no—
154C2×#102—1372no—
155C2×#41—2741no—
156C22×D4—21no—
157C22×D4C2×D13712yes—
158C2×D8—1371no—
159C2×D8—1372no—
160C22×C8—1372no—
161C4⋊Q8—1371no—
162#127C2×D13711yescharacteristic
163#123—2742no—
164C22⋊Q8—2741no—
165#121—21no—
166#121—1372no—
167#121C2×D13712yes—
168#120—22no—
169C4×D4—1371no—
170C4×D4C2×D13711yescharacteristic
171C4×D4C2×D13711yescharacteristic
172C2×C4⋊C4—1372no—
173#107—1371no—
174#107—1372no—
175C4⋊C8—1371no—
176#102—1371no—
177#102—1372no—
178#102—1372no—
179#102—1372no—
180#102—1372no—
181#102—1371no—
182C22⋊C8—1372no—
183C4×C8—1371no—
184C24—22no—
185#41—5484no—
186C2×Q8—2741no—
187C2×D4—24no—
188C2×D4—22no—
189C2×D4—22no—
190C2×D4—21no—
191C2×D4—21no—
192C2×D4—42no—
193C2×D4—44no—
194C2×D4—2742no—
195C2×D4—1371no—
196C2×D4C22⋊D13711yescharacteristic
197C2×D4C22×D13712yes—
198C2×D4C22×D13711yescharacteristic
199C22×C4—21no—
200C22×C4—2742no—
201C22×C4—2741no—
202C22×C4C22×D13712yes—
203D8—2742no—
204D8—5482no—
205C2×C8—5482no—
206C2×C8—1371no—
207C2×C8—1371no—
208C2×C8—1372no—
209C4⋊C4—2742no—
210C4⋊C4—2741no—
211C4⋊C4—1371no—
212C4⋊C4—1372no—
213C4⋊C4C22⋊D13711yescharacteristic
214C4⋊C4C22×D13711yescharacteristic
215C22⋊C4—21no—
216C22⋊C4—22no—
217C22⋊C4—2741no—
218C22⋊C4—2742no—
219C22⋊C4C22⋊D13712yes—
220C42C22×D13711yescharacteristic
221C23—21no—
222C23—22no—
223C23—44no—
224C23—44no—
225C23—82no—
226C23—2741no—
227C23C2×C22⋊D13712yes—
228Q8—5482no—
229D4—22no—
230D4—21no—
231D4—22no—
232D4—44no—
233D4—44no—
234D4—42no—
235D4—81no—
236D4—2741no—
237D4—2741no—
238D4—5484no—
239D4#1455549514yescomplemented
240D4#1455550912yescomplemented
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—21no—
244C2×C4—42no—
245C2×C4—2741no—
246C2×C4—2742no—
247C2×C4—5484no—
248C2×C4—5482no—
249C2×C4C2×C22⋊D13711yescharacteristic
250C2×C4C2×C22⋊D13712yes—
251C2×C4C2×C22⋊D13711yescharacteristic
252C2×C4C23×D13711yescharacteristic
253C8—2741no—
254C8—2741no—
255C8—2742no—
256C22—22no—
257C22—22no—
258C22—22no—
259C22—41no—
260C22—41no—
261C22—42no—
262C22—42no—
263C22—42no—
264C22—88no—
265C22—5481no—
266C22—5481no—
267C22—5481no—
268C22C2×#1455549514yes—
269C22C22×C22⋊D13711yescharacteristic
270C4—21no—
271C4—22no—
272C4—41no—
273C4—5481no—
274C4—5482no—
275C4C2×#1455549512yes—
276C4C2×#1455550911yescharacteristic
277C4#1798413311yescharacteristic
278C2—24no—
279C2—42no—
280C2—82no—
281C2—10961no—
282C2#2257804711yescharacteristic
283C2C22×#1455549511yescharacteristic
284C2#2257821411yescharacteristic
subgroupG/H
centerC22C22×C22⋊D137
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
FittingD42×C137C2
socleC22×C137C22×D4
hypercenterD42D137
hypocenterC137#3294
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C4×D4 ◁ D42

Complements

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

NH ≅ G/Ntype
1. #22578057C2semidirect
2. #22577749C2semidirect
3. #22577714C2semidirect
4. #22577683C2semidirect
5. #22577637C2semidirect
6. #22577595C2semidirect
7. #22577537C2semidirect
8. C4×#14555495C2semidirect
9. D4×#11764040C2semidirect
10. C2×#17984009C2semidirect
11. D42×C137C2semidirect
14. C2×#14555495C22semidirect
15. C22×#11764040C22semidirect
16. #17984088C22semidirect
19. #17984062C22semidirect
20. C4×D548C22semidirect
21. C2×C4⋊Dic137C22semidirect
22. #17984021C22semidirect
23. #17984021C22semidirect
24. C4⋊#11764040C22semidirect
27. #17984009C22semidirect
28. #17984009C22semidirect
29. #17984009C22semidirect
30. #17984008C22semidirect
31. C22⋊#11764040C22semidirect
32. C4×#11764040C22semidirect
103. D4×C137D8semidirect
104. D4×C137#41semidirect
142. C137#3294semidirect
145. D42D137semidirect
239. D4#14555495semidirect
240. D4#14555509semidirect

Maximal subgroups

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

subgroupindex
#225780572
#225777492
#225777142
#225776832
#225776372
#225775952
#225775372
C4×#145554952
D4×#117640402
C2×#179840092
D42×C1372
#3294137