Finite Group Databasebeta

← #31792596

Subgroups — #31792596

Order 17536.

494 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 494 conjugacy classes of subgroups — too many to draw. Only its 139 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
1D4×D548C211yesmaximal, characteristic, complemented
2#22577751C211yesmaximal, characteristic, complemented
3#22577716C212yesmaximal, complemented
4#22577680C211yesmaximal, characteristic, complemented
5#22577637C212yesmaximal, complemented
6#22577598C211yesmaximal, characteristic, complemented
7#22577536C212yesmaximal, complemented
8C4×#14555495C211yesmaximal, characteristic, complemented
9D4×#11764040C211yesmaximal, characteristic, complemented
10C2×#17984008C212yesmaximal, complemented
11#547×C137C211yesmaximal, characteristic, complemented
12C22×D548C2212yes—
13C2×D4×D137—21no—
14C2×#14555495C2211yescharacteristic
15C2×#14555495C2212yescomplemented
16C22×#11764040C2212yescomplemented
17#17984085C2211yescharacteristic
18#17984063—21no—
19#17984062C2212yes—
20#17984058—22no—
21C4×D548C2211yescharacteristic
22#17984021C2212yescomplemented
23#17984021C2211yescharacteristic, complemented
24C4⋊#11764040C2211yescharacteristic, complemented
25#17984009C2211yescharacteristic, complemented
26#17984008C2212yes—
27#17984008C2211yescharacteristic
28#17984008C2212yescomplemented
29#17984008C2212yescomplemented
30#17984008C2212yescomplemented
31C22⋊#11764040C2212yescomplemented
32C4×#11764040C2211yescharacteristic, complemented
33C2×#41×C137—21no—
34C4⋊Q8×C137C2211yescharacteristic, complemented
35#123×C137—22no—
36C22⋊Q8×C137—21no—
37#121×C137C2212yescomplemented
38C4×D4×C137C2211yescharacteristic
39C4×D4×C137C2211yescharacteristic, complemented
40C2×C4⋊C4×C137C2212yescomplemented
41C23×D137—22no—
42C2×D548—24no—
43C2×D548—22no—
44C2×D548—21no—
45C2×D548—21no—
46C2×D548—42no—
47C2×D548C2312yes—
48C2×D548C2311yescharacteristic
49C2×C22⋊D137—22no—
50D4×D137—44no—
51C2×C4×D137—21no—
52#14555495—22no—
53#14555495—42no—
54C2×#11764040—42no—
55C2×#11764040C2311yescharacteristic
56C2×#11764040C2312yes—
57C2×#11764040C2311yescharacteristic
58C4⋊Dic137C2311yescharacteristic
59#14555487—21no—
60#14555487—22no—
61#41×C137—44no—
62C2×Q8×C137—21no—
63C2×D4×C137—22no—
64C2×D4×C137D411yescharacteristic
65C2×D4×C137C2311yescharacteristic
66C22×C4×C137—22no—
67C22×C4×C137—21no—
68C22×C4×C137C2312yes—
69C4⋊C4×C137—22no—
70C4⋊C4×C137—21no—
71C4⋊C4×C137D411yescharacteristic
72C4⋊C4×C137C2311yescharacteristic
73C4⋊C4×C137C2312yes—
74C22⋊C4×C137—21no—
75C22⋊C4×C137—22no—
76C22⋊C4×C137D412yes—
77C42×C137C2311yescharacteristic
78C22×D137—21no—
79C22×D137—22no—
80C22×D137—44no—
81C22×D137—44no—
82C22×D137—82no—
83D548—22no—
84D548—21no—
85D548—22no—
86D548—44no—
87D548—42no—
88D548—81no—
89D548#4112yescomplemented
90C22⋊D137—44no—
91C2×Dic137—21no—
92C4×D137—42no—
93#11764040—21no—
94#11764040—21no—
95#11764040—22no—
96C23×C137—21no—
97C23×C137C2×D412yes—
98Q8×C137—42no—
99D4×C137—21no—
100D4×C137—21no—
101D4×C137—44no—
102D4×C137D814yescomplemented
103C2×C4×C137—22no—
104C2×C4×C137—22no—
105C2×C4×C137—21no—
106C2×C4×C137—22no—
107C2×C4×C137—44no—
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—41no—
116C22×C137—41no—
117C22×C137—41no—
118C22×C137C2×D814yes—
119C22×C137C22×D411yescharacteristic
120C2×D137—22no—
121C2×D137—41no—
122C2×D137—41no—
123C2×D137—42no—
124C2×D137—42no—
125C2×D137—42no—
126C2×D137—88no—
127C4×C137—22no—
128C4×C137—21no—
129C4×C137—41no—
130C4×C137—42no—
131C4×C137C2×D812yes—
132C4×C137C2×#4111yescharacteristic
133C4×C137H(5,𝔽2)11yescharacteristic
134Dic137—41no—
135C2×C137—24no—
136C2×C137—81no—
137C2×C137#54611yescharacteristic
138C2×C137C22×D811yescharacteristic
139C2×C137#57711yescharacteristic
140D137—42no—
141D137—82no—
142C137#329411yescharacteristic, complemented, Sylow
143#3294—1371nomaximal, Sylow
144#547D13711yescharacteristic, complemented
145D42—1371no—
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—21no—
156C22×D4—2741no—
157C22×D4—1372no—
158C2×D8—1371no—
159C2×D8—1372no—
160C22×C8—1372no—
161C4⋊Q8C2×D13711yescharacteristic, complemented
162#127—1371no—
163#123—22no—
164C22⋊Q8—21no—
165#121—2741no—
166#121—1372no—
167#121C2×D13712yescomplemented
168#120—2742no—
169C4×D4—1371no—
170C4×D4C2×D13711yescharacteristic
171C4×D4C2×D13711yescharacteristic, complemented
172C2×C4⋊C4C2×D13712yescomplemented
173#107—1371no—
174#107—1372no—
175C4⋊C8—1371no—
176#102—1371no—
177#102—1371no—
178#102—1372no—
179#102—1372no—
180#102—1372no—
181#102—1372no—
182C22⋊C8—1372no—
183C4×C8—1371no—
184C24—2742no—
185#41—44no—
186C2×Q8—21no—
187C2×D4—22no—
188C2×D4—2744no—
189C2×D4—2742no—
190C2×D4—2742no—
191C2×D4—2741no—
192C2×D4—2741no—
193C2×D4—5482no—
194C2×D4—5484no—
195C2×D4—1372no—
196C2×D4—1371no—
197C2×D4C22⋊D13711yescharacteristic
198C2×D4C22×D13711yescharacteristic
199C22×C4—22no—
200C22×C4—21no—
201C22×C4—2741no—
202C22×C4C22×D13712yes—
203D8—2742no—
204D8—5482no—
205C2×C8—5482no—
206C2×C8—1371no—
207C2×C8—1371no—
208C2×C8—1372no—
209C4⋊C4—22no—
210C4⋊C4—21no—
211C4⋊C4—1371no—
212C4⋊C4C22⋊D13711yescharacteristic
213C4⋊C4C22×D13711yescharacteristic
214C4⋊C4C22×D13712yes—
215C22⋊C4—21no—
216C22⋊C4—22no—
217C22⋊C4—2741no—
218C22⋊C4—2742no—
219C22⋊C4C22⋊D13712yes—
220C42C22×D13711yescharacteristic
221C23—21no—
222C23—2741no—
223C23—2742no—
224C23—5484no—
225C23—5484no—
226C23—10962no—
227C23C2×C22⋊D13712yes—
228Q8—42no—
229D4—21no—
230D4—21no—
231D4—44no—
232D4—2742no—
233D4—2741no—
234D4—2742no—
235D4—5484no—
236D4—5484no—
237D4—5482no—
238D4—10961no—
239D4—1372no—
240D4#1455549514yescomplemented
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—21no—
244C2×C4—22no—
245C2×C4—44no—
246C2×C4—42no—
247C2×C4—2741no—
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—41no—
259C22—41no—
260C22—41no—
261C22—2742no—
262C22—5481no—
263C22—5481no—
264C22—5482no—
265C22—5482no—
266C22—5482no—
267C22—10968no—
268C22C2×#1455549514yes—
269C22C22×C22⋊D13711yescharacteristic
270C4—21no—
271C4—22no—
272C4—41no—
273C4—42no—
274C4—5481no—
275C4C2×#1455549512yes—
276C4#41×D13711yescharacteristic
277C4#1798413411yescharacteristic
278C2—24no—
279C2—81no—
280C2—5482no—
281C2—10962no—
282C2#2257804111yescharacteristic
283C2C22×#1455549511yescharacteristic
284C2#2257821111yescharacteristic
subgroupG/H
centerC22C22×C22⋊D137
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
Fitting#547×C137C2
socleC22×C137C22×D4
hypercenter#547D137
hypocenterC137#3294
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C4×D4 ◁ #547

Complements

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

NH ≅ G/Ntype
1. D4×D548C2semidirect
2. #22577751C2semidirect
3. #22577716C2semidirect
4. #22577680C2semidirect
5. #22577637C2semidirect
6. #22577598C2semidirect
7. #22577536C2semidirect
8. C4×#14555495C2semidirect
9. D4×#11764040C2semidirect
10. C2×#17984008C2semidirect
11. #547×C137C2semidirect
15. C2×#14555495C22semidirect
16. C22×#11764040C22semidirect
22. #17984021C22semidirect
23. #17984021C22semidirect
24. C4⋊#11764040C22semidirect
25. #17984009C22semidirect
28. #17984008C22semidirect
29. #17984008C22semidirect
30. #17984008C22semidirect
31. C22⋊#11764040C22semidirect
32. C4×#11764040C22semidirect
34. C4⋊Q8×C137C22semidirect
37. #121×C137C22semidirect
39. C4×D4×C137C22semidirect
40. C2×C4⋊C4×C137C22semidirect
89. D548#41semidirect
102. D4×C137D8semidirect
142. C137#3294semidirect
144. #547D137semidirect
161. C4⋊Q8C2×D137semidirect
167. #121C2×D137semidirect
171. C4×D4C2×D137semidirect
172. C2×C4⋊C4C2×D137semidirect
240. D4#14555495semidirect

Maximal subgroups

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

subgroupindex
D4×D5482
#225777512
#225777162
#225776802
#225776372
#225775982
#225775362
C4×#145554952
D4×#117640402
C2×#179840082
#547×C1372
#3294137