Finite Group Databasebeta

← #31818218

Subgroups — #31818218

Order 17792.

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
1#22581607C211yesmaximal, characteristic, complemented
2D4×D556C211yesmaximal, characteristic, complemented
3#22581300C211yesmaximal, characteristic, complemented
4#22581267C212yesmaximal, complemented
5#22581234C211yesmaximal, characteristic, complemented
6#22581189C212yesmaximal, complemented
7#22581149C211yesmaximal, characteristic, complemented
8#22581089C212yesmaximal, complemented
9C4×D1112C211yesmaximal, characteristic, complemented
10C2×#17984764C212yesmaximal, complemented
11C8×D4×C139C211yesmaximal, characteristic, complemented
12C2×#14555784—21no—
13C22×D556C2212yes—
14C2×D4×D139—21no—
15C2×D1112C2211yescharacteristic
16C2×D1112C2212yescomplemented
17#17984842C2211yescharacteristic, complemented
18#17984839C2211yescharacteristic
19#17984826—22no—
20#17984822—21no—
21#17984817—21no—
22#17984816C2212yes—
23#17984816C2212yescomplemented
24#17984812—22no—
25C4×D556C2211yescharacteristic
26C4×D556C2211yescharacteristic, complemented
27C2×C4⋊Dic139C2212yescomplemented
28#17984776C2211yescharacteristic, complemented
29#17984776C2212yescomplemented
30#17984764C2211yescharacteristic
31#17984764C2212yes—
32#17984764C2212yescomplemented
33#17984764C2212yescomplemented
34#17984764C2211yescharacteristic, complemented
35#17984764C2212yescomplemented
36C22×C8×C139C2212yescomplemented
37C4×D4×C139C2211yescharacteristic
38C4⋊C8×C139C2211yescharacteristic, complemented
39C22⋊C8×C139C2212yescomplemented
40C4×C8×C139C2211yescharacteristic, complemented
41C23×D139—22no—
42#14555784—44no—
43C2×Dic278—21no—
44C2×D556—24no—
45C2×D556—22no—
46C2×D556—21no—
47C2×D556—21no—
48C2×D556—42no—
49C2×D556C2311yescharacteristic
50C2×D556C2312yes—
51C2×D556C2311yescharacteristic
52C2×C22⋊D139—22no—
53C2×C22⋊D139—22no—
54D4×D139—44no—
55C22×Dic139—22no—
56C2×C4×D139—21no—
57C2×C4×D139—21no—
58D1112—22no—
59D1112—42no—
60C4⋊Dic139—22no—
61C4⋊Dic139—21no—
62C4⋊Dic139C2311yescharacteristic
63C4⋊Dic139C2311yescharacteristic
64C4⋊Dic139C2312yes—
65(C2×D139)⋊C4—21no—
66(C2×D139)⋊C4—21no—
67(C2×D139)⋊C4—22no—
68(C2×D139)⋊C4—22no—
69C2×D4×C139D411yescharacteristic
70C22×C4×C139C2312yes—
71C2×C8×C139—42no—
72C2×C8×C139C2311yescharacteristic
73C2×C8×C139C2312yes—
74C2×C8×C139C2311yescharacteristic
75C4⋊C4×C139D411yescharacteristic
76C22⋊C4×C139D412yes—
77C42×C139C2311yescharacteristic
78C22×D139—21no—
79C22×D139—21no—
80C22×D139—22no—
81C22×D139—44no—
82C22×D139—44no—
83C22×D139—82no—
84Dic278—42no—
85D556—21no—
86D556—22no—
87D556—21no—
88D556—21no—
89D556—22no—
90D556—44no—
91D556—42no—
92D556—81no—
93D556#4112yescomplemented
94C22⋊D139—44no—
95C22⋊D139—44no—
96C2×Dic139—21no—
97C2×Dic139—21no—
98C2×Dic139—22no—
99C2×Dic139—44no—
100C4×D139—42no—
101C4×D139—42no—
102C23×C139C2×D412yes—
103D4×C139D814yescomplemented
104C2×C4×C139—22no—
105C2×C4×C139—22no—
106C2×C4×C139C2×D411yescharacteristic
107C2×C4×C139C2×D412yes—
108C2×C4×C139C2×D411yescharacteristic
109C2×C4×C139C2411yescharacteristic
110C8×C139—21no—
111C8×C139—21no—
112C8×C139—22no—
113C22×C139—22no—
114C22×C139—22no—
115C22×C139C2×D814yes—
116C22×C139C22×D411yescharacteristic
117C2×D139—22no—
118C2×D139—41no—
119C2×D139—41no—
120C2×D139—42no—
121C2×D139—41no—
122C2×D139—41no—
123C2×D139—42no—
124C2×D139—42no—
125C2×D139—41no—
126C2×D139—88no—
127C4×C139—22no—
128C4×C139—21no—
129C4×C139C2×D812yes—
130C4×C139C2×#4111yescharacteristic
131C4×C139H(5,𝔽2)11yescharacteristic
132Dic139—41no—
133Dic139—41no—
134Dic139—42no—
135C2×C139—24no—
136C2×C139#54611yescharacteristic
137C2×C139C22×D811yescharacteristic
138C2×C139#57711yescharacteristic
139D139—42no—
140D139—81no—
141D139—82no—
142C139#329411yescharacteristic, complemented, Sylow
143#3294—1391nomaximal, Sylow
144#547—1391no—
145D42—1391no—
146#486—1391no—
147#480—1392no—
148#474—1391no—
149#466—1392no—
150#459—1391no—
151#447—1392no—
152C4×D8—1391no—
153C8×D4D13911yescharacteristic, complemented
154C2×#102—1392no—
155C2×#41—2781no—
156C22×D4—2781no—
157C22×D4—1392no—
158C2×D8—1391no—
159C2×D8—1392no—
160C22×C8C2×D13912yescomplemented
161C4⋊Q8—1391no—
162#127—1391no—
163#123—2782no—
164C22⋊Q8—2781no—
165#121—2781no—
166#121—1392no—
167#121—1392no—
168#120—2782no—
169C4×D4—1391no—
170C4×D4—1391no—
171C4×D4C2×D13911yescharacteristic
172C2×C4⋊C4—1392no—
173#107—1391no—
174#107—1392no—
175C4⋊C8C2×D13911yescharacteristic, complemented
176#102—1391no—
177#102—1392no—
178#102—1392no—
179#102—1392no—
180#102—1391no—
181#102—1392no—
182C22⋊C8C2×D13912yescomplemented
183C4×C8C2×D13911yescharacteristic, complemented
184C24—2782no—
185#41—5564no—
186C2×Q8—2781no—
187C2×D4—2784no—
188C2×D4—2782no—
189C2×D4—2782no—
190C2×D4—2782no—
191C2×D4—2781no—
192C2×D4—2781no—
193C2×D4—5562no—
194C2×D4—5564no—
195C2×D4—1392no—
196C2×D4—1391no—
197C2×D4—1391no—
198C2×D4D55611yescharacteristic
199C22×C4—2782no—
200C22×C4—2781no—
201C22×C4—2781no—
202C22×C4C22×D13912yes—
203D8—2782no—
204D8—5562no—
205C2×C8—42no—
206C2×C8C22×D13911yescharacteristic
207C2×C8C22×D13911yescharacteristic
208C2×C8C22×D13912yes—
209C4⋊C4—2782no—
210C4⋊C4—2781no—
211C4⋊C4—1391no—
212C4⋊C4—1392no—
213C4⋊C4—1391no—
214C4⋊C4D55611yescharacteristic
215C22⋊C4—2781no—
216C22⋊C4—2781no—
217C22⋊C4—2782no—
218C22⋊C4—2782no—
219C22⋊C4D55612yes—
220C42C22×D13911yescharacteristic
221C23—2781no—
222C23—2781no—
223C23—2782no—
224C23—5564no—
225C23—5564no—
226C23—11122no—
227C23C2×D55612yes—
228Q8—5562no—
229D4—2781no—
230D4—2782no—
231D4—2781no—
232D4—2781no—
233D4—2782no—
234D4—5564no—
235D4—5564no—
236D4—5564no—
237D4—5562no—
238D4—11121no—
239D4—1392no—
240D4D111214yescomplemented
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—2781no—
244C2×C4—2781no—
245C2×C4—2782no—
246C2×C4—5564no—
247C2×C4—5562no—
248C2×C4—5562no—
249C2×C4C2×D55611yescharacteristic
250C2×C4C2×D55612yes—
251C2×C4C2×D55611yescharacteristic
252C2×C4C23×D13911yescharacteristic
253C8—21no—
254C8—21no—
255C8—22no—
256C22—22no—
257C22—22no—
258C22—2782no—
259C22—5561no—
260C22—5561no—
261C22—5562no—
262C22—5561no—
263C22—5562no—
264C22—5561no—
265C22—5561no—
266C22—5562no—
267C22—11128no—
268C22C2×D111214yes—
269C22C22×D55611yescharacteristic
270C4—21no—
271C4—22no—
272C4—5561no—
273C4—5561no—
274C4—5562no—
275C4C2×D111212yes—
276C4#41×D13911yescharacteristic
277C4#1798488711yescharacteristic
278C2—24no—
279C2—5562no—
280C2—11122no—
281C2—11121no—
282C2#2258159811yescharacteristic
283C2C22×D111211yescharacteristic
284C2#2258176511yescharacteristic
subgroupG/H
centerC22C22×D556
commutator, cosocleC2×C4×C139C24
FrattiniC2×C4C23×D139
FittingC8×D4×C139C2
socleC22×C139C22×D4
hypercenterC8×D4D139
hypocenterC139#3294
LCSG ▷ C2×C4×C139 ▷ C2×C139 ▷ C139
UCSC1 ◁ C22 ◁ C4×D4 ◁ C8×D4

Complements

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

NH ≅ G/Ntype
1. #22581607C2semidirect
2. D4×D556C2semidirect
3. #22581300C2semidirect
4. #22581267C2semidirect
5. #22581234C2semidirect
6. #22581189C2semidirect
7. #22581149C2semidirect
8. #22581089C2semidirect
9. C4×D1112C2semidirect
10. C2×#17984764C2semidirect
11. C8×D4×C139C2semidirect
16. C2×D1112C22semidirect
17. #17984842C22semidirect
23. #17984816C22semidirect
26. C4×D556C22semidirect
27. C2×C4⋊Dic139C22semidirect
28. #17984776C22semidirect
29. #17984776C22semidirect
32. #17984764C22semidirect
33. #17984764C22semidirect
34. #17984764C22semidirect
35. #17984764C22semidirect
36. C22×C8×C139C22semidirect
38. C4⋊C8×C139C22semidirect
39. C22⋊C8×C139C22semidirect
40. C4×C8×C139C22semidirect
93. D556#41semidirect
103. D4×C139D8semidirect
142. C139#3294semidirect
153. C8×D4D139semidirect
160. C22×C8C2×D139semidirect
175. C4⋊C8C2×D139semidirect
182. C22⋊C8C2×D139semidirect
183. C4×C8C2×D139semidirect
240. D4D1112semidirect

Maximal subgroups

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

subgroupindex
#225816072
D4×D5562
#225813002
#225812672
#225812342
#225811892
#225811492
#225810892
C4×D11122
C2×#179847642
C8×D4×C1392
#3294139