Finite Group Databasebeta

← #31818211

Subgroups — #31818211

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
1D4×D556C211yesmaximal, characteristic, complemented
2#22581301C211yesmaximal, characteristic, complemented
3#22581266C212yesmaximal, complemented
4#22581230C211yesmaximal, characteristic, complemented
5#22581187C212yesmaximal, complemented
6#22581148C211yesmaximal, characteristic, complemented
7#22581086C212yesmaximal, complemented
8C4×#14555770C211yesmaximal, characteristic, complemented
9D4×C139⋊C8C211yesmaximal, characteristic, complemented
10C2×#17984762C212yesmaximal, complemented
11#547×C139C211yesmaximal, characteristic, complemented
12C22×D556C2212yes—
13C2×D4×D139—21no—
14C2×#14555770C2211yescharacteristic
15C2×#14555770C2212yescomplemented
16C22×C139⋊C8C2212yescomplemented
17#17984839C2211yescharacteristic
18#17984817—21no—
19#17984816C2212yes—
20#17984812—22no—
21C4×D556C2211yescharacteristic
22#17984775C2211yescharacteristic, complemented
23#17984775C2212yescomplemented
24C4⋊(C139⋊C8)C2211yescharacteristic, complemented
25#17984763C2211yescharacteristic, complemented
26#17984762C2211yescharacteristic
27#17984762C2212yes—
28#17984762C2212yescomplemented
29#17984762C2212yescomplemented
30#17984762C2212yescomplemented
31C22⋊(C139⋊C8)C2212yescomplemented
32C4×C139⋊C8C2211yescharacteristic, complemented
33C2×#41×C139—21no—
34C4⋊Q8×C139C2211yescharacteristic, complemented
35#123×C139—22no—
36C22⋊Q8×C139—21no—
37#121×C139C2212yescomplemented
38C4×D4×C139C2211yescharacteristic
39C4×D4×C139C2211yescharacteristic, complemented
40C2×C4⋊C4×C139C2212yescomplemented
41C23×D139—22no—
42C2×D556—24no—
43C2×D556—22no—
44C2×D556—21no—
45C2×D556—21no—
46C2×D556—42no—
47C2×D556C2312yes—
48C2×D556C2311yescharacteristic
49C2×C22⋊D139—22no—
50D4×D139—44no—
51C2×C4×D139—21no—
52#14555770—22no—
53#14555770—42no—
54C2×C139⋊C8—42no—
55C2×C139⋊C8C2312yes—
56C2×C139⋊C8C2311yescharacteristic
57C2×C139⋊C8C2311yescharacteristic
58C4⋊Dic139C2311yescharacteristic
59(C2×D139)⋊C4—21no—
60(C2×D139)⋊C4—22no—
61#41×C139—44no—
62C2×Q8×C139—21no—
63C2×D4×C139—22no—
64C2×D4×C139D411yescharacteristic
65C2×D4×C139C2311yescharacteristic
66C22×C4×C139—22no—
67C22×C4×C139—21no—
68C22×C4×C139C2312yes—
69C4⋊C4×C139—22no—
70C4⋊C4×C139—21no—
71C4⋊C4×C139D411yescharacteristic
72C4⋊C4×C139C2311yescharacteristic
73C4⋊C4×C139C2312yes—
74C22⋊C4×C139—21no—
75C22⋊C4×C139—22no—
76C22⋊C4×C139D412yes—
77C42×C139C2311yescharacteristic
78C22×D139—21no—
79C22×D139—22no—
80C22×D139—44no—
81C22×D139—44no—
82C22×D139—82no—
83D556—22no—
84D556—21no—
85D556—22no—
86D556—44no—
87D556—42no—
88D556—81no—
89D556#4112yescomplemented
90C22⋊D139—44no—
91C2×Dic139—21no—
92C4×D139—42no—
93C139⋊C8—21no—
94C139⋊C8—21no—
95C139⋊C8—22no—
96C23×C139—21no—
97C23×C139C2×D412yes—
98Q8×C139—42no—
99D4×C139—21no—
100D4×C139—21no—
101D4×C139—44no—
102D4×C139D814yescomplemented
103C2×C4×C139—22no—
104C2×C4×C139—22no—
105C2×C4×C139—21no—
106C2×C4×C139—22no—
107C2×C4×C139—44no—
108C2×C4×C139—42no—
109C2×C4×C139C2×D411yescharacteristic
110C2×C4×C139C2×D412yes—
111C2×C4×C139C2×D411yescharacteristic
112C2×C4×C139C2411yescharacteristic
113C22×C139—22no—
114C22×C139—22no—
115C22×C139—41no—
116C22×C139—41no—
117C22×C139—41no—
118C22×C139C2×D814yes—
119C22×C139C22×D411yescharacteristic
120C2×D139—22no—
121C2×D139—41no—
122C2×D139—41no—
123C2×D139—42no—
124C2×D139—42no—
125C2×D139—42no—
126C2×D139—88no—
127C4×C139—22no—
128C4×C139—21no—
129C4×C139—41no—
130C4×C139—42no—
131C4×C139C2×D812yes—
132C4×C139C2×#4111yescharacteristic
133C4×C139H(5,𝔽2)11yescharacteristic
134Dic139—41no—
135C2×C139—24no—
136C2×C139—81no—
137C2×C139#54611yescharacteristic
138C2×C139C22×D811yescharacteristic
139C2×C139#57711yescharacteristic
140D139—42no—
141D139—82no—
142C139#329411yescharacteristic, complemented, Sylow
143#3294—1391nomaximal, Sylow
144#547D13911yescharacteristic, complemented
145D42—1391no—
146#486—1391no—
147#480—1392no—
148#474—1391no—
149#466—1392no—
150#459—1391no—
151#447—1392no—
152C4×D8—1391no—
153C8×D4—1391no—
154C2×#102—1392no—
155C2×#41—21no—
156C22×D4—2781no—
157C22×D4—1392no—
158C2×D8—1391no—
159C2×D8—1392no—
160C22×C8—1392no—
161C4⋊Q8C2×D13911yescharacteristic, complemented
162#127—1391no—
163#123—22no—
164C22⋊Q8—21no—
165#121—2781no—
166#121—1392no—
167#121C2×D13912yescomplemented
168#120—2782no—
169C4×D4—1391no—
170C4×D4C2×D13911yescharacteristic
171C4×D4C2×D13911yescharacteristic, complemented
172C2×C4⋊C4C2×D13912yescomplemented
173#107—1391no—
174#107—1392no—
175C4⋊C8—1391no—
176#102—1391no—
177#102—1391no—
178#102—1392no—
179#102—1392no—
180#102—1392no—
181#102—1392no—
182C22⋊C8—1392no—
183C4×C8—1391no—
184C24—2782no—
185#41—44no—
186C2×Q8—21no—
187C2×D4—22no—
188C2×D4—2784no—
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×D4C22⋊D13911yescharacteristic
198C2×D4C22×D13911yescharacteristic
199C22×C4—22no—
200C22×C4—21no—
201C22×C4—2781no—
202C22×C4C22×D13912yes—
203D8—2782no—
204D8—5562no—
205C2×C8—5562no—
206C2×C8—1391no—
207C2×C8—1391no—
208C2×C8—1392no—
209C4⋊C4—22no—
210C4⋊C4—21no—
211C4⋊C4—1391no—
212C4⋊C4C22⋊D13911yescharacteristic
213C4⋊C4C22×D13911yescharacteristic
214C4⋊C4C22×D13912yes—
215C22⋊C4—21no—
216C22⋊C4—22no—
217C22⋊C4—2781no—
218C22⋊C4—2782no—
219C22⋊C4C22⋊D13912yes—
220C42C22×D13911yescharacteristic
221C23—21no—
222C23—2781no—
223C23—2782no—
224C23—5564no—
225C23—5564no—
226C23—11122no—
227C23C2×C22⋊D13912yes—
228Q8—42no—
229D4—21no—
230D4—21no—
231D4—44no—
232D4—2782no—
233D4—2781no—
234D4—2782no—
235D4—5564no—
236D4—5564no—
237D4—5562no—
238D4—11121no—
239D4—1392no—
240D4#1455577014yescomplemented
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—21no—
244C2×C4—22no—
245C2×C4—44no—
246C2×C4—42no—
247C2×C4—2781no—
248C2×C4—5562no—
249C2×C4C2×C22⋊D13911yescharacteristic
250C2×C4C2×C22⋊D13912yes—
251C2×C4C2×C22⋊D13911yescharacteristic
252C2×C4C23×D13911yescharacteristic
253C8—2781no—
254C8—2781no—
255C8—2782no—
256C22—22no—
257C22—22no—
258C22—41no—
259C22—41no—
260C22—41no—
261C22—2782no—
262C22—5561no—
263C22—5561no—
264C22—5562no—
265C22—5562no—
266C22—5562no—
267C22—11128no—
268C22C2×#1455577014yes—
269C22C22×C22⋊D13911yescharacteristic
270C4—21no—
271C4—22no—
272C4—41no—
273C4—42no—
274C4—5561no—
275C4C2×#1455577012yes—
276C4#41×D13911yescharacteristic
277C4#1798488811yescharacteristic
278C2—24no—
279C2—81no—
280C2—5562no—
281C2—11122no—
282C2#2258159111yescharacteristic
283C2C22×#1455577011yescharacteristic
284C2#2258176111yescharacteristic
subgroupG/H
centerC22C22×C22⋊D139
commutator, cosocleC2×C4×C139C24
FrattiniC2×C4C23×D139
Fitting#547×C139C2
socleC22×C139C22×D4
hypercenter#547D139
hypocenterC139#3294
LCSG ▷ C2×C4×C139 ▷ C2×C139 ▷ C139
UCSC1 ◁ C22 ◁ C4×D4 ◁ #547

Complements

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

NH ≅ G/Ntype
1. D4×D556C2semidirect
2. #22581301C2semidirect
3. #22581266C2semidirect
4. #22581230C2semidirect
5. #22581187C2semidirect
6. #22581148C2semidirect
7. #22581086C2semidirect
8. C4×#14555770C2semidirect
9. D4×C139⋊C8C2semidirect
10. C2×#17984762C2semidirect
11. #547×C139C2semidirect
15. C2×#14555770C22semidirect
16. C22×C139⋊C8C22semidirect
22. #17984775C22semidirect
23. #17984775C22semidirect
24. C4⋊(C139⋊C8)C22semidirect
25. #17984763C22semidirect
28. #17984762C22semidirect
29. #17984762C22semidirect
30. #17984762C22semidirect
31. C22⋊(C139⋊C8)C22semidirect
32. C4×C139⋊C8C22semidirect
34. C4⋊Q8×C139C22semidirect
37. #121×C139C22semidirect
39. C4×D4×C139C22semidirect
40. C2×C4⋊C4×C139C22semidirect
89. D556#41semidirect
102. D4×C139D8semidirect
142. C139#3294semidirect
144. #547D139semidirect
161. C4⋊Q8C2×D139semidirect
167. #121C2×D139semidirect
171. C4×D4C2×D139semidirect
172. C2×C4⋊C4C2×D139semidirect
240. D4#14555770semidirect

Maximal subgroups

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

subgroupindex
D4×D5562
#225813012
#225812662
#225812302
#225811872
#225811482
#225810862
C4×#145557702
D4×C139⋊C82
C2×#179847622
#547×C1392
#3294139