Finite Group Databasebeta

← #31815680

Subgroups — #31815680

Order 17792.

752 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 752 conjugacy classes of subgroups — too many to draw. Only its 149 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
1C2×#17984888C211yesmaximal, characteristic
2C22×#14555772C211yesmaximal, characteristic, complemented
3#22581527C211yesmaximal, characteristic, complemented
4#22581114C214yesmaximal, complemented
5#22581102C214yesmaximal, complemented
6#22580959C211yesmaximal, characteristic, complemented
7#22580949C211yesmaximal, characteristic, complemented
8#22580921C211yesmaximal, characteristic, complemented
9C2×#137×C139C211yesmaximal, characteristic, complemented
10#17984888—24no—
11#17984888—24no—
12C2×#14555786C2212yes—
13C2×#14555785—21no—
14C2×#14555785C2211yescharacteristic
15C22×D556—21no—
16C22×D556C2211yescharacteristic
17C2×D4×D139—22no—
18C2×D4×D139C2212yes—
19C2×#14555772—24no—
20C2×#14555772C2214yescomplemented
21C22×C139⋊C8C2211yescharacteristic, complemented
22#17984840—21no—
23#17984831—21no—
24#17984825—22no—
25#17984818C2214yescomplemented
26#17984801C2211yescharacteristic, complemented
27#17984766C2212yescomplemented
28#17984764C2214yescomplemented
29#17984763C2212yescomplemented
30(C2×D139)⋊C8C2214yescomplemented
31C2×#41×C139C2211yescharacteristic
32C22×Q8×C139C2211yescharacteristic, complemented
33#137×C139—24no—
34C2×Q16×C139C2212yescomplemented
35C2×SD16×C139C2212yescomplemented
36C2×M16×C139C2211yescharacteristic, complemented
37C23×D139—22no—
38C23×D139—21no—
39#14555786—24no—
40#14555786—24no—
41#14555785—28no—
42#14555785—22no—
43#14555785—22no—
44C2×Dic278D411yescharacteristic
45C2×D556—28no—
46C2×D556—22no—
47C2×D556—21no—
48C2×D556—22no—
49C2×D556D411yescharacteristic
50C2×D556D414yes—
51C2×D556C2312yes—
52C2×C22⋊D139—22no—
53C2×C22⋊D139D412yes—
54D4×D139—216no—
55D4×D139—24no—
56D4×D139—24no—
57C2×C4×D139—22no—
58C2×C4×D139C2312yes—
59#14555772—28no—
60C2×C139⋊C8—22no—
61C2×C139⋊C8C2312yes—
62C4⋊Dic139C2312yes—
63Dic139⋊C4—24no—
64C22⋊Dic139—21no—
65(C2×D139)⋊C4—24no—
66C4×Dic139—21no—
67#41×C139—22no—
68#41×C139D414yes—
69C2×Q8×C139—21no—
70C2×Q8×C139—22no—
71C2×Q8×C139—22no—
72C2×Q8×C139C2312yes—
73C2×Q8×C139C2311yescharacteristic
74C2×D4×C139—21no—
75C2×D4×C139C2311yescharacteristic
76C22×C4×C139—21no—
77C22×C4×C139—21no—
78C22×C4×C139C2311yescharacteristic
79Q16×C139—24no—
80SD16×C139—24no—
81M16×C139—22no—
82C2×C8×C139C2312yes—
83C22×D139—22no—
84C22×D139—28no—
85C22×D139—24no—
86C22×D139—84no—
87C22×D139—82no—
88C22×D139C2×D412yes—
89Dic278—22no—
90D556—28no—
91D556—22no—
92D556—24no—
93D556—22no—
94D556C2×D414yes—
95C22⋊D139—28no—
96C22⋊D139—24no—
97C2×Dic139—22no—
98C2×Dic139C2×D412yes—
99C4×D139—28no—
100C4×D139—22no—
101C4×D139—22no—
102C139⋊C8—22no—
103C23×C139—21no—
104C23×C139C2×D411yescharacteristic
105Q8×C139—21no—
106Q8×C139—22no—
107Q8×C139—22no—
108Q8×C139—42no—
109Q8×C139C2×D412yes—
110D4×C139—24no—
111D4×C139—21no—
112D4×C139C2×D412yes—
113C2×C4×C139—21no—
114C2×C4×C139—24no—
115C2×C4×C139—22no—
116C2×C4×C139—42no—
117C2×C4×C139C2×D411yescharacteristic
118C2×C4×C139C2×D412yes—
119C2×C4×C139C2×D412yes—
120C2×C4×C139C2411yescharacteristic
121C8×C139—22no—
122C22×C139—22no—
123C22×C139—22no—
124C22×C139—42no—
125C22×C139#12012yes—
126C22×C139C22×D411yescharacteristic
127C2×D139—22no—
128C2×D139—24no—
129C2×D139—24no—
130C2×D139—44no—
131C2×D139—84no—
132C2×D139—88no—
133C4×C139—22no—
134C4×C139—42no—
135C4×C139#12012yes—
136C4×C139C22×D411yescharacteristic
137C4×C139C22×D411yescharacteristic
138Dic139—22no—
139Dic139—42no—
140C2×C139—22no—
141C2×C139—42no—
142C2×C139C2×#12011yescharacteristic
143C2×C139#57712yes—
144D139—42no—
145D139—44no—
146C139#300911yescharacteristic, complemented, Sylow
147#3009—1391nomaximal, Sylow
148C2×H(5,𝔽2)—1391no—
149C2×#137D13911yescharacteristic, complemented
150C22×SD16—1391no—
151#536—1391no—
152#452—1394no—
153#450—1394no—
154#418—1391no—
155#416—1391no—
156#408—1391no—
157H(5,𝔽2)—2784no—
158H(5,𝔽2)—2784no—
159C2×#41—2781no—
160C2×#41—1392no—
161C2×#41—1391no—
162C2×#41C2×D13911yescharacteristic
163C22×Q8C2×D13911yescharacteristic, complemented
164C22×D4—2781no—
165C22×D4—2782no—
166C22×D4—1392no—
167C22×D4—1391no—
168#137—24no—
169C2×Q16C2×D13912yescomplemented
170C2×SD16—2784no—
171C2×SD16—1394no—
172C2×SD16C2×D13912yescomplemented
173C2×M16C2×D13911yescharacteristic, complemented
174C22×C8—1391no—
175C4⋊Q8—2781no—
176#124—2781no—
177#123—2782no—
178C22⋊Q8—1394no—
179#117—1391no—
180#103—1392no—
181#102—1392no—
182#102—1394no—
183C22⋊C8—1394no—
184C24—2782no—
185C24—2781no—
186#41—22no—
187#41—2788no—
188#41—2784no—
189#41—2784no—
190#41—2782no—
191#41—2782no—
192#41C22⋊D13914yes—
193C2×Q8—21no—
194C2×Q8—22no—
195C2×Q8—22no—
196C2×Q8—1391no—
197C2×Q8C22×D13912yes—
198C2×Q8C22×D13911yescharacteristic
199C2×D4—21no—
200C2×D4—2788no—
201C2×D4—2782no—
202C2×D4—27816no—
203C2×D4—2784no—
204C2×D4—2784no—
205C2×D4—2782no—
206C2×D4—2781no—
207C2×D4—2782no—
208C2×D4—1392no—
209C2×D4—1391no—
210C2×D4—1394no—
211C2×D4—1392no—
212C2×D4C22×D13911yescharacteristic
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—2782no—
216C22×C4—1392no—
217C22×C4C22×D13911yescharacteristic
218Q16—24no—
219SD16—24no—
220SD16—2788no—
221M16—22no—
222C2×C8—2782no—
223C2×C8—1392no—
224C2×C8C22×D13912yes—
225C4⋊C4—2784no—
226C4⋊C4—1392no—
227C22⋊C4—2784no—
228C22⋊C4—2781no—
229C42—2781no—
230C23—21no—
231C23—2788no—
232C23—2782no—
233C23—2784no—
234C23—11122no—
235C23—11124no—
236C23—1392no—
237C23D4×D13911yescharacteristic
238Q8—21no—
239Q8—22no—
240Q8—22no—
241Q8—42no—
242Q8—2782no—
243Q8C2×C22⋊D13912yes—
244D4—24no—
245D4—21no—
246D4—2788no—
247D4—2788no—
248D4—2784no—
249D4—2782no—
250D4—2784no—
251D4—2782no—
252D4—1394no—
253D4C2×C22⋊D13912yes—
254C2×C4—21no—
255C2×C4—24no—
256C2×C4—22no—
257C2×C4—42no—
258C2×C4—2788no—
259C2×C4—2782no—
260C2×C4—2782no—
261C2×C4—2782no—
262C2×C4—1392no—
263C2×C4D4×D13911yescharacteristic
264C2×C4D4×D13912yes—
265C2×C4C2×C22⋊D13912yes—
266C2×C4C23×D13911yescharacteristic
267C8—22no—
268C8—2782no—
269C22—22no—
270C22—22no—
271C22—42no—
272C22—2784no—
273C22—2782no—
274C22—2784no—
275C22—5564no—
276C22—11124no—
277C22—11128no—
278C22#1798481012yes—
279C22C2×D4×D13911yescharacteristic
280C4—22no—
281C4—42no—
282C4—2782no—
283C4—5562no—
284C4#1798481012yes—
285C4C2×D4×D13911yescharacteristic
286C4C22×C22⋊D13911yescharacteristic
287C2—22no—
288C2—42no—
289C2—5562no—
290C2—5564no—
291C2C2×#1798481011yescharacteristic
292C2#2258176212yes—
subgroupG/H
centerC22C2×D4×D139
commutator, cosocleC2×C4×C139C24
FrattiniC2×C4C23×D139
FittingC2×#137×C139C2
socleC22×C139C22×D4
prime radicalC2×#17984888C2
hypercenterC2×#137D139
hypocenterC139#3009
LCSG ▷ C2×C4×C139 ▷ C2×C139 ▷ C139
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#137

Complements

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

NH ≅ G/Ntype
2. C22×#14555772C2semidirect
3. #22581527C2semidirect
4. #22581114C2semidirect
5. #22581102C2semidirect
6. #22580959C2semidirect
7. #22580949C2semidirect
8. #22580921C2semidirect
9. C2×#137×C139C2semidirect
20. C2×#14555772C22semidirect
21. C22×C139⋊C8C22semidirect
25. #17984818C22semidirect
26. #17984801C22semidirect
27. #17984766C22semidirect
28. #17984764C22semidirect
29. #17984763C22semidirect
30. (C2×D139)⋊C8C22semidirect
32. C22×Q8×C139C22semidirect
34. C2×Q16×C139C22semidirect
35. C2×SD16×C139C22semidirect
36. C2×M16×C139C22semidirect
146. C139#3009semidirect
149. C2×#137D139semidirect
163. C22×Q8C2×D139semidirect
169. C2×Q16C2×D139semidirect
172. C2×SD16C2×D139semidirect
173. C2×M16C2×D139semidirect

Maximal subgroups

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

subgroupindex
C2×#179848882
C22×#145557722
#225815272
#225811142
#225811022
#225809592
#225809492
#225809212
C2×#137×C1392
#3009139