Finite Group Databasebeta

← #31816787

Subgroups — #31816787

Order 17792.

400 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 400 conjugacy classes of subgroups — too many to draw. Only its 117 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#22581509C211yesmaximal, characteristic, complemented
2#124×D139C211yesmaximal, characteristic
3#22581314C212yesmaximal, complemented
4#22581311C212yesmaximal, complemented
5#22581310C211yesmaximal, characteristic, complemented
6#22581308C211yesmaximal, characteristic, complemented
7#22581307C211yesmaximal, characteristic, complemented
8#22580960C212yesmaximal, complemented
9#22580956C212yesmaximal
10C4×#14555768C211yesmaximal, characteristic, complemented
11#488×C139C211yesmaximal, characteristic, complemented
12C2×Q8×D139C2211yescharacteristic
13C2×D4×D139C2211yescharacteristic
14C2×#14555768C2212yes—
15#17984841C2212yescomplemented
16#17984834C2211yescharacteristic, complemented
17#17984831C2211yescharacteristic
18#17984830C2211yescharacteristic
19#17984829C2211yescharacteristic
20#17984828—22no—
21#17984820—22no—
22#17984807—22no—
23#17984800C2212yes—
24C22⋊C4×D139—22no—
25C42×D139C2211yescharacteristic
26#17984767C2212yescomplemented
27#17984766C2212yescomplemented
28#17984765C2212yescomplemented
29#17984764C2212yes—
30#17984763C2212yes—
31#17984762C2212yes—
32#17984752C2212yescomplemented
33C4×C139⋊C8C2211yescharacteristic, complemented
34#125×C139C2211yescharacteristic, complemented
35#124×C139C2211yescharacteristic
36#103×C139C2212yescomplemented
37#102×C139C2212yes—
38C4×C8×C139C2211yescharacteristic, complemented
39C23×D139—21no—
40C2×Dic278—22no—
41C2×Dic278—21no—
42C2×Dic278C2311yescharacteristic
43Q8×D139—24no—
44C2×D556C2311yescharacteristic
45C2×C22⋊D139—21no—
46D4×D139—24no—
47C2×C4×D139—21no—
48C2×C4×D139D412yes—
49C2×C4×D139C2311yescharacteristic
50#14555768—24no—
51C2×C139⋊C8C2312yes—
52C4⋊Dic139C2312yes—
53Dic139⋊C4—22no—
54Dic139⋊C4—22no—
55C22⋊Dic139—22no—
56(C2×D139)⋊C4—22no—
57(C2×D139)⋊C4—22no—
58(C2×D139)⋊C4—22no—
59C4×Dic139—22no—
60C4×Dic139D412yes—
61C4×Dic139C2311yescharacteristic
62C2×Q8×C139C2311yescharacteristic
63C2×D4×C139C2311yescharacteristic
64C2×C8×C139C2312yes—
65C4⋊C4×C139—22no—
66C4⋊C4×C139C2312yes—
67C22⋊C4×C139—22no—
68C42×C139C2311yescharacteristic
69C22×D139—21no—
70C22×D139—44no—
71C22×D139—42no—
72C22×D139C2×D411yescharacteristic
73Dic278—22no—
74Dic278—44no—
75Dic278—42no—
76D556—22no—
77C22⋊D139—42no—
78C2×Dic139—22no—
79C2×Dic139—21no—
80C2×Dic139C2×D412yes—
81C2×Dic139C2×D411yescharacteristic
82C4×D139—42no—
83C4×D139—42no—
84C4×D139#4114yes—
85C139⋊C8—22no—
86C23×C139—21no—
87Q8×C139—22no—
88D4×C139—22no—
89C2×C4×C139—21no—
90C2×C4×C139—22no—
91C2×C4×C139C2×D412yes—
92C2×C4×C139C2411yescharacteristic
93C8×C139—22no—
94C22×C139—41no—
95C22×C139—42no—
96C22×C139C22×D411yescharacteristic
97C2×D139—22no—
98C2×D139—41no—
99C2×D139—42no—
100C2×D139—84no—
101C2×D139#12412yes—
102C4×C139—22no—
103C4×C139—41no—
104C4×C139—42no—
105C4×C139C2×#4112yes—
106Dic139—22no—
107Dic139—41no—
108Dic139—42no—
109Dic139#12412yes—
110C2×C139—81no—
111C2×C139C2×#12411yescharacteristic
112C2×C139#57512yes—
113D139—22no—
114D139—81no—
115C139#314011yescharacteristic, complemented, Sylow
116#3140—1391nomaximal, Sylow
117#533—1391no—
118C2×#124—1391no—
119#489—1392no—
120#489—1392no—
121#488—1391no—
122#488—1391no—
123#488—1391no—
124#488D13911yescharacteristic, complemented
125#419—1392no—
126#418—1392no—
127C4×M16—1391no—
128C22×Q8—1391no—
129C22×D4—1391no—
130C2×M16—1392no—
131C4⋊Q8—1392no—
132#125—1391no—
133#125C2×D13911yescharacteristic, complemented
134#124—2782no—
135#124—1391no—
136#124—1391no—
137#124—1391no—
138#124C2×D13911yescharacteristic
139C22⋊Q8—2782no—
140C4×Q8—2782no—
141#117—1392no—
142C2×C22⋊C4—2782no—
143C2×C42—1391no—
144#103—1392no—
145#103—1392no—
146#103—1392no—
147#103C2×D13912yescomplemented
148#102—1392no—
149#102—1392no—
150#102—1392no—
151#102C2×D13912yes—
152#97—1392no—
153C4×C8—1391no—
154C4×C8C2×D13911yescharacteristic, complemented
155C24—2781no—
156C2×Q8—2782no—
157C2×Q8—2781no—
158C2×Q8—2784no—
159C2×Q8—1391no—
160C2×Q8C22×D13911yescharacteristic
161C2×D4—2784no—
162C2×D4—2781no—
163C2×D4—1391no—
164C2×D4C22×D13911yescharacteristic
165C22×C4—2781no—
166C22×C4—1392no—
167C22×C4—1391no—
168M16—2784no—
169C2×C8—1392no—
170C2×C8C22×D13912yes—
171C4⋊C4—22no—
172C4⋊C4—2782no—
173C4⋊C4—2782no—
174C4⋊C4—1392no—
175C4⋊C4C22×D13912yes—
176C22⋊C4—22no—
177C22⋊C4—2782no—
178C22⋊C4—2782no—
179C22⋊C4—2782no—
180C22⋊C4—2782no—
181C42—2782no—
182C42—1392no—
183C42—1391no—
184C42C22×D13911yescharacteristic
185C23—21no—
186C23—2781no—
187C23—5564no—
188C23—5562no—
189C23—1391no—
190Q8—22no—
191Q8—2782no—
192Q8—5564no—
193Q8—5562no—
194D4—22no—
195D4—2782no—
196D4—5562no—
197C2×C4—22no—
198C2×C4—21no—
199C2×C4—2782no—
200C2×C4—2781no—
201C2×C4—5562no—
202C2×C4—5562no—
203C2×C4—1392no—
204C2×C4—1391no—
205C2×C4—1394no—
206C2×C4D4×D13912yes—
207C2×C4C23×D13911yescharacteristic
208C8—22no—
209C8—2782no—
210C22—41no—
211C22—42no—
212C22—2782no—
213C22—5561no—
214C22—5562no—
215C22—11124no—
216C22—1392no—
217C22C2×D4×D13911yescharacteristic
218C4—22no—
219C4—41no—
220C4—42no—
221C4—2782no—
222C4—5561no—
223C4—5562no—
224C4—1392no—
225C4#41×D13912yes—
226C2—81no—
227C2—2782no—
228C2—11121no—
229C2#124×D13911yescharacteristic
230C2#2258175012yes—
subgroupG/H
centerC22C2×D4×D139
commutator, cosocleC2×C4×C139C24
FrattiniC2×C4C23×D139
Fitting#488×C139C2
socleC22×C139C22×D4
prime radicalC2×D4×D139C22
hypercenter#488D139
hypocenterC139#3140
LCSG ▷ C2×C4×C139 ▷ C2×C139 ▷ C139
UCSC1 ◁ C22 ◁ C42 ◁ #488

Complements

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

NH ≅ G/Ntype
1. #22581509C2semidirect
3. #22581314C2semidirect
4. #22581311C2semidirect
5. #22581310C2semidirect
6. #22581308C2semidirect
7. #22581307C2semidirect
8. #22580960C2semidirect
10. C4×#14555768C2semidirect
11. #488×C139C2semidirect
15. #17984841C22semidirect
16. #17984834C22semidirect
26. #17984767C22semidirect
27. #17984766C22semidirect
28. #17984765C22semidirect
32. #17984752C22semidirect
33. C4×C139⋊C8C22semidirect
34. #125×C139C22semidirect
36. #103×C139C22semidirect
38. C4×C8×C139C22semidirect
115. C139#3140semidirect
124. #488D139semidirect
133. #125C2×D139semidirect
147. #103C2×D139semidirect
154. C4×C8C2×D139semidirect

Maximal subgroups

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

subgroupindex
#225815092
#124×D1392
#225813142
#225813112
#225813102
#225813082
#225813072
#225809602
#225809562
C4×#145557682
#488×C1392
#3140139