Finite Group Databasebeta

← #31817009

Subgroups — #31817009

Order 17792.

364 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 364 conjugacy classes of subgroups — too many to draw. Only its 115 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#22581508C211yesmaximal, characteristic, complemented
2#22581488C211yesmaximal, characteristic
3#22581363C211yesmaximal, characteristic, complemented
4#22581345C211yesmaximal, characteristic, complemented
5#22581313C211yesmaximal, characteristic, complemented
6#22581210C212yesmaximal, complemented
7#22581202C212yesmaximal
8#22581045C212yesmaximal, complemented
9#22581038C212yesmaximal, complemented
10#22580905C211yesmaximal, characteristic, complemented
11#491×C139C211yesmaximal, characteristic, complemented
12C2×#14555775C2212yes—
13C2×#14555772C2212yes—
14C2×#14555768C2212yes—
15#17984842C2211yescharacteristic
16#17984837—21no—
17#17984833—21no—
18#17984830C2211yescharacteristic
19#17984827—21no—
20#17984822—21no—
21#17984822C2211yescharacteristic
22#17984818C2212yes—
23#17984817C2211yescharacteristic
24Q8×Dic139C2212yes—
25#17984807—21no—
26#17984807C2211yescharacteristic
27#17984804—21no—
28#17984804C2211yescharacteristic
29#17984800—21no—
30#17984800—21no—
31C42×D139C2211yescharacteristic
32#17984775C2212yes—
33#17984773C2212yes—
34#17984767C2212yes—
35#17984764C2212yes—
36#17984753C2211yescharacteristic
37C8×Dic139C2212yes—
38C4⋊Q8×C139C2211yescharacteristic
39#125×C139C2211yescharacteristic
40#103×C139C2212yes—
41#103×C139C2212yes—
42#97×C139C2211yescharacteristic
43C2×Dic278—21no—
44C2×Dic278C2311yescharacteristic
45C2×D556—21no—
46C2×D556C2311yescharacteristic
47C2×C4×D139—21no—
48C2×C4×D139D411yescharacteristic
49C2×C4×D139D411yescharacteristic
50C2×C4×D139C2311yescharacteristic
51#14555775—24no—
52#14555772—24no—
53#14555768—24no—
54C2×C139⋊C8C2312yes—
55C4⋊Dic139—21no—
56C4⋊Dic139—22no—
57C4⋊Dic139—21no—
58C4⋊Dic139C2312yes—
59Dic139⋊C4—22no—
60Dic139⋊C4—21no—
61(C2×D139)⋊C4—22no—
62(C2×D139)⋊C4—21no—
63(C2×D139)⋊C4—21no—
64(C2×D139)⋊C4—21no—
65(C2×D139)⋊C4—21no—
66(C2×D139)⋊C4—21no—
67C4×Dic139—21no—
68C4×Dic139—22no—
69C4×Dic139D411yescharacteristic
70C4×Dic139D411yescharacteristic
71C4×Dic139C2311yescharacteristic
72C2×Q8×C139C2312yes—
73C2×C8×C139C2312yes—
74C4⋊C4×C139—21no—
75C4⋊C4×C139—21no—
76C4⋊C4×C139—21no—
77C4⋊C4×C139—21no—
78C4⋊C4×C139C2311yescharacteristic
79C4⋊C4×C139C2311yescharacteristic
80C42×C139C2311yescharacteristic
81C22×D139—21no—
82C22×D139C2×D411yescharacteristic
83Dic278—22no—
84Dic278—42no—
85D556—22no—
86D556—42no—
87C2×Dic139—21no—
88C2×Dic139—22no—
89C2×Dic139C2×D411yescharacteristic
90C2×Dic139C2×D411yescharacteristic
91C2×Dic139C2×D411yescharacteristic
92C4×D139—22no—
93C4×D139—22no—
94C4×D139—41no—
95C4×D139—42no—
96C139⋊C8#4114yes—
97Q8×C139—22no—
98Q8×C139—22no—
99C2×C4×C139—21no—
100C2×C4×C139—21no—
101C2×C4×C139—22no—
102C2×C4×C139C2×D411yescharacteristic
103C2×C4×C139C2×D411yescharacteristic
104C2×C4×C139C2411yescharacteristic
105C8×C139—22no—
106C22×C139C22×D411yescharacteristic
107C2×D139—21no—
108C2×D139—21no—
109C2×D139—21no—
110C2×D139—41no—
111C2×D139—41no—
112C2×D139—41no—
113C4×C139—21no—
114C4×C139—21no—
115C4×C139—42no—
116C4×C139—41no—
117C4×C139—41no—
118C4×C139C2×#4112yes—
119Dic139—21no—
120Dic139—21no—
121Dic139—41no—
122Dic139—42no—
123Dic139#13712yes—
124C2×C139#53211yescharacteristic
125C2×C139C2×#13711yescharacteristic
126C2×C139#57511yescharacteristic
127D139—41no—
128D139—81no—
129C139#317211yescharacteristic, complemented, Sylow
130#3172—1391nomaximal, Sylow
131#533—1391no—
132#529—1391no—
133#501—1391no—
134#497—1391no—
135#491D13911yescharacteristic, complemented
136#489—1391no—
137#470—1392no—
138#468—1392no—
139C4×Q16—1392no—
140C4×SD16—1392no—
141C4×M16—1391no—
142C2×Q16—1392no—
143C2×SD16—1392no—
144C2×M16—1392no—
145C4⋊Q8—1391no—
146C4⋊Q8C2×D13911yescharacteristic
147#126—2781no—
148#125—2781no—
149#125C2×D13911yescharacteristic
150#124—1391no—
151#123—2781no—
152C22⋊Q8—2781no—
153C22⋊Q8—1392no—
154C22⋊Q8—1391no—
155#121—1391no—
156C4×Q8—2781no—
157C4×Q8—1392no—
158C4×Q8—1391no—
159C4×D4—2781no—
160C4×D4—1391no—
161#117—2781no—
162#117—2781no—
163C2×C42—1391no—
164#107—1392no—
165#106—1392no—
166#103—1392no—
167#103C2×D13912yes—
168#103C2×D13912yes—
169#102—1392no—
170#97—1391no—
171#97C2×D13911yescharacteristic
172C4×C8—1392no—
173C2×Q8—2781no—
174C2×Q8—1391no—
175C2×Q8C22×D13912yes—
176C2×D4—2781no—
177C2×D4—1391no—
178C22×C4—2781no—
179C22×C4—1391no—
180C22×C4—1391no—
181C22×C4—1391no—
182Q16—2784no—
183SD16—2784no—
184M16—2784no—
185C2×C8—1392no—
186C2×C8C22×D13912yes—
187C4⋊C4—21no—
188C4⋊C4—21no—
189C4⋊C4—21no—
190C4⋊C4—21no—
191C4⋊C4—2782no—
192C4⋊C4—2781no—
193C4⋊C4—2781no—
194C4⋊C4—2782no—
195C4⋊C4—2781no—
196C4⋊C4—1392no—
197C4⋊C4C22×D13911yescharacteristic
198C4⋊C4C22×D13911yescharacteristic
199C22⋊C4—2782no—
200C22⋊C4—2781no—
201C22⋊C4—2781no—
202C22⋊C4—2781no—
203C22⋊C4—2781no—
204C22⋊C4—2781no—
205C42—2781no—
206C42—2782no—
207C42—1391no—
208C42—1391no—
209C42—1391no—
210C42C22×D13911yescharacteristic
211C23—2781no—
212C23—1391no—
213Q8—22no—
214Q8—22no—
215Q8—2782no—
216Q8—5562no—
217D4—2782no—
218D4—5562no—
219C2×C4—22no—
220C2×C4—21no—
221C2×C4—21no—
222C2×C4—2782no—
223C2×C4—2782no—
224C2×C4—2781no—
225C2×C4—2782no—
226C2×C4—5561no—
227C2×C4—5562no—
228C2×C4—1391no—
229C2×C4—1391no—
230C2×C4—1391no—
231C2×C4D4×D13911yescharacteristic
232C2×C4D4×D13911yescharacteristic
233C2×C4C23×D13911yescharacteristic
234C8—22no—
235C8—1394no—
236C22—2781no—
237C22—2781no—
238C22—2781no—
239C22—5561no—
240C22—5561no—
241C22—5561no—
242C22C2×D4×D13911yescharacteristic
243C4—21no—
244C4—21no—
245C4—41no—
246C4—41no—
247C4—42no—
248C4—2781no—
249C4—2781no—
250C4—5562no—
251C4—5561no—
252C4—1392no—
253C4#41×D13912yes—
254C2—5561no—
255C2—11121no—
256C2#2258150211yescharacteristic
257C2#137×D13911yescharacteristic
258C2#2258175211yescharacteristic
subgroupG/H
centerC22C2×D4×D139
commutator, cosocleC2×C4×C139C24
FrattiniC2×C4C23×D139
Fitting#491×C139C2
socleC22×C139C22×D4
prime radical#17984817C22
hypercenter#491D139
hypocenterC139#3172
LCSG ▷ C2×C4×C139 ▷ C2×C139 ▷ C139
UCSC1 ◁ C22 ◁ C42 ◁ #491

Complements

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

NH ≅ G/Ntype
1. #22581508C2semidirect
3. #22581363C2semidirect
4. #22581345C2semidirect
5. #22581313C2semidirect
6. #22581210C2semidirect
8. #22581045C2semidirect
9. #22581038C2semidirect
10. #22580905C2semidirect
11. #491×C139C2semidirect
129. C139#3172semidirect
135. #491D139semidirect

Maximal subgroups

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

subgroupindex
#225815082
#225814882
#225813632
#225813452
#225813132
#225812102
#225812022
#225810452
#225810382
#225809052
#491×C1392
#3172139