Finite Group Databasebeta

← #31816578

Subgroups — #31816578

Order 17792.

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

normal subgroups only

This group has 386 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#22581489C211yesmaximal, characteristic
2C2×#17984807C211yesmaximal, characteristic, complemented
3#22581271C211yesmaximal, characteristic
4#22581259C212yesmaximal, complemented
5#22581178C212yesmaximal, complemented
6#22581108C211yesmaximal, characteristic, complemented
7#22581094C211yesmaximal, characteristic
8#22581057C212yesmaximal, complemented
9#22581052C212yesmaximal, complemented
10#22581007C211yesmaximal, characteristic, complemented
11#484×C139C211yesmaximal, characteristic, complemented
12C22×Dic278C2211yescharacteristic
13C2×#14555775C2212yes—
14C2×#14555774C2212yes—
15#17984837—21no—
16#17984833C2211yescharacteristic
17#17984831C2211yescharacteristic
18#17984826—21no—
19#17984816C2211yescharacteristic
20Q8×Dic139C2211yescharacteristic
21#17984807—21no—
22#17984807—21no—
23#17984807C2212yes—
24#17984807C2212yes—
25#17984804C2211yescharacteristic
26(C22⋊D139)⋊C4—21no—
27#17984800—21no—
28C2×Dic139⋊C4—21no—
29C2×C4×Dic139—21no—
30C2×C4×Dic139C2211yescharacteristic
31#17984773C2212yes—
32Dic139⋊C8C2212yes—
33#17984767C2212yes—
34#17984765C2212yes—
35#17984762C2212yes—
36C22⋊(C139⋊C8)C2211yescharacteristic
37#17984752C2212yes—
38C22⋊Q8×C139C2211yescharacteristic
39C2×C4⋊C4×C139C2211yescharacteristic
40#107×C139C2212yes—
41#103×C139C2212yes—
42C22⋊C8×C139C2211yescharacteristic
43C2×Dic278—24no—
44C2×Dic278—21no—
45C2×Dic278C2312yes—
46C2×D556C2311yescharacteristic
47C2×C22⋊D139—21no—
48C22×Dic139—21no—
49C22×Dic139D411yescharacteristic
50C22×Dic139D411yescharacteristic
51C2×C4×D139—21no—
52#14555775—24no—
53#14555774—24no—
54C2×C139⋊C8C2312yes—
55C4⋊Dic139—21no—
56C4⋊Dic139—21no—
57C4⋊Dic139C2311yescharacteristic
58Dic139⋊C4—22no—
59Dic139⋊C4—22no—
60Dic139⋊C4—21no—
61(C2×D139)⋊C4—21no—
62(C2×D139)⋊C4—21no—
63(C2×D139)⋊C4—21no—
64(C2×D139)⋊C4—21no—
65C4×Dic139—22no—
66C4×Dic139—22no—
67C4×Dic139—21no—
68C4×Dic139D411yescharacteristic
69C4×Dic139D411yescharacteristic
70C4×Dic139C2311yescharacteristic
71C4×Dic139C2311yescharacteristic
72C2×Q8×C139C2311yescharacteristic
73C22×C4×C139—21no—
74C22×C4×C139C2311yescharacteristic
75C2×C8×C139C2312yes—
76C4⋊C4×C139—21no—
77C4⋊C4×C139—21no—
78C4⋊C4×C139C2312yes—
79C4⋊C4×C139C2311yescharacteristic
80C22⋊C4×C139—21no—
81C22×D139—21no—
82Dic278—24no—
83Dic278—22no—
84Dic278#4114yes—
85D556—22no—
86C22⋊D139—42no—
87C2×Dic139—24no—
88C2×Dic139—22no—
89C2×Dic139—22no—
90C2×Dic139—21no—
91C2×Dic139—41no—
92C2×Dic139C2×D411yescharacteristic
93C2×Dic139C2×D411yescharacteristic
94C2×Dic139C2×D411yescharacteristic
95C2×Dic139C2×D411yescharacteristic
96C4×D139—42no—
97C139⋊C8—22no—
98C23×C139C2×D411yescharacteristic
99Q8×C139—22no—
100C2×C4×C139—22no—
101C2×C4×C139—21no—
102C2×C4×C139—21no—
103C2×C4×C139—22no—
104C2×C4×C139—42no—
105C2×C4×C139C2×D411yescharacteristic
106C2×C4×C139C2411yescharacteristic
107C8×C139—22no—
108C22×C139—21no—
109C22×C139—21no—
110C22×C139#13712yes—
111C22×C139C22×D411yescharacteristic
112C2×D139—41no—
113C2×D139—41no—
114C2×D139—41no—
115C4×C139—21no—
116C4×C139—41no—
117C4×C139—42no—
118C4×C139—41no—
119C4×C139C2×#4112yes—
120Dic139—24no—
121Dic139—21no—
122Dic139—21no—
123Dic139—21no—
124Dic139—21no—
125Dic139—41no—
126C2×C139—22no—
127C2×C139#52511yescharacteristic
128C2×C139C2×#13711yescharacteristic
129C2×C139#57511yescharacteristic
130D139—81no—
131C139#311711yescharacteristic, complemented, Sylow
132#3117—1391nomaximal, Sylow
133#529—1391no—
134C2×C4×Q8—1391no—
135#484D13911yescharacteristic, complemented
136#481—1391no—
137#479—1392no—
138#464—1392no—
139#451—1391no—
140#448—1391no—
141#441—1392no—
142#440—1392no—
143#432—1391no—
144C22×Q8—1391no—
145C2×Q16—1392no—
146C2×SD16—1392no—
147#126—2781no—
148#125—1391no—
149#124—1391no—
150#123—2781no—
151C22⋊Q8C2×D13911yescharacteristic
152#121—1391no—
153C4×Q8—2781no—
154C4×Q8—2781no—
155C4×Q8—1391no—
156C4×Q8—1392no—
157C4×Q8—1392no—
158C4×D4—2781no—
159C4×D4—1391no—
160#117—2781no—
161C2×C4⋊C4—2781no—
162C2×C4⋊C4C2×D13911yescharacteristic
163C2×C42—2781no—
164C2×C42—1391no—
165#107C2×D13912yes—
166#106—1392no—
167C4⋊C8—1392no—
168#103—1392no—
169#103—1392no—
170#103C2×D13912yes—
171#102—1392no—
172C22⋊C8—1391no—
173C22⋊C8C2×D13911yescharacteristic
174#97—1392no—
175C2×Q8—2784no—
176C2×Q8—2781no—
177C2×Q8—1392no—
178C2×Q8C22×D13911yescharacteristic
179C2×D4—2781no—
180C2×D4—1391no—
181C22×C4—21no—
182C22×C4—2781no—
183C22×C4—2781no—
184C22×C4—1391no—
185C22×C4—1391no—
186C22×C4C22×D13911yescharacteristic
187Q16—2784no—
188SD16—2784no—
189C2×C8—1392no—
190C2×C8C22×D13912yes—
191C4⋊C4—21no—
192C4⋊C4—21no—
193C4⋊C4—2782no—
194C4⋊C4—2782no—
195C4⋊C4—2781no—
196C4⋊C4—2781no—
197C4⋊C4—2781no—
198C4⋊C4—1391no—
199C4⋊C4C22×D13912yes—
200C4⋊C4C22×D13911yescharacteristic
201C22⋊C4—21no—
202C22⋊C4—2781no—
203C22⋊C4—2781no—
204C22⋊C4—2781no—
205C22⋊C4—2781no—
206C42—2782no—
207C42—2782no—
208C42—2781no—
209C42—1391no—
210C42—1391no—
211C42—1391no—
212C42—1391no—
213C23—2781no—
214C23D4×D13911yescharacteristic
215Q8—22no—
216Q8—2784no—
217Q8—2782no—
218Q8—1394no—
219D4—2782no—
220D4—5562no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—42no—
226C2×C4—2784no—
227C2×C4—2782no—
228C2×C4—2782no—
229C2×C4—2781no—
230C2×C4—5562no—
231C2×C4—5561no—
232C2×C4—1391no—
233C2×C4—1391no—
234C2×C4—1391no—
235C2×C4—1391no—
236C2×C4D4×D13911yescharacteristic
237C2×C4C23×D13911yescharacteristic
238C8—22no—
239C8—2782no—
240C22—21no—
241C22—21no—
242C22—5561no—
243C22—5561no—
244C22—5561no—
245C22#1798487412yes—
246C22C2×D4×D13911yescharacteristic
247C4—21no—
248C4—41no—
249C4—42no—
250C4—41no—
251C4—2784no—
252C4—2781no—
253C4—2781no—
254C4—2781no—
255C4—2781no—
256C4—5561no—
257C4#41×D13912yes—
258C2—22no—
259C2—11121no—
260C2#2258146111yescharacteristic
261C2C2×#1798487411yescharacteristic
262C2#2258174911yescharacteristic
subgroupG/H
centerC22C2×D4×D139
commutator, cosocleC2×C4×C139C24
FrattiniC2×C4C23×D139
Fitting#484×C139C2
socleC22×C139C22×D4
prime radical#17984816C22
hypercenter#484D139
hypocenterC139#3117
LCSG ▷ C2×C4×C139 ▷ C2×C139 ▷ C139
UCSC1 ◁ C22 ◁ C22×C4 ◁ #484

Complements

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

NH ≅ G/Ntype
2. C2×#17984807C2semidirect
4. #22581259C2semidirect
5. #22581178C2semidirect
6. #22581108C2semidirect
8. #22581057C2semidirect
9. #22581052C2semidirect
10. #22581007C2semidirect
11. #484×C139C2semidirect
131. C139#3117semidirect
135. #484D139semidirect

Maximal subgroups

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

subgroupindex
#225814892
C2×#179848072
#225812712
#225812592
#225811782
#225811082
#225810942
#225810572
#225810522
#225810072
#484×C1392
#3117139