Finite Group Databasebeta

← #31815782

Subgroups — #31815782

Order 17792.

704 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 704 conjugacy classes of subgroups — too many to draw. Only its 145 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#22581789C211yesmaximal, characteristic
2C2×#17984873C212yesmaximal, complemented
3#22581135C214yesmaximal, complemented
4#22581130C214yesmaximal, complemented
5#22580971C212yesmaximal, complemented
6#22580937C211yesmaximal, characteristic, complemented
7#536×C139C211yesmaximal, characteristic, complemented
8#17984890C2212yes—
9#17984889—22no—
10#17984888—22no—
11#17984887—22no—
12#17984887C2212yes—
13C2×#14555785—22no—
14C2×#14555785C2212yes—
15#41×D139—24no—
16#17984873—24no—
17#17984873C2218yescomplemented
18C2×#14555775—22no—
19C2×#14555772—22no—
20C2×#14555769C2212yescomplemented
21#17984770C2218yescomplemented
22#17984761C2212yescomplemented
23#17984759C2212yescomplemented
24C2×#41×C139C2211yescharacteristic
25C22×Q8×C139C2211yescharacteristic, complemented
26C4⋊Q8×C139C2212yescomplemented
27#124×C139C2212yescomplemented
28#123×C139—22no—
29C22⋊Q8×C139—22no—
30#117×C139C2211yescharacteristic, complemented
31#14555786—24no—
32#14555785—24no—
33#14555785—24no—
34#14555785—22no—
35#14555785—24no—
36#14555785—22no—
37#14555785D414yes—
38#14555785C2314yes—
39#14555784—28no—
40#14555784—24no—
41C2×Dic278—22no—
42C2×Dic278D412yes—
43Q8×D139—24no—
44C2×D556—22no—
45C2×D556D412yes—
46C2×C22⋊D139—24no—
47C2×C22⋊D139—22no—
48D4×D139—28no—
49D4×D139—24no—
50C2×C4×D139—24no—
51C2×C4×D139—22no—
52#14555775—28no—
53#14555772—28no—
54#14555769—22no—
55#14555769C2314yes—
56C2×C139⋊C8—22no—
57#41×C139—22no—
58#41×C139—22no—
59C2×Q8×C139—21no—
60C2×Q8×C139—22no—
61C2×Q8×C139—22no—
62C2×Q8×C139C2312yes—
63C2×Q8×C139C2311yescharacteristic
64C2×D4×C139D412yes—
65C2×D4×C139C2311yescharacteristic
66C22×C4×C139—21no—
67C22×C4×C139D412yes—
68C22×C4×C139C2311yescharacteristic
69C4⋊C4×C139—21no—
70C4⋊C4×C139—24no—
71C22⋊C4×C139—21no—
72C22⋊C4×C139—24no—
73C42×C139C2312yes—
74C22×D139—24no—
75C22×D139—22no—
76Dic278—24no—
77Dic278—22no—
78Dic278C2×D414yes—
79D556—24no—
80D556—22no—
81D556C2×D414yes—
82C22⋊D139—28no—
83C22⋊D139—24no—
84C22⋊D139—24no—
85C2×Dic139—24no—
86C2×Dic139—22no—
87C4×D139—28no—
88C4×D139—24no—
89C4×D139—24no—
90C139⋊C8—24no—
91C23×C139C2×D412yes—
92C23×C139C2×D411yescharacteristic
93Q8×C139—22no—
94Q8×C139—24no—
95Q8×C139—41no—
96Q8×C139—41no—
97D4×C139—24no—
98D4×C139—22no—
99C2×C4×C139—22no—
100C2×C4×C139—22no—
101C2×C4×C139—22no—
102C2×C4×C139—22no—
103C2×C4×C139—42no—
104C2×C4×C139C2×D412yes—
105C2×C4×C139C2×D412yes—
106C2×C4×C139C2×D412yes—
107C2×C4×C139C2×D411yescharacteristic
108C2×C4×C139C2411yescharacteristic
109C22×C139—22no—
110C22×C139—41no—
111C22×C139—42no—
112C22×C139#12012yes—
113C22×C139C22×D411yescharacteristic
114C2×D139—24no—
115C2×D139—82no—
116C2×D139—84no—
117C4×C139—22no—
118C4×C139—42no—
119C4×C139—42no—
120C4×C139#12011yescharacteristic
121C4×C139#12011yescharacteristic
122C4×C139C22×D412yes—
123Dic139—24no—
124C2×C139—21no—
125C2×C139—22no—
126C2×C139—42no—
127C2×C139C2×#12011yescharacteristic
128D139—44no—
129C139#302211yescharacteristic, complemented, Sylow
130#3022—1391nomaximal, Sylow
131#585—1391no—
132C2×#137—1392no—
133#536D13911yescharacteristic, complemented
134#456—1394no—
135#455—1394no—
136#421—1392no—
137#413—1391no—
1382−1+4—2782no—
1392−1+4—1392no—
140H(5,𝔽2)—2782no—
141H(5,𝔽2)—2782no—
142H(5,𝔽2)—1392no—
143C2×#41—2784no—
144C2×#41—2782no—
145C2×#41—1392no—
146C2×#41C2×D13911yescharacteristic
147C22×Q8C2×D13911yescharacteristic, complemented
148#137—2784no—
149#137—1398no—
150C2×Q16—2782no—
151C2×SD16—2782no—
152C2×M16—1392no—
153C4⋊Q8C2×D13912yescomplemented
154#124C2×D13912yescomplemented
155#123—22no—
156C22⋊Q8—22no—
157#117C2×D13911yescharacteristic, complemented
158#104—1398no—
159#101—1392no—
160#100—1392no—
161#41—22no—
162#41—22no—
163#41—2784no—
164#41—2784no—
165#41—2788no—
166#41—2784no—
167#41—2784no—
168#41—2782no—
169#41—2782no—
170#41—2784no—
171#41—1394no—
172#41—1394no—
173C2×Q8—21no—
174C2×Q8—22no—
175C2×Q8—22no—
176C2×Q8—2782no—
177C2×Q8—2784no—
178C2×Q8—1392no—
179C2×Q8C22×D13912yes—
180C2×Q8C22×D13911yescharacteristic
181C2×D4—2782no—
182C2×D4—2788no—
183C2×D4—2784no—
184C2×D4—2784no—
185C2×D4—2782no—
186C2×D4—1392no—
187C2×D4C22⋊D13912yes—
188C2×D4C22×D13911yescharacteristic
189C22×C4—21no—
190C22×C4—2784no—
191C22×C4—2782no—
192C22×C4C22⋊D13912yes—
193C22×C4C22×D13911yescharacteristic
194Q16—2788no—
195SD16—2788no—
196M16—2782no—
197M16—1394no—
198C2×C8—2782no—
199C4⋊C4—21no—
200C4⋊C4—24no—
201C22⋊C4—21no—
202C22⋊C4—24no—
203C42C22×D13912yes—
204C23—2784no—
205C23—2782no—
206C23C2×C22⋊D13912yes—
207C23C2×C22⋊D13911yescharacteristic
208Q8—22no—
209Q8—24no—
210Q8—41no—
211Q8—41no—
212Q8—2784no—
213Q8—2782no—
214Q8—1394no—
215D4—24no—
216D4—22no—
217D4—2784no—
218D4—2788no—
219D4—2784no—
220D4—2784no—
221D4—2782no—
222D4—1394no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—42no—
228C2×C4—2788no—
229C2×C4—2784no—
230C2×C4—2782no—
231C2×C4—2784no—
232C2×C4—2784no—
233C2×C4D4×D13912yes—
234C2×C4D4×D13912yes—
235C2×C4C2×C22⋊D13912yes—
236C2×C4C2×C22⋊D13911yescharacteristic
237C2×C4C23×D13911yescharacteristic
238C8—2784no—
239C22—22no—
240C22—41no—
241C22—42no—
242C22—2784no—
243C22—11122no—
244C22—11124no—
245C22#1798481012yes—
246C22C22×C22⋊D13911yescharacteristic
247C4—22no—
248C4—42no—
249C4—42no—
250C4—2784no—
251C4#1798481011yescharacteristic
252C4#1798481011yescharacteristic
253C4C2×D4×D13912yes—
254C2—21no—
255C2—22no—
256C2—42no—
257C2—5564no—
258C2C2×#1798481011yescharacteristic
subgroupG/H
centerC2C2×#17984810
commutator, cosocleC2×C4×C139C24
FrattiniC2×C4C23×D139
Fitting#536×C139C2
socleC2×C139C2×#120
prime radical#22581789C2
hypercenter#536D139
hypocenterC139#3022
LCSG ▷ C2×C4×C139 ▷ C2×C139 ▷ C139
UCSC1 ◁ C2 ◁ C22×C4 ◁ #536

Complements

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

NH ≅ G/Ntype
2. C2×#17984873C2semidirect
3. #22581135C2semidirect
4. #22581130C2semidirect
5. #22580971C2semidirect
6. #22580937C2semidirect
7. #536×C139C2semidirect
17. #17984873C22semidirect
20. C2×#14555769C22semidirect
21. #17984770C22semidirect
22. #17984761C22semidirect
23. #17984759C22semidirect
25. C22×Q8×C139C22semidirect
26. C4⋊Q8×C139C22semidirect
27. #124×C139C22semidirect
30. #117×C139C22semidirect
129. C139#3022semidirect
133. #536D139semidirect
147. C22×Q8C2×D139semidirect
153. C4⋊Q8C2×D139semidirect
154. #124C2×D139semidirect
157. #117C2×D139semidirect

Maximal subgroups

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

subgroupindex
#225817892
C2×#179848732
#225811352
#225811302
#225809712
#225809372
#536×C1392
#3022139