Finite Group Databasebeta

← #31815767

Subgroups — #31815767

Order 17792.

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

normal subgroups only

This group has 698 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
1C2×#17984890C211yesmaximal, characteristic, complemented
2#22581751C212yesmaximal, complemented
3#22581135C214yesmaximal, complemented
4#22581125C214yesmaximal, complemented
5#22580971C212yesmaximal, complemented
6C2×#17984761C211yesmaximal, characteristic, complemented
7#536×C139C211yesmaximal, characteristic, complemented
8#17984890—24no—
9#17984890—22no—
10#17984890C2214yes—
11C2×#14555786—22no—
12C2×#14555785—22no—
13C2×#14555785C2212yes—
14C2×Q8×D139—22no—
15#17984873C2214yescomplemented
16#17984872—22no—
17#17984868C2214yescomplemented
18#17984867—22no—
19#17984862—24no—
20C2×#14555769C2212yes—
21#17984770C2218yescomplemented
22#17984761C2214yescomplemented
23C2×#41×C139C2211yescharacteristic
24C22×Q8×C139C2211yescharacteristic
25C4⋊Q8×C139C2212yescomplemented
26#124×C139C2212yescomplemented
27#123×C139—22no—
28C22⋊Q8×C139—22no—
29#117×C139C2211yescharacteristic
30#14555786—28no—
31#14555786—28no—
32#14555785—28no—
33#14555785—28no—
34#14555785—22no—
35#14555785—22no—
36#14555785D414yes—
37#14555785C2314yes—
38C2×Dic278—22no—
39C2×Dic278D412yes—
40Q8×D139—28no—
41Q8×D139—28no—
42C2×D556—22no—
43C2×D556D412yes—
44C2×C22⋊D139—22no—
45C2×C4×D139—24no—
46C2×C4×D139—22no—
47#14555775—24no—
48#14555773—24no—
49#14555772—24no—
50#14555770—24no—
51#14555769—22no—
52#14555769C2314yes—
53C2×C139⋊C8—22no—
54#41×C139—22no—
55#41×C139—22no—
56C2×Q8×C139—22no—
57C2×Q8×C139—22no—
58C2×Q8×C139D412yes—
59C2×Q8×C139C2311yescharacteristic
60C2×Q8×C139C2312yes—
61C2×D4×C139—21no—
62C2×D4×C139C2311yescharacteristic
63C22×C4×C139—21no—
64C22×C4×C139D412yes—
65C22×C4×C139C2311yescharacteristic
66C4⋊C4×C139—21no—
67C4⋊C4×C139—24no—
68C22⋊C4×C139—21no—
69C22⋊C4×C139—24no—
70C42×C139C2312yes—
71C22×D139—22no—
72Dic278—28no—
73Dic278—22no—
74Dic278C2×D414yes—
75D556—28no—
76D556—22no—
77D556C2×D414yes—
78C22⋊D139—24no—
79C22⋊D139—24no—
80C2×Dic139—22no—
81C4×D139—216no—
82C4×D139—24no—
83C4×D139—24no—
84C139⋊C8—24no—
85C23×C139—21no—
86C23×C139C2×D411yescharacteristic
87Q8×C139—24no—
88Q8×C139—24no—
89Q8×C139—22no—
90D4×C139—22no—
91D4×C139—41no—
92D4×C139—41no—
93C2×C4×C139—22no—
94C2×C4×C139—22no—
95C2×C4×C139—22no—
96C2×C4×C139—21no—
97C2×C4×C139—42no—
98C2×C4×C139C2×D412yes—
99C2×C4×C139C2×D412yes—
100C2×C4×C139C2×D414yes—
101C2×C4×C139C2×D411yescharacteristic
102C2×C4×C139C2411yescharacteristic
103C22×C139—41no—
104C22×C139—42no—
105C22×C139—41no—
106C22×C139#12011yescharacteristic
107C22×C139#12011yescharacteristic
108C22×C139C22×D411yescharacteristic
109C2×D139—24no—
110C2×D139—44no—
111C4×C139—24no—
112C4×C139—42no—
113C4×C139—41no—
114C4×C139#12012yes—
115C4×C139C22×D412yes—
116Dic139—24no—
117C2×C139—21no—
118C2×C139—21no—
119C2×C139—21no—
120C2×C139—81no—
121C2×C139C2×#12011yescharacteristic
122D139—44no—
123C139#302011yescharacteristic, complemented, Sylow
124#3020—1391nomaximal, Sylow
125C2×2−1+4—1391no—
126#575—1392no—
127#536D13911yescharacteristic, complemented
128#456—1394no—
129#454—1394no—
130#421—1392no—
131C2×#101—1391no—
1322−1+4—2784no—
1332−1+4—2782no—
1342−1+4—1394no—
135C2×#41—2782no—
136C2×#41—2782no—
137C2×#41—1392no—
138C2×#41C2×D13911yescharacteristic
139C22×Q8—2782no—
140C22×Q8C2×D13911yescharacteristic
141#137—2782no—
142#137—1394no—
143AGL(1,ℤ/8)—2782no—
144AGL(1,ℤ/8)—1394no—
145#135—2784no—
146C2×M16—1392no—
147C4⋊Q8C2×D13912yescomplemented
148#124C2×D13912yescomplemented
149#123—22no—
150C22⋊Q8—22no—
151#117C2×D13911yescharacteristic
152#104—1398no—
153#101—1394no—
154#41—22no—
155#41—22no—
156#41—2788no—
157#41—2788no—
158#41—2788no—
159#41—2782no—
160#41—2788no—
161#41—2782no—
162#41—1394no—
163#41—1394no—
164C2×Q8—22no—
165C2×Q8—22no—
166C2×Q8—2782no—
167C2×Q8—2788no—
168C2×Q8—2788no—
169C2×Q8—1392no—
170C2×Q8C22⋊D13912yes—
171C2×Q8C22×D13911yescharacteristic
172C2×Q8C22×D13912yes—
173C2×D4—21no—
174C2×D4—2782no—
175C2×D4—2782no—
176C2×D4—1392no—
177C2×D4C22×D13911yescharacteristic
178C22×C4—21no—
179C22×C4—2784no—
180C22×C4—2782no—
181C22×C4C22⋊D13912yes—
182C22×C4C22×D13911yescharacteristic
183Q16—2784no—
184SD16—2784no—
185SD16—2784no—
186D8—2784no—
187M16—2782no—
188M16—1394no—
189C2×C8—2782no—
190C4⋊C4—21no—
191C4⋊C4—24no—
192C22⋊C4—21no—
193C22⋊C4—24no—
194C42C22×D13912yes—
195C23—21no—
196C23—2782no—
197C23C2×C22⋊D13911yescharacteristic
198Q8—24no—
199Q8—24no—
200Q8—22no—
201Q8—2788no—
202Q8—2782no—
203Q8—1394no—
204D4—22no—
205D4—41no—
206D4—41no—
207D4—2788no—
208D4—2784no—
209D4—2784no—
210D4—2782no—
211D4—1394no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—21no—
216C2×C4—42no—
217C2×C4—27816no—
218C2×C4—2782no—
219C2×C4—2784no—
220C2×C4—2784no—
221C2×C4D4×D13912yes—
222C2×C4D4×D13912yes—
223C2×C4C2×C22⋊D13914yes—
224C2×C4C2×C22⋊D13911yescharacteristic
225C2×C4C23×D13911yescharacteristic
226C8—2784no—
227C22—41no—
228C22—42no—
229C22—41no—
230C22—2784no—
231C22—5564no—
232C22#1798481011yescharacteristic
233C22#1798481011yescharacteristic
234C22C22×C22⋊D13911yescharacteristic
235C4—24no—
236C4—42no—
237C4—41no—
238C4—2784no—
239C4#1798481012yes—
240C4C2×D4×D13912yes—
241C2—21no—
242C2—21no—
243C2—21no—
244C2—81no—
245C2—5564no—
246C2C2×#1798481011yescharacteristic
subgroupG/H
centerC2C2×#17984810
commutator, cosocleC2×C4×C139C24
FrattiniC2×C4C23×D139
Fitting#536×C139C2
socleC2×C139C2×#120
hypercenter#536D139
hypocenterC139#3020
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
1. C2×#17984890C2semidirect
2. #22581751C2semidirect
3. #22581135C2semidirect
4. #22581125C2semidirect
5. #22580971C2semidirect
6. C2×#17984761C2semidirect
7. #536×C139C2semidirect
15. #17984873C22semidirect
17. #17984868C22semidirect
21. #17984770C22semidirect
22. #17984761C22semidirect
25. C4⋊Q8×C139C22semidirect
26. #124×C139C22semidirect
123. C139#3020semidirect
127. #536D139semidirect
147. C4⋊Q8C2×D139semidirect
148. #124C2×D139semidirect

Maximal subgroups

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

subgroupindex
C2×#179848902
#225817512
#225811352
#225811252
#225809712
C2×#179847612
#536×C1392
#3020139