Finite Group Databasebeta

← #31815768

Subgroups — #31815768

Order 17792.

698 conjugacy classes of subgroups; only the 161 normal subgroups are shown.

normal subgroups only

This group has 698 conjugacy classes of subgroups — too many to draw. Only its 161 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#22581751C212yesmaximal, complemented
2#22581527C211yesmaximal, characteristic, complemented
3#22581136C214yesmaximal, complemented
4#22581126C214yesmaximal, complemented
5#22580970C212yesmaximal, complemented
6C2×#17984761C211yesmaximal, characteristic, complemented
7C2×2−1+4×C139C211yesmaximal, characteristic, complemented
8C2×#14555785C2211yescharacteristic
9#17984873—22no—
10#17984872C2214yescomplemented
11#17984868—22no—
12#17984867C2214yescomplemented
13#17984862—24no—
14C2×#14555769C2212yes—
15#17984840C2212yescomplemented
16#17984831C2212yescomplemented
17#17984825—22no—
18#17984818—22no—
19#17984801C2211yescharacteristic
20#17984769C2218yescomplemented
21#17984761C2214yescomplemented
222−1+4×C139—24no—
232−1+4×C139—22no—
242−1+4×C139C2214yes—
25C2×#41×C139—22no—
26C2×#41×C139—22no—
27C2×#41×C139C2212yes—
28C22×Q8×C139—22no—
29C22×Q8×C139C2211yescharacteristic
30#14555785—22no—
31#14555785—22no—
32C2×Dic278C2311yescharacteristic
33C2×D556C2311yescharacteristic
34C2×C22⋊D139—21no—
35C2×C4×D139—21no—
36#14555775—24no—
37#14555773—24no—
38#14555772—24no—
39#14555770—24no—
40#14555769—22no—
41#14555769C2314yes—
42C2×C139⋊C8—22no—
43C4⋊Dic139—21no—
44Dic139⋊C4—24no—
45C22⋊Dic139—21no—
46(C2×D139)⋊C4—24no—
47C4×Dic139C2312yes—
48#41×C139—28no—
49#41×C139—28no—
50#41×C139—28no—
51#41×C139—28no—
52#41×C139—22no—
53#41×C139—22no—
54#41×C139D414yes—
55#41×C139C2314yes—
56C2×Q8×C139—22no—
57C2×Q8×C139—28no—
58C2×Q8×C139—28no—
59C2×Q8×C139—22no—
60C2×Q8×C139—22no—
61C2×Q8×C139D412yes—
62C2×Q8×C139D412yes—
63C2×Q8×C139C2312yes—
64C2×D4×C139—22no—
65C2×D4×C139—22no—
66C2×D4×C139D412yes—
67C22×C4×C139—24no—
68C22×C4×C139—22no—
69C22×C4×C139D412yes—
70C22×C4×C139C2311yescharacteristic
71C22×D139—21no—
72Dic278—22no—
73D556—22no—
74C22⋊D139—41no—
75C22⋊D139—41no—
76C2×Dic139—22no—
77C2×Dic139—21no—
78C4×D139—42no—
79C139⋊C8—24no—
80C23×C139—22no—
81C23×C139C2×D411yescharacteristic
82Q8×C139—28no—
83Q8×C139—22no—
84Q8×C139—24no—
85Q8×C139—24no—
86Q8×C139C2×D414yes—
87D4×C139—28no—
88D4×C139—24no—
89D4×C139—24no—
90D4×C139—22no—
91D4×C139C2×D414yes—
92C2×C4×C139—216no—
93C2×C4×C139—22no—
94C2×C4×C139—22no—
95C2×C4×C139—22no—
96C2×C4×C139—24no—
97C2×C4×C139—24no—
98C2×C4×C139C2×D414yes—
99C2×C4×C139C2×D411yescharacteristic
100C2×C4×C139C2×D412yes—
101C2×C4×C139C2×D412yes—
102C2×C4×C139C2411yescharacteristic
103C22×C139—24no—
104C22×C139—41no—
105C22×C139—44no—
106C22×C139#12011yescharacteristic
107C22×C139#12011yescharacteristic
108C22×C139C22×D411yescharacteristic
109C2×D139—42no—
110C2×D139—41no—
111C4×C139—24no—
112C4×C139—24no—
113C4×C139#12012yes—
114C4×C139C22×D412yes—
115Dic139—42no—
116Dic139—41no—
117C2×C139—21no—
118C2×C139—21no—
119C2×C139—21no—
120C2×C139—44no—
121C2×C139C2×#12011yescharacteristic
122D139—81no—
123C139#302011yescharacteristic, complemented, Sylow
124#3020—1391nomaximal, Sylow
125C2×2−1+4D13911yescharacteristic, complemented
126#575—1392no—
127#536—1391no—
128#456—1394no—
129#454—1394no—
130#421—1392no—
131C2×#101—1391no—
1322−1+4—24no—
1332−1+4—22no—
1342−1+4C2×D13914yes—
135C2×#41—22no—
136C2×#41—22no—
137C2×#41—1391no—
138C2×#41C2×D13912yes—
139C22×Q8—22no—
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⋊Q8—1392no—
148#124—1392no—
149#123—2782no—
150C22⋊Q8—2782no—
151#117—1391no—
152#104—1398no—
153#101—1394no—
154#41—28no—
155#41—28no—
156#41—28no—
157#41—28no—
158#41—22no—
159#41—22no—
160#41—2782no—
161#41—2782no—
162#41C22⋊D13914yes—
163#41C22×D13914yes—
164C2×Q8—22no—
165C2×Q8—28no—
166C2×Q8—28no—
167C2×Q8—22no—
168C2×Q8—22no—
169C2×Q8—1391no—
170C2×Q8C22⋊D13912yes—
171C2×Q8C22⋊D13912yes—
172C2×Q8C22×D13912yes—
173C2×D4—22no—
174C2×D4—22no—
175C2×D4—2781no—
176C2×D4—1391no—
177C2×D4C22⋊D13912yes—
178C22×C4—24no—
179C22×C4—22no—
180C22×C4—2781no—
181C22×C4C22⋊D13912yes—
182C22×C4C22×D13911yescharacteristic
183Q16—2784no—
184SD16—2784no—
185SD16—2784no—
186D8—2784no—
187M16—2782no—
188M16—1394no—
189C2×C8—2782no—
190C4⋊C4—2784no—
191C4⋊C4—2781no—
192C22⋊C4—2781no—
193C22⋊C4—2784no—
194C42—1392no—
195C23—22no—
196C23—2781no—
197C23C2×C22⋊D13911yescharacteristic
198Q8—28no—
199Q8—22no—
200Q8—24no—
201Q8—24no—
202Q8—2782no—
203Q8C2×C22⋊D13914yes—
204D4—28no—
205D4—24no—
206D4—24no—
207D4—22no—
208D4—2782no—
209D4—5561no—
210D4—5561no—
211D4C2×C22⋊D13914yes—
212C2×C4—216no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—24no—
217C2×C4—24no—
218C2×C4—2782no—
219C2×C4—2781no—
220C2×C4—5562no—
221C2×C4C2×C22⋊D13914yes—
222C2×C4C2×C22⋊D13911yescharacteristic
223C2×C4C2×C22⋊D13912yes—
224C2×C4C2×C22⋊D13912yes—
225C2×C4C23×D13911yescharacteristic
226C8—2784no—
227C22—24no—
228C22—41no—
229C22—44no—
230C22—5562no—
231C22—5561no—
232C22#1798481111yescharacteristic
233C22#1798481111yescharacteristic
234C22C22×C22⋊D13911yescharacteristic
235C4—24no—
236C4—24no—
237C4—5562no—
238C4—5561no—
239C4#1798481112yes—
240C4C22×C22⋊D13912yes—
241C2—21no—
242C2—21no—
243C2—21no—
244C2—44no—
245C2—11121no—
246C2C2×#1798481111yescharacteristic
subgroupG/H
centerC2C2×#17984811
commutator, cosocleC2×C4×C139C24
FrattiniC2×C4C23×D139
FittingC2×2−1+4×C139C2
socleC2×C139C2×#120
hypercenterC2×2−1+4D139
hypocenterC139#3020
LCSG ▷ C2×C4×C139 ▷ C2×C139 ▷ C139
UCSC1 ◁ C2 ◁ C22×C4 ◁ C2×2−1+4

Complements

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

NH ≅ G/Ntype
1. #22581751C2semidirect
2. #22581527C2semidirect
3. #22581136C2semidirect
4. #22581126C2semidirect
5. #22580970C2semidirect
6. C2×#17984761C2semidirect
7. C2×2−1+4×C139C2semidirect
10. #17984872C22semidirect
12. #17984867C22semidirect
15. #17984840C22semidirect
16. #17984831C22semidirect
20. #17984769C22semidirect
21. #17984761C22semidirect
123. C139#3020semidirect
125. C2×2−1+4D139semidirect

Maximal subgroups

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

subgroupindex
#225817512
#225815272
#225811362
#225811262
#225809702
C2×#179847612
C2×2−1+4×C1392
#3020139