Finite Group Databasebeta

← #31816870

Subgroups — #31816870

Order 17792.

572 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 572 conjugacy classes of subgroups — too many to draw. Only its 165 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×#17984875C212yesmaximal, complemented
2C2×#17984870C212yesmaximal, complemented
3#22581503C211yesmaximal, characteristic, complemented
4C2×#17984830C211yesmaximal, characteristic, complemented
5#22581347C212yesmaximal, complemented
6#22581342C212yesmaximal, complemented
7#22581337C214yesmaximal, complemented
8C4×M16×C139C211yesmaximal, characteristic, complemented
9C2×#14555785C2212yes—
10C22×Dic278C2211yescharacteristic
11C22×D556C2211yescharacteristic
12#17984875—28no—
13#17984870—28no—
14C2×Dic556C2214yescomplemented
15C2×#14555774C2214yes—
16C2×#14555774C2214yescomplemented
17C2×D1112C2214yes—
18#17984842C2211yescharacteristic, complemented
19#17984839C2211yescharacteristic, complemented
20#17984830—22no—
21#17984830C2212yes—
22#17984830C2212yes—
23#17984830C2212yescomplemented
24#17984816—22no—
25C4×D556—22no—
26C2×(C2×D139)⋊C4—22no—
27C2×M16×C139C2212yes—
28C2×C42×C139C2211yescharacteristic
29#97×C139C2212yescomplemented
30C4×C8×C139C2212yescomplemented
31C23×D139—21no—
32#14555785—24no—
33#14555785—24no—
34C2×Dic278—21no—
35C2×Dic278—24no—
36C2×Dic278C2312yes—
37C2×Dic278C2312yes—
38C2×D556—24no—
39C2×D556—22no—
40C2×D556—21no—
41C2×D556C2312yes—
42C2×D556C2312yes—
43C2×C22⋊D139—22no—
44C22×Dic139—21no—
45C2×C4×D139—22no—
46Dic556—28no—
47#14555774—28no—
48#14555774—28no—
49D1112—28no—
50C4⋊Dic139—22no—
51(C2×D139)⋊C4—24no—
52(C2×D139)⋊C4—24no—
53(C2×D139)⋊C4—24no—
54C22×C4×C139D412yes—
55C22×C4×C139C2311yescharacteristic
56M16×C139D418yescomplemented
57C2×C8×C139C2314yes—
58C42×C139D412yes—
59C42×C139C2311yescharacteristic
60C42×C139C2311yescharacteristic
61C22×D139—22no—
62C22×D139—22no—
63C22×D139—44no—
64C22×D139—42no—
65Dic278—22no—
66Dic278—24no—
67Dic278—22no—
68Dic278—42no—
69D556—22no—
70D556—24no—
71D556—22no—
72D556—44no—
73D556—42no—
74C22⋊D139—44no—
75C2×Dic139—22no—
76C2×Dic139—22no—
77C2×Dic139—42no—
78C4×D139—44no—
79C23×C139C2×D411yescharacteristic
80C2×C4×C139—42no—
81C2×C4×C139C2×D412yes—
82C2×C4×C139C2×D412yes—
83C2×C4×C139C2×D411yescharacteristic
84C2×C4×C139C2×D414yes—
85C2×C4×C139C2411yescharacteristic
86C8×C139C2×D418yes—
87C22×C139—22no—
88C22×C139#12712yes—
89C22×C139C22×D411yescharacteristic
90C2×D139—42no—
91C2×D139—42no—
92C2×D139—44no—
93C2×D139—44no—
94C2×D139—84no—
95C4×C139—22no—
96C4×C139—22no—
97C4×C139#12712yes—
98C4×C139C22×D412yes—
99Dic139—42no—
100Dic139—42no—
101C2×C139—22no—
102C2×C139C2×#12711yescharacteristic
103C2×C139#57512yes—
104D139—82no—
105D139—82no—
106C139#315311yescharacteristic, complemented, Sylow
107#3153—1391nomaximal, Sylow
108C2×#137—1392no—
109C2×AGL(1,ℤ/8)—1392no—
110#532—1391no—
111C2×#124—1391no—
112#497—1392no—
113#496—1392no—
114#495—1394no—
115C4×M16D13911yescharacteristic, complemented
116C2×#41—1392no—
117C22×Q8—1391no—
118C22×D4—1391no—
119#137—2788no—
120AGL(1,ℤ/8)—2788no—
121C2×Q16—1394no—
122C2×SD16—1394no—
123C2×SD16—1394no—
124C2×D8—1394no—
125C2×M16C2×D13912yes—
126C4⋊Q8—1391no—
127#127—1391no—
128#124—2782no—
129#124—1392no—
130#124—1392no—
131#124—1392no—
132#121—2782no—
133C4×D4—2782no—
134C2×C22⋊C4—2782no—
135C2×C42C2×D13911yescharacteristic
136#97C2×D13912yescomplemented
137C4×C8C2×D13912yescomplemented
138C24—2781no—
139#41—2784no—
140#41—2784no—
141C2×Q8—2781no—
142C2×Q8—2784no—
143C2×Q8—1392no—
144C2×Q8—1392no—
145C2×D4—2782no—
146C2×D4—2784no—
147C2×D4—2782no—
148C2×D4—2781no—
149C2×D4—1392no—
150C2×D4—1392no—
151C22×C4—2781no—
152C22×C4—2782no—
153C22×C4D55612yes—
154C22×C4C22×D13911yescharacteristic
155Q16—2788no—
156SD16—2788no—
157SD16—2788no—
158D8—2788no—
159M16D55618yescomplemented
160C2×C8C22×D13914yes—
161C4⋊C4—2782no—
162C22⋊C4—2784no—
163C22⋊C4—2784no—
164C22⋊C4—2784no—
165C42D55612yes—
166C42C22×D13911yescharacteristic
167C42C22×D13911yescharacteristic
168C23—2782no—
169C23—2782no—
170C23—5564no—
171C23—5562no—
172C23C2×D55611yescharacteristic
173Q8—2782no—
174Q8—2784no—
175Q8—2782no—
176Q8—5562no—
177D4—2782no—
178D4—2784no—
179D4—2782no—
180D4—5564no—
181D4—5564no—
182D4—5562no—
183C2×C4—42no—
184C2×C4—2782no—
185C2×C4—2782no—
186C2×C4—5562no—
187C2×C4—5564no—
188C2×C4C2×D55612yes—
189C2×C4C2×D55612yes—
190C2×C4C2×D55611yescharacteristic
191C2×C4C2×D55614yes—
192C2×C4C23×D13911yescharacteristic
193C8C2×D55618yes—
194C22—22no—
195C22—5562no—
196C22—5562no—
197C22—5564no—
198C22—5564no—
199C22—11124no—
200C22#1798483912yes—
201C22C22×D55611yescharacteristic
202C4—22no—
203C4—22no—
204C4—5562no—
205C4—5562no—
206C4#1798483912yes—
207C4C22×D55612yes—
208C2—22no—
209C2—11122no—
210C2—11122no—
211C2C2×#1798483911yescharacteristic
212C2#2258175312yes—
subgroupG/H
centerC22C22×D556
commutator, cosocleC2×C4×C139C24
FrattiniC2×C4C23×D139
FittingC4×M16×C139C2
socleC22×C139C22×D4
hypercenterC4×M16D139
hypocenterC139#3153
LCSG ▷ C2×C4×C139 ▷ C2×C139 ▷ C139
UCSC1 ◁ C22 ◁ C2×C42 ◁ C4×M16

Complements

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

NH ≅ G/Ntype
1. C2×#17984875C2semidirect
2. C2×#17984870C2semidirect
3. #22581503C2semidirect
4. C2×#17984830C2semidirect
5. #22581347C2semidirect
6. #22581342C2semidirect
7. #22581337C2semidirect
8. C4×M16×C139C2semidirect
14. C2×Dic556C22semidirect
16. C2×#14555774C22semidirect
18. #17984842C22semidirect
19. #17984839C22semidirect
23. #17984830C22semidirect
29. #97×C139C22semidirect
30. C4×C8×C139C22semidirect
56. M16×C139D4semidirect
106. C139#3153semidirect
115. C4×M16D139semidirect
136. #97C2×D139semidirect
137. C4×C8C2×D139semidirect
159. M16D556semidirect

Maximal subgroups

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

subgroupindex
C2×#179848752
C2×#179848702
#225815032
C2×#179848302
#225813472
#225813422
#225813372
C4×M16×C1392
#3153139