Finite Group Databasebeta

← #31808405

Subgroups — #31808405

Order 17792.

270 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 270 conjugacy classes of subgroups — too many to draw. Only its 45 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#22581139C212yesmaximal, complemented
2#22581130C211yesmaximal, characteristic, complemented
3#22580748C211yesmaximal, characteristic, complemented
4#22580740C212yesmaximal, complemented
5H(3,𝔽4)×C139C211yesmaximal, characteristic, complemented
6#17984887C2211yescharacteristic
7#17984873—21no—
8#17984811—22no—
9#17984770—21no—
10#17984759C2211yescharacteristic
11#17984757C2212yes—
12#17984756—22no—
13#124×C139—21no—
14#124×C139—22no—
15#124×C139—21no—
16#124×C139C2211yescharacteristic
17#120×C139—22no—
18#120×C139C2212yes—
19#14555785—21no—
20#14555784—41no—
21C2×C22⋊D139—22no—
22C2×C22⋊D139—42no—
23D4×D139—41no—
24#14555775—41no—
25#14555772—41no—
26#14555769—21no—
27C22⋊Dic139—22no—
28C22⋊Dic139—42no—
29C24×C139D412yescomplemented
30C2×Q8×C139—21no—
31C2×Q8×C139D411yescharacteristic, complemented
32C2×D4×C139—21no—
33C2×D4×C139—21no—
34C2×D4×C139—22no—
35C2×D4×C139C2311yescharacteristic
36C22⋊C4×C139—22no—
37C22⋊C4×C139—22no—
38C22⋊C4×C139—22no—
39C22⋊C4×C139—22no—
40C22⋊C4×C139D412yescomplemented
41C42×C139—21no—
42C42×C139D411yescharacteristic, complemented
43C22×D139—22no—
44Dic278—21no—
45D556—21no—
46C22⋊D139—42no—
47C22⋊D139—41no—
48C22⋊D139—84no—
49C2×Dic139—22no—
50C2×Dic139—42no—
51C4×D139—41no—
52C139⋊C8—41no—
53C23×C139—22no—
54C23×C139—42no—
55C23×C139—82no—
56C23×C139C2×D412yes—
57Q8×C139—41no—
58Q8×C139—81no—
59D4×C139—41no—
60D4×C139—82no—
61D4×C139—81no—
62D4×C139—81no—
63C2×C4×C139—21no—
64C2×C4×C139—22no—
65C2×C4×C139—21no—
66C2×C4×C139C2×D411yescharacteristic
67C22×C139—22no—
68C22×C139—42no—
69C22×C139—42no—
70C22×C139—42no—
71C22×C139—44no—
72C22×C139—42no—
73C22×C139—82no—
74C22×C139#12011yescharacteristic
75C2×D139—41no—
76C2×D139—82no—
77C4×C139—21no—
78C4×C139—42no—
79C4×C139—41no—
80C4×C139—41no—
81Dic139—41no—
82Dic139—82no—
83C2×C139—21no—
84C2×C139—42no—
85C2×C139—82no—
86C2×C139B(4,𝔽2)11yescharacteristic
87D139—81no—
88C139#219911yescharacteristic, complemented, Sylow
89#2199—1391nomaximal, Sylow
90H(3,𝔽4)D13911yescharacteristic, complemented
91B(4,𝔽2)—1392no—
92#455—1391no—
93#354—1391no—
94AΣL(1,𝔽16)—1392no—
95H(5,𝔽2)—1391no—
96#137—2781no—
97#124—21no—
98#124—22no—
99#124—21no—
100#124C2×D13911yescharacteristic
101#120—22no—
102#120—2782no—
103#120C2×D13912yes—
104#104—2781no—
105#100—1391no—
106#99—2782no—
107#99—1392no—
108C24C22⋊D13912yescomplemented
109#41—2781no—
110#41—5561no—
111C2×Q8—21no—
112C2×Q8C22⋊D13911yescharacteristic, complemented
113C2×D4—21no—
114C2×D4—21no—
115C2×D4—22no—
116C2×D4—2782no—
117C2×D4—5561no—
118C2×D4—5562no—
119C2×D4C22×D13911yescharacteristic
120Q16—5561no—
121SD16—5561no—
122M16—2781no—
123C22⋊C4—22no—
124C22⋊C4—22no—
125C22⋊C4—22no—
126C22⋊C4—22no—
127C22⋊C4—2782no—
128C22⋊C4—5562no—
129C22⋊C4C22⋊D13912yescomplemented
130C42—21no—
131C42C22⋊D13911yescharacteristic, complemented
132C23—22no—
133C23—42no—
134C23—82no—
135C23—2782no—
136C23C2×C22⋊D13912yes—
137Q8—41no—
138Q8—81no—
139Q8—2781no—
140D4—41no—
141D4—82no—
142D4—81no—
143D4—81no—
144D4—2781no—
145D4—5562no—
146D4—5561no—
147D4—11124no—
148C2×C4—21no—
149C2×C4—22no—
150C2×C4—21no—
151C2×C4—2782no—
152C2×C4—5561no—
153C2×C4—5562no—
154C2×C4C2×C22⋊D13911yescharacteristic
155C8—5561no—
156C22—22no—
157C22—42no—
158C22—42no—
159C22—42no—
160C22—44no—
161C22—42no—
162C22—82no—
163C22—5561no—
164C22—11122no—
165C22#1798481111yescharacteristic
166C4—21no—
167C4—42no—
168C4—41no—
169C4—41no—
170C4—5561no—
171C4—11122no—
172C2—21no—
173C2—42no—
174C2—82no—
175C2—11121no—
176C2#2258114011yescharacteristic
subgroupG/H
centerC2#22581140
commutator, cosocleC2×D4×C139C23
FrattiniC2×D4C22×D139
FittingH(3,𝔽4)×C139C2
socleC2×C139B(4,𝔽2)
hypercenterH(3,𝔽4)D139
hypocenterC139#2199
LCSG ▷ C2×D4×C139 ▷ C22×C139 ▷ C2×C139 ▷ C139
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ H(3,𝔽4)

Complements

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

NH ≅ G/Ntype
1. #22581139C2semidirect
2. #22581130C2semidirect
3. #22580748C2semidirect
4. #22580740C2semidirect
5. H(3,𝔽4)×C139C2semidirect
29. C24×C139D4semidirect
31. C2×Q8×C139D4semidirect
40. C22⋊C4×C139D4semidirect
42. C42×C139D4semidirect
88. C139#2199semidirect
90. H(3,𝔽4)D139semidirect
108. C24C22⋊D139semidirect
112. C2×Q8C22⋊D139semidirect
129. C22⋊C4C22⋊D139semidirect
131. C42C22⋊D139semidirect

Maximal subgroups

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

subgroupindex
#225811392
#225811302
#225807482
#225807402
H(3,𝔽4)×C1392
#2199139