Finite Group Databasebeta

← #31806842

Subgroups — #31806842

Order 17792.

412 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 412 conjugacy classes of subgroups — too many to draw. Only its 109 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×#17984816C211yesmaximal, characteristic
2C2×#17984764C212yesmaximal, complemented
3#22580813C211yesmaximal, characteristic, complemented
4#22580716C212yesmaximal, complemented
5C22×M16×C139C211yesmaximal, characteristic, complemented
6C22×D556C2211yescharacteristic
7C22×C22⋊D139—21no—
8#17984816—22no—
9#17984816C414yescomplemented
10C2×C4⋊Dic139C2212yes—
11C2×C4⋊Dic139C2211yescharacteristic
12C2×C22⋊Dic139—21no—
13C2×(C2×D139)⋊C4—21no—
14#17984764—24no—
15C139⋊#95—21no—
16C23×C4×C139C2211yescharacteristic
17C2×M16×C139—22no—
18C2×M16×C139—24no—
19C22×C8×C139C2212yes—
20C23×D139—21no—
21C2×D556—22no—
22C2×D556—21no—
23C2×D556—22no—
24C2×D556C2×C412yes—
25C2×C22⋊D139—24no—
26C2×C22⋊D139—44no—
27C22×Dic139—21no—
28C22×Dic139—22no—
29C4⋊Dic139—21no—
30C4⋊Dic139—22no—
31C4⋊Dic139—22no—
32C4⋊Dic139C2×C412yes—
33C22⋊Dic139—42no—
34(C2×D139)⋊C4—24no—
35C24×C139D411yescharacteristic
36C22×C4×C139—22no—
37C22×C4×C139—22no—
38C22×C4×C139C2×C412yes—
39C22×C4×C139D412yes—
40C22×C4×C139D411yescharacteristic
41C22×C4×C139C2311yescharacteristic
42M16×C139—28no—
43C2×C8×C139—22no—
44C2×C8×C139—22no—
45C2×C8×C139D414yescomplemented
46C22×D139—22no—
47C22×D139—42no—
48C22×D139—42no—
49C22×D139—41no—
50C22×D139—41no—
51D556—24no—
52D556—41no—
53D556—41no—
54C22⋊D139—48no—
55C2×Dic139—22no—
56C2×Dic139—42no—
57C2×Dic139—42no—
58C2×Dic139—42no—
59C2×Dic139—42no—
60C23×C139—22no—
61C23×C139—21no—
62C23×C139—22no—
63C23×C139—21no—
64C23×C139C22⋊C412yes—
65C23×C139C2×D411yescharacteristic
66C2×C4×C139—22no—
67C2×C4×C139—24no—
68C2×C4×C139—22no—
69C2×C4×C139—22no—
70C2×C4×C139C22⋊C412yes—
71C2×C4×C139C22×C411yescharacteristic
72C2×C4×C139C2×D412yes—
73C2×C4×C139C2×D411yescharacteristic
74C2×C4×C139#4112yes—
75C8×C139—24no—
76C22×C139—22no—
77C22×C139—22no—
78C22×C139—22no—
79C22×C139—42no—
80C22×C139—42no—
81C22×C139C2×C22⋊C411yescharacteristic
82C22×C139#12112yes—
83C22×C139AGL(1,ℤ/8)11yescharacteristic
84C22×C139AGL(1,ℤ/8)12yes—
85C22×C139#13711yescharacteristic
86C2×D139—42no—
87C2×D139—42no—
88C2×D139—42no—
89C2×D139—82no—
90C2×D139—82no—
91C4×C139—22no—
92C4×C139C4×D412yes—
93C4×C139#12011yescharacteristic
94C4×C139#12311yescharacteristic
95Dic139—42no—
96Dic139—82no—
97C2×C139—42no—
98C2×C139#38611yescharacteristic
99C2×C139#41711yescharacteristic
100C2×C139#41811yescharacteristic
101C2×C139#46812yes—
102C2×C139#46912yes—
103D139—82no—
104C139#189511yescharacteristic, complemented, Sylow
105#1895—1391nomaximal, Sylow
106C22×M16D13911yescharacteristic, complemented
107C2×#121—1391no—
108C2×#102—1392no—
109#380—1391no—
110#340—1392no—
111C22×D4—2781no—
112C22×D4—1391no—
113C23×C4C2×D13911yescharacteristic
114C2×M16—22no—
115C2×M16—24no—
116C22×C8C2×D13912yes—
117#121—2782no—
118#121—1394no—
119C2×C4⋊C4—1392no—
120C2×C4⋊C4—1391no—
121C2×C22⋊C4—2781no—
122C2×C22⋊C4—2781no—
123#102—2784no—
124#95—2781no—
125C24—2781no—
126C24D55611yescharacteristic
127C2×D4—2784no—
128C2×D4—2782no—
129C2×D4—2781no—
130C2×D4—2782no—
131C2×D4—5564no—
132C2×D4—1392no—
133C22×C4—22no—
134C22×C4—22no—
135C22×C4—2781no—
136C22×C4—2782no—
137C22×C4C4×D13912yes—
138C22×C4C22⋊D13912yes—
139C22×C4D55611yescharacteristic
140C22×C4C22×D13911yescharacteristic
141M16—28no—
142C2×C8—22no—
143C2×C8—22no—
144C2×C8C22⋊D13914yescomplemented
145C4⋊C4—2781no—
146C4⋊C4—2782no—
147C4⋊C4—2782no—
148C4⋊C4—1392no—
149C22⋊C4—2784no—
150C22⋊C4—5562no—
151C23—22no—
152C23—21no—
153C23—22no—
154C23—21no—
155C23—2782no—
156C23—5562no—
157C23—5562no—
158C23—5561no—
159C23—5561no—
160C23(C2×D139)⋊C412yes—
161C23C2×D55611yescharacteristic
162D4—2784no—
163D4—5568no—
164D4—5561no—
165D4—5561no—
166C2×C4—22no—
167C2×C4—24no—
168C2×C4—22no—
169C2×C4—22no—
170C2×C4—2782no—
171C2×C4—5562no—
172C2×C4—5562no—
173C2×C4—5562no—
174C2×C4—5562no—
175C2×C4(C2×D139)⋊C412yes—
176C2×C4C2×C4×D13911yescharacteristic
177C2×C4C2×C22⋊D13912yes—
178C2×C4C2×C22⋊D13911yescharacteristic
179C2×C4#1455578512yes—
180C8—24no—
181C22—22no—
182C22—22no—
183C22—22no—
184C22—42no—
185C22—42no—
186C22—5562no—
187C22—5562no—
188C22—5562no—
189C22—11122no—
190C22—11122no—
191C22C2×(C2×D139)⋊C411yescharacteristic
192C22#1798481612yes—
193C22#1798487012yes—
194C22#1798487011yescharacteristic
195C22#1798487511yescharacteristic
196C4—22no—
197C4—5562no—
198C4—11122no—
199C4C4×C22⋊D13912yes—
200C4#1798481111yescharacteristic
201C4#1798482511yescharacteristic
202C2—42no—
203C2—11122no—
204C2#2258083211yescharacteristic
205C2#2258095511yescharacteristic
206C2#2258095911yescharacteristic
207C2#2258119912yes—
208C2#2258120612yes—
subgroupG/H
centerC23C2×D556
commutatorC2×C4×C139C22×C4
FrattiniC22×C4C22×D139
FittingC22×M16×C139C2
socleC23×C139C2×D4
prime radicalC2×#17984816C2
cosocleC22×C4×C139C23
hypercenterC22×M16D139
hypocenterC139#1895
LCSG ▷ C2×C4×C139 ▷ C2×C139 ▷ C139
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×M16

Complements

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

NH ≅ G/Ntype
2. C2×#17984764C2semidirect
3. #22580813C2semidirect
4. #22580716C2semidirect
5. C22×M16×C139C2semidirect
9. #17984816C4semidirect
45. C2×C8×C139D4semidirect
104. C139#1895semidirect
106. C22×M16D139semidirect
144. C2×C8C22⋊D139semidirect

Maximal subgroups

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

subgroupindex
C2×#179848162
C2×#179847642
#225808132
#225807162
C22×M16×C1392
#1895139