Finite Group Databasebeta

← #1486×C239 #44480256

Subgroups — #1486×C239 #44480256

Order 30592.

312 conjugacy classes of subgroups; only the 116 normal subgroups are shown.

normal subgroups only

This group has 312 conjugacy classes of subgroups — too many to draw. Only its 116 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×C4×D4×C239C211yesmaximal, characteristic
2#431×C239C211yesmaximal, characteristic, complemented
3#423×C239C211yesmaximal, characteristic, complemented
4D4⋊C8×C239C214yesmaximal, complemented
5C22×D4×C239C411yescharacteristic
6C23×C4×C239—21no—
7C2×M16×C239—21no—
8C4×D4×C239—21no—
9C4×D4×C239—21no—
10C4×D4×C239C412yes—
11C4×D4×C239C2212yes—
12C2×C4⋊C4×C239C411yescharacteristic
13C2×C22⋊C4×C239—21no—
14C2×C42×C239C2211yescharacteristic
15C4⋊C8×C239—21no—
16C4⋊C8×C239—21no—
17C4⋊C8×C239C2212yescomplemented
18C22⋊C8×C239—21no—
19C4×C8×C239C2212yescomplemented
20C24×C239—21no—
21C2×D4×C239—24no—
22C2×D4×C239—21no—
23C2×D4×C239C2×C412yes—
24C22×C4×C239—21no—
25C22×C4×C239—22no—
26C22×C4×C239—22no—
27C22×C4×C239—42no—
28C22×C4×C239—42no—
29C22×C4×C239C2×C411yescharacteristic
30C22×C4×C239D411yescharacteristic
31C22×C4×C239D411yescharacteristic
32M16×C239—42no—
33C2×C8×C239—22no—
34C2×C8×C239—22no—
35C4⋊C4×C239—21no—
36C4⋊C4×C239C2×C412yes—
37C22⋊C4×C239—22no—
38C22⋊C4×C239—22no—
39C42×C239C2×C411yescharacteristic
40C42×C239D411yescharacteristic
41C42×C239D411yescharacteristic
42C42×C239C2311yescharacteristic
43C23×C239—24no—
44C23×C239—22no—
45C23×C239—42no—
46C23×C239—42no—
47C23×C239C22⋊C411yescharacteristic
48D4×C239—24no—
49D4×C239—22no—
50D4×C239M1614yescomplemented
51C2×C4×C239—22no—
52C2×C4×C239—22no—
53C2×C4×C239—41no—
54C2×C4×C239—42no—
55C2×C4×C239—44no—
56C2×C4×C239—42no—
57C2×C4×C239—42no—
58C2×C4×C239C22⋊C411yescharacteristic
59C2×C4×C239C22⋊C411yescharacteristic
60C2×C4×C239C22⋊C411yescharacteristic
61C2×C4×C239D812yes—
62C2×C4×C239SD1612yes—
63C2×C4×C239C22×C411yescharacteristic
64C2×C4×C239C2×D411yescharacteristic
65C2×C4×C239C2×D411yescharacteristic
66C8×C239—42no—
67C8×C239—42no—
68C22×C239—21no—
69C22×C239—21no—
70C22×C239—24no—
71C22×C239—48no—
72C22×C239—42no—
73C22×C239—42no—
74C22×C239#10212yes—
75C22×C239C2×C22⋊C411yescharacteristic
76C4×C239—21no—
77C4×C239—21no—
78C4×C239—21no—
79C4×C239—42no—
80C4×C239#10212yes—
81C4×C239C2×M1612yes—
82C4×C239C2×D811yescharacteristic
83C4×C239C2×SD1611yescharacteristic
84C2×C239—22no—
85C2×C239—44no—
86C2×C239#40711yescharacteristic
87C2×C239C2×#10211yescharacteristic
88C2×C239#42111yescharacteristic
89C239#148611yescharacteristic, direct product factor, Sylow
90#1486C23911yesmaximal, characteristic, direct product factor, Sylow
91C2×C4×D4C2×C23911yescharacteristic
92#431C2×C23911yescharacteristic, complemented
93#423C2×C23911yescharacteristic, complemented
94D4⋊C8C2×C23914yescomplemented
95C22×D4C4×C23911yescharacteristic
96C23×C4—21no—
97C2×M16—21no—
98C4×D4—21no—
99C4×D4—21no—
100C4×D4C4×C23912yes—
101C4×D4C22×C23912yes—
102C2×C4⋊C4C4×C23911yescharacteristic
103C2×C22⋊C4—21no—
104C2×C42C22×C23911yescharacteristic
105C4⋊C8—21no—
106C4⋊C8—21no—
107C4⋊C8C22×C23912yescomplemented
108C22⋊C8—21no—
109C4×C8C22×C23912yescomplemented
110C24—21no—
111C2×D4—24no—
112C2×D4—21no—
113C2×D4C2×C4×C23912yes—
114C22×C4—21no—
115C22×C4—22no—
116C22×C4—22no—
117C22×C4—42no—
118C22×C4—42no—
119C22×C4C2×C4×C23911yescharacteristic
120C22×C4D4×C23911yescharacteristic
121C22×C4D4×C23911yescharacteristic
122M16—42no—
123C2×C8—22no—
124C2×C8—22no—
125C4⋊C4—21no—
126C4⋊C4C2×C4×C23912yes—
127C22⋊C4—22no—
128C22⋊C4—22no—
129C42C2×C4×C23911yescharacteristic
130C42D4×C23911yescharacteristic
131C42D4×C23911yescharacteristic
132C42C23×C23911yescharacteristic
133C23—24no—
134C23—22no—
135C23—42no—
136C23—42no—
137C23C22⋊C4×C23911yescharacteristic
138D4—24no—
139D4—22no—
140D4M16×C23914yescomplemented
141C2×C4—22no—
142C2×C4—22no—
143C2×C4—41no—
144C2×C4—42no—
145C2×C4—44no—
146C2×C4—42no—
147C2×C4—42no—
148C2×C4C22⋊C4×C23911yescharacteristic
149C2×C4C22⋊C4×C23911yescharacteristic
150C2×C4C22⋊C4×C23911yescharacteristic
151C2×C4D8×C23912yes—
152C2×C4SD16×C23912yes—
153C2×C4C22×C4×C23911yescharacteristic
154C2×C4C2×D4×C23911yescharacteristic
155C2×C4C2×D4×C23911yescharacteristic
156C8—42no—
157C8—42no—
158C22—21no—
159C22—21no—
160C22—24no—
161C22—48no—
162C22—42no—
163C22—42no—
164C22#102×C23912yes—
165C22C2×C22⋊C4×C23911yescharacteristic
166C4—21no—
167C4—21no—
168C4—21no—
169C4—42no—
170C4#102×C23912yes—
171C4C2×M16×C23912yes—
172C4C2×D8×C23911yescharacteristic
173C4C2×SD16×C23911yescharacteristic
174C2—22no—
175C2—44no—
176C2#407×C23911yescharacteristic
177C2C2×#102×C23911yescharacteristic
178C2#421×C23911yescharacteristic
subgroupG/H
centerC2×C4×C239C2×D4
commutatorC2×C4C22×C4×C239
Frattini, cosocleC42C23×C239
socleC22×C239C2×C22⋊C4
prime radicalC22×D4×C239C4
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4×C239 ◁ C2×C42×C239 ◁ G

Complements

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

NH ≅ G/Ntype
2. #431×C239C2semidirect
3. #423×C239C2semidirect
4. D4⋊C8×C239C2semidirect
17. C4⋊C8×C239C22semidirect
19. C4×C8×C239C22semidirect
50. D4×C239M16semidirect
89. C239#1486direct (N × H)
90. #1486C239direct (N × H)
92. #431C2×C239semidirect
93. #423C2×C239semidirect
94. D4⋊C8C2×C239semidirect
107. C4⋊C8C22×C239semidirect
109. C4×C8C22×C239semidirect
140. D4M16×C239semidirect

Maximal subgroups

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

subgroupindex
C2×C4×D4×C2392
#431×C2392
#423×C2392
D4⋊C8×C2392
#1486239