Finite Group Databasebeta

← #1394×C211 #41009014

Subgroups — #1394×C211 #41009014

Order 27008.

172 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 172 conjugacy classes of subgroups — too many to draw. Only its 64 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×#101×C211C212yesmaximal
2#391×C211C211yesmaximal, characteristic
3C22×Q8×C211C2211yescharacteristic
4C2×M16×C211—22no—
5C2×C4⋊C4×C211C411yescharacteristic
6C2×C42×C211C411yescharacteristic
7#101×C211—22no—
8#101×C211C414yescomplemented
9#95×C211—22no—
10C2×Q8×C211—41no—
11C2×Q8×C211—41no—
12C2×Q8×C211C2×C412yes—
13C2×Q8×C211D411yescharacteristic
14C2×Q8×C211Q811yescharacteristic
15C22×C4×C211—21no—
16C22×C4×C211—21no—
17C22×C4×C211C2×C411yescharacteristic
18C22×C4×C211D412yes—
19M16×C211—24no—
20M16×C211—42no—
21C2×C8×C211—42no—
22C4⋊C4×C211—41no—
23C42×C211—41no—
24C23×C211C22⋊C411yescharacteristic
25Q8×C211—42no—
26Q8×C211—41no—
27Q8×C211—41no—
28C2×C4×C211—21no—
29C2×C4×C211—22no—
30C2×C4×C211—21no—
31C2×C4×C211—22no—
32C2×C4×C211—22no—
33C2×C4×C211—41no—
34C2×C4×C211—41no—
35C2×C4×C211—42no—
36C2×C4×C211—41no—
37C2×C4×C211C4211yescharacteristic
38C2×C4×C211C22⋊C412yes—
39C2×C4×C211C4⋊C411yescharacteristic
40C2×C4×C211C4⋊C412yes—
41C8×C211—44no—
42C22×C211—21no—
43C22×C211—21no—
44C22×C211#9511yescharacteristic
45C22×C211#9911yescharacteristic
46C22×C211#9911yescharacteristic
47C4×C211—22no—
48C4×C211—21no—
49C4×C211—21no—
50C4×C211—22no—
51C4×C211—42no—
52C4×C211—81no—
53C2×C211—21no—
54C2×C211—21no—
55C2×C211#34211yescharacteristic
56C2×C211#35511yescharacteristic
57C2×C211#35611yescharacteristic
58C211#139411yescharacteristic, direct product factor, Sylow
59#1394C21111yesmaximal, characteristic, direct product factor, Sylow
60C2×#101C2×C21112yes—
61#391C2×C21111yescharacteristic
62C22×Q8C22×C21111yescharacteristic
63C2×M16—22no—
64C2×C4⋊C4C4×C21111yescharacteristic
65C2×C42C4×C21111yescharacteristic
66#101—22no—
67#101C4×C21114yescomplemented
68#95—22no—
69C2×Q8—41no—
70C2×Q8—41no—
71C2×Q8C2×C4×C21112yes—
72C2×Q8D4×C21111yescharacteristic
73C2×Q8Q8×C21111yescharacteristic
74C22×C4—21no—
75C22×C4—21no—
76C22×C4C2×C4×C21111yescharacteristic
77C22×C4D4×C21112yes—
78M16—24no—
79M16—42no—
80C2×C8—42no—
81C4⋊C4—41no—
82C42—41no—
83C23C22⋊C4×C21111yescharacteristic
84Q8—42no—
85Q8—41no—
86Q8—41no—
87C2×C4—21no—
88C2×C4—21no—
89C2×C4—22no—
90C2×C4—22no—
91C2×C4—22no—
92C2×C4—41no—
93C2×C4—41no—
94C2×C4—42no—
95C2×C4—41no—
96C2×C4C42×C21111yescharacteristic
97C2×C4C22⋊C4×C21112yes—
98C2×C4C4⋊C4×C21111yescharacteristic
99C2×C4C4⋊C4×C21112yes—
100C8—44no—
101C22—21no—
102C22—21no—
103C22#95×C21111yescharacteristic
104C22#99×C21111yescharacteristic
105C22#99×C21111yescharacteristic
106C4—22no—
107C4—21no—
108C4—21no—
109C4—22no—
110C4—42no—
111C4—81no—
112C2—21no—
113C2—21no—
114C2#342×C21111yescharacteristic
115C2#355×C21111yescharacteristic
116C2#356×C21111yescharacteristic
subgroupG/H
center, socleC22×C211#99
commutatorC2×C4C42×C211
Frattini, cosocleC22×Q8C22×C211
prime radicalC23×C211C22⋊C4
LCSG ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C211 ◁ C23×C211 ◁ C22×Q8×C211 ◁ G

Complements

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

NH ≅ G/Ntype
8. #101×C211C4semidirect
58. C211#1394direct (N × H)
59. #1394C211direct (N × H)
67. #101C4×C211semidirect

Maximal subgroups

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

subgroupindex
C2×#101×C2112
#391×C2112
#1394211