Finite Group Databasebeta

← #41003031

Subgroups — #41003031

Order 26944.

160 conjugacy classes of subgroups; only the 39 normal subgroups are shown.

normal subgroups only

This group has 160 conjugacy classes of subgroups — too many to draw. Only its 39 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#27212411C211yesmaximal, characteristic
2#27212350C212yesmaximal, complemented
3#27212347C211yesmaximal, characteristic, complemented
4#27212280C212yesmaximal, complemented
5#99×C421C211yesmaximal, characteristic, complemented
6#20269771—21no—
7#20269770—22no—
8C2×C22⋊D421—21no—
9C2×C22⋊D421C2212yes—
10D4×D421—22no—
11C22×Dic421C2211yescharacteristic, complemented
12C4⋊Dic421—22no—
13#20269750—21no—
14C22⋊Dic421—21no—
15C22⋊Dic421C2211yescharacteristic, complemented
16#20269748—22no—
17C2×D4×C421C2211yescharacteristic
18C22⋊C4×C421C2212yescomplemented
19C22×D421—21no—
20C22×D421D412yes—
21Dic842—21no—
22D1684—21no—
23C22⋊D421—22no—
24C22⋊D421—22no—
25C22⋊D421—22no—
26C2×Dic421—21no—
27C2×Dic421—22no—
28C2×Dic421—21no—
29C2×Dic421—21no—
30C2×Dic421D412yes—
31C4×D421—22no—
32C23×C421D411yescharacteristic
33C23×C421C2311yescharacteristic
34D4×C421—22no—
35C2×C4×C421—22no—
36C2×C4×C421D411yescharacteristic
37C22×C421—21no—
38C22×C421—41no—
39C22×C421—41no—
40C22×C421C2×D412yes—
41C22×C421C2×D411yescharacteristic
42C2×D421—22no—
43C2×D421—42no—
44C2×D421—81no—
45C4×C421—21no—
46C4×C421—42no—
47Dic421—22no—
48Dic421—41no—
49Dic421—41no—
50C2×C421—21no—
51C2×C421—22no—
52C2×C421—41no—
53C2×C421#12011yescharacteristic
54D421—42no—
55C421#45811yescharacteristic, complemented, Sylow
56#458—4211nomaximal, Sylow
57H(5,𝔽2)—4211no—
58#123—4211no—
59#123—4212no—
60#99—4212no—
61#99D42111yescharacteristic, complemented
62#41—8422no—
63#41—8421no—
64C2×D4—8422no—
65C2×D4—8421no—
66C2×D4—4212no—
67C2×D4C2×D42111yescharacteristic
68C22×C4—4211no—
69C4⋊C4—8422no—
70C4⋊C4—8421no—
71C22⋊C4—8422no—
72C22⋊C4—8421no—
73C22⋊C4—4211no—
74C22⋊C4C2×D42112yescomplemented
75C23—8421no—
76C23—4212no—
77C23D168411yescharacteristic
78C23C22×D42111yescharacteristic
79Q8—8421no—
80D4—22no—
81D4—8421no—
82D4—8422no—
83D4—8422no—
84D4—8422no—
85C2×C4—22no—
86C2×C4—8422no—
87C2×C4—8421no—
88C2×C4—8422no—
89C2×C4—8421no—
90C2×C4—8421no—
91C2×C4—4212no—
92C2×C4D168411yescharacteristic
93C22—21no—
94C22—41no—
95C22—41no—
96C22—33681no—
97C22—8422no—
98C22—16842no—
99C22D4×D42112yes—
100C22C2×D168411yescharacteristic
101C4—21no—
102C4—42no—
103C4—8422no—
104C4—16841no—
105C4—16841no—
106C2—22no—
107C2—21no—
108C2—41no—
109C2—16842no—
110C2#2721233611yescharacteristic
subgroupG/H
centerC2#27212336
commutator, cosocleC23×C421C23
FrattiniC23C22×D421
Fitting#99×C421C2
socleC2×C421#120
prime radical#27212411C2
hypercenter#99D421
hypocenterC421#458
LCSG ▷ C23×C421 ▷ C2×C421 ▷ C421
UCSC1 ◁ C2 ◁ C23 ◁ #99

Complements

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

NH ≅ G/Ntype
2. #27212350C2semidirect
3. #27212347C2semidirect
4. #27212280C2semidirect
5. #99×C421C2semidirect
11. C22×Dic421C22semidirect
15. C22⋊Dic421C22semidirect
18. C22⋊C4×C421C22semidirect
55. C421#458semidirect
61. #99D421semidirect
74. C22⋊C4C2×D421semidirect

Maximal subgroups

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

subgroupindex
#272124112
#272123502
#272123472
#272122802
#99×C4212
#458421