Finite Group Databasebeta

← #27170441

Subgroups — #27170441

Order 12928.

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

normal subgroups only

This group has 196 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#20265927C211yesmaximal, characteristic
2#20265583C212yesmaximal, complemented
3#20264918C212yesmaximal, complemented
4#20264917C211yesmaximal, characteristic
5#497×C101C211yesmaximal, characteristic
6#15692253—21no—
7#15692248—22no—
8#15692247C2212yes—
9#15692236—21no—
10#15692232—22no—
11#15692188—22no—
12#15692186—22no—
13#15692166C2212yes—
14C2×Q16×C101—21no—
15C2×SD16×C101—21no—
16C4⋊Q8×C101C2211yescharacteristic
17#124×C101C2211yescharacteristic
18#97×C101C2211yescharacteristic
19C2×Dic202—21no—
20C22×Dic101—21no—
21C2×#11717579—22no—
22C4⋊Dic101D412yes—
23C4⋊Dic101D412yes—
24#13108053—22no—
25#13108053—22no—
26#13108053—22no—
27C22⋊Dic101—22no—
28C22⋊Dic101—21no—
29C4×Dic101—21no—
30C2×Q8×C101—21no—
31C2×Q8×C101D411yescharacteristic
32C2×D4×C101D411yescharacteristic
33Q16×C101—42no—
34SD16×C101—42no—
35C2×C8×C101—21no—
36C4⋊C4×C101—22no—
37C22⋊C4×C101—22no—
38C42×C101C2311yescharacteristic
39C23×C101—21no—
40Q8×C101—41no—
41Q8×C101—41no—
42Q8×C101—41no—
43D4×C101—41no—
44C2×C4×C101—21no—
45C2×C4×C101—41no—
46C2×C4×C101C2×D412yes—
47C2×C4×C101C2×D411yescharacteristic
48C2×Dic101—22no—
49C2×Dic101—22no—
50C2×Dic101—81no—
51Dic202—81no—
52C8×C101—22no—
53#11717579—24no—
54C22×C101—41no—
55C22×C101—42no—
56C22×C101#12011yescharacteristic
57C4×C101—21no—
58C4×C101—22no—
59C4×C101—41no—
60C4×C101—81no—
61Dic101—42no—
62Dic101—42no—
63C2×C101—81no—
64C2×C101#45311yescharacteristic
65C2×C101#45612yes—
66#1660—1011nomaximal, Sylow
67C101#166011yescharacteristic, complemented, Sylow
68#563—1011no—
69#501—1012no—
70#497D10111yescharacteristic
71#330—1012no—
72#329—1011no—
73C2×Q16—21no—
74C2×SD16—21no—
75C4⋊Q8—2021no—
76C4⋊Q8C2×D10111yescharacteristic
77#126—2022no—
78#125—1012no—
79#124C2×D10111yescharacteristic
80#123—2021no—
81C22⋊Q8—2022no—
82#107—2022no—
83#106—2022no—
84#97—1012no—
85#97C2×D10111yescharacteristic
86C2×Q8—21no—
87C2×Q8—2021no—
88C2×Q8C22⋊D10111yescharacteristic
89C2×D4C22⋊D10111yescharacteristic
90C22×C4—2021no—
91Q16—42no—
92SD16—42no—
93C2×C8—21no—
94C2×C8—2022no—
95C4⋊C4—22no—
96C4⋊C4—1012no—
97C4⋊C4—1012no—
98C4⋊C4—2022no—
99C4⋊C4—2022no—
100C4⋊C4—2022no—
101C22⋊C4—22no—
102C22⋊C4—2022no—
103C22⋊C4—2021no—
104C42—2021no—
105C42C22×D10111yescharacteristic
106C23—21no—
107Q8—41no—
108Q8—41no—
109Q8—41no—
110Q8—8081no—
111D4—41no—
112C2×C4—21no—
113C2×C4—41no—
114C2×C4—8081no—
115C2×C4—2022no—
116C2×C4—2022no—
117C2×C4D4×D10112yes—
118C2×C4C2×C22⋊D10111yescharacteristic
119C8—22no—
120C8—2024no—
121C22—41no—
122C22—42no—
123C22#1569222311yescharacteristic
124C4—21no—
125C4—22no—
126C4—41no—
127C4—81no—
128C4—4042no—
129C4—4042no—
130C2—81no—
131C2#2026534411yescharacteristic
132C2#2026535812yes—
subgroupG/H
centerC22#15692223
commutator, cosocleC42×C101C23
FrattiniC42C22×D101
Fitting#497×C101C2
socleC22×C101#120
prime radicalC2×D4×C101D4
hypercenter#497D101
hypocenterC101#1660
LCSG ▷ C42×C101 ▷ C22×C101 ▷ C101
UCSC1 ◁ C22 ◁ C42 ◁ #497

Complements

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

NH ≅ G/Ntype
2. #20265583C2semidirect
3. #20264918C2semidirect
67. C101#1660semidirect

Maximal subgroups

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

subgroupindex
#202659272
#202655832
#202649182
#202649172
#497×C1012
#1660101