Finite Group Databasebeta

← #27170428

Subgroups — #27170428

Order 12928.

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

normal subgroups only

This group has 206 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, complemented
2#20265541C212yesmaximal, complemented
3#20264918C212yesmaximal, complemented
4#20264917C211yesmaximal, characteristic, complemented
5#496×C101C211yesmaximal, characteristic
6#15692253—21no—
7#15692248—22no—
8#15692247C2212yes—
9#15692236—21no—
10#15692232—22no—
11#15692176—22no—
12#15692176—22no—
13#15692166C2212yes—
14C2×SD16×C101—21no—
15C2×D8×C101—21no—
16#127×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×C101D411yescharacteristic
31C2×D4×C101—21no—
32C2×D4×C101—22no—
33C2×D4×C101D411yescharacteristic
34SD16×C101—42no—
35D8×C101—42no—
36C2×C8×C101—21no—
37C22⋊C4×C101—22no—
38C42×C101C2311yescharacteristic
39C23×C101—21no—
40C23×C101—41no—
41Q8×C101—41no—
42D4×C101—41no—
43D4×C101—41no—
44D4×C101—41no—
45D4×C101—44no—
46C2×C4×C101—21no—
47C2×C4×C101C2×D412yes—
48C2×C4×C101C2×D411yescharacteristic
49C2×Dic101—22no—
50C2×Dic101—22no—
51C2×Dic101—81no—
52Dic202—81no—
53C8×C101—22no—
54#11717579—42no—
55C22×C101—41no—
56C22×C101—42no—
57C22×C101—81no—
58C22×C101—82no—
59C22×C101#12011yescharacteristic
60C4×C101—21no—
61C4×C101—22no—
62C4×C101—41no—
63Dic101—42no—
64Dic101—42no—
65C2×C101—81no—
66C2×C101—161no—
67C2×C101#45311yescharacteristic
68C2×C101#45412yes—
69#1657—1011nomaximal, Sylow
70C101#165711yescharacteristic, complemented, Sylow
71#563—1011no—
72#496D10111yescharacteristic
73#490—1012no—
74#330—1012no—
75#329—1011no—
76C2×SD16—21no—
77C2×D8—21no—
78C4⋊Q8—2021no—
79#127C2×D10111yescharacteristic
80#126—2022no—
81#125—1012no—
82#124C2×D10111yescharacteristic
83#123—2021no—
84C22⋊Q8—2022no—
85#102—2022no—
86#102—2022no—
87#97—1012no—
88#97C2×D10111yescharacteristic
89C2×Q8—2021no—
90C2×Q8C22⋊D10111yescharacteristic
91C2×D4—22no—
92C2×D4—21no—
93C2×D4C22⋊D10111yescharacteristic
94C22×C4—2021no—
95SD16—42no—
96D8—42no—
97C2×C8—21no—
98C2×C8—2022no—
99C4⋊C4—1012no—
100C4⋊C4—1012no—
101C4⋊C4—2022no—
102C4⋊C4—2022no—
103C4⋊C4—2022no—
104C22⋊C4—22no—
105C22⋊C4—2022no—
106C22⋊C4—2021no—
107C42—2021no—
108C42C22×D10111yescharacteristic
109C23—21no—
110C23—41no—
111Q8—41no—
112Q8—8081no—
113D4—41no—
114D4—41no—
115D4—41no—
116D4—44no—
117C2×C4—21no—
118C2×C4—8081no—
119C2×C4—2022no—
120C2×C4—2022no—
121C2×C4D4×D10112yes—
122C2×C4C2×C22⋊D10111yescharacteristic
123C8—22no—
124C8—4042no—
125C22—41no—
126C22—42no—
127C22—81no—
128C22—82no—
129C22#1569222311yescharacteristic
130C4—21no—
131C4—22no—
132C4—41no—
133C4—4042no—
134C4—4042no—
135C2—81no—
136C2—161no—
137C2#2026534411yescharacteristic
138C2#2026534812yes—
subgroupG/H
centerC22#15692223
commutator, cosocleC42×C101C23
FrattiniC42C22×D101
Fitting, prime radical#496×C101C2
socleC22×C101#120
hypercenter#496D101
hypocenterC101#1657
LCSG ▷ C42×C101 ▷ C22×C101 ▷ C101
UCSC1 ◁ C22 ◁ C42 ◁ #496

Complements

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

NH ≅ G/Ntype
1. #20265927C2semidirect
2. #20265541C2semidirect
3. #20264918C2semidirect
4. #20264917C2semidirect
70. C101#1657semidirect

Maximal subgroups

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

subgroupindex
#202659272
#202655412
#202649182
#202649172
#496×C1012
#1657101