Finite Group Databasebeta

← #27170436

Subgroups — #27170436

Order 12928.

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

normal subgroups only

This group has 228 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#20265919C211yesmaximal, characteristic
2#20265539C212yesmaximal, complemented
3#20264917C211yesmaximal, characteristic, complemented
4#20264916C212yesmaximal, complemented
5#497×C101C211yesmaximal, characteristic, complemented
6#15692244—21no—
7#15692243C2212yes—
8#15692241—22no—
9#15692225—22no—
10#15692223—21no—
11#15692178—22no—
12#15692175—22no—
13#15692166C2212yescomplemented
14C2×Q16×C101—21no—
15C2×SD16×C101—21no—
16C4⋊Q8×C101C2211yescharacteristic, complemented
17#124×C101C2211yescharacteristic
18#97×C101C2211yescharacteristic, complemented
19C23×D101—21no—
20C2×Dic202D412yes—
21C2×D404—21no—
22C2×D404D412yes—
23C2×C22⋊D101—22no—
24C2×#11717579—22no—
25C22⋊Dic101—21no—
26#13108051—22no—
27#13108051—22no—
28#13108051—22no—
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—
40C22×D101—22no—
41C22×D101—81no—
42C22×D101—82no—
43Q8×C101—41no—
44Q8×C101—41no—
45Q8×C101—41no—
46D4×C101—41no—
47C2×C4×C101—21no—
48C2×C4×C101—41no—
49C2×C4×C101C2×D412yes—
50C2×C4×C101C2×D411yescharacteristic
51C22⋊D101—82no—
52C2×Dic101—22no—
53D404—42no—
54D404—81no—
55Dic202—42no—
56C8×C101—22no—
57#11717579—42no—
58C22×C101—41no—
59C22×C101—42no—
60C22×C101#12011yescharacteristic
61C2×D101—42no—
62C2×D101—42no—
63C2×D101—42no—
64C2×D101—84no—
65C4×C101—21no—
66C4×C101—22no—
67C4×C101—41no—
68C4×C101—81no—
69Dic101—42no—
70C2×C101—81no—
71C2×C101#45311yescharacteristic
72C2×C101#45512yes—
73D101—82no—
74#1659—1011nomaximal, Sylow
75C101#165911yescharacteristic, complemented, Sylow
76H(3,𝔽4)—1011no—
77#497D10111yescharacteristic, complemented
78#489—1012no—
79#329—1011no—
80#329—1012no—
81C2×Q16—21no—
82C2×SD16—21no—
83C4⋊Q8C2×D10111yescharacteristic, complemented
84#124—1012no—
85#124—2022no—
86#124—2021no—
87#124C2×D10111yescharacteristic
88#120—2022no—
89#120—2021no—
90#103—2022no—
91#102—2022no—
92#97—1012no—
93#97C2×D10111yescharacteristic, complemented
94C24—2021no—
95C2×Q8—21no—
96C2×Q8—1012no—
97C2×Q8C22⋊D10111yescharacteristic
98C2×D4—1012no—
99C2×D4—2021no—
100C2×D4—2022no—
101C2×D4C22⋊D10111yescharacteristic
102Q16—42no—
103SD16—42no—
104C2×C8—21no—
105C2×C8—2022no—
106C4⋊C4—22no—
107C22⋊C4—22no—
108C22⋊C4—2022no—
109C22⋊C4—2021no—
110C22⋊C4—2022no—
111C22⋊C4—2022no—
112C42—2021no—
113C42C22×D10111yescharacteristic
114C23—21no—
115C23—8081no—
116C23—8082no—
117C23—2022no—
118Q8—41no—
119Q8—41no—
120Q8—41no—
121Q8—4042no—
122D4—41no—
123D4—8081no—
124D4—8082no—
125D4—4042no—
126C2×C4—21no—
127C2×C4—41no—
128C2×C4—2022no—
129C2×C4D4×D10112yes—
130C2×C4C2×C22⋊D10111yescharacteristic
131C8—22no—
132C8—4042no—
133C22—41no—
134C22—42no—
135C22—8084no—
136C22—4042no—
137C22—4042no—
138C22—4042no—
139C22#1569222311yescharacteristic
140C4—21no—
141C4—22no—
142C4—41no—
143C4—81no—
144C4—4042no—
145C2—81no—
146C2—8082no—
147C2#2026534411yescharacteristic
148C2#2026535312yes—
subgroupG/H
centerC22#15692223
commutator, cosocleC42×C101C23
FrattiniC42C22×D101
Fitting#497×C101C2
socleC22×C101#120
prime radical#20265919C2
hypercenter#497D101
hypocenterC101#1659
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. #20265539C2semidirect
3. #20264917C2semidirect
4. #20264916C2semidirect
5. #497×C101C2semidirect
13. #15692166C22semidirect
16. C4⋊Q8×C101C22semidirect
18. #97×C101C22semidirect
75. C101#1659semidirect
77. #497D101semidirect
83. C4⋊Q8C2×D101semidirect
93. #97C2×D101semidirect

Maximal subgroups

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

subgroupindex
#202659192
#202655392
#202649172
#202649162
#497×C1012
#1659101