Finite Group Databasebeta

← #27170430

Subgroups — #27170430

Order 12928.

206 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 206 conjugacy classes of subgroups — too many to draw. Only its 49 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#20265567C211yesmaximal, characteristic
2#20265543C212yesmaximal, complemented
3#20264919C212yesmaximal, complemented
4#20264917C211yesmaximal, characteristic, complemented
5#563×C101C211yesmaximal, characteristic, complemented
6C2×#13108061—21no—
7C2×#13108059—21no—
8#15692252C2211yescharacteristic
9#15692175—22no—
10#15692175—22no—
11#15692166C2212yes—
12#15692166C2211yescharacteristic
13C4⋊Q8×C101—21no—
14#126×C101—22no—
15#125×C101C2212yes—
16#124×C101C2211yescharacteristic
17#123×C101—21no—
18C22⋊Q8×C101—22no—
19C2×D404—22no—
20C2×D404—21no—
21#13108061—42no—
22#13108059—42no—
23C2×#11717579—21no—
24C2×#11717579—22no—
25C2×Q8×C101—21no—
26C2×Q8×C101D411yescharacteristic
27C2×D4×C101D411yescharacteristic
28C22×C4×C101—21no—
29C4⋊C4×C101—22no—
30C4⋊C4×C101—22no—
31C4⋊C4×C101—22no—
32C4⋊C4×C101D412yes—
33C4⋊C4×C101D412yes—
34C22⋊C4×C101—22no—
35C22⋊C4×C101—21no—
36C22⋊C4×C101—22no—
37C42×C101—21no—
38C42×C101C2311yescharacteristic
39C23×C101—21no—
40C22×D101—41no—
41Q8×C101—41no—
42Q8×C101—81no—
43D4×C101—41no—
44C2×C4×C101—22no—
45C2×C4×C101—21no—
46C2×C4×C101—22no—
47C2×C4×C101—81no—
48C2×C4×C101C2×D412yes—
49C2×C4×C101C2×D411yescharacteristic
50D404—41no—
51D404—41no—
52D404—44no—
53#11717579—22no—
54#11717579—42no—
55C22×C101—41no—
56C22×C101—42no—
57C22×C101#12011yescharacteristic
58C2×D101—81no—
59C2×D101—82no—
60C4×C101—21no—
61C4×C101—22no—
62C4×C101—42no—
63C4×C101—41no—
64C4×C101—42no—
65C2×C101—81no—
66C2×C101#45311yescharacteristic
67C2×C101#45412yes—
68D101—161no—
69#1657—1011nomaximal, Sylow
70C101#165711yescharacteristic, complemented, Sylow
71#563D10111yescharacteristic, complemented
72#496—1011no—
73#490—1012no—
74#330—1012no—
75#329—1011no—
76C2×SD16—2021no—
77C2×D8—2021no—
78C4⋊Q8—21no—
79#127—1011no—
80#126—22no—
81#125C2×D10112yes—
82#124C2×D10111yescharacteristic
83#123—21no—
84C22⋊Q8—22no—
85#102—2022no—
86#102—2022no—
87#97—1012no—
88#97—1011no—
89C2×Q8—21no—
90C2×Q8C22⋊D10111yescharacteristic
91C2×D4—2022no—
92C2×D4—2021no—
93C2×D4C22⋊D10111yescharacteristic
94C22×C4—21no—
95SD16—4042no—
96D8—4042no—
97C2×C8—2022no—
98C2×C8—2021no—
99C4⋊C4—22no—
100C4⋊C4—22no—
101C4⋊C4—22no—
102C4⋊C4C22⋊D10112yes—
103C4⋊C4C22⋊D10112yes—
104C22⋊C4—22no—
105C22⋊C4—21no—
106C22⋊C4—22no—
107C42—21no—
108C42C22×D10111yescharacteristic
109C23—21no—
110C23—4041no—
111Q8—41no—
112Q8—81no—
113D4—41no—
114D4—4041no—
115D4—4041no—
116D4—4044no—
117C2×C4—22no—
118C2×C4—21no—
119C2×C4—22no—
120C2×C4—81no—
121C2×C4C2×C22⋊D10112yes—
122C2×C4C2×C22⋊D10111yescharacteristic
123C8—4042no—
124C8—2022no—
125C22—41no—
126C22—42no—
127C22—8081no—
128C22—8082no—
129C22#1569222411yescharacteristic
130C4—21no—
131C4—22no—
132C4—42no—
133C4—41no—
134C4—42no—
135C2—81no—
136C2—16161no—
137C2#2026534511yescharacteristic
138C2#2026535012yes—
subgroupG/H
centerC22#15692224
commutator, cosocleC42×C101C23
FrattiniC42C22×D101
Fitting#563×C101C2
socleC22×C101#120
prime radical#20265567C2
hypercenter#563D101
hypocenterC101#1657
LCSG ▷ C42×C101 ▷ C22×C101 ▷ C101
UCSC1 ◁ C22 ◁ C42 ◁ #563

Complements

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

NH ≅ G/Ntype
2. #20265543C2semidirect
3. #20264919C2semidirect
4. #20264917C2semidirect
5. #563×C101C2semidirect
70. C101#1657semidirect
71. #563D101semidirect

Maximal subgroups

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

subgroupindex
#202655672
#202655432
#202649192
#202649172
#563×C1012
#1657101