Finite Group Databasebeta

← #27173207

Subgroups — #27173207

Order 12928.

170 conjugacy classes of subgroups; only the 43 normal subgroups are shown.

normal subgroups only

This group has 170 conjugacy classes of subgroups — too many to draw. Only its 43 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#20265615C211yesmaximal, characteristic, complemented
2#20265613C211yesmaximal, characteristic, complemented
3#20265552C211yesmaximal, characteristic, complemented
4#20265293C211yesmaximal, characteristic, complemented
5#20264997C211yesmaximal, characteristic, complemented
6#20264989C211yesmaximal, characteristic, complemented
7#361×C101C211yesmaximal, characteristic, complemented
8#15692277C2211yescharacteristic
9C2×D808C2211yescharacteristic
10C2×#13108059—21no—
11#15692263—21no—
12#15692251—21no—
13#15692200—21no—
14#15692199—21no—
15D1616—21no—
16#15692196—21no—
17#15692195C2211yescharacteristic
18#15692181—21no—
19C8×Dic101C2211yescharacteristic
20C2×D8×C101C2211yescharacteristic
21M32×C101C2211yescharacteristic
22#108×C101C2211yescharacteristic
23#13108074—21no—
24C2×D404—21no—
25C2×C22⋊D101—42no—
26Dic404D411yescharacteristic, complemented
27#13108063—21no—
28D808—21no—
29D808—21no—
30D808D411yescharacteristic, complemented
31#13108059—41no—
32#13108059—41no—
33#13108057—41no—
34C2×#11717579D411yescharacteristic
35C8×D101—41no—
36C4×Dic101D411yescharacteristic
37#13108049—21no—
38C2×D4×C101—21no—
39D8×C101—21no—
40D8×C101—21no—
41M16×C101—21no—
42C2×C8×C101C2311yescharacteristic
43C16×C101—21no—
44C23×C101—41no—
45C22×D101—41no—
46D4×C101—41no—
47D4×C101—41no—
48C2×C4×C101C2×D411yescharacteristic
49C22⋊D101—41no—
50C22⋊D101—84no—
51C2×Dic101—21no—
52D404—21no—
53D404—41no—
54D404—41no—
55C4×D101—41no—
56Dic202—21no—
57C8×C101—41no—
58C8×C101C2×D411yescharacteristic
59C8×C101#4111yescharacteristic
60#11717579D811yescharacteristic
61#11717579D811yescharacteristic
62C22×C101—82no—
63C22×C101—81no—
64C22×C101AGL(1,ℤ/8)11yescharacteristic
65C2×D101—41no—
66C2×D101—82no—
67C2×D101—81no—
68C4×C101#12111yescharacteristic
69C4×C101C2×D811yescharacteristic
70Dic101—22no—
71Dic101—41no—
72C2×C101—21no—
73C2×C101—161no—
74C2×C101#45911yescharacteristic
75D101—81no—
76D101—161no—
77#2213—1011nomaximal, Sylow
78C101#221311yescharacteristic, complemented, Sylow
79#509—1011no—
80#509—1011no—
81#493—1011no—
82#443—1011no—
83#364—1011no—
84#361—1011no—
85#361D10111yescharacteristic, complemented
86#135—1011no—
87C2×D8—1011no—
88C2×D8—2021no—
89C2×D8C2×D10111yescharacteristic
90#131—2021no—
91#127—2021no—
92SD32—2021no—
93SD32—2021no—
94D16—2021no—
95D16—2021no—
96M32—1011no—
97M32C2×D10111yescharacteristic
98#108C2×D10111yescharacteristic
99#104—2021no—
100C4×C8—1011no—
101#41—2021no—
102C2×D4—21no—
103C2×D4—4042no—
104C2×D4—2021no—
105Q16—1011no—
106SD16—2021no—
107D8—21no—
108D8—21no—
109D8—1011no—
110D8—4041no—
111D8—4041no—
112D8—2021no—
113D8—2021no—
114M16—21no—
115M16—4041no—
116C2×C8—1011no—
117C2×C8—4041no—
118C2×C8C22×D10111yescharacteristic
119C42—1011no—
120C16—21no—
121C16—2021no—
122C23—41no—
123C23—4041no—
124Q8—2021no—
125D4—41no—
126D4—41no—
127D4—8084no—
128D4—4041no—
129D4—4041no—
130D4—4041no—
131D4—2021no—
132C2×C4—4041no—
133C2×C4—2021no—
134C2×C4D4×D10111yescharacteristic
135C8—41no—
136C8—1011no—
137C8—1011no—
138C8D4×D10111yescharacteristic
139C8#1310807411yescharacteristic
140C22—82no—
141C22—81no—
142C22—8082no—
143C22—8081no—
144C22—4041no—
145C22#1569228211yescharacteristic
146C4—4041no—
147C4—2022no—
148C4#1569222811yescharacteristic
149C4D8×D10111yescharacteristic
150C2—21no—
151C2—161no—
152C2—8081no—
153C2—16161no—
154C2#2026537111yescharacteristic
subgroupG/H
centerC2#20265371
commutator, cosocleC2×C8×C101C23
FrattiniC2×C8C22×D101
Fitting#361×C101C2
socleC2×C101#459
hypercenter#361D101
hypocenterC101#2213
LCSG ▷ C2×C8×C101 ▷ C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ #361

Complements

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

NH ≅ G/Ntype
1. #20265615C2semidirect
2. #20265613C2semidirect
3. #20265552C2semidirect
4. #20265293C2semidirect
5. #20264997C2semidirect
6. #20264989C2semidirect
7. #361×C101C2semidirect
26. Dic404D4semidirect
30. D808D4semidirect
78. C101#2213semidirect
85. #361D101semidirect

Maximal subgroups

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

subgroupindex
#202656152
#202656132
#202655522
#202652932
#202649972
#202649892
#361×C1012
#2213101