Finite Group Databasebeta

← #27173086

Subgroups — #27173086

Order 12928.

220 conjugacy classes of subgroups; only the 47 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 47 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#20265979C211yesmaximal, characteristic, complemented
2C2×#15692196C211yesmaximal, characteristic, complemented
3#20265444C211yesmaximal, characteristic, complemented
4#20264991C211yesmaximal, characteristic, complemented
5#20264984C211yesmaximal, characteristic, complemented
6#20264962C211yesmaximal, characteristic, complemented
7#509×C101C211yesmaximal, characteristic, complemented
8#15692301—21no—
9#15692282—22no—
10#15692281—21no—
11#15692280—21no—
12#15692278—21no—
13C2×D808C2211yescharacteristic
14D8×D101—21no—
15#15692261C2211yescharacteristic
16#15692196—22no—
17C2×#13108049C2211yescharacteristic, complemented
18#15692191C2211yescharacteristic, complemented
19#15692172—21no—
20#135×C101C2211yescharacteristic
21C2×D8×C101C2211yescharacteristic, complemented
22SD32×C101—21no—
23D16×C101—21no—
24M32×C101C2211yescharacteristic, complemented
25#13108075—22no—
26#13108074—41no—
27#13108074D411yescharacteristic
28C2×D404—21no—
29C2×D404—41no—
30D4×D101—22no—
31D4×D101—42no—
32#13108062—41no—
33#13108061—22no—
34D808—21no—
35D808D412yes—
36#13108059—22no—
37#13108059—41no—
38#13108058—21no—
39#13108058D411yescharacteristic
40#13108057—21no—
41C8×D101—21no—
42#13108049—21no—
43#41×C101—21no—
44C2×D4×C101—21no—
45Q16×C101D411yescharacteristic
46SD16×C101—21no—
47D8×C101—21no—
48D8×C101—21no—
49D8×C101D411yescharacteristic
50C2×C8×C101C2311yescharacteristic
51C16×C101—21no—
52C23×C101—41no—
53C22×D101—42no—
54C22×D101—41no—
55Q8×C101—21no—
56D4×C101—21no—
57D4×C101—41no—
58D4×C101—41no—
59D4×C101—41no—
60C2×C4×C101—41no—
61C2×C4×C101C2×D411yescharacteristic
62C22⋊D101—21no—
63C22⋊D101—42no—
64D404—22no—
65D404—42no—
66D404—41no—
67D404D811yescharacteristic
68C4×D101—21no—
69C4×D101—42no—
70Dic202D811yescharacteristic
71C8×C101C2×D411yescharacteristic
72C8×C101C2×D411yescharacteristic
73#11717579—21no—
74#11717579—41no—
75C22×C101—41no—
76C22×C101—81no—
77C22×C101—161no—
78C22×C101AGL(1,ℤ/8)11yescharacteristic
79C2×D101—21no—
80C2×D101—42no—
81C2×D101—162no—
82C2×D101—161no—
83C4×C101—41no—
84C4×C101#12011yescharacteristic
85C4×C101C2×D811yescharacteristic
86Dic101—21no—
87C2×C101—21no—
88C2×C101—81no—
89C2×C101—161no—
90C2×C101#44711yescharacteristic
91D101—41no—
92D101—82no—
93#2192—1011nomaximal, Sylow
94C101#219211yescharacteristic, complemented, Sylow
95#576—1011no—
96#509D10111yescharacteristic, complemented
97C2×D16—1011no—
98#472—1011no—
99#361—1011no—
100#359—1011no—
101#350—1011no—
102H(5,𝔽2)—2021no—
103AGL(1,ℤ/8)—2021no—
104AGL(1,ℤ/8)—2022no—
105AGL(1,ℤ/8)—2021no—
106#135—2021no—
107#135C2×D10111yescharacteristic
108C2×D8—1011no—
109C2×D8—2021no—
110C2×D8C2×D10111yescharacteristic, complemented
111#131—1011no—
112SD32—21no—
113D16—21no—
114D16—2022no—
115M32C2×D10111yescharacteristic, complemented
116C2×C16—1011no—
117#108—1011no—
118#100—2021no—
119#41—21no—
120#41—1011no—
121#41—4041no—
122#41—2022no—
123C2×D4—21no—
124C2×D4—4041no—
125C2×D4—4042no—
126C2×D4—2022no—
127C2×D4—2021no—
128Q16C22⋊D10111yescharacteristic
129SD16—21no—
130SD16—4041no—
131SD16—2022no—
132D8—21no—
133D8—21no—
134D8—1012no—
135D8—4041no—
136D8—2022no—
137D8—2021no—
138D8C22⋊D10111yescharacteristic
139M16—1011no—
140M16—2021no—
141M16—2021no—
142C2×C8—2021no—
143C2×C8C22×D10111yescharacteristic
144C16—21no—
145C16—2021no—
146C23—41no—
147C23—4042no—
148C23—4041no—
149Q8—21no—
150Q8—1011no—
151D4—21no—
152D4—41no—
153D4—41no—
154D4—41no—
155D4—1011no—
156D4—4042no—
157D4—4042no—
158D4—4041no—
159D4—2021no—
160D4—2022no—
161C2×C4—41no—
162C2×C4—4042no—
163C2×C4—2021no—
164C2×C4D4×D10111yescharacteristic
165C8—4041no—
166C8—2021no—
167C8D4×D10111yescharacteristic
168C8C2×C22⋊D10111yescharacteristic
169C22—41no—
170C22—81no—
171C22—161no—
172C22—16161no—
173C22—16162no—
174C22—4042no—
175C22—2021no—
176C22#1569227911yescharacteristic
177C4—41no—
178C4—2021no—
179C4#1569222311yescharacteristic
180C4D8×D10111yescharacteristic
181C2—21no—
182C2—81no—
183C2—161no—
184C2—8082no—
185C2—4041no—
186C2#2026530911yescharacteristic
subgroupG/H
centerC2#20265309
commutator, cosocleC2×C8×C101C23
FrattiniC2×C8C22×D101
Fitting#509×C101C2
socleC2×C101#447
hypercenter#509D101
hypocenterC101#2192
LCSG ▷ C2×C8×C101 ▷ C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ #509

Complements

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

NH ≅ G/Ntype
1. #20265979C2semidirect
2. C2×#15692196C2semidirect
3. #20265444C2semidirect
4. #20264991C2semidirect
5. #20264984C2semidirect
6. #20264962C2semidirect
7. #509×C101C2semidirect
17. C2×#13108049C22semidirect
18. #15692191C22semidirect
21. C2×D8×C101C22semidirect
24. M32×C101C22semidirect
94. C101#2192semidirect
96. #509D101semidirect
110. C2×D8C2×D101semidirect
115. M32C2×D101semidirect

Maximal subgroups

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

subgroupindex
#202659792
C2×#156921962
#202654442
#202649912
#202649842
#202649622
#509×C1012
#2192101