Finite Group Databasebeta

← #27173100

Subgroups — #27173100

Order 12928.

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

normal subgroups only

This group has 216 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
2C2×#15692198C211yesmaximal, characteristic, complemented
3#20265449C211yesmaximal, characteristic, complemented
4#20264991C211yesmaximal, characteristic, complemented
5#20264984C211yesmaximal, characteristic, complemented
6#20264962C211yesmaximal, characteristic, complemented
7#510×C101C211yesmaximal, characteristic, complemented
8#15692301—21no—
9#15692286—21no—
10#15692282—22no—
11#15692280—21no—
12#15692278—21no—
13C2×D808C2211yescharacteristic
14D8×D101—21no—
15#15692261C2211yescharacteristic
16#15692198—22no—
17C2×#13108049C2211yescharacteristic, complemented
18#15692191C2211yescharacteristic, complemented
19#15692174—21no—
20#135×C101C2211yescharacteristic
21C2×Q16×C101C2211yescharacteristic, complemented
22Q32×C101—21no—
23SD32×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#13108064—41no—
33#13108061—22no—
34#13108061—41no—
35D808—21no—
36D808D412yes—
37#13108059—22no—
38#13108058—21no—
39#13108058D411yescharacteristic
40#13108057—21no—
41C8×D101—21no—
42#13108049—21no—
43#41×C101—21no—
44C2×Q8×C101—21no—
45Q16×C101—21no—
46Q16×C101—21no—
47Q16×C101D411yescharacteristic
48SD16×C101—21no—
49D8×C101D411yescharacteristic
50C2×C8×C101C2311yescharacteristic
51C16×C101—21no—
52C22×D101—42no—
53C22×D101—41no—
54Q8×C101—21no—
55Q8×C101—41no—
56Q8×C101—41no—
57D4×C101—21no—
58D4×C101—41no—
59C2×C4×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×C101AGL(1,ℤ/8)11yescharacteristic
77C2×D101—21no—
78C2×D101—42no—
79C2×D101—162no—
80C2×D101—161no—
81C4×C101—41no—
82C4×C101—81no—
83C4×C101#12011yescharacteristic
84C4×C101C2×D811yescharacteristic
85Dic101—21no—
86C2×C101—21no—
87C2×C101—81no—
88C2×C101#44711yescharacteristic
89D101—41no—
90D101—82no—
91#2194—1011nomaximal, Sylow
92C101#219411yescharacteristic, complemented, Sylow
93#576—1011no—
94#510D10111yescharacteristic, complemented
95C2×SD32—1011no—
96#473—1011no—
97#361—1011no—
98#359—1011no—
99#350—1011no—
100H(5,𝔽2)—2021no—
101#137—2021no—
102AGL(1,ℤ/8)—2021no—
103AGL(1,ℤ/8)—2022no—
104#135—2021no—
105#135C2×D10111yescharacteristic
106C2×Q16C2×D10111yescharacteristic, complemented
107C2×D8—1011no—
108C2×D8—2021no—
109#131—1011no—
110Q32—21no—
111SD32—21no—
112SD32—2022no—
113M32C2×D10111yescharacteristic, complemented
114C2×C16—1011no—
115#108—1011no—
116#101—2021no—
117#41—21no—
118#41—1011no—
119#41—4041no—
120#41—2022no—
121C2×Q8—21no—
122C2×D4—4041no—
123C2×D4—4042no—
124C2×D4—2022no—
125C2×D4—2021no—
126Q16—21no—
127Q16—21no—
128Q16—4041no—
129Q16C22⋊D10111yescharacteristic
130SD16—21no—
131SD16—4041no—
132SD16—2022no—
133D8—1012no—
134D8—2022no—
135D8—2021no—
136D8C22⋊D10111yescharacteristic
137M16—1011no—
138M16—2021no—
139M16—2021no—
140C2×C8—2021no—
141C2×C8C22×D10111yescharacteristic
142C16—21no—
143C16—2021no—
144C23—4042no—
145C23—4041no—
146Q8—21no—
147Q8—41no—
148Q8—41no—
149Q8—1011no—
150D4—21no—
151D4—41no—
152D4—1011no—
153D4—4042no—
154D4—4042no—
155D4—4041no—
156D4—2021no—
157D4—2022no—
158C2×C4—41no—
159C2×C4—41no—
160C2×C4—4042no—
161C2×C4—2021no—
162C2×C4D4×D10111yescharacteristic
163C8—4041no—
164C8—2021no—
165C8D4×D10111yescharacteristic
166C8C2×C22⋊D10111yescharacteristic
167C22—41no—
168C22—16161no—
169C22—16162no—
170C22—4042no—
171C22—2021no—
172C22#1569227911yescharacteristic
173C4—41no—
174C4—81no—
175C4—2021no—
176C4#1569222311yescharacteristic
177C4D8×D10111yescharacteristic
178C2—21no—
179C2—81no—
180C2—8082no—
181C2—4041no—
182C2#2026530911yescharacteristic
subgroupG/H
centerC2#20265309
commutator, cosocleC2×C8×C101C23
FrattiniC2×C8C22×D101
Fitting#510×C101C2
socleC2×C101#447
prime radical#20265979C2
hypercenter#510D101
hypocenterC101#2194
LCSG ▷ C2×C8×C101 ▷ C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ #510

Complements

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

NH ≅ G/Ntype
2. C2×#15692198C2semidirect
3. #20265449C2semidirect
4. #20264991C2semidirect
5. #20264984C2semidirect
6. #20264962C2semidirect
7. #510×C101C2semidirect
17. C2×#13108049C22semidirect
18. #15692191C22semidirect
21. C2×Q16×C101C22semidirect
24. M32×C101C22semidirect
92. C101#2194semidirect
94. #510D101semidirect
106. C2×Q16C2×D101semidirect
113. M32C2×D101semidirect

Maximal subgroups

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

subgroupindex
#202659792
C2×#156921982
#202654492
#202649912
#202649842
#202649622
#510×C1012
#2194101