Finite Group Databasebeta

← #27173103

Subgroups — #27173103

Order 12928.

216 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 53 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#20265617C211yesmaximal, characteristic, complemented
2C2×#15692199C211yesmaximal, characteristic, complemented
3#20265452C211yesmaximal, characteristic, complemented
4#20264990C211yesmaximal, characteristic, complemented
5#20264984C211yesmaximal, characteristic, complemented
6#20264961C211yesmaximal, characteristic, complemented
7#576×C101C211yesmaximal, characteristic
8#15692285—21no—
9C2×Dic404C2211yescharacteristic, complemented
10#15692201—21no—
11#15692199—22no—
12#15692199—21no—
13#15692195C2211yescharacteristic, complemented
14C2×#13108049C2211yescharacteristic, complemented
15#15692191C2211yescharacteristic, complemented
16#15692173—21no—
17H(5,𝔽2)×C101—21no—
18AGL(1,ℤ/8)×C101—22no—
19AGL(1,ℤ/8)×C101—21no—
20#135×C101—21no—
21#135×C101C2211yescharacteristic
22C2×D8×C101—21no—
23C2×D8×C101C2211yescharacteristic
24#131×C101C2211yescharacteristic
25C2×Dic202—21no—
26Dic404—21no—
27Dic404—21no—
28#13108064—41no—
29#13108062—41no—
30#13108058—21no—
31#13108049—21no—
32#13108049—21no—
33#41×C101—22no—
34#41×C101—21no—
35#41×C101—41no—
36#41×C101D411yescharacteristic
37C2×D4×C101—22no—
38C2×D4×C101—21no—
39C2×D4×C101—42no—
40C2×D4×C101—41no—
41Q16×C101D411yescharacteristic
42SD16×C101—22no—
43SD16×C101—21no—
44D8×C101—22no—
45D8×C101—21no—
46D8×C101D411yescharacteristic
47D8×C101D412yes—
48M16×C101—21no—
49M16×C101D411yescharacteristic
50C2×C8×C101—21no—
51C2×C8×C101C2311yescharacteristic
52C23×C101—42no—
53C23×C101—41no—
54Q8×C101—21no—
55Q8×C101D811yescharacteristic
56D4×C101—21no—
57D4×C101—21no—
58D4×C101—22no—
59D4×C101—42no—
60D4×C101—42no—
61D4×C101—41no—
62D4×C101—41no—
63D4×C101D811yescharacteristic
64C2×C4×C101—21no—
65C2×C4×C101—42no—
66C2×C4×C101—41no—
67C2×C4×C101C2×D411yescharacteristic
68C2×Dic101—41no—
69Dic202—41no—
70Dic202—41no—
71C8×C101—21no—
72C8×C101C2×D411yescharacteristic
73C8×C101C2×D411yescharacteristic
74#11717579—41no—
75C22×C101—21no—
76C22×C101—42no—
77C22×C101—41no—
78C22×C101—161no—
79C22×C101—162no—
80C22×C101AGL(1,ℤ/8)11yescharacteristic
81C4×C101—21no—
82C4×C101—41no—
83C4×C101#12011yescharacteristic
84C4×C101C2×D811yescharacteristic
85Dic101—81no—
86C2×C101—21no—
87C2×C101—41no—
88C2×C101—82no—
89C2×C101—81no—
90C2×C101#44711yescharacteristic
91#2194—1011nomaximal, Sylow
92C101#219411yescharacteristic, complemented, Sylow
93#576D10111yescharacteristic
94#510—1011no—
95C2×SD32—1011no—
96#473—1011no—
97#361—1011no—
98#359—1011no—
99#350—1011no—
100H(5,𝔽2)—21no—
101#137—2021no—
102AGL(1,ℤ/8)—22no—
103AGL(1,ℤ/8)—21no—
104#135—21no—
105#135C2×D10111yescharacteristic
106C2×Q16—1011no—
107C2×D8—21no—
108C2×D8C2×D10111yescharacteristic
109#131C2×D10111yescharacteristic
110Q32—2021no—
111SD32—2022no—
112SD32—2021no—
113M32—1011no—
114C2×C16—1011no—
115#108—1011no—
116#101—2021no—
117#41—22no—
118#41—21no—
119#41—41no—
120#41C22⋊D10111yescharacteristic
121C2×Q8—2021no—
122C2×D4—22no—
123C2×D4—21no—
124C2×D4—42no—
125C2×D4—41no—
126Q16—4041no—
127Q16—2021no—
128Q16—2021no—
129Q16C22⋊D10111yescharacteristic
130SD16—22no—
131SD16—21no—
132SD16—4041no—
133D8—22no—
134D8—21no—
135D8C22⋊D10111yescharacteristic
136D8C22⋊D10112yes—
137M16—21no—
138M16—2021no—
139M16C22⋊D10111yescharacteristic
140C2×C8—21no—
141C2×C8C22×D10111yescharacteristic
142C16—2021no—
143C16—2021no—
144C23—42no—
145C23—41no—
146Q8—21no—
147Q8—4041no—
148Q8—4041no—
149Q8#1310805911yescharacteristic
150D4—21no—
151D4—21no—
152D4—22no—
153D4—42no—
154D4—42no—
155D4—41no—
156D4—41no—
157D4#1310805911yescharacteristic
158C2×C4—21no—
159C2×C4—42no—
160C2×C4—41no—
161C2×C4—4041no—
162C2×C4C2×C22⋊D10111yescharacteristic
163C8—21no—
164C8—4041no—
165C8C2×C22⋊D10111yescharacteristic
166C8C2×C22⋊D10111yescharacteristic
167C22—21no—
168C22—42no—
169C22—41no—
170C22—161no—
171C22—162no—
172C22#1569228111yescharacteristic
173C4—21no—
174C4—41no—
175C4—8081no—
176C4#1569222411yescharacteristic
177C4C2×#1310805911yescharacteristic
178C2—21no—
179C2—41no—
180C2—82no—
181C2—81no—
182C2#2026531111yescharacteristic
subgroupG/H
centerC2#20265311
commutator, cosocleC2×C8×C101C23
FrattiniC2×C8C22×D101
Fitting, prime radical#576×C101C2
socleC2×C101#447
hypercenter#576D101
hypocenterC101#2194
LCSG ▷ C2×C8×C101 ▷ C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C2 ◁ C2×C4 ◁ #131 ◁ #576

Complements

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

NH ≅ G/Ntype
1. #20265617C2semidirect
2. C2×#15692199C2semidirect
3. #20265452C2semidirect
4. #20264990C2semidirect
5. #20264984C2semidirect
6. #20264961C2semidirect
9. C2×Dic404C22semidirect
13. #15692195C22semidirect
14. C2×#13108049C22semidirect
15. #15692191C22semidirect
92. C101#2194semidirect

Maximal subgroups

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

subgroupindex
#202656172
C2×#156921992
#202654522
#202649902
#202649842
#202649612
#576×C1012
#2194101