Finite Group Databasebeta

← #27173115

Subgroups — #27173115

Order 12928.

354 conjugacy classes of subgroups; only the 41 normal subgroups are shown.

normal subgroups only

This group has 354 conjugacy classes of subgroups — too many to draw. Only its 41 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#20265812C211yesmaximal, characteristic, complemented
2#20265364C212yesmaximal, complemented
3#20264970C211yesmaximal, characteristic, complemented
4#20264965C212yesmaximal, complemented
5#453×C101C211yesmaximal, characteristic, complemented
6C2×D4×D101—22no—
7#15692226—22no—
8#15692223—22no—
9#15692223—21no—
10#15692223C2212yes—
11C4×D404—21no—
12#15692170—22no—
13#15692170C2212yescomplemented
14H(5,𝔽2)×C101C2211yescharacteristic, complemented
15AGL(1,ℤ/8)×C101—21no—
16#127×C101C2211yescharacteristic
17#104×C101—21no—
18#100×C101C2211yescharacteristic, complemented
19C23×D101—21no—
20C23×D101D412yescomplemented
21C2×D404—21no—
22C2×C22⋊D101—22no—
23C2×C22⋊D101—22no—
24C2×C22⋊D101—42no—
25D4×D101—48no—
26C2×C4×D101—22no—
27C4⋊Dic101—21no—
28C22⋊Dic101—21no—
29C22⋊Dic101—22no—
30C22⋊Dic101D412yescomplemented
31#13108051—22no—
32#13108051—42no—
33#41×C101—21no—
34#41×C101—41no—
35C2×D4×C101—22no—
36C2×D4×C101—22no—
37C2×D4×C101—41no—
38C2×D4×C101D411yescharacteristic, complemented
39C2×D4×C101C2311yescharacteristic
40SD16×C101—41no—
41D8×C101—41no—
42M16×C101—21no—
43C42×C101D411yescharacteristic, complemented
44C23×C101—22no—
45C23×C101—21no—
46C23×C101C2×D412yes—
47C22×D101—22no—
48C22×D101—44no—
49C22×D101—44no—
50C22×D101—42no—
51C22×D101—81no—
52Q8×C101—21no—
53D4×C101—21no—
54D4×C101—41no—
55D4×C101—41no—
56D4×C101—44no—
57D4×C101—42no—
58D4×C101—41no—
59C2×C4×C101—21no—
60C2×C4×C101—22no—
61C2×C4×C101—41no—
62C2×C4×C101C2×D411yescharacteristic
63C22⋊D101—44no—
64C22⋊D101—44no—
65C22⋊D101—84no—
66C2×Dic101—22no—
67C2×Dic101—42no—
68D404—24no—
69C4×D101—44no—
70C8×C101—41no—
71C22×C101—22no—
72C22×C101—42no—
73C22×C101—42no—
74C22×C101—41no—
75C22×C101—41no—
76C22×C101—82no—
77C22×C101#12011yescharacteristic
78C2×D101—24no—
79C2×D101—42no—
80C2×D101—42no—
81C2×D101—44no—
82C2×D101—82no—
83C2×D101—84no—
84C4×C101—22no—
85C4×C101—21no—
86C4×C101—41no—
87Dic101—42no—
88Dic101—82no—
89C2×C101—21no—
90C2×C101—42no—
91C2×C101—81no—
92C2×C101—81no—
93C2×C101B(4,𝔽2)11yescharacteristic
94D101—44no—
95#2196—1011nomaximal, Sylow
96C101#219611yescharacteristic, complemented, Sylow
97D42—1011no—
98B(4,𝔽2)—1012no—
99#453D10111yescharacteristic, complemented
100#353—1011no—
101AΣL(1,𝔽16)—1012no—
102H(5,𝔽2)C2×D10111yescharacteristic, complemented
103C22×D4—2022no—
104AGL(1,ℤ/8)—21no—
105#127C2×D10111yescharacteristic
106#121—2022no—
107#120—1012no—
108#120—2022no—
109#120—2021no—
110C4×D4—2021no—
111#104—21no—
112#100C2×D10111yescharacteristic, complemented
113#99—1012no—
114#99—2022no—
115C24—1012no—
116C24—2021no—
117#41—21no—
118#41—41no—
119C2×D4—22no—
120C2×D4—22no—
121C2×D4—41no—
122C2×D4—4048no—
123C2×D4—4042no—
124C2×D4—2021no—
125C2×D4—2022no—
126C2×D4—2022no—
127C2×D4C22⋊D10111yescharacteristic, complemented
128C2×D4C22×D10111yescharacteristic
129C22×C4—2022no—
130SD16—41no—
131D8—41no—
132M16—21no—
133C4⋊C4—2021no—
134C22⋊C4—1012no—
135C22⋊C4—4042no—
136C22⋊C4—2022no—
137C22⋊C4—2021no—
138C22⋊C4—2022no—
139C42C22⋊D10111yescharacteristic, complemented
140C23—22no—
141C23—21no—
142C23—8081no—
143C23—4042no—
144C23—4044no—
145C23—4044no—
146C23—2022no—
147C23D4×D10112yes—
148Q8—21no—
149D4—21no—
150D4—41no—
151D4—41no—
152D4—44no—
153D4—42no—
154D4—41no—
155D4—8084no—
156D4—4044no—
157D4—4044no—
158D4—2024no—
159C2×C4—21no—
160C2×C4—22no—
161C2×C4—41no—
162C2×C4—4044no—
163C2×C4—4042no—
164C2×C4—2022no—
165C2×C4C2×C22⋊D10111yescharacteristic
166C8—41no—
167C22—22no—
168C22—42no—
169C22—42no—
170C22—41no—
171C22—41no—
172C22—82no—
173C22—8082no—
174C22—8084no—
175C22—4042no—
176C22—4044no—
177C22—4042no—
178C22—2024no—
179C22#1569222311yescharacteristic
180C4—22no—
181C4—21no—
182C4—41no—
183C4—8082no—
184C4—4042no—
185C2—21no—
186C2—42no—
187C2—81no—
188C2—81no—
189C2—4044no—
190C2#2026536311yescharacteristic
subgroupG/H
centerC2#20265363
commutator, cosocleC2×D4×C101C23
FrattiniC2×D4C22×D101
Fitting#453×C101C2
socleC2×C101B(4,𝔽2)
hypercenter#453D101
hypocenterC101#2196
LCSG ▷ C2×D4×C101 ▷ C22×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #453

Complements

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

NH ≅ G/Ntype
1. #20265812C2semidirect
2. #20265364C2semidirect
3. #20264970C2semidirect
4. #20264965C2semidirect
5. #453×C101C2semidirect
13. #15692170C22semidirect
14. H(5,𝔽2)×C101C22semidirect
18. #100×C101C22semidirect
20. C23×D101D4semidirect
30. C22⋊Dic101D4semidirect
38. C2×D4×C101D4semidirect
43. C42×C101D4semidirect
96. C101#2196semidirect
99. #453D101semidirect
102. H(5,𝔽2)C2×D101semidirect
112. #100C2×D101semidirect
127. C2×D4C22⋊D101semidirect
139. C42C22⋊D101semidirect

Maximal subgroups

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

subgroupindex
#202658122
#202653642
#202649702
#202649652
#453×C1012
#2196101