Finite Group Databasebeta

← #27173132

Subgroups — #27173132

Order 12928.

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

normal subgroups only

This group has 270 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#20265919C211yesmaximal, characteristic, complemented
2#20265364C212yesmaximal, complemented
3#20264972C211yesmaximal, characteristic, complemented
4#20264965C212yesmaximal, complemented
5#455×C101C211yesmaximal, characteristic, complemented
6#15692244—21no—
7#15692243—21no—
8#15692241—22no—
9#15692225—22no—
10#15692223—22no—
11#15692223C2212yes—
12#15692170—22no—
13#15692170C2212yes—
14H(5,𝔽2)×C101C2211yescharacteristic
15#137×C101—21no—
16#124×C101C2211yescharacteristic
17#104×C101—21no—
18#100×C101C2211yescharacteristic
19C23×D101D412yescomplemented
20C2×Dic202—21no—
21C2×D404—21no—
22C2×D404—21no—
23C2×C22⋊D101—22no—
24C2×C22⋊D101—42no—
25C22⋊Dic101—22no—
26C22⋊Dic101D412yescomplemented
27#13108051—22no—
28#13108051—22no—
29#13108051—22no—
30#13108051—42no—
31C4×Dic101—21no—
32#41×C101—21no—
33#41×C101—41no—
34C2×Q8×C101D411yescharacteristic, complemented
35C2×D4×C101—22no—
36C2×D4×C101—41no—
37C2×D4×C101C2311yescharacteristic
38Q16×C101—41no—
39SD16×C101—41no—
40M16×C101—21no—
41C22⋊C4×C101—22no—
42C42×C101D411yescharacteristic, complemented
43C23×C101—22no—
44C23×C101C2×D412yes—
45C22×D101—22no—
46C22×D101—42no—
47C22×D101—82no—
48Q8×C101—21no—
49Q8×C101—41no—
50D4×C101—21no—
51D4×C101—41no—
52D4×C101—42no—
53D4×C101—41no—
54C2×C4×C101—22no—
55C2×C4×C101—21no—
56C2×C4×C101—21no—
57C2×C4×C101—41no—
58C2×C4×C101C2×D411yescharacteristic
59C22⋊D101—82no—
60C22⋊D101—84no—
61C2×Dic101—22no—
62C2×Dic101—42no—
63D404—81no—
64D404—81no—
65Dic202—81no—
66C8×C101—41no—
67C22×C101—22no—
68C22×C101—42no—
69C22×C101—41no—
70C22×C101—82no—
71C22×C101#12011yescharacteristic
72C2×D101—42no—
73C2×D101—42no—
74C2×D101—44no—
75C2×D101—42no—
76C2×D101—82no—
77C4×C101—21no—
78C4×C101—41no—
79C4×C101—41no—
80C4×C101—41no—
81Dic101—42no—
82Dic101—82no—
83C2×C101—21no—
84C2×C101—42no—
85C2×C101—81no—
86C2×C101B(4,𝔽2)11yescharacteristic
87D101—82no—
88#2199—1011nomaximal, Sylow
89C101#219911yescharacteristic, complemented, Sylow
90H(3,𝔽4)—1011no—
91B(4,𝔽2)—1012no—
92#455D10111yescharacteristic, complemented
93#354—1011no—
94AΣL(1,𝔽16)—1012no—
95H(5,𝔽2)C2×D10111yescharacteristic
96#137—21no—
97#124—2022no—
98#124—2021no—
99#124—2021no—
100#124C2×D10111yescharacteristic
101#120—1012no—
102#120—2022no—
103#120—2022no—
104#104—21no—
105#100C2×D10111yescharacteristic
106#99—1012no—
107#99—2022no—
108C24—1012no—
109#41—21no—
110#41—41no—
111C2×Q8—2021no—
112C2×Q8C22⋊D10111yescharacteristic, complemented
113C2×D4—22no—
114C2×D4—41no—
115C2×D4—4042no—
116C2×D4—2021no—
117C2×D4—2021no—
118C2×D4—2022no—
119C2×D4C22×D10111yescharacteristic
120Q16—41no—
121SD16—41no—
122M16—21no—
123C22⋊C4—22no—
124C22⋊C4—1012no—
125C22⋊C4—4042no—
126C22⋊C4—2022no—
127C22⋊C4—2022no—
128C22⋊C4—2022no—
129C22⋊C4—2022no—
130C42—2021no—
131C42C22⋊D10111yescharacteristic, complemented
132C23—22no—
133C23—8082no—
134C23—4042no—
135C23—2022no—
136C23D4×D10112yes—
137Q8—21no—
138Q8—41no—
139Q8—8081no—
140D4—21no—
141D4—41no—
142D4—42no—
143D4—41no—
144D4—8081no—
145D4—8081no—
146D4—8084no—
147D4—8082no—
148C2×C4—22no—
149C2×C4—21no—
150C2×C4—21no—
151C2×C4—41no—
152C2×C4—4042no—
153C2×C4—2022no—
154C2×C4C2×C22⋊D10111yescharacteristic
155C8—41no—
156C22—22no—
157C22—42no—
158C22—41no—
159C22—82no—
160C22—8082no—
161C22—4042no—
162C22—4042no—
163C22—4044no—
164C22—4042no—
165C22#1569222311yescharacteristic
166C4—21no—
167C4—41no—
168C4—41no—
169C4—41no—
170C4—8082no—
171C4—4042no—
172C2—21no—
173C2—42no—
174C2—81no—
175C2—8082no—
176C2#2026536311yescharacteristic
subgroupG/H
centerC2#20265363
commutator, cosocleC2×D4×C101C23
FrattiniC2×D4C22×D101
Fitting#455×C101C2
socleC2×C101B(4,𝔽2)
hypercenter#455D101
hypocenterC101#2199
LCSG ▷ C2×D4×C101 ▷ C22×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #455

Complements

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

NH ≅ G/Ntype
1. #20265919C2semidirect
2. #20265364C2semidirect
3. #20264972C2semidirect
4. #20264965C2semidirect
5. #455×C101C2semidirect
19. C23×D101D4semidirect
26. C22⋊Dic101D4semidirect
34. C2×Q8×C101D4semidirect
42. C42×C101D4semidirect
89. C101#2199semidirect
92. #455D101semidirect
112. C2×Q8C22⋊D101semidirect
131. C42C22⋊D101semidirect

Maximal subgroups

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

subgroupindex
#202659192
#202653642
#202649722
#202649652
#455×C1012
#2199101