Finite Group Databasebeta

← #27173133

Subgroups — #27173133

Order 12928.

270 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 270 conjugacy classes of subgroups — too many to draw. Only its 45 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#20265363C212yesmaximal, complemented
2#20265354C211yesmaximal, characteristic, complemented
3#20264972C211yesmaximal, characteristic, complemented
4#20264964C212yesmaximal, complemented
5H(3,𝔽4)×C101C211yesmaximal, characteristic, complemented
6#15692300C2211yescharacteristic
7#15692286—21no—
8#15692224—22no—
9#15692183—21no—
10#15692172C2211yescharacteristic
11#15692170C2212yes—
12#15692169—22no—
13#124×C101—21no—
14#124×C101—22no—
15#124×C101—21no—
16#124×C101C2211yescharacteristic
17#120×C101—22no—
18#120×C101C2212yes—
19#13108074—21no—
20#13108073—41no—
21C2×C22⋊D101—22no—
22C2×C22⋊D101—42no—
23D4×D101—41no—
24#13108064—41no—
25#13108061—41no—
26#13108058—21no—
27C22⋊Dic101—22no—
28C22⋊Dic101—42no—
29C24×C101D412yescomplemented
30C2×Q8×C101—21no—
31C2×Q8×C101D411yescharacteristic, complemented
32C2×D4×C101—21no—
33C2×D4×C101—21no—
34C2×D4×C101—22no—
35C2×D4×C101C2311yescharacteristic
36C22⋊C4×C101—22no—
37C22⋊C4×C101—22no—
38C22⋊C4×C101—22no—
39C22⋊C4×C101—22no—
40C22⋊C4×C101D412yescomplemented
41C42×C101—21no—
42C42×C101D411yescharacteristic, complemented
43C23×C101—22no—
44C23×C101—42no—
45C23×C101—82no—
46C23×C101C2×D412yes—
47C22×D101—22no—
48Q8×C101—41no—
49Q8×C101—81no—
50D4×C101—41no—
51D4×C101—82no—
52D4×C101—81no—
53D4×C101—81no—
54C2×C4×C101—21no—
55C2×C4×C101—22no—
56C2×C4×C101—21no—
57C2×C4×C101C2×D411yescharacteristic
58C22⋊D101—42no—
59C22⋊D101—41no—
60C22⋊D101—84no—
61C2×Dic101—22no—
62C2×Dic101—42no—
63D404—21no—
64C4×D101—41no—
65Dic202—21no—
66#11717579—41no—
67C22×C101—22no—
68C22×C101—42no—
69C22×C101—42no—
70C22×C101—42no—
71C22×C101—44no—
72C22×C101—42no—
73C22×C101—82no—
74C22×C101#12011yescharacteristic
75C2×D101—41no—
76C2×D101—82no—
77C4×C101—21no—
78C4×C101—42no—
79C4×C101—41no—
80C4×C101—41no—
81Dic101—41no—
82Dic101—82no—
83C2×C101—21no—
84C2×C101—42no—
85C2×C101—82no—
86C2×C101B(4,𝔽2)11yescharacteristic
87D101—81no—
88#2199—1011nomaximal, Sylow
89C101#219911yescharacteristic, complemented, Sylow
90H(3,𝔽4)D10111yescharacteristic, complemented
91B(4,𝔽2)—1012no—
92#455—1011no—
93#354—1011no—
94AΣL(1,𝔽16)—1012no—
95H(5,𝔽2)—1011no—
96#137—2021no—
97#124—21no—
98#124—22no—
99#124—21no—
100#124C2×D10111yescharacteristic
101#120—22no—
102#120—2022no—
103#120C2×D10112yes—
104#104—2021no—
105#100—1011no—
106#99—1012no—
107#99—2022no—
108C24C22⋊D10112yescomplemented
109#41—4041no—
110#41—2021no—
111C2×Q8—21no—
112C2×Q8C22⋊D10111yescharacteristic, complemented
113C2×D4—21no—
114C2×D4—21no—
115C2×D4—22no—
116C2×D4—4041no—
117C2×D4—4042no—
118C2×D4—2022no—
119C2×D4C22×D10111yescharacteristic
120Q16—4041no—
121SD16—4041no—
122M16—2021no—
123C22⋊C4—22no—
124C22⋊C4—22no—
125C22⋊C4—22no—
126C22⋊C4—22no—
127C22⋊C4—4042no—
128C22⋊C4—2022no—
129C22⋊C4C22⋊D10112yescomplemented
130C42—21no—
131C42C22⋊D10111yescharacteristic, complemented
132C23—22no—
133C23—42no—
134C23—82no—
135C23—2022no—
136C23C2×C22⋊D10112yes—
137Q8—41no—
138Q8—81no—
139Q8—2021no—
140D4—41no—
141D4—82no—
142D4—81no—
143D4—81no—
144D4—8084no—
145D4—4042no—
146D4—4041no—
147D4—2021no—
148C2×C4—21no—
149C2×C4—22no—
150C2×C4—21no—
151C2×C4—4041no—
152C2×C4—4042no—
153C2×C4—2022no—
154C2×C4C2×C22⋊D10111yescharacteristic
155C8—4041no—
156C22—22no—
157C22—42no—
158C22—42no—
159C22—42no—
160C22—44no—
161C22—42no—
162C22—82no—
163C22—8082no—
164C22—4041no—
165C22#1569222411yescharacteristic
166C4—21no—
167C4—42no—
168C4—41no—
169C4—41no—
170C4—8082no—
171C4—4041no—
172C2—21no—
173C2—42no—
174C2—82no—
175C2—8081no—
176C2#2026536411yescharacteristic
subgroupG/H
centerC2#20265364
commutator, cosocleC2×D4×C101C23
FrattiniC2×D4C22×D101
FittingH(3,𝔽4)×C101C2
socleC2×C101B(4,𝔽2)
hypercenterH(3,𝔽4)D101
hypocenterC101#2199
LCSG ▷ C2×D4×C101 ▷ C22×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ H(3,𝔽4)

Complements

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

NH ≅ G/Ntype
1. #20265363C2semidirect
2. #20265354C2semidirect
3. #20264972C2semidirect
4. #20264964C2semidirect
5. H(3,𝔽4)×C101C2semidirect
29. C24×C101D4semidirect
31. C2×Q8×C101D4semidirect
40. C22⋊C4×C101D4semidirect
42. C42×C101D4semidirect
89. C101#2199semidirect
90. H(3,𝔽4)D101semidirect
108. C24C22⋊D101semidirect
112. C2×Q8C22⋊D101semidirect
129. C22⋊C4C22⋊D101semidirect
131. C42C22⋊D101semidirect

Maximal subgroups

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

subgroupindex
#202653632
#202653542
#202649722
#202649642
H(3,𝔽4)×C1012
#2199101