Finite Group Databasebeta

← #27171531

Subgroups — #27171531

Order 12928.

444 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 97 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#20265652C211yesmaximal, characteristic, complemented
2#20265182C212yesmaximal, complemented
3C2×#15692172C211yesmaximal, characteristic, complemented
4#20264937C212yesmaximal, complemented
5C2×#127×C101C211yesmaximal, characteristic
6C2×#13108058C2212yes—
7D4×Dic101—24no—
8#15692214C2212yes—
9C2×C22⋊Dic101—22no—
10#15692176—24no—
11#15692172—22no—
12C22×D4×C101—22no—
13C22×D4×C101C2211yescharacteristic
14C22×D4×C101C2211yescharacteristic
15#127×C101—22no—
16#127×C101C414yescomplemented
17C2×C42×C101C2211yescharacteristic
18C22×Dic101—24no—
19#13108058—22no—
20#13108058—22no—
21C2×#11717579—22no—
22C4⋊Dic101D414yescomplemented
23C22⋊Dic101—24no—
24C22⋊Dic101—22no—
25C4×Dic101—22no—
26C24×C101—21no—
27C24×C101D412yes—
28C2×D4×C101—28no—
29C2×D4×C101—22no—
30C2×D4×C101—22no—
31C2×D4×C101—21no—
32C2×D4×C101—22no—
33C2×D4×C101—22no—
34C2×D4×C101—48no—
35C2×D4×C101C2×C412yes—
36C2×D4×C101C2×C412yescomplemented
37C2×D4×C101D412yes—
38C22×C4×C101—21no—
39C22×C4×C101C2311yescharacteristic
40C42×C101—21no—
41C42×C101C2×C412yescomplemented
42C23×C101—22no—
43C23×C101—22no—
44C23×C101—22no—
45C23×C101—44no—
46C23×C101—44no—
47C23×C101—41no—
48C23×C101—41no—
49C23×C101C22⋊C414yes—
50C23×C101C2×D411yescharacteristic
51D4×C101—24no—
52D4×C101—24no—
53D4×C101—24no—
54D4×C101—416no—
55D4×C101—42no—
56C2×C4×C101—24no—
57C2×C4×C101—22no—
58C2×C4×C101C22×C411yescharacteristic
59C2×C4×C101C2×D412yes—
60C2×C4×C101C2×D411yescharacteristic
61C2×C4×C101#4112yes—
62C2×Dic101—24no—
63C2×Dic101—84no—
64#11717579—42no—
65C22×C101—22no—
66C22×C101—24no—
67C22×C101—44no—
68C22×C101—44no—
69C22×C101—44no—
70C22×C101—42no—
71C22×C101—82no—
72C22×C101—82no—
73C22×C101—82no—
74C22×C101C2×C22⋊C411yescharacteristic
75C22×C101#12011yescharacteristic
76C22×C101#12311yescharacteristic
77C4×C101—24no—
78C4×C101C4×D412yes—
79C4×C101#12112yes—
80Dic101—44no—
81C2×C101—21no—
82C2×C101—21no—
83C2×C101—44no—
84C2×C101—82no—
85C2×C101#38611yescharacteristic
86C2×C101#45312yes—
87#1887—1011nomaximal, Sylow
88C101#188711yescharacteristic, complemented, Sylow
89C2×#127D10111yescharacteristic
90#518—1011no—
91#418—1012no—
92C2×#100—1011no—
93#339—1012no—
94C22×D4—22no—
95C22×D4C2×D10111yescharacteristic
96C22×D4C2×D10111yescharacteristic
97C2×M16—1012no—
98#127—22no—
99#127Dic10114yescomplemented
100C4×D4—2024no—
101#117—1012no—
102C2×C22⋊C4—2022no—
103C2×C42C2×D10111yescharacteristic
104#102—2024no—
105#100—2022no—
106C24—21no—
107C24C22⋊D10112yes—
108C2×D4—28no—
109C2×D4—22no—
110C2×D4—22no—
111C2×D4—21no—
112C2×D4—22no—
113C2×D4—22no—
114C2×D4—48no—
115C2×D4C2×Dic10112yes—
116C2×D4C2×Dic10112yescomplemented
117C2×D4C22⋊D10112yes—
118C22×C4—21no—
119C22×C4—2024no—
120C22×C4C22×D10111yescharacteristic
121M16—2022no—
122M16—2022no—
123C2×C8—2022no—
124C4⋊C4—1014no—
125C22⋊C4—2024no—
126C22⋊C4—2022no—
127C42—21no—
128C42—2022no—
129C42C2×Dic10112yescomplemented
130C23—22no—
131C23—22no—
132C23—22no—
133C23—44no—
134C23—44no—
135C23—41no—
136C23—41no—
137C23C22⋊Dic10114yes—
138C23C2×C22⋊D10111yescharacteristic
139D4—24no—
140D4—24no—
141D4—24no—
142D4—416no—
143D4—42no—
144C2×C4—24no—
145C2×C4—22no—
146C2×C4—8084no—
147C2×C4—2024no—
148C2×C4C22×Dic10111yescharacteristic
149C2×C4D4×D10112yes—
150C2×C4C2×C22⋊D10111yescharacteristic
151C2×C4#1310807312yes—
152C8—4042no—
153C22—22no—
154C22—24no—
155C22—44no—
156C22—44no—
157C22—44no—
158C22—42no—
159C22—82no—
160C22—82no—
161C22—82no—
162C22C2×C22⋊Dic10111yescharacteristic
163C22#1569222311yescharacteristic
164C22#1569223611yescharacteristic
165C4—24no—
166C4—4044no—
167C4D4×Dic10112yes—
168C4#1569222712yes—
169C2—21no—
170C2—21no—
171C2—44no—
172C2—82no—
173C2#2026505711yescharacteristic
174C2#2026534412yes—
subgroupG/H
centerC22#15692223
commutatorC2×C4×C101C22×C4
FrattiniC22×C4C22×D101
Fitting, prime radicalC2×#127×C101C2
socleC22×C101#120
cosocleC22×C4×C101C23
hypercenterC2×#127D101
hypocenterC101#1887
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#127

Complements

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

NH ≅ G/Ntype
1. #20265652C2semidirect
2. #20265182C2semidirect
3. C2×#15692172C2semidirect
4. #20264937C2semidirect
16. #127×C101C4semidirect
22. C4⋊Dic101D4semidirect
36. C2×D4×C101C2×C4semidirect
41. C42×C101C2×C4semidirect
88. C101#1887semidirect
99. #127Dic101semidirect
116. C2×D4C2×Dic101semidirect
129. C42C2×Dic101semidirect

Maximal subgroups

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

subgroupindex
#202656522
#202651822
C2×#156921722
#202649372
C2×#127×C1012
#1887101