Finite Group Databasebeta

← #27171570

Subgroups — #27171570

Order 12928.

412 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 412 conjugacy classes of subgroups — too many to draw. Only its 109 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
1C2×#15692229C211yesmaximal, characteristic
2C2×#15692177C212yesmaximal, complemented
3#20265037C211yesmaximal, characteristic, complemented
4#20264940C212yesmaximal, complemented
5C22×M16×C101C211yesmaximal, characteristic, complemented
6C22×D404C2211yescharacteristic
7C22×C22⋊D101—21no—
8#15692229—22no—
9#15692229C414yescomplemented
10C2×C4⋊Dic101C2212yes—
11C2×C4⋊Dic101C2211yescharacteristic
12C2×C22⋊Dic101—21no—
13C2×#13108051—21no—
14#15692177—24no—
15#15692162—21no—
16C23×C4×C101C2211yescharacteristic
17C2×M16×C101—22no—
18C2×M16×C101—24no—
19C22×C8×C101C2212yes—
20C23×D101—21no—
21C2×D404—22no—
22C2×D404—22no—
23C2×D404—21no—
24C2×D404C2×C412yes—
25C2×C22⋊D101—24no—
26C2×C22⋊D101—44no—
27C22×Dic101—21no—
28C22×Dic101—22no—
29C4⋊Dic101—21no—
30C4⋊Dic101—22no—
31C4⋊Dic101—22no—
32C4⋊Dic101C2×C412yes—
33C22⋊Dic101—42no—
34#13108051—24no—
35C24×C101D411yescharacteristic
36C22×C4×C101—22no—
37C22×C4×C101—22no—
38C22×C4×C101C2×C412yes—
39C22×C4×C101D411yescharacteristic
40C22×C4×C101D412yes—
41C22×C4×C101C2311yescharacteristic
42M16×C101—28no—
43C2×C8×C101—22no—
44C2×C8×C101—22no—
45C2×C8×C101D414yescomplemented
46C23×C101—22no—
47C23×C101—21no—
48C23×C101—22no—
49C23×C101—21no—
50C23×C101C22⋊C412yes—
51C23×C101C2×D411yescharacteristic
52C22×D101—22no—
53C22×D101—42no—
54C22×D101—42no—
55C22×D101—41no—
56C22×D101—41no—
57C2×C4×C101—22no—
58C2×C4×C101—24no—
59C2×C4×C101—22no—
60C2×C4×C101—22no—
61C2×C4×C101C22⋊C412yes—
62C2×C4×C101C22×C411yescharacteristic
63C2×C4×C101C2×D412yes—
64C2×C4×C101C2×D411yescharacteristic
65C2×C4×C101#4112yes—
66C22⋊D101—48no—
67C2×Dic101—22no—
68C2×Dic101—42no—
69C2×Dic101—42no—
70C2×Dic101—42no—
71C2×Dic101—42no—
72D404—24no—
73D404—41no—
74D404—41no—
75C8×C101—24no—
76C22×C101—22no—
77C22×C101—22no—
78C22×C101—22no—
79C22×C101—42no—
80C22×C101—42no—
81C22×C101C2×C22⋊C411yescharacteristic
82C22×C101#12112yes—
83C22×C101AGL(1,ℤ/8)11yescharacteristic
84C22×C101AGL(1,ℤ/8)12yes—
85C22×C101#13711yescharacteristic
86C2×D101—42no—
87C2×D101—42no—
88C2×D101—42no—
89C2×D101—82no—
90C2×D101—82no—
91C4×C101—22no—
92C4×C101C4×D412yes—
93C4×C101#12011yescharacteristic
94C4×C101#12311yescharacteristic
95Dic101—42no—
96Dic101—82no—
97C2×C101—42no—
98C2×C101#38611yescharacteristic
99C2×C101#41711yescharacteristic
100C2×C101#41811yescharacteristic
101C2×C101#46812yes—
102C2×C101#46912yes—
103D101—82no—
104#1895—1011nomaximal, Sylow
105C101#189511yescharacteristic, complemented, Sylow
106C22×M16D10111yescharacteristic, complemented
107C2×#121—1011no—
108C2×#102—1012no—
109#380—1011no—
110#340—1012no—
111C22×D4—1011no—
112C22×D4—2021no—
113C23×C4C2×D10111yescharacteristic
114C2×M16—22no—
115C2×M16—24no—
116C22×C8C2×D10112yes—
117#121—1014no—
118#121—2022no—
119C2×C4⋊C4—1011no—
120C2×C4⋊C4—1012no—
121C2×C22⋊C4—2021no—
122C2×C22⋊C4—2021no—
123#102—2024no—
124#95—2021no—
125C24—2021no—
126C24D40411yescharacteristic
127C2×D4—1012no—
128C2×D4—4044no—
129C2×D4—2024no—
130C2×D4—2021no—
131C2×D4—2022no—
132C2×D4—2022no—
133C22×C4—22no—
134C22×C4—22no—
135C22×C4—2022no—
136C22×C4—2021no—
137C22×C4C4×D10112yes—
138C22×C4D40411yescharacteristic
139C22×C4C22⋊D10112yes—
140C22×C4C22×D10111yescharacteristic
141M16—28no—
142C2×C8—22no—
143C2×C8—22no—
144C2×C8C22⋊D10114yescomplemented
145C4⋊C4—1012no—
146C4⋊C4—2022no—
147C4⋊C4—2022no—
148C4⋊C4—2021no—
149C22⋊C4—4042no—
150C22⋊C4—2024no—
151C23—21no—
152C23—22no—
153C23—21no—
154C23—22no—
155C23—4041no—
156C23—4042no—
157C23—4042no—
158C23—4041no—
159C23—2022no—
160C23#1310805112yes—
161C23C2×D40411yescharacteristic
162D4—4048no—
163D4—4041no—
164D4—4041no—
165D4—2024no—
166C2×C4—22no—
167C2×C4—24no—
168C2×C4—22no—
169C2×C4—22no—
170C2×C4—4042no—
171C2×C4—4042no—
172C2×C4—4042no—
173C2×C4—4042no—
174C2×C4—2022no—
175C2×C4#1310805112yes—
176C2×C4C2×C4×D10111yescharacteristic
177C2×C4C2×C22⋊D10112yes—
178C2×C4C2×C22⋊D10111yescharacteristic
179C2×C4#1310807412yes—
180C8—24no—
181C22—22no—
182C22—22no—
183C22—22no—
184C22—42no—
185C22—42no—
186C22—8082no—
187C22—8082no—
188C22—4042no—
189C22—4042no—
190C22—4042no—
191C22C2×#1310805111yescharacteristic
192C22#1569222912yes—
193C22#1569228312yes—
194C22#1569228311yescharacteristic
195C22#1569228811yescharacteristic
196C4—22no—
197C4—8082no—
198C4—4042no—
199C4C4×C22⋊D10112yes—
200C4#1569222411yescharacteristic
201C4#1569223811yescharacteristic
202C2—42no—
203C2—8082no—
204C2#2026505611yescharacteristic
205C2#2026517911yescharacteristic
206C2#2026518311yescharacteristic
207C2#2026542312yes—
208C2#2026543012yes—
subgroupG/H
centerC23C2×D404
commutatorC2×C4×C101C22×C4
FrattiniC22×C4C22×D101
FittingC22×M16×C101C2
socleC23×C101C2×D4
prime radicalC2×#15692229C2
cosocleC22×C4×C101C23
hypercenterC22×M16D101
hypocenterC101#1895
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×M16

Complements

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

NH ≅ G/Ntype
2. C2×#15692177C2semidirect
3. #20265037C2semidirect
4. #20264940C2semidirect
5. C22×M16×C101C2semidirect
9. #15692229C4semidirect
45. C2×C8×C101D4semidirect
105. C101#1895semidirect
106. C22×M16D101semidirect
144. C2×C8C22⋊D101semidirect

Maximal subgroups

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

subgroupindex
C2×#156922292
C2×#156921772
#202650372
#202649402
C22×M16×C1012
#1895101