Finite Group Databasebeta

← #1547×C211 #41009167

Subgroups — #1547×C211 #41009167

Order 27008.

260 conjugacy classes of subgroups; only the 108 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 108 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#532×C211C211yesmaximal, characteristic
2#423×C211C211yesmaximal, characteristic, complemented
3C2×C4⋊C8×C211C211yesmaximal, characteristic, complemented
4#332×C211C212yesmaximal, complemented
5#331×C211C212yesmaximal, complemented
6C2×#41×C211—21no—
7C2×M16×C211—21no—
8C22×C8×C211—21no—
9C4⋊Q8×C211C2211yescharacteristic
10#127×C211C2211yescharacteristic
11#124×C211—21no—
12#121×C211—21no—
13#121×C211C412yes—
14C4×D4×C211—21no—
15C4×D4×C211C412yes—
16C2×C42×C211C2211yescharacteristic
17C4⋊C8×C211—21no—
18C4⋊C8×C211—21no—
19C4⋊C8×C211C2212yes—
20C4⋊C8×C211C2212yes—
21#41×C211—42no—
22#41×C211—42no—
23C2×Q8×C211—21no—
24C2×D4×C211—22no—
25C2×D4×C211—21no—
26C2×D4×C211—21no—
27C2×D4×C211C2×C412yes—
28C22×C4×C211—22no—
29C22×C4×C211C2×C411yescharacteristic
30C22×C4×C211D411yescharacteristic
31C22×C4×C211D411yescharacteristic
32M16×C211—42no—
33C2×C8×C211—22no—
34C2×C8×C211—22no—
35C2×C8×C211—42no—
36C4⋊C4×C211—21no—
37C4⋊C4×C211C2×C412yes—
38C22⋊C4×C211—22no—
39C22⋊C4×C211—22no—
40C42×C211C2×C411yescharacteristic
41C42×C211D411yescharacteristic
42C42×C211D411yescharacteristic
43C42×C211C2311yescharacteristic
44C23×C211—22no—
45C23×C211C22⋊C411yescharacteristic
46Q8×C211—41no—
47Q8×C211—41no—
48D4×C211—42no—
49D4×C211—44no—
50D4×C211—41no—
51D4×C211—41no—
52D4×C211—41no—
53D4×C211—41no—
54C2×C4×C211—21no—
55C2×C4×C211—21no—
56C2×C4×C211—22no—
57C2×C4×C211—41no—
58C2×C4×C211—44no—
59C2×C4×C211C22⋊C411yescharacteristic
60C2×C4×C211C22⋊C411yescharacteristic
61C2×C4×C211C22⋊C411yescharacteristic
62C2×C4×C211D812yes—
63C2×C4×C211SD1612yes—
64C2×C4×C211C22×C411yescharacteristic
65C2×C4×C211C2×D411yescharacteristic
66C2×C4×C211C2×D411yescharacteristic
67C8×C211—42no—
68C8×C211—42no—
69C22×C211—21no—
70C22×C211—21no—
71C22×C211—42no—
72C22×C211—42no—
73C22×C211—42no—
74C22×C211#10212yes—
75C22×C211C2×C22⋊C411yescharacteristic
76C4×C211—21no—
77C4×C211—21no—
78C4×C211—21no—
79C4×C211—42no—
80C4×C211#10212yes—
81C4×C211C2×D811yescharacteristic
82C4×C211C2×SD1611yescharacteristic
83C4×C211AGL(1,ℤ/8)11yescharacteristic
84C4×C211#13711yescharacteristic
85C2×C211—22no—
86C2×C211—82no—
87C2×C211#41311yescharacteristic
88C2×C211C2×#10211yescharacteristic
89C2×C211#41711yescharacteristic
90C211#154711yescharacteristic, direct product factor, Sylow
91#1547C21111yesmaximal, characteristic, direct product factor, Sylow
92#532C2×C21111yescharacteristic
93#423C2×C21111yescharacteristic, complemented
94C2×C4⋊C8C2×C21111yescharacteristic, complemented
95#332C2×C21112yescomplemented
96#331C2×C21112yescomplemented
97C2×#41—21no—
98C2×M16—21no—
99C22×C8—21no—
100C4⋊Q8C22×C21111yescharacteristic
101#127C22×C21111yescharacteristic
102#124—21no—
103#121—21no—
104#121C4×C21112yes—
105C4×D4—21no—
106C4×D4C4×C21112yes—
107C2×C42C22×C21111yescharacteristic
108C4⋊C8—21no—
109C4⋊C8—21no—
110C4⋊C8C22×C21112yes—
111C4⋊C8C22×C21112yes—
112#41—42no—
113#41—42no—
114C2×Q8—21no—
115C2×D4—22no—
116C2×D4—21no—
117C2×D4—21no—
118C2×D4C2×C4×C21112yes—
119C22×C4—22no—
120C22×C4C2×C4×C21111yescharacteristic
121C22×C4D4×C21111yescharacteristic
122C22×C4D4×C21111yescharacteristic
123M16—42no—
124C2×C8—22no—
125C2×C8—22no—
126C2×C8—42no—
127C4⋊C4—21no—
128C4⋊C4C2×C4×C21112yes—
129C22⋊C4—22no—
130C22⋊C4—22no—
131C42C2×C4×C21111yescharacteristic
132C42D4×C21111yescharacteristic
133C42D4×C21111yescharacteristic
134C42C23×C21111yescharacteristic
135C23—22no—
136C23C22⋊C4×C21111yescharacteristic
137Q8—41no—
138Q8—41no—
139D4—42no—
140D4—44no—
141D4—41no—
142D4—41no—
143D4—41no—
144D4—41no—
145C2×C4—21no—
146C2×C4—21no—
147C2×C4—22no—
148C2×C4—41no—
149C2×C4—44no—
150C2×C4C22⋊C4×C21111yescharacteristic
151C2×C4C22⋊C4×C21111yescharacteristic
152C2×C4C22⋊C4×C21111yescharacteristic
153C2×C4D8×C21112yes—
154C2×C4SD16×C21112yes—
155C2×C4C22×C4×C21111yescharacteristic
156C2×C4C2×D4×C21111yescharacteristic
157C2×C4C2×D4×C21111yescharacteristic
158C8—42no—
159C8—42no—
160C22—21no—
161C22—21no—
162C22—42no—
163C22—42no—
164C22—42no—
165C22#102×C21112yes—
166C22C2×C22⋊C4×C21111yescharacteristic
167C4—21no—
168C4—21no—
169C4—21no—
170C4—42no—
171C4#102×C21112yes—
172C4C2×D8×C21111yescharacteristic
173C4C2×SD16×C21111yescharacteristic
174C4AGL(1,ℤ/8)×C21111yescharacteristic
175C4#137×C21111yescharacteristic
176C2—22no—
177C2—82no—
178C2#413×C21111yescharacteristic
179C2C2×#102×C21111yescharacteristic
180C2#417×C21111yescharacteristic
subgroupG/H
center, socleC22×C211C2×C22⋊C4
commutatorC2×C4C22×C4×C211
Frattini, cosocleC42C23×C211
prime radical#532×C211C2
LCSG ▷ C2×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C211 ◁ C2×C42×C211 ◁ G

Complements

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

NH ≅ G/Ntype
2. #423×C211C2semidirect
3. C2×C4⋊C8×C211C2semidirect
4. #332×C211C2semidirect
5. #331×C211C2semidirect
90. C211#1547direct (N × H)
91. #1547C211direct (N × H)
93. #423C2×C211semidirect
94. C2×C4⋊C8C2×C211semidirect
95. #332C2×C211semidirect
96. #331C2×C211semidirect

Maximal subgroups

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

subgroupindex
#532×C2112
#423×C2112
C2×C4⋊C8×C2112
#332×C2112
#331×C2112
#1547211