Finite Group Databasebeta

← #1887×C211 #41009507

Subgroups — #1887×C211 #41009507

Order 27008.

444 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 112 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×#127×C211C211yesmaximal, characteristic
2#518×C211C211yesmaximal, characteristic, complemented
3#418×C211C212yesmaximal, complemented
4C2×#100×C211C211yesmaximal, characteristic, complemented
5#339×C211C212yesmaximal, complemented
6C22×D4×C211—22no—
7C22×D4×C211C2211yescharacteristic
8C22×D4×C211C2211yescharacteristic
9C2×M16×C211C2212yes—
10#127×C211—22no—
11#127×C211C414yescomplemented
12C4×D4×C211—24no—
13#117×C211C2212yes—
14C2×C22⋊C4×C211—22no—
15C2×C42×C211C2211yescharacteristic
16#102×C211—24no—
17#100×C211—22no—
18C24×C211—21no—
19C24×C211D412yes—
20C2×D4×C211—28no—
21C2×D4×C211—22no—
22C2×D4×C211—22no—
23C2×D4×C211—21no—
24C2×D4×C211—22no—
25C2×D4×C211—22no—
26C2×D4×C211—48no—
27C2×D4×C211C2×C412yes—
28C2×D4×C211C2×C412yescomplemented
29C2×D4×C211D412yes—
30C22×C4×C211—21no—
31C22×C4×C211—24no—
32C22×C4×C211C2311yescharacteristic
33M16×C211—22no—
34M16×C211—22no—
35C2×C8×C211—22no—
36C4⋊C4×C211D414yescomplemented
37C22⋊C4×C211—24no—
38C22⋊C4×C211—22no—
39C42×C211—21no—
40C42×C211—22no—
41C42×C211C2×C412yescomplemented
42C23×C211—22no—
43C23×C211—22no—
44C23×C211—22no—
45C23×C211—44no—
46C23×C211—44no—
47C23×C211—41no—
48C23×C211—41no—
49C23×C211C22⋊C414yes—
50C23×C211C2×D411yescharacteristic
51D4×C211—24no—
52D4×C211—24no—
53D4×C211—24no—
54D4×C211—416no—
55D4×C211—42no—
56C2×C4×C211—24no—
57C2×C4×C211—22no—
58C2×C4×C211—24no—
59C2×C4×C211—84no—
60C2×C4×C211C22×C411yescharacteristic
61C2×C4×C211C2×D412yes—
62C2×C4×C211C2×D411yescharacteristic
63C2×C4×C211#4112yes—
64C8×C211—42no—
65C22×C211—22no—
66C22×C211—24no—
67C22×C211—44no—
68C22×C211—44no—
69C22×C211—44no—
70C22×C211—42no—
71C22×C211—82no—
72C22×C211—82no—
73C22×C211—82no—
74C22×C211C2×C22⋊C411yescharacteristic
75C22×C211#12011yescharacteristic
76C22×C211#12311yescharacteristic
77C4×C211—24no—
78C4×C211—44no—
79C4×C211C4×D412yes—
80C4×C211#12112yes—
81C2×C211—21no—
82C2×C211—21no—
83C2×C211—44no—
84C2×C211—82no—
85C2×C211#38611yescharacteristic
86C2×C211#45312yes—
87C211#188711yescharacteristic, direct product factor, Sylow
88#1887C21111yesmaximal, characteristic, direct product factor, Sylow
89C2×#127C2×C21111yescharacteristic
90#518C2×C21111yescharacteristic, complemented
91#418C2×C21112yescomplemented
92C2×#100C2×C21111yescharacteristic, complemented
93#339C2×C21112yescomplemented
94C22×D4—22no—
95C22×D4C22×C21111yescharacteristic
96C22×D4C22×C21111yescharacteristic
97C2×M16C22×C21112yes—
98#127—22no—
99#127C4×C21114yescomplemented
100C4×D4—24no—
101#117C22×C21112yes—
102C2×C22⋊C4—22no—
103C2×C42C22×C21111yescharacteristic
104#102—24no—
105#100—22no—
106C24—21no—
107C24D4×C21112yes—
108C2×D4—28no—
109C2×D4—22no—
110C2×D4—22no—
111C2×D4—21no—
112C2×D4—22no—
113C2×D4—22no—
114C2×D4—48no—
115C2×D4C2×C4×C21112yes—
116C2×D4C2×C4×C21112yescomplemented
117C2×D4D4×C21112yes—
118C22×C4—21no—
119C22×C4—24no—
120C22×C4C23×C21111yescharacteristic
121M16—22no—
122M16—22no—
123C2×C8—22no—
124C4⋊C4D4×C21114yescomplemented
125C22⋊C4—24no—
126C22⋊C4—22no—
127C42—21no—
128C42—22no—
129C42C2×C4×C21112yescomplemented
130C23—22no—
131C23—22no—
132C23—22no—
133C23—44no—
134C23—44no—
135C23—41no—
136C23—41no—
137C23C22⋊C4×C21114yes—
138C23C2×D4×C21111yescharacteristic
139D4—24no—
140D4—24no—
141D4—24no—
142D4—416no—
143D4—42no—
144C2×C4—24no—
145C2×C4—22no—
146C2×C4—24no—
147C2×C4—84no—
148C2×C4C22×C4×C21111yescharacteristic
149C2×C4C2×D4×C21112yes—
150C2×C4C2×D4×C21111yescharacteristic
151C2×C4#41×C21112yes—
152C8—42no—
153C22—22no—
154C22—24no—
155C22—44no—
156C22—44no—
157C22—44no—
158C22—42no—
159C22—82no—
160C22—82no—
161C22—82no—
162C22C2×C22⋊C4×C21111yescharacteristic
163C22#120×C21111yescharacteristic
164C22#123×C21111yescharacteristic
165C4—24no—
166C4—44no—
167C4C4×D4×C21112yes—
168C4#121×C21112yes—
169C2—21no—
170C2—21no—
171C2—44no—
172C2—82no—
173C2#386×C21111yescharacteristic
174C2#453×C21112yes—
subgroupG/H
center, socleC22×C211#120
commutatorC2×C4C22×C4×C211
Frattini, cosocleC22×C4C23×C211
prime radicalC2×#127×C211C2
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C211 ◁ C22×C4×C211 ◁ G

Complements

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

NH ≅ G/Ntype
2. #518×C211C2semidirect
3. #418×C211C2semidirect
4. C2×#100×C211C2semidirect
5. #339×C211C2semidirect
11. #127×C211C4semidirect
28. C2×D4×C211C2×C4semidirect
36. C4⋊C4×C211D4semidirect
41. C42×C211C2×C4semidirect
87. C211#1887direct (N × H)
88. #1887C211direct (N × H)
90. #518C2×C211semidirect
91. #418C2×C211semidirect
92. C2×#100C2×C211semidirect
93. #339C2×C211semidirect
99. #127C4×C211semidirect
116. C2×D4C2×C4×C211semidirect
124. C4⋊C4D4×C211semidirect
129. C42C2×C4×C211semidirect

Maximal subgroups

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

subgroupindex
C2×#127×C2112
#518×C2112
#418×C2112
C2×#100×C2112
#339×C2112
#1887211