Finite Group Databasebeta

← #1553×C211 #41009173

Subgroups — #1553×C211 #41009173

Order 27008.

242 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 242 conjugacy classes of subgroups — too many to draw. Only its 92 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#432×C211C211yesmaximal, characteristic, complemented
3#431×C211C211yesmaximal, characteristic, complemented
4#332×C211C212yesmaximal, complemented
5#331×C211C212yesmaximal, complemented
6C2×#41×C211—21no—
7C4⋊Q8×C211C2211yescharacteristic
8#127×C211C2211yescharacteristic
9#124×C211—21no—
10#121×C211—21no—
11#121×C211C412yes—
12C4×D4×C211—21no—
13C4×D4×C211C412yes—
14C2×C42×C211C2211yescharacteristic
15C4⋊C8×C211C2212yes—
16C4⋊C8×C211C2212yes—
17C22⋊C8×C211—21no—
18C22⋊C8×C211—21no—
19#97×C211—21no—
20C4×C8×C211—21no—
21#41×C211—44no—
22C2×Q8×C211—21no—
23C2×D4×C211—22no—
24C2×D4×C211—21no—
25C2×D4×C211—21no—
26C2×D4×C211C2×C412yes—
27C22×C4×C211—22no—
28C22×C4×C211C2×C411yescharacteristic
29C22×C4×C211D411yescharacteristic
30C22×C4×C211D411yescharacteristic
31C2×C8×C211—22no—
32C2×C8×C211—22no—
33C4⋊C4×C211—21no—
34C4⋊C4×C211C2×C412yes—
35C22⋊C4×C211—22no—
36C22⋊C4×C211—22no—
37C42×C211C2×C411yescharacteristic
38C42×C211D411yescharacteristic
39C42×C211D411yescharacteristic
40C42×C211C2311yescharacteristic
41C23×C211—22no—
42C23×C211C22⋊C411yescharacteristic
43Q8×C211—42no—
44D4×C211—42no—
45D4×C211—44no—
46D4×C211—42no—
47D4×C211—42no—
48C2×C4×C211—22no—
49C2×C4×C211—22no—
50C2×C4×C211—22no—
51C2×C4×C211—41no—
52C2×C4×C211—44no—
53C2×C4×C211C22⋊C411yescharacteristic
54C2×C4×C211C22⋊C411yescharacteristic
55C2×C4×C211C22⋊C411yescharacteristic
56C2×C4×C211C22×C411yescharacteristic
57C2×C4×C211C2×D411yescharacteristic
58C2×C4×C211C2×D411yescharacteristic
59C8×C211—42no—
60C8×C211—42no—
61C22×C211—21no—
62C22×C211—21no—
63C22×C211—21no—
64C22×C211—42no—
65C22×C211—42no—
66C22×C211—42no—
67C22×C211C2×C22⋊C411yescharacteristic
68C4×C211—21no—
69C4×C211—21no—
70C4×C211—21no—
71C4×C211—21no—
72C4×C211—42no—
73C4×C211#13512yes—
74C4×C211AGL(1,ℤ/8)12yes—
75C2×C211—41no—
76C2×C211—82no—
77C2×C211#41311yescharacteristic
78C2×C211#41611yescharacteristic
79C2×C211#41811yescharacteristic
80C211#155311yescharacteristic, direct product factor, Sylow
81#1553C21111yesmaximal, characteristic, direct product factor, Sylow
82#532C2×C21111yescharacteristic
83#432C2×C21111yescharacteristic, complemented
84#431C2×C21111yescharacteristic, complemented
85#332C2×C21112yescomplemented
86#331C2×C21112yescomplemented
87C2×#41—21no—
88C4⋊Q8C22×C21111yescharacteristic
89#127C22×C21111yescharacteristic
90#124—21no—
91#121—21no—
92#121C4×C21112yes—
93C4×D4—21no—
94C4×D4C4×C21112yes—
95C2×C42C22×C21111yescharacteristic
96C4⋊C8C22×C21112yes—
97C4⋊C8C22×C21112yes—
98C22⋊C8—21no—
99C22⋊C8—21no—
100#97—21no—
101C4×C8—21no—
102#41—44no—
103C2×Q8—21no—
104C2×D4—22no—
105C2×D4—21no—
106C2×D4—21no—
107C2×D4C2×C4×C21112yes—
108C22×C4—22no—
109C22×C4C2×C4×C21111yescharacteristic
110C22×C4D4×C21111yescharacteristic
111C22×C4D4×C21111yescharacteristic
112C2×C8—22no—
113C2×C8—22no—
114C4⋊C4—21no—
115C4⋊C4C2×C4×C21112yes—
116C22⋊C4—22no—
117C22⋊C4—22no—
118C42C2×C4×C21111yescharacteristic
119C42D4×C21111yescharacteristic
120C42D4×C21111yescharacteristic
121C42C23×C21111yescharacteristic
122C23—22no—
123C23C22⋊C4×C21111yescharacteristic
124Q8—42no—
125D4—42no—
126D4—44no—
127D4—42no—
128D4—42no—
129C2×C4—22no—
130C2×C4—22no—
131C2×C4—22no—
132C2×C4—41no—
133C2×C4—44no—
134C2×C4C22⋊C4×C21111yescharacteristic
135C2×C4C22⋊C4×C21111yescharacteristic
136C2×C4C22⋊C4×C21111yescharacteristic
137C2×C4C22×C4×C21111yescharacteristic
138C2×C4C2×D4×C21111yescharacteristic
139C2×C4C2×D4×C21111yescharacteristic
140C8—42no—
141C8—42no—
142C22—21no—
143C22—21no—
144C22—21no—
145C22—42no—
146C22—42no—
147C22—42no—
148C22C2×C22⋊C4×C21111yescharacteristic
149C4—21no—
150C4—21no—
151C4—21no—
152C4—21no—
153C4—42no—
154C4#135×C21112yes—
155C4AGL(1,ℤ/8)×C21112yes—
156C2—41no—
157C2—82no—
158C2#413×C21111yescharacteristic
159C2#416×C21111yescharacteristic
160C2#418×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. #432×C211C2semidirect
3. #431×C211C2semidirect
4. #332×C211C2semidirect
5. #331×C211C2semidirect
80. C211#1553direct (N × H)
81. #1553C211direct (N × H)
83. #432C2×C211semidirect
84. #431C2×C211semidirect
85. #332C2×C211semidirect
86. #331C2×C211semidirect

Maximal subgroups

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

subgroupindex
#532×C2112
#432×C2112
#431×C2112
#332×C2112
#331×C2112
#1553211