Finite Group Databasebeta

← C67×#1514 #22555945

Subgroups — C67×#1514 #22555945

Order 8576.

252 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 252 conjugacy classes of subgroups — too many to draw. Only its 88 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×C67C211yesmaximal, characteristic
2#432×C67C211yesmaximal, characteristic, complemented
3#383×C67C211yesmaximal, characteristic, complemented
4#327×C67C214yesmaximal, complemented
5C2×#41×C67—21no—
6C4⋊Q8×C67C411yescharacteristic, complemented
7#127×C67C411yescharacteristic, complemented
8#124×C67—21no—
9#121×C67—21no—
10#121×C67C2212yes—
11C4×D4×C67—21no—
12C4×D4×C67C412yescomplemented
13C2×C42×C67C2211yescharacteristic
14C4⋊C8×C67—21no—
15C22⋊C8×C67C2212yes—
16#97×C67—21no—
17#95×C67—21no—
18#95×C67—21no—
19#95×C67C2212yes—
20#41×C67—44no—
21C2×Q8×C67—21no—
22C2×D4×C67—22no—
23C2×D4×C67—21no—
24C2×D4×C67—21no—
25C2×D4×C67C2×C412yes—
26C22×C4×C67—22no—
27C22×C4×C67—22no—
28C22×C4×C67D411yescharacteristic
29C22×C4×C67D411yescharacteristic
30C22×C4×C67C2311yescharacteristic
31C2×C8×C67—22no—
32C4⋊C4×C67—21no—
33C4⋊C4×C67C2×C412yes—
34C22⋊C4×C67—22no—
35C22⋊C4×C67—22no—
36C42×C67C2×C411yescharacteristic
37C42×C67C2×C411yescharacteristic
38C42×C67D411yescharacteristic
39C42×C67D411yescharacteristic
40C23×C67—22no—
41C23×C67C2×D411yescharacteristic
42Q8×C67—42no—
43D4×C67—42no—
44D4×C67—42no—
45D4×C67—42no—
46D4×C67—44no—
47C2×C4×C67—22no—
48C2×C4×C67—22no—
49C2×C4×C67—41no—
50C2×C4×C67—41no—
51C2×C4×C67—44no—
52C2×C4×C67—42no—
53C2×C4×C67—42no—
54C2×C4×C67—42no—
55C2×C4×C67C22⋊C411yescharacteristic
56C2×C4×C67C22⋊C411yescharacteristic
57C2×C4×C67C22⋊C411yescharacteristic
58C2×C4×C67C22⋊C411yescharacteristic
59C2×C4×C67C22×C411yescharacteristic
60C2×C4×C67C2×D411yescharacteristic
61C8×C67—42no—
62C22×C67—21no—
63C22×C67—21no—
64C22×C67—42no—
65C22×C67—42no—
66C22×C67—42no—
67C22×C67C2×C22⋊C411yescharacteristic
68C22×C67AGL(1,ℤ/8)12yes—
69C4×C67—21no—
70C4×C67—21no—
71C4×C67—21no—
72C4×C67—21no—
73C4×C67—21no—
74C4×C67—21no—
75C4×C67—42no—
76C4×C67—82no—
77C2×C67—22no—
78C2×C67—82no—
79C2×C67#41011yescharacteristic
80C2×C67#41811yescharacteristic
81C2×C67#42111yescharacteristic
82#1514C6711yesmaximal, characteristic, direct product factor, Sylow
83C67#151411yescharacteristic, direct product factor, Sylow
84#532C2×C6711yescharacteristic
85#432C2×C6711yescharacteristic, complemented
86#383C2×C6711yescharacteristic, complemented
87#327C2×C6714yescomplemented
88C2×#41—21no—
89C4⋊Q8C4×C6711yescharacteristic, complemented
90#127C4×C6711yescharacteristic, complemented
91#124—21no—
92#121—21no—
93#121C22×C6712yes—
94C4×D4—21no—
95C4×D4C4×C6712yescomplemented
96C2×C42C22×C6711yescharacteristic
97C4⋊C8—21no—
98C22⋊C8C22×C6712yes—
99#97—21no—
100#95—21no—
101#95—21no—
102#95C22×C6712yes—
103#41—44no—
104C2×Q8—21no—
105C2×D4—22no—
106C2×D4—21no—
107C2×D4—21no—
108C2×D4C2×C4×C6712yes—
109C22×C4—22no—
110C22×C4—22no—
111C22×C4D4×C6711yescharacteristic
112C22×C4D4×C6711yescharacteristic
113C22×C4C23×C6711yescharacteristic
114C2×C8—22no—
115C4⋊C4—21no—
116C4⋊C4C2×C4×C6712yes—
117C22⋊C4—22no—
118C22⋊C4—22no—
119C42C2×C4×C6711yescharacteristic
120C42C2×C4×C6711yescharacteristic
121C42D4×C6711yescharacteristic
122C42D4×C6711yescharacteristic
123C23—22no—
124C23C2×D4×C6711yescharacteristic
125Q8—42no—
126D4—42no—
127D4—42no—
128D4—42no—
129D4—44no—
130C2×C4—22no—
131C2×C4—22no—
132C2×C4—41no—
133C2×C4—41no—
134C2×C4—44no—
135C2×C4—42no—
136C2×C4—42no—
137C2×C4—42no—
138C2×C4C22⋊C4×C6711yescharacteristic
139C2×C4C22⋊C4×C6711yescharacteristic
140C2×C4C22⋊C4×C6711yescharacteristic
141C2×C4C22⋊C4×C6711yescharacteristic
142C2×C4C22×C4×C6711yescharacteristic
143C2×C4C2×D4×C6711yescharacteristic
144C8—42no—
145C22—21no—
146C22—21no—
147C22—42no—
148C22—42no—
149C22—42no—
150C22C2×C22⋊C4×C6711yescharacteristic
151C22AGL(1,ℤ/8)×C6712yes—
152C4—21no—
153C4—21no—
154C4—21no—
155C4—21no—
156C4—21no—
157C4—21no—
158C4—42no—
159C4—82no—
160C2—22no—
161C2—82no—
162C2#410×C6711yescharacteristic
163C2#418×C6711yescharacteristic
164C2#421×C6711yescharacteristic
subgroupG/H
center, socleC22×C67C2×C22⋊C4
commutatorC2×C4C22×C4×C67
Frattini, cosocleC22×C4C23×C67
prime radical#532×C67C2
LCSG ▷ C2×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C67 ◁ C2×C42×C67 ◁ G

Complements

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

NH ≅ G/Ntype
2. #432×C67C2semidirect
3. #383×C67C2semidirect
4. #327×C67C2semidirect
6. C4⋊Q8×C67C4semidirect
7. #127×C67C4semidirect
12. C4×D4×C67C4semidirect
82. #1514C67direct (N × H)
83. C67#1514direct (N × H)
85. #432C2×C67semidirect
86. #383C2×C67semidirect
87. #327C2×C67semidirect
89. C4⋊Q8C4×C67semidirect
90. #127C4×C67semidirect
95. C4×D4C4×C67semidirect

Maximal subgroups

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

subgroupindex
#532×C672
#432×C672
#383×C672
#327×C672
#151467