Finite Group Databasebeta

← C67×#7506 #30686757

Subgroups — C67×#7506 #30686757

Order 17152.

274 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 274 conjugacy classes of subgroups — too many to draw. Only its 66 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
1C67×#2126C212yesmaximal
2C67×#1963C211yesmaximal, characteristic
3#532×C67C2211yescharacteristic
4#423×C67C411yescharacteristic, complemented
5C2×#104×C67—22no—
6#410×C67—22no—
7#376×C67C411yescharacteristic, complemented
8#354×C67C414yescomplemented
9#353×C67—22no—
10C2×#41×C67D412yes—
11C2×M16×C67—21no—
12C4⋊Q8×C67Q811yescharacteristic
13#127×C67D411yescharacteristic
14#124×C67C2×C412yes—
15#121×C67—41no—
16C4×D4×C67—41no—
17#117×C67—42no—
18C2×C42×C67—21no—
19C2×C42×C67C2×C411yescharacteristic
20C4⋊C8×C67—41no—
21#104×C67—44no—
22#99×C67—24no—
23#99×C67—42no—
24#95×C67—41no—
25#41×C67—42no—
26#41×C67—42no—
27C2×Q8×C67C4⋊C412yes—
28C2×D4×C67—22no—
29C2×D4×C67—41no—
30C2×D4×C67C22⋊C412yes—
31C22×C4×C67—21no—
32C22×C4×C67—22no—
33C22×C4×C67—41no—
34C22×C4×C67C22⋊C411yescharacteristic
35M16×C67—42no—
36C2×C8×C67—41no—
37C4⋊C4×C67—41no—
38C4⋊C4×C67—82no—
39C22⋊C4×C67—42no—
40C22⋊C4×C67—44no—
41C42×C67—21no—
42C42×C67—22no—
43C42×C67—41no—
44C42×C67—82no—
45C42×C67C4211yescharacteristic
46C42×C67C4⋊C411yescharacteristic
47C23×C67—22no—
48C23×C67#9911yescharacteristic
49Q8×C67—42no—
50D4×C67—42no—
51D4×C67—82no—
52D4×C67—82no—
53C2×C4×C67—21no—
54C2×C4×C67—21no—
55C2×C4×C67—21no—
56C2×C4×C67—21no—
57C2×C4×C67—22no—
58C2×C4×C67—42no—
59C2×C4×C67—42no—
60C2×C4×C67—42no—
61C2×C4×C67—81no—
62C2×C4×C67—82no—
63C2×C4×C67—84no—
64C2×C4×C67#9511yescharacteristic
65C2×C4×C67#9911yescharacteristic
66C8×C67—81no—
67C22×C67—21no—
68C22×C67—41no—
69C22×C67—42no—
70C22×C67—82no—
71C22×C67#34211yescharacteristic
72C4×C67—21no—
73C4×C67—41no—
74C4×C67—41no—
75C4×C67—42no—
76C4×C67—82no—
77C4×C67—164no—
78C4×C67#35311yescharacteristic
79C4×C67#35411yescharacteristic
80#7506C6711yesmaximal, characteristic, direct product factor, Sylow
81C2×C67—21no—
82C2×C67—41no—
83C2×C67—82no—
84C2×C67#139311yescharacteristic
85#2126C2×C6712yes—
86#1963C2×C6711yescharacteristic
87C67#750611yescharacteristic, direct product factor, Sylow
88#532C22×C6711yescharacteristic
89#423C4×C6711yescharacteristic, complemented
90C2×#104—22no—
91#410—22no—
92#376C4×C6711yescharacteristic, complemented
93#354C4×C6714yescomplemented
94#353—22no—
95C2×#41D4×C6712yes—
96C2×M16—21no—
97C4⋊Q8Q8×C6711yescharacteristic
98#127D4×C6711yescharacteristic
99#124C2×C4×C6712yes—
100#121—41no—
101C4×D4—41no—
102#117—42no—
103C2×C42—21no—
104C2×C42C2×C4×C6711yescharacteristic
105C4⋊C8—41no—
106#104—44no—
107#99—24no—
108#99—42no—
109#95—41no—
110#41—42no—
111#41—42no—
112C2×Q8C4⋊C4×C6712yes—
113C2×D4—22no—
114C2×D4—41no—
115C2×D4C22⋊C4×C6712yes—
116C22×C4—21no—
117C22×C4—22no—
118C22×C4—41no—
119C22×C4C22⋊C4×C6711yescharacteristic
120M16—42no—
121C2×C8—41no—
122C4⋊C4—41no—
123C4⋊C4—82no—
124C22⋊C4—42no—
125C22⋊C4—44no—
126C42—21no—
127C42—22no—
128C42—41no—
129C42—82no—
130C42C42×C6711yescharacteristic
131C42C4⋊C4×C6711yescharacteristic
132C23—22no—
133C23#99×C6711yescharacteristic
134Q8—42no—
135D4—42no—
136D4—82no—
137D4—82no—
138C2×C4—21no—
139C2×C4—21no—
140C2×C4—21no—
141C2×C4—21no—
142C2×C4—22no—
143C2×C4—42no—
144C2×C4—42no—
145C2×C4—42no—
146C2×C4—81no—
147C2×C4—82no—
148C2×C4—84no—
149C2×C4#95×C6711yescharacteristic
150C2×C4#99×C6711yescharacteristic
151C8—81no—
152C22—21no—
153C22—41no—
154C22—42no—
155C22—82no—
156C22#342×C6711yescharacteristic
157C4—21no—
158C4—41no—
159C4—41no—
160C4—42no—
161C4—82no—
162C4—164no—
163C4#353×C6711yescharacteristic
164C4#354×C6711yescharacteristic
165C2—21no—
166C2—41no—
167C2—82no—
168C2C67×#139311yescharacteristic
subgroupG/H
centerC4×C67#353
commutatorC42C42×C67
Frattini, cosocle#532C22×C67
socleC2×C67#1393
prime radical#532×C67C22
LCSG ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C4×C67 ◁ C2×C4×C67 ◁ C22×C4×C67 ◁ #532×C67 ◁ G

Complements

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

NH ≅ G/Ntype
4. #423×C67C4semidirect
7. #376×C67C4semidirect
8. #354×C67C4semidirect
80. #7506C67direct (N × H)
87. C67#7506direct (N × H)
89. #423C4×C67semidirect
92. #376C4×C67semidirect
93. #354C4×C67semidirect

Maximal subgroups

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

subgroupindex
C67×#21262
C67×#19632
#750667