Finite Group Databasebeta

← C67×#13662 #30692913

Subgroups — C67×#13662 #30692913

Order 17152.

294 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 294 conjugacy classes of subgroups — too many to draw. Only its 82 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
1C22×Q32×C67C211yesmaximal, characteristic, complemented
2C2×SD64×C67C212yesmaximal, complemented
3C67×#2215C211yesmaximal, characteristic
4C67×#1416C212yesmaximal, complemented
5C67×C22⋊C32C211yesmaximal, characteristic, complemented
6C22×Q16×C67—21no—
7C2×Q32×C67—24no—
8C2×Q32×C67—21no—
9C2×Q32×C67C2212yes—
10C2×D16×C67C2211yescharacteristic
11C22×C16×C67C2211yescharacteristic
12#466×C67—21no—
13SD64×C67—24no—
14C2×C32×C67C2212yes—
15#366×C67C2211yescharacteristic
16#357×C67—21no—
17C22×Q8×C67—41no—
18C2×Q16×C67—24no—
19C2×Q16×C67—22no—
20C2×Q16×C67—42no—
21C2×Q16×C67—41no—
22C2×D8×C67—21no—
23C22×C8×C67D411yescharacteristic
24#121×C67—41no—
25Q32×C67—24no—
26Q32×C67—22no—
27Q32×C67D414yescomplemented
28D16×C67—22no—
29C2×C16×C67—22no—
30C2×C16×C67D411yescharacteristic
31C2×C16×C67C2311yescharacteristic
32#107×C67—21no—
33#102×C67—41no—
34C32×C67—22no—
35C2×Q8×C67—44no—
36C2×Q8×C67—42no—
37C2×Q8×C67—82no—
38C2×Q8×C67—81no—
39C2×D4×C67—41no—
40C2×D4×C67—81no—
41C22×C4×C67—81no—
42C22×C4×C67D811yescharacteristic
43Q16×C67—24no—
44Q16×C67—44no—
45Q16×C67—42no—
46D8×C67—41no—
47D8×C67—41no—
48C2×C8×C67—21no—
49C2×C8×C67—21no—
50C2×C8×C67D811yescharacteristic
51C2×C8×C67C2×D411yescharacteristic
52C4⋊C4×C67—41no—
53C22⋊C4×C67—81no—
54C16×C67—21no—
55C16×C67C2×D412yes—
56C23×C67—81no—
57C23×C67D1611yescharacteristic
58Q8×C67—44no—
59Q8×C67—84no—
60Q8×C67—82no—
61D4×C67—81no—
62D4×C67—81no—
63D4×C67—162no—
64C2×C4×C67—21no—
65C2×C4×C67—21no—
66C2×C4×C67—84no—
67C2×C4×C67—82no—
68C2×C4×C67—81no—
69C2×C4×C67D1611yescharacteristic
70C2×C4×C67C2×D811yescharacteristic
71C8×C67—21no—
72C8×C67#12011yescharacteristic
73C8×C67AGL(1,ℤ/8)11yescharacteristic
74C22×C67—21no—
75C22×C67—21no—
76C22×C67—161no—
77C22×C67—161no—
78C22×C67—161no—
79C22×C67SD6412yescomplemented
80C22×C67C2×D1611yescharacteristic
81C4×C67—21no—
82C4×C67—84no—
83C4×C67—161no—
84C4×C67#44711yescharacteristic
85C4×C67#50911yescharacteristic
86#13662C6711yesmaximal, characteristic, direct product factor, Sylow
87C2×C67—22no—
88C2×C67—321no—
89C2×C67#218411yescharacteristic
90C2×C67C2×SD6411yescharacteristic
91C2×C67#226411yescharacteristic
92C22×Q32C2×C6711yescharacteristic, complemented
93C2×SD64C2×C6712yescomplemented
94#2215C2×C6711yescharacteristic
95#1416C2×C6712yescomplemented
96C22⋊C32C2×C6711yescharacteristic, complemented
97C67#1366211yescharacteristic, direct product factor, Sylow
98C22×Q16—21no—
99C2×Q32—24no—
100C2×Q32—21no—
101C2×Q32C22×C6712yes—
102C2×D16C22×C6711yescharacteristic
103C22×C16C22×C6711yescharacteristic
104#466—21no—
105SD64—24no—
106C2×C32C22×C6712yes—
107#366C22×C6711yescharacteristic
108#357—21no—
109C22×Q8—41no—
110C2×Q16—24no—
111C2×Q16—22no—
112C2×Q16—42no—
113C2×Q16—41no—
114C2×D8—21no—
115C22×C8D4×C6711yescharacteristic
116#121—41no—
117Q32—24no—
118Q32—22no—
119Q32D4×C6714yescomplemented
120D16—22no—
121C2×C16—22no—
122C2×C16D4×C6711yescharacteristic
123C2×C16C23×C6711yescharacteristic
124#107—21no—
125#102—41no—
126C32—22no—
127C2×Q8—44no—
128C2×Q8—42no—
129C2×Q8—82no—
130C2×Q8—81no—
131C2×D4—41no—
132C2×D4—81no—
133C22×C4—81no—
134C22×C4D8×C6711yescharacteristic
135Q16—24no—
136Q16—44no—
137Q16—42no—
138D8—41no—
139D8—41no—
140C2×C8—21no—
141C2×C8—21no—
142C2×C8D8×C6711yescharacteristic
143C2×C8C2×D4×C6711yescharacteristic
144C4⋊C4—41no—
145C22⋊C4—81no—
146C16—21no—
147C16C2×D4×C6712yes—
148C23—81no—
149C23D16×C6711yescharacteristic
150Q8—44no—
151Q8—84no—
152Q8—82no—
153D4—81no—
154D4—81no—
155D4—162no—
156C2×C4—21no—
157C2×C4—21no—
158C2×C4—84no—
159C2×C4—82no—
160C2×C4—81no—
161C2×C4D16×C6711yescharacteristic
162C2×C4C2×D8×C6711yescharacteristic
163C8—21no—
164C8#120×C6711yescharacteristic
165C8AGL(1,ℤ/8)×C6711yescharacteristic
166C22—21no—
167C22—21no—
168C22—161no—
169C22—161no—
170C22—161no—
171C22SD64×C6712yescomplemented
172C22C2×D16×C6711yescharacteristic
173C4—21no—
174C4—84no—
175C4—161no—
176C4#447×C6711yescharacteristic
177C4#509×C6711yescharacteristic
178C2—22no—
179C2—321no—
180C2C67×#218411yescharacteristic
181C2C2×SD64×C6711yescharacteristic
182C2C67×#226411yescharacteristic
subgroupG/H
center, socleC22×C67C2×D16
commutator, Frattini, cosocleC2×C16C23×C67
prime radicalC67×#2215C2
LCSG ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C67 ◁ C22×C4×C67 ◁ C22×C8×C67 ◁ C22×C16×C67 ◁ G

Complements

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

NH ≅ G/Ntype
1. C22×Q32×C67C2semidirect
2. C2×SD64×C67C2semidirect
4. C67×#1416C2semidirect
5. C67×C22⋊C32C2semidirect
27. Q32×C67D4semidirect
79. C22×C67SD64semidirect
86. #13662C67direct (N × H)
92. C22×Q32C2×C67semidirect
93. C2×SD64C2×C67semidirect
95. #1416C2×C67semidirect
96. C22⋊C32C2×C67semidirect
97. C67#13662direct (N × H)
119. Q32D4×C67semidirect
171. C22SD64×C67semidirect

Maximal subgroups

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

subgroupindex
C22×Q32×C672
C2×SD64×C672
C67×#22152
C67×#14162
C67×C22⋊C322
#1366267