Finite Group Databasebeta

← C67×#33983 #30713234

Subgroups — C67×#33983 #30713234

Order 17152.

462 conjugacy classes of subgroups; only the 202 normal subgroups are shown.

normal subgroups only

This group has 462 conjugacy classes of subgroups — too many to draw. Only its 202 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×#3417C211yesmaximal, characteristic, complemented
2C67×#3415C211yesmaximal, characteristic, complemented
3C67×#2264C213yesmaximal, complemented
4C67×#2263C213yesmaximal, complemented
5C67×#2262C213yesmaximal, complemented
6C2×SD64×C67C213yesmaximal, complemented
7C67×#2258C211yesmaximal, characteristic, complemented
8#578×C67—21no—
9#576×C67—21no—
10#510×C67—23no—
11#509×C67—23no—
12#508×C67C2213yes—
13#508×C67C2213yescomplemented
14C2×Q32×C67C2213yescomplemented
15C2×D16×C67C2213yes—
16#504×C67C2211yescharacteristic
17Q64×C67C2213yescomplemented
18SD64×C67C2213yescomplemented
19SD64×C67C2216yescomplemented
20SD64×C67C2211yescharacteristic, complemented
21D32×C67C2213yescomplemented
22M64×C67C2213yescomplemented
23C2×C32×C67C2213yescomplemented
242−1+4×C67—41no—
25H(5,𝔽2)×C67—41no—
26#137×C67—43no—
27AGL(1,ℤ/8)×C67—43no—
28#135×C67—23no—
29#135×C67—23no—
30C2×Q16×C67—23no—
31C2×D8×C67—23no—
32#131×C67D411yescharacteristic
33Q32×C67—23no—
34Q32×C67C2313yes—
35Q32×C67C2311yescharacteristic
36SD32×C67—23no—
37SD32×C67—23no—
38D16×C67—23no—
39D16×C67C2313yes—
40D16×C67C2311yescharacteristic
41M32×C67D413yes—
42C2×C16×C67C2313yes—
43C32×C67C2311yescharacteristic
44C32×C67C2313yes—
45#41×C67—43no—
46#41×C67—43no—
47#41×C67—81no—
48#41×C67—83no—
49#41×C67D811yescharacteristic
50C2×Q8×C67—43no—
51C2×Q8×C67—81no—
52C2×D4×C67—43no—
53C2×D4×C67—83no—
54Q16×C67—21no—
55Q16×C67—23no—
56Q16×C67—43no—
57SD16×C67—43no—
58SD16×C67—43no—
59D8×C67—23no—
60D8×C67—21no—
61D8×C67—43no—
62M16×C67D813yes—
63C2×C8×C67C2×D413yes—
64C16×C67C2×D413yes—
65C16×C67C2411yescharacteristic
66C23×C67—83no—
67Q8×C67—41no—
68Q8×C67—43no—
69Q8×C67—83no—
70Q8×C67D1611yescharacteristic
71D4×C67—43no—
72D4×C67—41no—
73D4×C67—83no—
74D4×C67—83no—
75D4×C67—83no—
76D4×C67D1613yes—
77C2×C4×C67—83no—
78C2×C4×C67—83no—
79C2×C4×C67—83no—
80C2×C4×C67C2×D813yes—
81C8×C67C2×D813yes—
82C8×C67C22×D411yescharacteristic
83C22×C67—81no—
84C22×C67—83no—
85C22×C67—323no—
86C22×C67C2×D1613yes—
87C4×C67—81no—
88C4×C67—83no—
89C4×C67C2×D1613yes—
90C4×C67C22×D811yescharacteristic
91#33983C6711yesmaximal, characteristic, direct product factor, Sylow
92C2×C67—23no—
93C2×C67—161no—
94C2×C67—163no—
95C2×C67C22×D1611yescharacteristic
96#3417C2×C6711yescharacteristic, complemented
97#3415C2×C6711yescharacteristic, complemented
98#2264C2×C6713yescomplemented
99#2263C2×C6713yescomplemented
100#2262C2×C6713yescomplemented
101C2×SD64C2×C6713yescomplemented
102#2258C2×C6711yescharacteristic, complemented
103C67#3398311yescharacteristic, direct product factor, Sylow
104#578—21no—
105#576—21no—
106#510—23no—
107#509—23no—
108#508C22×C6713yes—
109#508C22×C6713yescomplemented
110C2×Q32C22×C6713yescomplemented
111C2×D16C22×C6713yes—
112#504C22×C6711yescharacteristic
113Q64C22×C6713yescomplemented
114SD64C22×C6713yescomplemented
115SD64C22×C6716yescomplemented
116SD64C22×C6711yescharacteristic, complemented
117D32C22×C6713yescomplemented
118M64C22×C6713yescomplemented
119C2×C32C22×C6713yescomplemented
1202−1+4—41no—
121H(5,𝔽2)—41no—
122#137—43no—
123AGL(1,ℤ/8)—43no—
124#135—23no—
125#135—23no—
126C2×Q16—23no—
127C2×D8—23no—
128#131D4×C6711yescharacteristic
129Q32—23no—
130Q32C23×C6713yes—
131Q32C23×C6711yescharacteristic
132SD32—23no—
133SD32—23no—
134D16—23no—
135D16C23×C6711yescharacteristic
136D16C23×C6713yes—
137M32D4×C6713yes—
138C2×C16C23×C6713yes—
139C32C23×C6711yescharacteristic
140C32C23×C6713yes—
141#41—43no—
142#41—43no—
143#41—81no—
144#41—83no—
145#41D8×C6711yescharacteristic
146C2×Q8—43no—
147C2×Q8—81no—
148C2×D4—43no—
149C2×D4—83no—
150Q16—21no—
151Q16—23no—
152Q16—43no—
153SD16—43no—
154SD16—43no—
155D8—23no—
156D8—21no—
157D8—43no—
158M16D8×C6713yes—
159C2×C8C2×D4×C6713yes—
160C16C2×D4×C6713yes—
161C16C24×C6711yescharacteristic
162C23—83no—
163Q8—41no—
164Q8—43no—
165Q8—83no—
166Q8D16×C6711yescharacteristic
167D4—43no—
168D4—41no—
169D4—83no—
170D4—83no—
171D4—83no—
172D4D16×C6713yes—
173C2×C4—83no—
174C2×C4—83no—
175C2×C4—83no—
176C2×C4C2×D8×C6713yes—
177C8C2×D8×C6713yes—
178C8C22×D4×C6711yescharacteristic
179C22—81no—
180C22—83no—
181C22—323no—
182C22C2×D16×C6713yes—
183C4—81no—
184C4—83no—
185C4C2×D16×C6713yes—
186C4C22×D8×C6711yescharacteristic
187C2—23no—
188C2—163no—
189C2—161no—
190C2C22×D16×C6711yescharacteristic
subgroupG/H
center, socleC2×C67C22×D16
commutator, Frattini, cosocleC16C24×C67
LCSG ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C67 ◁ #41×C67 ◁ #131×C67 ◁ #504×C67 ◁ G

Complements

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

NH ≅ G/Ntype
1. C67×#3417C2semidirect
2. C67×#3415C2semidirect
3. C67×#2264C2semidirect
4. C67×#2263C2semidirect
5. C67×#2262C2semidirect
6. C2×SD64×C67C2semidirect
7. C67×#2258C2semidirect
13. #508×C67C22semidirect
14. C2×Q32×C67C22semidirect
17. Q64×C67C22semidirect
18. SD64×C67C22semidirect
19. SD64×C67C22semidirect
20. SD64×C67C22semidirect
21. D32×C67C22semidirect
22. M64×C67C22semidirect
23. C2×C32×C67C22semidirect
91. #33983C67direct (N × H)
96. #3417C2×C67semidirect
97. #3415C2×C67semidirect
98. #2264C2×C67semidirect
99. #2263C2×C67semidirect
100. #2262C2×C67semidirect
101. C2×SD64C2×C67semidirect
102. #2258C2×C67semidirect
103. C67#33983direct (N × H)
109. #508C22×C67semidirect
110. C2×Q32C22×C67semidirect
113. Q64C22×C67semidirect
114. SD64C22×C67semidirect
115. SD64C22×C67semidirect
116. SD64C22×C67semidirect
117. D32C22×C67semidirect
118. M64C22×C67semidirect
119. C2×C32C22×C67semidirect

Maximal subgroups

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

subgroupindex
C67×#34172
C67×#34152
C67×#22642
C67×#22632
C67×#22622
C2×SD64×C672
C67×#22582
#3398367